(PDF) Boris Computational Spintronics User Manual

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BORIS Computational Spintronics

User manual, version 3.80

Dr. Serban Lepadatu, 22nd February 2023

Abstract

This manual describes BORIS Computational Spintronics, a multi-physics and multi-scale

research software designed to solve three-dimensional magnetization dynamics problems,

coupled with a self-consistent charge and spin transport solver, heat flow solver with

temperature-dependent material parameters, and elastodynamics solver including

thermoelastic and magnetoelastic/magnetostriction effects, in arbitrary multi-layered

structures and shapes. The computational routines run both on central processors and

graphics processors using the CUDA platform. In addition to simple user control, advanced

simulation configurations are made possible using Python scripts. The software is open

source and currently runs on Windows 7, Windows 10, and Linux-based 64-bit operating

systems, and was programmed using C++17 and Python.

boris-spintronics.uk/download

https://github.com/SerbanL/Boris2

https://groups.google.com/forum/#!forum/boris-computational-spintronics

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Disclaimer

BORIS Computational Spintronics is a freely available research, design and educational

software. The author assumes no responsibility whatsoever for its use by other parties, and

makes no guarantees, expressed or implied, about its quality, reliability, or any other

characteristic. If using Boris for published research please reference it using: S. Lepadatu,

“Boris Computational Spintronics - High Performance Multi-Mesh Magnetic and Spin

Transport Modelling Software” Journal of Applied Physics 128, 243902 (2020).

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Contents

Installation - Windows ............................................................................................................. 5

Installation – Linux-based OS ................................................................................................ 6

Version Updates ..................................................................................................................... 11

Overview ................................................................................................................................. 12

Tutorial 0 – Quick-Start ........................................................................................................ 13

Tutorial 1 – Introduction ...................................................................................................... 23

Tutorial 2 – Data Output ....................................................................................................... 30

Tutorial 3 – Further Data Output ........................................................................................ 37

Tutorial 4 – Domain Walls .................................................................................................... 41

Tutorial 5 – Domain Wall Movement and Data Processing .............................................. 46

Tutorial 6 – ODE Control and Setting Shapes .................................................................... 50

Tutorial 7 – Magnetocrystalline Anisotropy ....................................................................... 54

Tutorial 8 – Anisotropic Magneto-Resistance ..................................................................... 57

Tutorial 9 – Scripting using Python ..................................................................................... 63

Tutorial 10 – Current-Induced Domain Wall Movement .................................................. 72

Tutorial 11 – Oersted Fields .................................................................................................. 75

Tutorial 12 – Surface Exchange, Multi-Layered Demagnetization and CUDA............... 78

Tutorial 13 – Dzyaloshinskii-Moriya Exchange .................................................................. 84

Tutorial 14 – Simulations with non-zero temperature ....................................................... 87

Tutorial 15 – Thermal Fields ................................................................................................ 90

Tutorial 16 – Heat Flow Solver and Joule Heating ............................................................. 91

Tutorial 17 – Spin Transport Solver .................................................................................... 96

Tutorial 18 – Spin Hall Effect ............................................................................................. 101

Tutorial 19 – Spin Pumping and Inverse Spin Hall Effect .............................................. 104

Tutorial 20 – Ferromagnetic Resonance ............................................................................ 107

Tutorial 21 – Ferromagnetic Resonance with Spin Torques ........................................... 113

Tutorial 22 – CPP-GMR ..................................................................................................... 117

Tutorial 23 – Skyrmion Movement with Spin Currents .................................................. 120

Tutorial 24 – Roughness and Staircase Corrections ......................................................... 124

Tutorial 25 – Defects and Impurities ................................................................................. 130

Tutorial 26 – Polycrystalline and Granular Films ............................................................ 133

Tutorial 27 – Periodic Boundary Conditions .................................................................... 136

Tutorial 28 – Ultrafast Demagnetization ........................................................................... 138

Tutorial 29 – Magneto-Optical Effect ................................................................................ 142

Tutorial 30 – Spin-Wave Dispersion .................................................................................. 143

Tutorial 31 – Two-Sublattice Model .................................................................................. 146

Tutorial 32 – Exchange Bias ............................................................................................... 152

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Tutorial 33 – Magneto-Elastic Effect and Elastodynamics Solver .................................. 153

Shapes and Regions.............................................................................................................. 158

User-Defined Text Equations .............................................................................................. 164

Working with OVF2 Files ................................................................................................... 170

Materials Database .............................................................................................................. 173

Meshes ................................................................................................................................... 176

Differential Equations ......................................................................................................... 182

Modules ................................................................................................................................. 193

Material Parameters ............................................................................................................ 234

Parameters Temperature Dependence .............................................................................. 243

Parameters Spatial Variation ............................................................................................. 244

Simulation Schedules ........................................................................................................... 250

Monte Carlo Algorithms ..................................................................................................... 256

Simulation Algorithms......................................................................................................... 259

Data Extraction Algorithms ................................................................................................ 260

Demagnetizing Field Polynomial Extrapolation ............................................................... 262

Output Data .......................................................................................................................... 263

Command Buffering ............................................................................................................ 272

Effective Fields and Energies Spatially Resolved Output Data ....................................... 274

Mesh Display ........................................................................................................................ 275

Commands ............................................................................................................................ 279

References ............................................................................................................................. 321

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Installation - Windows

An installer has been provided with the program and instructions therein should be followed.

The program is designed to run on Windows 7, Windows 10, and Windows 11, 64 bit

versions, and requires Microsoft Visual C++ 2017 Redistributable (x64) – included with the

installer. On Windows 10 the executable (Boris.exe) must be run in compatibility mode – the

installer sets this.

CUDA

To enable CUDA computations Boris requires a CUDA-enabled graphics card with CUDA

compute capability 5.0 or greater. The installer detects the CUDA compute capability of the

graphics card and launches the required program version. You should always run the

installed Boris.exe program, not the separate CUDA versions found in the same directory.

The following architectures are supported by the installer package: Maxwell (sm_50, sm_52,

sm_53), Pascal (sm_60, sm_61, sm_62), Volta (sm_70, sm_72), Turing (sm_75), Ampere

(sm_80, sm_86), Ada and Hopper (sm_90). CUDA Toolkit 12.0 has been used to compile.

Directories

BORIS is installed in C:\Program Files (x86)\Boris

BORIS user data is contained in C:\Users\(UserName)\Documents\Boris Data

Boris Data folder contains:

• BorisPythonScripts folder, which contains required Python files for simulation

scripting, in particular NetSocks.py which is required for all Python scripts.

• Examples folder, with worked solutions for the Tutorials in this Manual.

• Simulations folder, which is the default folder for saving and loading simulation files.

• Manual, BorisMDB.txt materials database, errorlog.txt showing the errors log,

startup.txt with BORIS startup configuration options.

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Installation – Linux-based OS

Extract the archive. On Linux-based OS the program needs to be compiled from source using

the provided makefile in the extracted BorisLin directory.

Make sure you have all the required updates and dependencies:

Step 0: Updates.

1. Get latest g++ compiler: $ sudo apt install build-essential

2. Get OpenMP: $ sudo apt-get install libomp-dev

3. Get LibTBB: $ sudo apt install libtbb-dev

4. Get latest CUDA Toolkit by following instructions at: CUDA Toolkit Downloads

For example using the deb(network) method, the following instructions may be used:

$ wget https://developer.download.nvidia.com/compute/cuda/repos/ubuntu2004/x86_64/cuda-

keyring_1.0-1_all.deb

$ sudo dpkg -i cuda-keyring_1.0-1_all.deb

$ sudo apt-get update

$ sudo apt-get -y install cuda

NOTES:

If at any point it is necessary to re-install the CUDA toolkit, it is recommended any

existing CUDA installation is removed. This may be done using the following:

$ sudo apt-get purge nvidia*

$ sudo apt-get autoremove

$ sudo apt-get autoclean

$ sudo rm -rf /usr/local/cuda*

FURTHER NOTES (IMPORTANT):

It may be necessary to set the export path for the CUDA dynamically linked library.

This may be done by editing the bash script:

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$ nano /home/$USER/.bashrc

When bash file opens, add there the following lines (e.g. at the end), but replace 12.0

with correct CUDA version you installed:

export PATH="/usr/local/cuda-12.0/bin:$PATH"

export LD_LIBRARY_PATH="/usr/local/cuda-12.0/lib64:$LD_LIBRARY_PATH"

Ctrl + o : Save

Enter or Return key : Accept changes

Ctrl + x : Close editor

Close terminal and start another one.

Now nvcc --version should show correct version.

The include and library directories for CUDA are as follows, and these are expected

by makefile:

Include directory for compilation:

-I/usr/local/cuda-12.0/targets/x86_64-linux/include

Library directory for linking:

-L/usr/local/cuda-12.0/targets/x86_64-linux/lib

TROUBLESHOOTING:

There are known incompatibility issues between certain CUDA Toolkit versions and

gcc compiler versions. For example the combination gcc 11.3 and CUDA 11.5 will

not compile. In this case it is necessary to upgrade to latest CUDA version.

Please report any incompatibilities and issues here:

https://groups.google.com/forum/#!forum/boris-computational-spintronics

5. Get and install FFTW3: Instructions at http://www.fftw.org/fftw2_doc/fftw_6.html

6. Get Python3 development version, required for running Python scripts in embedded

mode. To get Python3 development version:

$ sudo apt-get install python-dev

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NOTES:

The following directories are expected by makefile, e.g. for Python3.8 version:

Include directory for compilation:

-I/usr/include/python3.8

Open terminal and go to extracted BorisLin directory.

Step 1: Configuration.

$ make configure (arch=xx) (sprec=0/1) (python=x.x) (cuda=x.x)

Before compiling you need to set the correct CUDA architecture for your NVidia GPU.

For a list of architectures and more details see: https://en.wikipedia.org/wiki/CUDA.

Possible values for arch are:

• arch=50 is required for Maxwell architecture; translates to

-arch=sm_50 in nvcc compilation.

• arch=60 is required for Pascal architecture; translates to

-arch=sm_60 in nvcc compilation.

• arch=70 is required for Volta (and Turing) architecture; translates to

-arch=sm_70 in nvcc compilation.

• arch=80 is required for Ampere architecture; translates to

-arch=sm_80 in nvcc compilation.

• arch=90 is required for Ada (and Hopper) architecture; translates to

-arch=sm_90 in nvcc compilation.

sprec sets either single precision (1) or double precision (0) for CUDA computations.

python is the Python version installed, e.g. 3.8

cuda is the CUDA Toolkit version installed, e.g. 12.0.

Example: $ make configure arch=80 sprec=1 python=3.8 cuda=12.0

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If arch is not specified a default value of 50 is used.

You can also compile CUDA code with single or double precision floating point. The default

value, if not specified, is sprec=1 (single precision – recommended for most users). If you

have a GPU capable of handling double precision floating point efficiently you can configure

with sprec=0.

Step 2: Compilation.

$ make compile -j N

(replace N with the number of logical cores on your CPU for multi-processor compilation,

e.g. $ make compile -j 16)

Step 3: Installation.

$ make install

Step4: Run.

$ ./BorisLin

Notes:

In the current version Boris runs in text mode only on Linux-based OS, thus the GRAPHICS

0 value needs to be kept in CompileFlags.h file. In a future version the graphical interface

will also be available under Linux-based OS.

Directories

BORIS is installed in the extracted BorisLin directory. This folder contains the compiled

executable, makefile, required Python files for simulation scripting, in particular NetSocks.py

which is required for all Python scripts. It also contains a number of configuration files.

The code is contained in Boris folder, with additional library files in BorisLib and

BorisCUDALib folders.

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BORIS user data is contained in: …\BorisLin\Boris_Data

Boris_Data folder contains:

• Examples folder, with worked solutions for the Tutorials in this Manual.

• Simulations folder, which is the default folder for saving and loading simulation files.

Manual, BorisMDB.txt materials database, errorlog.txt showing the errors log, startup.txt

with BORIS startup configuration options.

CompileFlags.h file:

There are a number of compilation options, described there, found in the Boris folder. In most

cases this shouldn’t be changed. Some special cases include:

• Set PYTHON_EMBEDDING to 0 if embedded Python scripting is not required. With

this setting Python3 headers and libraries will not be required for compilation and

linking. This could be useful for troubleshooting, but otherwise leave this value to 1.

• Set COMPILECUDA to 0 if CUDA computations are not required. In this mode only

CPU computations will be available. This could be useful for troubleshooting, but

otherwise leave this value to 1.

• GRAPHICS is set to 0 for compiling under Linux-based OS (this should already be

set, so does not need to be changed).

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Version Updates

The program version may be updated from the console using the versionupdate

command, without having to download the entire installer again. Instead, incremental patches

are downloaded and applied. Currently this only works for Windows. This is intended mainly

for small updates, e.g. bug fix patches. The latest version 3.8 requires full download and

install.

1) Check if incremental updates are available, but take no other action:

versionupdate check

2) Download any available and required incremental updates, but do not install:

versionupdate download

3) Download any available and required incremental updates, and install the latest version:

versionupdate install

Using this command automatically downloads, installs all required incremental patches, then

automatically restarts the program which should now be the latest available version.

4) If you want to only update to a given version, which may not be the latest version then

specify it as a parameter, e.g. v3.5 would be 350, etc:

versionupdate install 350

5) If you want to roll back to an earlier version, e.g. back to v3.4 then specify it as:

versionupdate rollback 340

Note this command assumes v3.4 is present on the working machine, which typically means

the software was previously updated from this version to a newer one.

6) If you want to simply check if your version is the latest version then use the checkupdates

command. To have the program automatically check for updates whenever it is started then

use (from v3.4 this is disabled by default, but if you’ve previously installed an earlier version

this might still be enabled in the local startup.txt configuration file):

startupupdatecheck 1

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Overview

This manual contains a set of self-teaching tutorials that guide the user through most of its

functionality. The tutorials contain a number of exercises designed for users without a

background in micromagnetics, and may be skipped by more advanced users. A number of

examples that accompany the tutorials have also been provided in the accompanying

Examples folder.

You can use Tutorial 0 as a quick-start. This tutorial contains a number of Python scripts as

examples but doesn’t contain in-depth explanations.

Tutorials 1 to 7 cover the basics and it is recommended all users read through them. After

these you can skip to the required tutorials as needed. Tutorial 9 covers the basics of

automating simulations using Python and should be used as a starting point if required. For

the transport solver Tutorials 8 and 10 should be used as a starting point. Tutorials 17 to 23

cover the spin transport solver. Tutorials 14 to 16 cover the heat solver.

The equations solved are given in the Differential Equations and Modules sections. All

material parameters used in these equations have been given in the Material Parameters

section. All the important parts of the program have been documented in the sections after the

Tutorials, including output data and advanced features.

A full list of commands has been provided in the Commands section in alphabetical order.

The most commonly used commands have also been outlined.

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Tutorial 0 – Quick-Start

This tutorial contains a set of Python scripts for a selected number of micromagnetics

problems, and are intended as a crash-course for experienced users who want to get things

going very quickly. No detailed comments are provided, except for comments in the Python

scripts – if you want in-depth explanations you need to read the other tutorials. All scripts are

found in the Examples/Tutorial 0 folder.

The Python scripts below can be run either locally (connect through NSClient as ‘localhost’ –

this is the default), or remotely (connect using the IP address of machine running Boris).

Alternatively Python scripts may be executed in embedded mode, where Boris loads the

script and calls the Python interpreter, not requiring communication through network sockets.

For the Python scripts below you need the NetSocks.py module. Before executing the script a

Boris instance should be running.

The default program start-up state is a permalloy rectangle with dimensions 80 nm × 80 nm ×

10 nm, and cubic 5 nm cellsize. The demag, exchange, and Zeeman modules are enabled. The

LLG equation is set with the RKF45 evaluation method. To restore the program to default

state at any time use the default console command (or alternatively send it as ns.default()

from a Python script).

The default simulation stage is a Relax stage with |mxh| < 10-4 stopping condition and no data

saving condition.

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Hysteresis Loop

import matplotlib.pyplot as plt

import numpy as np

from NetSocks import NSClient

ns = NSClient(); ns.configure(True)

########################################

#Make a ferromagnetic mesh with given rectangle, cubic cellsize, and set modules

Py = ns.Ferromagnet([160e-9, 80e-9, 10e-9], [5e-9])

Py.modules(['demag', 'exchange', 'Zeeman'])

#configure output data : applied field and average magnetisation

ns.setsavedata('hysteresis.txt', ['Ha', Py], ['<M>', Py])

########################################

#run two stages to sweep field up and down between -100 kA/m and +100kA/m in 100

steps, slightly off-axis

#each field step relaxed to mxh < 1e-5, and data saved every field step

ns.setode('LLGStatic', 'SDesc')

ns.Hpolar_seq([Py, [-100e3, 90, 1, +100e3, 90, 1, 100], 'mxh', 1e-5, 'step'])

ns.Hpolar_seq([Py, [100e3, 90, 1, -100e3, 90, 1, 100], 'mxh', 1e-5, 'step'])

########################################

#output file has field (x, y, z components) in columns 0, 1, 2, and average

magnetisation (x, y, z components) in columns 3, 4, 5

hysteresis_data = ns.Get_Data_Columns('hysteresis.txt', [0, 3])

#plot Mx vs Hx

plt.axes(xlabel = 'H (kA/m)', ylabel = '<M> (kA/m)')

plt.plot(np.array(hysteresis_data[0])/1e3, np.array(hysteresis_data[1])/1e3)

plt.show()

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Hysteresis Loop (embedded script version)

import matplotlib.pyplot as plt

import numpy as np

from NetSocks import NSClient

ns = NSClient(embedded = True); ns.configure(True)

########################################

#Make a ferromagnetic mesh with given rectangle, cubic cellsize, and set modules

Py = ns.Ferromagnet([160e-9, 80e-9, 10e-9], [5e-9])

Py.modules(['demag', 'exchange', 'Zeeman'])

#configure output data : applied field and average magnetisation

ns.setsavedata('hysteresis.txt', ['Ha', Py], ['<M>', Py])

########################################

#This method is executed every simulation iteration, called by BORIS

def save_coercivity_image(Mx_previous):

M = Py.showdata('<M>')

if M[0] * Mx_previous < 0.0:

if M[0] > 0.0: ns.savemeshimage('Coercive_up.png')

else: ns.savemeshimage('Coercive_dn.png')

return 1, M[0]

#run two stages to sweep field up and down between -100 kA/m and +100kA/m in 100

steps, slightly off-axis

#each field step relaxed to mxh < 1e-5, and data saved every field step

ns.setode('LLGStatic', 'SDesc')

ns.Hpolar_seq([Py, [-100e3, 90, 1, +100e3, 90, 1, 100], 'mxh', 1e-5, 'step'],

save_coercivity_image, -800e3)

ns.Hpolar_seq([Py, [100e3, 90, 1, -100e3, 90, 1, 100], 'mxh', 1e-5, 'step'],

save_coercivity_image, 800e3)

########################################

#output file has field (x, y, z components) in columns 0, 1, 2, and average

magnetisation (x, y, z components) in columns 3, 4, 5

hysteresis_data = ns.Get_Data_Columns('hysteresis.txt', [0, 3])

#plot Mx vs Hx

plt.axes(xlabel = 'H (kA/m)', ylabel = '<M> (kA/m)')

plt.plot(np.array(hysteresis_data[0])/1e3, np.array(hysteresis_data[1])/1e3)

plt.show()

16

Domain Wall Movement

from NetSocks import NSClient

import matplotlib.pyplot as plt

import numpy as np

ns = NSClient(); ns.configure(True)

########################################

#This is based on Exercise 5.1, done entirely using a Python script

Py = ns.Ferromagnet([320e-9, 80e-9, 20e-9], [5e-9, 5e-9, 5e-9])

Py.modules(['demag', 'exchange', 'Zeeman'])

#setup the moving mesh algorithm for a transverse domain wall along the x axis:

#1. remove end magnetic charges using two dipole meshes (enables strayfield

module)

#2. freeze x-axis ends spins

#3. set a transverse domain wall

#4. enable moving mesh algorithm which keeps the dw centered

ns.preparemovingmesh()

#relax dw in zero field to |mxh| < 10^-5

ns.setode('LLGStatic', 'SDesc')

ns.Relax(['mxh', 1e-5])

ns.reset()

#set fixed time-step RK4 method with 500fs time step

ns.setode('LLG', 'RK4')

ns.setdt(500e-15)

#save time (s) and dw shift (m) data

ns.setsavedata('dwmovement_temp.txt', ['stime'], ['dwshift'])

Hrange = np.arange(100, 2400, 200)

dwvelocity = np.array([])

for H in Hrange:

#first stage achieves steady state movement

ns.Hxyz([Py, [H, 0, 0], 'time', 5e-9])

#second stage captures data

ns.Hxyz([Py, [H, 0, 0], 'time', 10e-9, 'time', 1e-12])

ns.reset()

#process data to extract domain wall velocity

ns.dp_load('dwmovement_temp.txt', [0, 1, 0, 1])

ns.dp_replacerepeats(1)

dwdata = ns.dp_linreg(0, 1)

dwvelocity = np.append(dwvelocity, dwdata[0])

print('H (A/m) = %f, DW velocity (m/s) = %0.4f' % (H, dwdata[0]))

plt.axes(xlabel = 'H (A/m)', ylabel = 'DW Velocity (m/s)', title = 'DW Velocity

and Walker Breakdown')

plt.plot(Hrange, dwvelocity, 'o-')

plt.show()

17

Anisotropic Magnetoresistance

from NetSocks import NSClient

import matplotlib.pyplot as plt

import numpy as np

ns = NSClient(); ns.configure(True)

########################################

#This is based on Exercise 8.3, done entirely using a Python script

Py = ns.Ferromagnet([160e-9, 80e-9, 10e-9], [5e-9, 5e-9, 5e-9])

Py.modules(['Zeeman', 'demag', 'exchange', 'transport'])

#set amr percentage of 2%

Py.param.amr = 2.0

#amr loop angle (deg.)

direction_deg = 5

ns.setangle(90, 180.0 + direction_deg)

#set electrodes at x-axis ends with a 1 mV potential drop

ns.setdefaultelectrodes()

ns.setpotential(1e-3)

#save applied field (A/m) and resistance (Ohms)

ns.setsavedata('amr_rawdata.txt', ['Ha', Py], ['R'])

#Run hysteresis loop

ns.setode('LLGStatic', 'SDesc')

ns.Hpolar_seq(

[Py, [-100e3, 90, direction_deg, 100e3, 90, direction_deg, 200],

'mxh', 1e-6, 'step'])

ns.Hpolar_seq(

[Py, [100e3, 90, direction_deg, -100e3, 90, direction_deg, 200],

'mxh', 1e-6, 'step'])

########################################

#Plot loop

#load all columns from file (0, 1, 2, 3) into internal arrays (0, 1, 2, 4)

ns.dp_load('amr_rawdata.txt', [0, 1, 2, 3, 0, 1, 2, 4])

#get field strength along loop direction and save it in internal array 3

ns.dp_dotprod(0, np.cos(np.radians(direction_deg)),

np.sin(np.radians(direction_deg)), 0, 3)

#save field strength and resistance in processed file, then plot it here

ns.dp_save('amr_loop.txt', [3, 4])

amr_data = ns.Get_Data_Columns('amr_loop.txt', [0, 1])

plt.axes(xlabel = 'H (kA/m)', ylabel = 'R (Ohms)', title = 'AMR Loop')

plt.plot(np.array(amr_data[0])/1e3, amr_data[1])

plt.show()

18

RKKY Simulation (multi-mesh demonstration)

from NetSocks import NSClient

import matplotlib.pyplot as plt

import numpy as np

ns = NSClient(); ns.configure(True)

########################################

direction_deg = 1.0

FM1 = ns.Ferromagnet([320e-9, 160e-9, 10e-9], [5e-9, 5e-9, 5e-9])

FM1.modules(['Zeeman', 'demag', 'exchange', 'surfexchange'])

#shape mesh as an ellipse (mask file is stretched to mesh aspect ratios)

FM1.loadmaskfile('Circle')

#add a new ferromagnetic mesh (permalloy by default) above the first one with 1 nm

separation

FM2 = ns.Ferromagnet([0.0, 0.0, 11e-9, 320e-9, 160e-9, 31e-9], [5e-9, 5e-9, 5e-9])

FM2.modules(['Zeeman', 'demag', 'exchange', 'surfexchange'])

FM2.loadmaskfile('Circle')

#enable multilayered demag field calculation allowing exact and efficient

calculation of demag fields, even though the separation between meshes is 1 nm

ns.addmodule('supermesh', 'sdemag')

ns.setsavedata('rkky_hysteresis.txt', ['Ha', FM1], ['<M>', FM1], ['<M>', FM2])

ns.setode('LLGStatic', 'SDesc')

ns.cuda(1)

ns.Hpolar_seq(['supermesh', [-300e3, 90, direction_deg, 300e3, 90, direction_deg,

300], 'mxh', 1e-5, 'step'])

ns.Hpolar_seq(['supermesh', [300e3, 90, direction_deg, -300e3, 90, direction_deg,

300], 'mxh', 1e-5, 'step'])

########################################

u = [np.cos(np.radians(direction_deg)), np.sin(np.radians(direction_deg)), 0]

ns.dp_load('rkky_hysteresis', [0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 3, 4, 5, 6, 7,

8])

ns.dp_dotprod(0, u[0], u[1], u[2], 10)

ns.dp_dotprod(3, u[0], u[1], u[2], 11)

ns.dp_dotprod(6, u[0], u[1], u[2], 12)

ns.dp_mul(11, 1.0/3)

ns.dp_mul(12, 2.0/3)

ns.dp_adddp(11, 12, 13)

ns.dp_save('rkky_hysteresis_loop.txt', [10, 13])

loop = ns.Get_Data_Columns('rkky_hysteresis_loop.txt', [0, 1])

plt.axes(xlabel = 'H (kA/m)', ylabel = 'M (kA/m)', title = 'RKKY Hysteresis Loop')

plt.plot(np.array(loop[0])/1e3, np.array(loop[1])/1e3)

plt.show()

19

Setting/Editing Parameter Variations Programatically

Parameter spatial variations can be set using ovf2 files in the most general case. For many

cases it suffices to use the Shape class, imported from NetSocks – see chapter on Shapes and

Regions. This allows defining regions for setting material parameter values easily.

from NetSocks import NSClient

ns = NSClient(); ns.configure(True)

########################################

ns.meshrect([800e-9, 800e-9, 5e-9])

#setup a grain structure, marking the grains with values in uniform random number

distribution between 0 and 1

ns.setparamvar('permalloy', 'J1', 'vor2D', [0.0, 1.0, 20e-9, 1])

#save it to a file and load it here so we can edit it

ns.saveovf2param(paramname = 'J1', filename = 'grains.ovf')

vec, n, rect = ns.Read_OVF2('grains.ovf')

#snap grain values to 0 or 1 : on average half will be 0, the other half 1

for idx in range(len(vec)):

if vec[idx] < 0.5: vec[idx] = 0.0

else: vec[idx] = 1.0

#write it to a file, then load the new grain values

ns.Write_OVF2('grains_snapped.ovf', vec, n, rect)

ns.setparamvar('permalloy', 'J1', 'ovf2', 'grains_snapped.ovf')

#setup display so we can see the grain structure

ns.display('ParamVar')

ns.setdisplayedparamsvar('permalloy', 'J1')

ns.displaydetail(5e-9)

20

Exchange Bias

from NetSocks import NSClient

import matplotlib.pyplot as plt

import numpy as np

ns = NSClient(); ns.configure(True)

########################################

AFM = ns.AntiFerromagnet([320e-9, 320e-9, 10e-9], [5e-9, 5e-9, 5e-9])

AFM.modules(['Zeeman', 'exchange', 'aniuni', 'surfexchange'])

#set sub-lattice A magnetisation to result in biasing towards +ve side

AFM.setangle(90, 180)

#Add Fe mesh on top of the antiferromagnet

FM = ns.Material('Fe', [0, 0, 10e-9, 320e-9, 320e-9, 12e-9], [2.5e-9, 2.5e-9, 2e-

9])

FM.modules(['Zeeman', 'demag', 'exchange', 'anicubi', 'surfexchange'])

FM.pbc('x', 10)

FM.pbc('y', 10)

#set bilinear surface exchange coupling value - exchange bias is proportional to

this

FM.param.J1 = 0.2e-3

########################################

ns.setsavedata('exchangebias.txt', ['Ha', FM], ['<M>', FM])

ns.setode('LLGStatic', 'SDesc')

ns.cuda(1)

ns.Hpolar_seq(['supermesh', [-50e3, 90, 5, 100e3, 90, 5, 50], 'mxh', 1e-5,

'step'])

ns.Hpolar_seq(['supermesh', [100e3, 90, 5, -50e3, 90, 5, 50], 'mxh', 1e-5,

'step'])

########################################

#we should really project along the 5 degree direction, but will keep this simple

data = ns.Get_Data_Columns('exchangebias.txt', [0, 3])

plt.axes(xlabel = 'H (kA/m)', ylabel = 'M (kA/m)', title = 'Exchange bias')

plt.plot(np.array(data[0])/1e3, np.array(data[1])/1e3)

plt.show()

21

Ultrafast Demagnetization and Skyrmion Creation in a Co/Pt/SiO2 Trilayer (Advanced)

from NetSocks import NSClient

import matplotlib.pyplot as plt

import numpy as np

ns = NSClient(); ns.configure(True)

########################################

#Setup trilayer

#Co layer

FM = ns.Material('Co/Pt', [512e-9, 512e-9, 2e-9], [1e-9, 1e-9, 2e-9])

FM.modules(['Zeeman', 'demag', 'iDMexchange', 'aniuni', 'heat'])

FM.scellsize([4e-9, 4e-9, 2e-9])

FM.tcellsize([2e-9, 2e-9, 1e-9])

FM.tmodel(2)

FM.curietemperature(500)

FM.pbc('x', 10)

FM.pbc('y', 10)

FM.setangle(0, 0)

FM.setfield(100e3, 0, 0)

#Pt layer

HM = ns.Material('Pt', [0.0, 0.0, -8e-9, 512e-9, 512e-9, 0.0], [4e-9, 4e-9, 4e-9])

HM.modules(['heat'])

HM.tmodel(2)

#SiO2 layer

Substrate = ns.Material('SiO2', [0.0, 0.0, -48e-9, 512e-9, 512e-9, -8e-9], [8e-9,

8e-9, 8e-9])

Substrate.modules(['heat'])

ns.temperature(300)

########################################

#Solver

#RK4 with quartic polynomial extrapolation of demag field

ns.setode('sLLB', 'RK4')

ns.evalspeedup(5)

ns.setdtstoch(20e-15)

########################################

#Set-up display

FM.display('M')

FM.vecrep(3)

ns.displaydetail(1e-9)

########################################

#Ouput data and heat source

ns.setsavedata('ufsky.txt', ['time'], ['Q_topo', FM], ['<T>', FM, [255e-9, 255e-9,

0.0, 256e-9, 256e-9, 2e-9]])

#heat source due to laser spot in Co layer

FM.param.Q = 2e21

FM.param.Q.setparamvar('equation', 'exp(-sqrt((x/Lx - 0.5)^2 + (y/Ly - 0.5)^2) /

((d0/Lx)^2/(4*ln(2)))) * exp(-(t-2*tau)^2/(tau^2/(4*ln(2))))')

22

ns.equationconstants('d0', 400e-9)

ns.equationconstants('tau', 100e-15)

########################################

#Simulate

ns.cuda(1)

#0 to 10ps

ns.setdt(1e-15)

ns.setheatdt(1e-15)

ns.Relax(['time', 10e-12, 'time', 10e-15])

#10ps to 20ps

ns.setdt(4e-15)

ns.setheatdt(2e-15)

ns.Relax(['time', 20e-12, 'time', 10e-15])

#20ps to 60ps

#mid-simulation re-meshing! 1nm cellsize only needed around the Curie temperature.

FM.cellsize([2e-9, 2e-9, 2e-9])

ns.setdt(20e-15)

ns.setheatdt(2e-15)

ns.Relax(['time', 60e-12, 'time', 10e-15])

#60ps to 800ps

FM.tcellsize([4e-9, 4e-9, 1e-9])

ns.setdt(100e-15)

ns.setheatdt(5e-15)

ns.Relax(['time', 800e-12, 'time', 250e-15])

########################################

#Plotting

ns.savemeshimage('ufsky_final')

#now plot |Q| as a function of time

data = ns.Get_Data_Columns('ufsky.txt', [0, 1])

time_ps = [t/1e-12 for t in data[0]]

Qmod = [np.abs(Qval) for Qval in data[1]]

plt.axes(xlabel = 'Time (ps)', ylabel = '|Q|')

plt.xscale('log')

plt.yscale('log')

plt.plot(time_ps, Qmod)

plt.xlim(0.1)

plt.ylim(1e-3)

plt.savefig('ufsky_plot.png', dpi = 600)

plt.show()

23

Tutorial 1 – Introduction

Basics

All commands are entered using the console (top-left black box) in Figure 1.1. Each console

command has a case-sensitive syntax and may have a number of parameters separated by

spaces. If a command is entered with wrong parameters a help prompt will be displayed

explaining the command and full syntax. Alternatively a command may be immediately

preceded by a question mark in order to display the command help. For example try it for the

run command:

?run

Note, the program auto-completes commands entered out of the list of possible inputs – and

equally stops any wrong inputs from being entered.

The main display shows the magnetization configuration (other vector and scalar quantities

may be displayed, but magnetization is the default setting).

Figure 1.1 – Boris interface

24

The magnetization display may be controlled using the mouse: left-click and drag to re-

position the display mesh, middle-click and move mouse to rotate the camera view about the

center of the displayed (focused) mesh, right-click and move mouse up/down to zoom in/out,

or left/right to rotate the camera about its axis; finally the wheel may be used to set the

coarseness of magnetization representation: for large mesh dimensions, each arrow represents

an average of the magnetization in that area.

Various simulation data may be displayed in the data box (top-right box) for convenience –

more on these later. The displayed windows may be resized by dragging their outlines – the

outline appears if you hover the mouse over the edge of a window.

The magnetization display can be reset to the default view using:

center

Simulation Mesh Control

To display the current problem size enter the following command (see Figure 1.2 for

expected output):

mesh

Figure 1.2 – Default mesh configuration

The simulation space consists of one or more named meshes – the default configuration

consists of a single ferromagnetic mesh named permalloy. This has a rectangle with lower-

left corner coordinates of (0, 0, 0) and upper-right corner coordinates of (80 nm, 80 nm, 10

nm). The magnetic discretization cellsize is a rectangular prism with dimensions (dx, dy, dz) =

(5 nm, 5 nm, 5 nm) – a cubic cellsize by default. Thus the permalloy mesh is discretized with

the integer number of cells (16, 16, 2).

To adjust the mesh dimensions you can use the meshrect command. An easy way to bring up

this command, with current fields already entered, is to double-click on the outlined text

containing the mesh rectangle dimensions:

25

This type of outlined text is a special console text called an interactive console object,

allowing a number of user interactions depending on the particular object, including left or

right click, double-click, or drag. The text is also automatically updated to display currently

set values. You can find out what an interactive object does by using shift-click.

Try to resize the permalloy mesh so it has the dimensions (300 nm, 100 nm, 15 nm). Values

may be entered without specifying the units, in which case the applicable S.I. unit is assumed,

or the applicable unit may be entered together with its magnitude specifier (e.g. for a meter

the currently available units are designated as am, fm, pm, nm, um, mm, m, km, Mm, Gm, Tm,

Pm). If entering the unit, do not leave a space between the number and unit.

The magnetic cellsize may be adjusted by double-clicking on the magnetic cell interactive

object, which brings up the cellsize command. Try to adjust the cellsize so it has dimensions

(6 nm, 6 nm, 5 nm). After changing the cellsize its dimensions are automatically adjusted in

order to satisfy the requirement of integer number of discretization cells in each dimension.

Similarly the permalloy mesh may be renamed by double-clicking on the mesh name

interactive object, which brings up the renamemesh command.

In Figure 1.2 you can also see an entry for the supermesh. Its rectangle is not controlled

directly, but depends on the currently set meshes (however you can adjust the supermesh

cellsize). The magnetic supermesh is the smallest simulation space containing all the

currently set magnetic meshes and is useful to compute long-range interactions over several

independently discretized meshes (e.g. supermesh demagnetizing field) – more on this in a

dedicated tutorial.

Basic Simulation Control

The simulation is started and stopped using:

run

stop

26

The stop command simply pauses the simulation without resetting it. To continue from the

stop point simply type run again. To reset the simulation use:

reset

The display refresh frequency can be set using (iter is the number of iterations – remember

you can query to command for full details: ?iterupdate):

iterupdate iter

The simulation may be saved at any point using (do not use a termination, the .bsm

termination is added by default):

savesim filename

If a directory path is not specified, the default directory path is used. To set a default

directory use:

chdir directory

To load a previously saved simulation use:

loadsim filename

Alternatively a simulation file may be dragged into the console area. At any point you can

return to the default program state by using:

default

A uniform magnetization configuration can be set using the following (theta is the polar

angle, phi is the azimuthal angle in spherical polar coordinates):

setangle theta phi

27

A uniform magnetic field can be set using (again use spherical polar coordinates):

setfield Hmag Htheta Hphi

Simulation Modules

Simulation modules typically correspond to effective field terms. These can be managed

using interactive objects by typing the following command:

modules

The default configuration includes the demagnetizing field (demag), direct exchange

interaction (exchange), and applied field (zeeman) – see Figure 1.3.

Figure 1.3 – Default simulation modules

Currently added modules are displayed in green. To add or remove a module left or right-

click on the respective interactive object. Individual modules will be explored in future

tutorials.

Material Parameters

Default simulation parameters are set for permalloy (Ni80Fe20), as Ms = 8e5 A/m, A

= 1.3e-11 J/m, and α = 0.02. To see a list of currently set parameter values use the command:

params

Values may be modified by double-clicking on the respective interactive objects. By default

parameters are constant for each mesh, however they can be assigned temperature and spatial

dependence for advanced simulations (more on this later).

28

Simulation Flow

A basic simulation flow can be programmed by setting a number of stages. Each stage has an

identifier, parameters depending on the identifier, a stopping condition and data save

condition. To show the currently set simulation stages use:

stages

By default the Relax stage type is used (no simulation values changed), with a stopping

condition based on the normalized torque |mxh| < 10-4 (mxh: 0.0001), and no data saving

configured. New stages may be added by double-clicking on the interactive objects at the

bottom – see Figure 1.4.

Figure 1.4 – Simulation stage types

You can delete added stages by right-clicking on them, change the stage type by double-

clicking and editing, and re-arrange the stage order by dragging.

Available stopping conditions include: nostop, iter (stop after a number of iterations), mxh

(stop when |mxh| falls below the set threshold), dmdt (stop when the normalized |∂m/∂t| value

falls below the set threshold), or time (stop after an elapsed simulation time). When a stage

reaches the stopping condition the next stage starts or the simulation finishes.

Some stage types are broken down into several sub-stages (referred to as steps), for example

a field sequence using Hpolar_seq. Try to add a field sequence stage by double-clicking on

the Hpolar_seq interactive object. The default parameters for a field sequence are shown in

Figure 1.5. This consists of a field sequence starting from a field of -100 kA/m along the (90,

0) direction, and stopping at +100 kA/m along the (90, 0) direction – this is given by the

polar and azimuthal angles in degrees, thus (90, 0) is along the x axis. The sequence consists

of 100 field steps, thus the field step is 2 kA/m. The simulation proceeds to the next step

when the |mxh| < 10-4 stopping condition is satisfied.

29

Figure 1.5 – Default parameters for the Hpolar_seq stage type

Exercise 1.1

Set a 160×80×5 nm permalloy mesh (with cubic cellsize of 5 nm) starting from a saturated

magnetization state along the negative x-axis direction. Set a field sequence from -60 kA/m

to +60 kA/m along the x-axis using a step of 2 kA/m and mxh stopping condition of 10-5.

Note: additionally you should configure a static magnetization problem – more on this in

future tutorials. Type ode in the console, then select the LLGStatic micromagnetic equation

by clicking on it, and make sure the SDesc evaluation method is enabled.

Display the applied field and average magnetization in the data box. To do this use the

command:

data

You will see a list of interactive console objects representing possible output data which can

be displayed in the data box or saved to a file. For now just display the applied field (Ha) and

average magnetization (<M>) by right-clicking on the interactive objects (or dragging them

to the data box).

Run the simulation - the magnetization should switch during this field sequence.

30

Tutorial 2 – Data Output

Saving Numerical Data

In order to save numerical simulation data (automated saving of images will be explored in a

future tutorial) you need to set a data saving file, a list of output data, and a saving schedule.

To set output data and a save file use the data command.

Figure 2.1 – Default output data

The default output data file is called out_data.txt and its name may be modified by double-

clicking on the interactive object. The default saving directory is the path to the program

executable file and may be modified by double-clicking on the interactive object.

The default output data includes sstep (stage and step), iter (iteration), time (simulation time),

Ha (applied field), and <M> (average magnetization). This is the order the output data will

appear in the output file as numerical columns. The order may be modified by dragging the

respective interactive objects in the list of output data. New output data may be added by

double-clicking on the interactive objects at the bottom, and set output data may be deleted

by right-clicking on the respective interactive objects in the output list.

Some output data (such as ha and <M>) may be saved in a particular mesh – in this case the

permalloy mesh which is specified using the notation <permalloy>, whilst other output data

do not depend on any particular mesh. Some output data (such as <M>) may also be saved in

a particular rectangle of the named mesh (the rectangle is relative to the named mesh) – by

31

default the entire mesh rectangle is saved, but this can be modified by double-clicking on the

respective interactive object and editing.

Finally, a saving schedule may be set in the simulation stages: use the stages command. Each

stage has a list of possible saving conditions: none (default – do not save), stage (save at the

end of the stage), step (save at the end of each step in the current stage), iter (save every n

iterations), and time (save every t simulation seconds); for iter and time the parameters may

be edited by double-clicking the respective interactive objects.

Exercise 2.1

Set a 160×80×10 nm permalloy mesh (with cubic cellsize of 5 nm) starting from a saturated

magnetization state along the negative x-axis direction. Set a field sequence from -100 kA/m

to +100 kA/m and then back to -100 kA/m in the xy plane, and at 1 degree from the x-axis.

Use a step of 2 kA/m and mxh stopping condition of 10-5. Note: additionally you should

configure a static magnetization problem as in exercise 1.1.

Configure the simulation so it saves output data for a hysteresis loop (applied field and

average magnetization saved after every step).

Before running the simulation save the simulation file. Once the simulation file is saved using

a specified name (and directory path if needed), the next time you don’t need to specify the

file name – simply use savesim without a file name and the previously used file name will be

saved. Note, correct commands previously entered in the console can be recalled using the

arrow keys (invalid commands are not saved). You can also use Ctrl^v to paste text in the

console.

Run the simulation and plot the hysteresis loop (magnetization along the applied field

direction) at the end.

32

Figure 2.2 – Hysteresis loop obtained in exercise 2.1

Further Data Box Control

As introduced in the previous tutorial, output data may also be displayed in the data box for

convenience. The possible output data may be listed as interactive objects by using the data

command, and the listed interactive objects may be displayed in the data box by dragging

them there, or right-clicking on them. Data box entries may be removed by right-clicking on

them in the data box, and they may be re-arranged by dragging them.

If you just want to quickly see the current values of particular data without displaying them in

the data box, bring up an interactive object list using the showdata command and double-

click on the respective interactive objects.

Exercise 2.2

In this exercise you will run the µMAG standard problem #4:

https://www.ctcms.nist.gov/~rdm/std4/spec4.html

a) For this problem we need a permalloy mesh with dimensions 500x125x3 nm. First

initialize the magnetization configuration to a so-called s-state: this may be obtained

by reducing a large applied field to zero along the [1,1,1] direction. For example set a

field sequence starting from 1 MA/m along the [1,1,1] direction, reducing to zero in

33

20 steps – you should use the stricter |mxh| < 10-5 condition this time. Save an image

of the obtained magnetization configuration – see Figure 2.3.

The easiest way to setup Exercise 2.2a is to use a polar field sequence: Hpolar_seq. This

specifies the starting and ending field values using polar coordinates: magnitude, polar angle

and azimuthal angle – thus a starting field of 1 MA/m along the [1,1,1] direction would be

specified (roughly) as 1MA/m, 55, 45. Make sure to specify the ending field value as 0, 55, 45

to keep the field values in the sequence along the same direction. You should also set the

starting magnetization state along the [1,1,1] direction: setangle 55 45.

To save an image of the currently displayed mesh, use the command:

savemeshimage (directory\)filename

Figure 2.3 – Starting s-state for micromagnetics standard problem #4

Exercise 2.2 continued

b) Starting from the s-state, apply a fixed field with magnetic flux density (B-field) of 25

mT directed 170° counterclockwise from the x-axis in the x-y plane. Simulate the

switching event for a duration of 5 ns, saving output data (in particular the average

magnetization and simulation time are required) every 5 ps. Plot the 3 components of

magnetization against time. Remember to save the simulation before starting it. How

do these results compare with published solutions ?

(see https://www.ctcms.nist.gov/~rdm/std4/results.html)

34

c) Repeat part b) but this time for a B-field of 36 mT directed 190 degress

counterclockwise from the x-axis in the x-y plane.

d) For parts b) and c) obtain images of the magnetization configuration when the average

magnetization (x component) first crosses zero – use the output data to determine the

time when this occurs, then run the simulation to stop at this particular time.

Figure 2.4 – a) Results obtained after running the µMAG standard problem #4 with field 1,

and b) magnetization configuration when the average magnetization (x component) first

crosses zero.

a)

b)

35

Figure 2.5 – a) Results obtained after running the µMAG standard problem #4 with field 2,

and b) magnetization configuration when the average magnetization (x component) first

crosses zero.

a)

b)

Making a video from an image sequence

A video may be encoded from a sequence of .png files (e.g. as produced from a simulation

with an image saving schedule) – this functionality is built into the program for convenience;

for advanced image processing you should use an external program. To produce a video file

from an image sequence, use:

makevideo (directory\)filebase fps quality

This makes a video from all .png files which start with the filebase name, including the

directory, at the given fps (frames per second). The quality parameter sets the bit-rate of the

36

output video: 0 for worst quality but smallest size, 5 for best quality but largest size. For the

makevideo command, the files are sorted by their creation time, not alphabetically.

To enable mesh image saving, use the data command then click on the respective interactive

object. You can also edit the mesh image filebase name. The mesh images are saved using the

same save conditions as output data.

Exercise 2.3

For the switching event in exercise 2.2b, set the problem to save mesh images every 10 ps;

also reduce the simulation time to 3 ns and disable data saving. Make a video of the switching

event (40 fps and quality level 3 works well).

If you want to capture only a part of the mesh display window you can set cropping factors.

These are specified as normalized values and are applied whenever an image is saved (either

manually or during a simulation). This is set using:

imagecropping left bottom right top

The left-bottom of the mesh display is (0, 0), whilst the right-top of the mesh display is (1,1).

37

Tutorial 3 – Further Data Output

Exercise 3.1

In this exercise you will compare coercive fields obtained using the full micromagnetics

model with the predictions of the simpler Stoner-Wohlfarth model.

a) Simulate an in-plane hysteresis loop along a 10° direction in a permalloy rectangle

with dimensions 250x50x5 nm using the full micromagnetics model (demag, exch,

and zeeman modules for permalloy) and obtain the coercive field. In order to speed up

the simulation you only need to simulate one branch of the hysteresis loop, e.g.

negative to positive field only, and you should also set a coarse field step up to zero

field (e.g. 10 kA/m), then a fine field step in order to obtain a more accurate switching

field value (use a 500 A/m fine field step or less); use the mxh stopping condition with

a 10-5 threshold. As before you should also configure a static micromagnetics problem

(LLGStatic equation with SDesc evaluation – type ode command in console).

Solution: use two polar field sequences (Hpolar_seq) along the (polar, azimuthal) = (90°,

10°) direction. Start at -200 kA/m and finish at 60 kA/m. This range is just enough to start

from a saturated state and capture the switching field.

i.e. : 1) Hpolar_seq -200kA/m 90 10 0kA/m 90 10 20, 2) Hpolar_seq 0kA/m 90 10 60kA/m 90

10 120

b) Compute the anisotropy energy density (shape anisotropy) and hence obtain the

switching field predicted by the Stoner-Wohlfarth model.

Note, the Stoner-Wohlfarth model predicts the switching field:

2

42

01

1

2

t

tt

M

K

H

S

u

S+

+−

=

µ

, where

( )

3

tan

θ

=t

.

38

Here θ is the angle between the applied field and anisotropy easy axis (formula applicable for

θ between 0 and 45°).

To compute the energy density for a given magnetization orientation, set the magnetization

orientation (setangle) and calculate the energy density terms using the command:

computefields

This command runs the simulation for a single iteration and does not advance the simulation

time – only the currently set simulation modules are refreshed. After running this command

the required energy density term (e_demag) will be available.

c) For the same geometry and applied field direction obtain the hysteresis loop using the

Stoner-Wohlfarth model. Does the coercive field agree with that obtained in part b) ?

To run this you will need to use the demag_N module instead of the full demag

module; the exchange module (exch) is not needed.

The demag_N module computes the demagnetizing field using the simple

approximation Hd, i = -Ni Mi (i = x, y, z). You will need to enter correct values for the

demagnetizing factors Nx and Ny (remembering that Nx + Ny + Nz = 1). These can be

entered using the command params, then editing the values under the Nxy interactive

object.

Calculate Nx, Ny and Nz directly from the demagnetizing field (obtained using the full

micromagnetics model) and also using the demagnetizing energy density values

obtained in part b). Do the values agree, and does the relationship Nx + Ny + Nz = 1

hold?

To obtain the demagnetizing field you will need to update the field using the computefields

command with only the demag module enabled. After this you can display the demagnetizing

field using the command:

39

display

Using this command brings up a list of interactive objects with display options. Click on the

Heff option under the permalloy mesh. This will display the computed effective field. Using

the average effective field value you can obtain a value for the demagnetizing factor along

the set magnetization direction using the expression Hd = -N M.

In order to obtain the average value of the demagnetizing field you can use the command:

averagemeshrect (sx sy sz ex ey ez)

This command returns the average value for the displayed quantity in the currently focused

mesh (the permalloy mesh in this case) when used without parameters. The parameters

specify a mesh rectangle (start and end Cartesian coordinates) which is relative to the

currently focused mesh.

d) Repeat this exercise using the 100x25x5 nm.

e) Repeat this exercise using the 50x25x3 nm.

Another way to calculate the demagnetizing factors is to use the formula:

dd

HM.

2

0

µ

ε

−=

Thus for M along i = x, y, z, you can obtain e_demag, then use:

),,(

2

0

,

zyxiMN

Siid

==

µ

ε

40

Exercise 3.2

Here you will obtain the magnetization dynamics during a switching event and investigate the

effect of the cellsize value on the simulation.

a) Set a 320x160x10 nm permalloy rectangle with magnetization along the length of the

rectangle (set the magnetization towards the left, thus blue coloured). Obtain the

stable magnetization configuration at zero field by reducing the magnetic field from a

large saturation value along x to zero in a number of steps. (e.g. from -50 kA/m to

0). For now use a cellsize of 5 nm.

b) Starting from the magnetization configuration set-up in part a) set a single stage

where you apply a large field along the x direction, opposing the magnetization – use

50 kA/m. As stopping condition using a time interval of 4 ns. Set a saving schedule to

save the simulation time and average magnetization components every 10 ps. For now

use a cellsize of 5 nm. From the saved data plot <Mx> and <My> versus time.

c) Repeat the simulation in b) with cellsize values of 10 nm and 2.5 nm. Plot <Mx> and

<My> versus stage time for the 3 cellsize values. How do the results compare ? Which

cellsize would you recommend to use ?

41

Tutorial 4 – Domain Walls

Generating Domain Walls

An idealized domain wall (using a tanh profile) along the x direction can be generated using:

dwall longitudinal transverse width position

For an in-plane domain wall the longitudinal parameter determines if the wall is head-to-head

(longitudinal = x) or tail-to-tail (longitudinal = -x); Bloch walls may be generated using z or –

z for the longitudinal component. The transverse parameter determines the rotation direction

through the wall – see Figure 4.1 for examples. The width value is the total domain wall

width and position is the starting left-hand-side coordinate of the wall (along the x axis),

relative to the focused mesh rectangle.

Figure 4.1 – Domain walls generated using the dwall command for a mesh with 240nm

length, and:

a) in-plane head-to-head transverse domain wall as dwall x y 240nm 0

b) in-plane tail-to-tail transverse domain wall as dwall -x -y 240nm 0

42

c) Bloch domain wall as dwall z y 240nm 0

d) Néel domain wall as dwall z x 240nm 0

Exercise 4.1

Set a mesh for a permalloy rectangle of dimensions 640x80x5 nm with 5 nm cellsize.

Generate a head-to-head domain wall over this wire. Run the simulation without a stopping

condition and observe how the domain wall is relaxed. You will observe the domain wall

does not remain in the center, but eventually drifts towards one side until it is expelled at one

of the edges. Can you explain why this happens?

Setting Stray Fields

For domain wall mobility calculations and domain wall configuration relaxation problems in

very long wires, it is possible to extend the wires outside of the ferromagnetic mesh by using

external uniformly magnetized magnetic bodies, and to calculate the stray field inside the

mesh, thereby allowing a smaller mesh size – this eliminates the domain wall drift problem

noted in Exercise 4.1. This is done by adding dipole meshes at the left and right-hand-side of

the permalloy mesh with magnetization direction set as a continuation of the magnetization

inside the permalloy mesh, and enabling the strayfield module.

To add a dipole mesh use the following:

adddipole name rectangle

43

A dipole mesh has a uniform magnetization orientation which is not evolved by the ODE

solver but may be handled in a similar manner to ferromagnetic meshes (such as the

permalloy mesh). Thus modules may be added for computation (modules), mesh parameters

edited (params), quantities displayed (display), etc.

You should also exchange couple the ends of the wires to the dipole meshes, thus completing

the approximation of a long wire with a domain wall in the center. To enable this use the

command:

coupletodipoles

Click on the interactive object to enable exchange coupling to the On state – with this flag

turned on all magnetic cells in a ferromagnetic meshes, at the interface with a dipole mesh,

are exchange coupled to the fixed dipole magnetization direction.

Exercise 4.2

a) Set two dipole meshes to the left and right of the permalloy mesh from the previous

exercise, with lengths of 2.56 µm (but same width and thickness). These dipole

meshes should now be visible when using the mesh command – see Figure 4.2. Set

their magnetization orientation in order to extend the head-to-head domain wall

configuration from Exercise 4.1 (use setangle remembering to specify the mesh name

– see ?setangle for details) – note, the dipole meshes do not display anything by

default, you will need to use the display command and click the M interactive object

in order to see their magnetization orientation.

Run the simulation to relax the domain wall configuration to |mxh| < 10-5. What does

the stray field from the dipole meshes look like? (use the display command)

Figure 4.2 – Configuration of dipole meshes for exercise 4.2

44

b) Change the permalloy mesh from part a) to a new size of 640x160x30 nm, making

sure the dipole meshes are also scaled accordingly (to 160nm width and 30 nm

thickness). For the purposes of this exercise you may use a 2D approximation by

setting a cellsize of 5x5x30 nm.

Run the simulation to relax the domain wall configuration to |mxh| < 10-5. What type

of domain wall results?

When changing mesh dimensions with multiple meshes added to the simulation, there are two

options available: i) change just the mesh rectangle required, ii) change the mesh rectangle

required and resize/translate all other meshes in proportion. To change the behaviour of the

program use the following command and click the interactive object to the On state:

scalemeshrects

With multiple meshes, clicking on the mesh name interactive object (e.g. as displayed using

the mesh command) changes the display focus to that mesh and resets camera orientation –

try it. To quickly focus on a mesh without changing the camera orientation you can double-

click on a mesh in the display window.

Exercise 4.3

In this exercise you will calculate the domain wall width for a symmetric transverse domain

wall as a function of wire width and compare it to the values obtained using the domain wall

formula (A is the exchange stiffness – see the params command – and Ku is the anisotropy

energy density).

u

dw

K

A

π

=∆

45

a) Relax domain walls as a function of wire width for a permalloy mesh with dimensions

640 x Width x 5nm, where Width ranges from 40 nm up to 160 nm (simulate at least

4 different values of width). Obtain the domain wall width defined as the half-Ms

width value – i.e. the distance it takes for the longitudinal component to change from

+MS/2 to –MS/2 for a head-to-head domain wall – and compare it to the value

obtained using the formula above (when calculating Ku remember the longitudinal

demagnetizing energy is assumed to be negligible as for a very long wire)

To obtain the domain wall profile you can use the following command:

dp_getprofile start end dp_index

The above command saves numerical data from the currently displayed mesh quantities

(magnetization in this case) along a line starting from the start up to the end Cartesian

coordinates (absolute position values, i.e. not relative to any mesh). The data is saved in

internal data processing arrays – more on these in a separate tutorial. For now just obtain a

magnetization profile through the middle of the wire as (e.g. for the 80nm wire width):

dp_getprofile 0 40nm 0 640nm 40nm 0 0.

The magnetization components are saved in the data processing arrays with indexes starting

at 1 (so 1, 2, and 3), whilst data processing array 0 contains the position value. These can be

saved to a file as numerical columns using the dp_save command as dp_save

(directory\)filename.txt 0 1 2 3. The file will contain the 4 columns as position along the

profile (so x coordinate), Mx, My, Mz.

b) Repeat the exercise for a thickness of 10nm using both a 3D simulation (cubic cellsize

of 5x5x5 nm) and a 2D approximation (cellsize of 5x5x10 nm). How do the width

values compare to those predicted by the formula and are the results obtained using

the 2D and 3D model similar?

46

Tutorial 5 – Domain Wall Movement and Data Processing

In this tutorial you will learn how to obtain a domain wall field-driven mobility curve.

In order to simulate domain wall movement, in addition to setting up a domain wall and

dipole meshes as in the previous exercise, the moving mesh algorithm must be enabled by

setting a “triggering mesh” as:

movingmesh mesh_name

When moving mesh is enabled the magnetization is shifted either to the left or to the right by

one notch at a time in order to keep the average x component of magnetization in the

triggering mesh, <Mx>, within set boundaries. This keeps the domain wall roughly in the

centre of the mesh. When the mesh is shifted to the left or to the right, the data parameter

dwshift is changed. This data parameter is available for saving to file – see list of data

parameters using the data command, as discussed previously. Note, this can also be

displayed in the console using showdata dwshift, or displayed in the data box. By saving the

simulation time and domain wall shift, the domain wall velocity can be calculated using

linear regression.

Since this type of computation is common, there is a shortcut command which sets-up

everything required (adding dipole meshes with exchange coupling to the ferromagnetic

mesh, enabling stray field computation, setting a domain wall and a triggering mesh):

preparemovingmesh (meshname)

Exercise 5.1

a) Set a 320 × 80 × 20 nm permalloy rectangle with 5x5x20 nm cellsize (2D problem).

Enable the moving mesh algorithm and let the domain wall relax in zero field.

b) Set a simulation stage with a field sequence starting from 500 A/m to 2000 A/m in

500 A/m steps, keeping each field step for exactly 2 ns. Set a data save file, and make

sure you save the stage time and dwshift parameters every 50 ps.

47

c) For each field step extract the gradient of the dwshift vs time and plot the wall

velocity as a function of field. How does the velocity compare with that predicted by

the formula below? (Ku is the in-plane anisotropy energy density)

(m/As)221276where

K

A

Hv

e

u

dw

≅==

γµγ

α

γ

0

,

Console Data Processing

There are a number of built-in commands which allow for a number of operations to be

performed on data processing arrays.

First of all, it is possible to load tab-spaced data from a file (such as the data files produced

by a Boris simulation) into the internal data processing arrays. This is done using the

following command:

dp_load filename filecol1 … dp_arr1 …

The above command loads entire columns from the specified file. Thus if the file has a

number of tab-spaced data columns, we can load the column with number filecol1 from the

file into the internal data processing array with number dp_arr1 (these indexes are numbered

from 0 up). Multiple columns can be loaded in one command.

Some common data processing commands are listed below.

To multiply a data processing array by a constant value use the following command:

dp_mul dp_source value dp_dest

The above command multiplies the data processing array with index dp_source by the

specified value and stores the result in the dp_dest data processing array (this can be the same

as dp_source). This can be used to normalize data (e.g. a hysteresis loop).

48

We can use the built-in linear regression command to extract the velocity values:

dp_linreg dp_x dp_y (dp_z dp_out)

The above command performs linear regression on the data stored in dp_x and dp_y data

processing arrays and outputs the extracted gradient values and intercepts together with their

uncertainties. If dp_z is specified multiple linear regressions are performed by using the

values in the dp_z array to identify adjacent points to be included in a single linear regression;

e.g. dp_z would contain the applied field values. In this case the outputs are placed in 5 data

processing arrays starting at dp_out as follows: 1) unique dp_z values, 2) gradient, 3) gradient

error, 4) intercept, 5) intercept error.

We can also save our processed data, e.g. the domain wall velocity curve, using:

dp_save filename dp_arr1 …

Exercise 5.2

Use the console data processing commands to process the output data from Exercise 5.1 and

save a domain wall velocity curve.

Other notable commands include:

dp_coercivity dp_x dp_y

dp_remanence dp_x dp_y

These commands can be used on data from a simulated hysteresis loop in order to extract

coercivity and remanence values.

The data processing arrays may be cleared using:

dp_clear dp_arr1 …

49

This clears data in the specified data processing arrays. If no parameters are included all data

processing arrays are cleared.

Exercise 5.3

Continuing Exercise 1, find the Walker breakdown threshold with a resolution of 100 A/m

starting at 100 A/m. Compare the velocity values with that predicted by the formula in

Exercise 5.1c for the steady domain wall movement regime. What is the Walker breakdown

threshold?

Exercise 5.4

Calculate the field-driven domain wall mobility curve for permalloy, with a resolution of 100

A/m, as a function of Gilbert damping, for values of damping 0.005, 0.01 and 0.015. How

does the Walker breakdown threshold compare with the value predicted by the formula

below? (Ku,op is the out-of-plane anisotropy energy density)

)/(

20

,mA

M

K

H

S

opu

W

µ

α

=

50

Tutorial 6 – ODE Control and Setting Shapes

Setting an ODE solver

The differential equation to solve and its evaluation method is configured using the following

command:

ode

The default equation is the Landau-Lifshitz-Gilbert (LLG) equation which you have been

using so far. Other equations may be set, e.g. LLB for temperature-dependent simulations,

which will be covered in other tutorials.

There are a number of evaluation methods which you can select. The fixed-step methods

available are: Euler (1st order), trapezoidal Euler (TEuler – 2nd order) and Runge-Kuta (RK4 -

4th order). The adaptive time-step methods are the adaptive Heun (AHeun – 2nd order), the

multi-step Adams-Bashforth-Moulton (ABM – 2nd order), Runge-Kutta-Bogacki-Shampine

(RK23 – 3rd order with embedded 2nd order error estimator), Runge-Kutta-Fehlberg (RKF45 –

4th order with embedded 5th order error estimator), Runge-Kutta-Fehlberg (RKF56 – 5th order

with embedded 6th order error estimator), Runge-Kutta-Cash-Karp (RKCK45 – 4th order with

embedded 5th order error estimator), and Runge-Kutta-Dormand-Prince (RKDP54 – 5th order

with embedded 4th order error estimator).

There is also a Steepest Descent solver (SDesc) used for static problems where we don’t need

the magnetization dynamics, but are merely interested in calculating the ground state (e.g.

relaxing a magnetization configuration and hysteresis loops). The steepest descent solver is

only enabled for the LLGStatic equation, which is the LLG equation with the precession term

disabled and damping value set to 1. In this case the SDesc solver is typically at least an order

of magnitude faster in computing the ground state compared to the other methods.

For most methods the calculated mxh and dmdt stopping condition values are the maximum

|mxh| value (normalized torque) in any given iteration. The exceptions are the Euler, TEuler,

and AHeun methods, which are only ever used in practice for stochastic equations when

51

including a thermal field (more on this later); additionally RK4 may also be used for

stochastic equations. For this reason the |mxh| values for these are the mesh averages in any

given iteration. This allows using these methods with an mxh stopping condition for

stochastic equations.

As an exercise we will briefly investigate here the stability of the fixed-step methods as the

time step is changed. The time step may be set using:

setdt dt

Here dt is the time value in seconds. For the adaptive time step methods this command sets

the starting time step.

Exercise 6.1

a) Set a 320 × 160 x 20 nm permalloy mesh with a 5 nm cubic cell and relax the

magnetization to |mxh| < 10-5.

b) Set a stage with a magnetic field with components (40 kA/m, 5 kA/m, 0) for 5 ns and

save data every 10 ps. Use the RKF45 method. Record the actual computation time

required to complete the simulations. Plot the magnetization switching dynamics.

c) Repeat the simulation using the RK4 method for fixed time steps of 0.5 ps, 0.7 ps, 0.9

ps and 1.1 ps. Record the actual computation time required to complete the

simulations. Compare the results with the reference results from the RKF45 method.

How do the results change and why?

d) Compare the computation times. Which method is more efficient whilst still

maintaining accuracy?

e) Investigate the computation time required to complete the same problem with TEuler

with a time step of 50fs and Euler with a time step of 30as.

52

Setting shapes

Until now we’ve mostly considered meshes which are filled with magnetic cells. In general,

complex shapes may be generated by masking the mesh using a shape in an image file. A

more advanced method of setting shapes using numerical ovf2 files is covered in a later

tutorial. This is achieved using the following command:

loadmaskfile (zDepth) (directory\)maskfile

In the simplest case the image file defines a 2D shape in black and the void cells in white –

see Figure 6.1 for an example. Instead of using the command you may also drag and drop the

image in the mesh viewer window. The zDepth value defines the depth the mesh is voided to

from top down (if zDepth > 0), or the height the mesh is voided to from bottom up (if zDepth

< 0); this may be used to define 3D shapes with the maskfile being a grayscale image.

Figure 6.1 – Setting a mesh shape using a mask from a png file

Exercise 6.2

a) Load an ellipse into a 320×160×10 nm mesh. Try not to leave void cells at the sides.

You should use a circle mask as this will be stretched over the defined mesh

rectangle.

b) Obtain hysteresis loops between -50 kA/m and 50 kA/m for the ellipse along the x

axis and along the y axis separately using a cellsize of 5x5x10 nm (2D simulations).

Drag png image file

to mesh viewer to

apply mask

53

You may use the relaxation condition |mxh| < 10-4 in order to speed up the exercise.

Use the RKF45 evaluation method. How do the two hysteresis loops compare ?

c) Obtain further hysteresis loops for ellipses with dimensions 260x160x10 nm and

200x160x10 nm. Compare results for all the simulated ellipses and explain the

changes in the hysteresis loops.

There is a modifier for how shapes are applied to a mesh, accessed using the individualshape

command. By default this flag is Off, and any shape applied to the mesh is set for all relevant

computational quantities. Thus M (magnetization) is a computational quantity which may be

shaped, but also T (temperature), and σ (electrical conductivity) may be shaped for the

relevant solvers (micromagnetics, heat equation, and transport solver respectively). Note that

in order to shape T and σ you need the relevant solvers enabled. If instead you only want to

apply the shape to one of these quantities, or even have different shapes for all of them, you

need to enabled the individualshape flag, then apply the mask. This is useful for example if

you want to include non-magnetic components (e.g. contact leads) in the magnetic mesh.

You may reset the mesh back to its solid shape using:

resetmesh

This command is also useful to recover the mesh following a wrongly-posed computation –

e.g. if too large a time-step is used the magnetization values will become NaN (not a number)

and must be reset. Other methods to shape a mesh include setting and deleting rectangles

using:

delrect rectangle (meshname)

addrect rectangle (meshname)

54

Tutorial 7 – Magnetocrystalline Anisotropy

In this tutorial you will learn how to use the anisotropy module and simulate hysteresis loops

for different magneto-crystalline anisotropy configurations. You need to be familiar with all

the basic tutorials.

There are two options available for adding magneto-crystalline anisotropy to the

computations: uniaxial or cubic. These are enabled by choosing the aniuni or anicubi

modules from the list displayed using the modules command. The modules are mutually

exclusive, thus enabling one will delete the other one from the list of active modules.

The strength of the anisotropy is controlled using the K1 and K2 parameters (K1 and K2 are

the anisotropy energy density constants) from the list displayed using the params command.

These are the constants that appear in the anisotropy energy formulas:

Uniaxial anisotropy:

( )

[ ]

( )

[ ]

2

1.1.

1a

aKK e

m

em −+

−=

ε

Cubic anisotropy:

[] [ ]

222

2

222222

1

γ

βαγβγαβα

ε

KK +++=

, where

( )

2121

.,.,. eememem ×===

γβα

.

We also need to define the anisotropy symmetry axes. For uniaxial anisotropy we only have

one symmetry axis and this is set using the ea1 parameter by giving the Cartesian

components of the unit vector ea (e.g. default is 1, 0, 0 for easy axis along the x-axis). For

cubic anisotropy we need two symmetry axes directions, ea1 and ea2 which should normally

be orthogonal.

In the following you will investigate the effect of magnetocrystalline anisotropy on hysteresis

loops in circular dots.

55

Exercise 7.1

a) Set a 160 × 160 × 5 nm permalloy circle with 5 nm cellsize using a mask file. Set a

uniform magnetization along the x direction towards the left (blue state). Set uniaxial

anisotropy with K1 = 10 kJ/m3, K2 = 0 J/m3 and easy axis along x direction.

b) Simulate hysteresis loops along the x-axis (easy axis), y-axis (hard axis) and in

between along a 45° in-plane direction (remember you will need to use a polar field

sequence for this). You will need to determine appropriate field sweep ranges so the

loops start from a saturated magnetization state.

c) Plot the hysteresis loops using the normalized magnetization (divide by MS value –

the saturation magnetization constant). What are the coercivity and normalized

remanence values? Explain the difference between the loops.

Exercise 7.2

Repeat the simulations in Exercise 7.1, but this time set cubic anisotropy with K1 = 20 kJ/m3

and K2 = 0 J/m3, with two perpendicular easy axes in the plane (e.g. x-axis and y-axis).

Plot the resulting hysteresis loops and compare them with the previous results.

Hints:

For the 45° direction you will need to project the magnetization along the applied field

direction. You can do this by taking the dot product of M with the applied field direction unit

vector:

Mh.

ˆ

=

H

M

You can do this using console data processing with the command:

56

dp_dotprod dp_vector ux uy uz dp_out

Here dp_vector is the dp array index such that the dp arrays dp_vector, dp_vector + 1,

dp_vector + 2 hold the x, y, and z components of the magnetization, (ux, uy, uz) are the

components of a vector (dot product taken with this vector), and dp_out is the dp array where

the output is placed.

Finally, you can normalize the magnetization using the dp_div command where you will

need to divide by MS. To see the value of MS you can look it up in the list displayed using the

params command.

Remember you will first need to load into dp arrays the appropriate columns for the saved

hysteresis loop data file using the dp_load command. You can save the contents of dp arrays

after processing data using the dp_save command. As always you can find more details about

a command by preceding it with the ? symbol, e.g. ?dp_load.

57

Tutorial 8 – Anisotropic Magneto-Resistance

In this tutorial you will learn how to simulate magneto-resistance loops and calculate charge

current densities using the transport module.

Transport Module Basics

The transport module is a complex spin and charge current solver (electron transport),

allowing for a number of physical effects to be included in the magnetization dynamics

problem, including Zhang-Li spin-transfer torques based on calculated charge currents, spin

torques based on computed spin accumulations in multilayers, direct and inverse spin Hall

effects (SHE and ISHE), spin pumping torques, anisotropic magneto-resistance (AMR),

current-perpendicular-to-plane giant magneto-resistance (CPP-GMR), Oersted fields and

Joule heating.

Here we will look at how a simple charge current density may be computed and AMR

included in the simulation.

You will first need to enable the transport module from the list displayed using the modules

command for the meshes where you want a charge current density to be computed. In the

simplest case the computation is reduced to obtaining J, the charge current density, using

Ohm’s law : J = σE, where σ is the electrical conductivity and E = -∇V is the electrical field

with V being the electrical potential. If σ is constant this reduces to a Laplace equation for V.

The electrical conductivity, potential and charge current density are available as display

outputs under the display command (elC, V and Jc).

The base electrical conductivity value may be changed by editing the elC mesh parameter

displayed using the params command.

Similarly AMR may be enabled by editing the amr mesh parameter (0% by default which

disables it). A typical value for permalloy is 2%. Enabling AMR results in a non-uniform

electrical conductivity and now the equation for V becomes a Poisson equation.

58

Before starting a computation you will need to define at least 2 electrodes – these set

Dirichlet boundary conditions for V (fixed potential values) in the Laplace/Poisson solvers.

The most common electrode configuration is to define two electrodes at the x-axis ends of the

mesh (so in the y-z plane). You can do this using the command:

setdefaultelectrodes

To see which electrodes have been defined use the command:

electrodes

You will see two electrode rectangles. The electrode rectangles are in absolute values, so not

relative to any particular mesh. You can add new electrodes but they must always be placed

at the edges of a mesh rectangle – when initializing the simulation Dirichlet boundary

conditions will be flagged for the boundary cells of the mesh intersecting the electrode

rectangle. Each electrode has a fixed potential which may be edited. Exactly one of the

electrodes has to be designated as the ground – this is the electrode where the outgoing total

electrical current is calculated.

You can edit the individual electrode potential values, however a more common scenario is

the set a single electrical potential drop from the ground to the other electrodes using:

setpotential potential

This sets a single inversely-symmetrical potential drop (i.e. +potential/2 to -

potential/2). The inversely-symmetrical potential drop minimizes floating point errors (as

opposed to setting a potential drop of potential to 0).

Simulations may use the constant-voltage or the constant-current mode (the interactive object

displayed when using the electrodes command may be toggled between these two states).

Normally you would use the constant-voltage mode; with constant-current the electrode

potentials are adjusted during the simulation to maintain a constant current (which may be set

using the setcurrent command). In this tutorial we will be using the constant-voltage mode.

59

Exercise 8.1

Set a permalloy mesh with dimensions 320x320x10 nm and mask it using a circle shape (in

the image file make sure to not leave any white spaces at the left and right sides).

Enable the transport module, set the default electrode configuration and a potential of 10 mV.

In the data box display the average current density (Jc), set potential (V), total current (I) and

resistance (R). In the mesh display the current density. Run the simulation. The calculated

current density should look similar to that in Figure 8.1.

Figure 8.1 – Computed charge current density for Exercise 8.1

For more advanced simulations you can add electrodes using the addelectrode command.

Electodes may be deleted using the delelectrode command (or more simply by right-clicking

on an existing electrode in the list displayed by the electrodes command. All electrodes may

be deleted by using the clearelectrodes command.

Further info:

When displaying the meshes (use the mesh command) you will now notice a value for the

electric cell. This is the discretization cellsize used by the transport solvers. Normally this

should be equal to the magnetic cellsize (default setting) but can be controlled separately for

more advanced simulations (decrease computation time or increase computation accuracy as

required). Multiple meshes with transport modules enabled may be configured. If the meshes

60

are touching, composite media boundary conditions will automatically be inserted in the

computation, however we still typically require 2 electrodes for a well-posed problem.

Exercise 8.2

Set a permalloy rectangle with dimensions 300x100x20 nm. Calculate the current density for

the default electrodes setting by using a potential drop of 1 V.

What is the computed sample resistance and does it agree with that predicted by the formula:

A

l

R

ρ

=

,

where l is the length, A the cross-sectional area and

ρ

is the resistivity.

What is the total current, and does it agree with the expected value for a 1 V potential drop?

What is the average current density and does it agree with the expected value for the total

current?

Further info:

For advanced simulations the accuracy of the transport solver may be controlled by using the

command:

tsolverconfig

This command displays the set convergence error (a value around 10-6) is normally a good

compromise between accuracy and computational speed. In this case the Laplace/Poisson

solvers stop iterating when the maximum change in V from one iteration to another,

normalized to the set potential drop, drops below this set convergence value. You can display

the ts_iter (current number of transport solver iterations) and ts_err (current transport solver

error) data in the data box. If the convergence error threshold is too low the transport solver

will take a large number of iterations during computations – if AMR, GMR, ISHE or

61

temperature-dependent transport parameters are enabled the transport solver must update

after every magnetization and/or heat solver time step.

The transport solver may be used to calculate magneto-resistance loops when an AMR

(anisotropic magneto-resistance) value is set. Since this is a static problem it is best to set the

LLGStatic equation with the SDesc solver. When using the SDesc solver the stopping

condition (mxh or dmdt) should be much lower than usual, typically at least 10-6 or even

lower. You should also set the static transport solver flag to On (see tsolverconfig command

output). Setting this flag to On will only iterate the transport solver at the end of a schedule

step, thus resulting in faster computation. In this case you should also increase the iterations

timeout to at least 5000 or more to ensure the transport solver converge threshold is met.

For problems where the current density is uniform a transport solver converge threshold of

10-6 is sufficient, however this may have to be decreased to 10-8 or even 10-9 if the current

density is highly non-uniform.

In the following problem you should use both the static LLG equation and static transport

solver as explained above.

Exercise 8.3

Here you will obtain longitudinal and transverse magneto-resistance loops.

a) Set a permalloy rectangle with dimensions 160x80x10 nm. You can use a 2D

simulation with magnetic cellsize 5x5x10 nm, but the transport solver should be left

with cubic electric cellsize of 5 nm.

Sweep the field from -100 kA/m to +100 kA/m strength and back, using a field step of

at most 1 kA/m, along a nearly longitudinal direction (use 5° from the x-axis). You

should use a Hpolar_seq sequence, and an mxh stopping condition of 10-7.

Set the transport solver with default electrodes, a non-zero potential drop (e.g. 1 mV),

and amr = 2% (see params).

62

In the output data make sure to save the applied field and sample resistance every

step. You should also save the transport solver error to check the converge threshold

has been met at the end of each step (ts_err).

Run the simulation and plot the obtained MR loop. Explain your results.

b) Repeat the simulation along the in-plane nearly transverse direction (use 85° from the

x-axis) by sweeping the field between -100 kA/m and +100 kA/m strength with a step

of 1 kA/m.

Estimate the AMR percentage from the obtained H vs R loops from parts a) and b) –

does it agree with the set 2% value?

Figure 8.2 – Longitudinal and transverse MR loops due to AMR as calculated in Exercise

8.3.

63

Tutorial 9 – Scripting using Python

In this tutorial you will learn how to use scripts to automate multiple simulations. Boris can

communicate with an external program via network sockets, so both local and remote script-

based control is possible. For this purpose a Python module (NetSocks) is provided, allowing

use of Python scripts to communicate with Boris. This is contained in the NetSocks.py file

and requires Python 3.7 or above.

It is strongly recommended Python scripts are run using a Python IDE and not directly from

the terminal (although see below for embedded Python script execution). This should be

configured with a search path to find NetSocks.py (on Windows this is located in

Documents/Boris Data/BorisPythonScripts folder, on Linux-based operating systems

NetSocks.py is located in the executable folder).

Alternatively Boris can load a Python script directly in emebbed mode and invoke the Python

interpreter. Whilst this is an advanced user mode, it has a number of advantages: 1) Scripts

may be launcher from the terminal directly, 2) Python methods may be injected into the

simulation, which are called every simulation iteration efficiently, allowing the user to define

custom data saving and simulation flow control algorithms, 3) script execution in the setup

phase is much faster (runtime execution speed not affected), 4) when multiple GPUs are

present, Python for loops may be parallelized easily when running in CUDA mode, which is

useful for parameter-scan simulations.

The scripts work by invoking console commands, with syntax identical to that you would use

when typing commands directly in the console. Some console commands also return values,

which can be read by Python scripts. You can find out if a console command is set to return

values by looking at the USAGE help for it. For example type the following in the console:

dp_coercivity

In the USAGE help you can see this command will return the calculated coercivity values

and uncertainties.

64

NetSocks Python Commands Usage

Look through the examples in Tutorial 0. To create a new simulation script:

1. Place the NetSocks.py file in the Python include path and import NSClient from

NetSocks.

2. Next a NSClient object, ns, must be created as:

ns = NSClient()

The default configuration is for a local host (‘localhost’), i.e. the script must be on the

same computer as the Boris running instance. For remote clients and other options see

below. Now all communication is done through methods in the ns object.

3. After creating the NSClient object, it should normally be configured as:

ns.configure(True)

This function sets the working directory the same as the Python script directory, so

any simulation outputs are saved in this directory. If calling the function with True,

the program also resets back to the default starting state. If you don’t want to reset to

default call with False.

See BorisScriptTemplate_Simple.py script in this tutorial folder. All commands can be sent

through the ns object as ns.command(parameters…), for example ns.setangle(90, 0) sets

magnetization direction along the x axis, etc.

You can obtain returned parameters as data = ns.command(parameters…), for example Mav

= ns.showdata(‘<M>’) returns the average magnetization as a list with 3 elements.

In many cases the script must wait for a simulation to finish before proceeding. This is

achieved by using the Run function:

65

• ns.Run()

This is a blocking function call, which sends the run command, then waits for the simulation

to finish by listening to messages from the running program.

NSClient also has a useful method for writing data to a file:

• ns.SaveDataToFile(‘Results.txt’, [data1, data2, …])

This command appends a new line in the Results.txt file (in the script directory) which

contains two numbers: the x and y components of magnetization.

Another useful built-in command is Get_Data_Columns, which loads tab-spaced values from

a file into a list (not numpy array), as they would be saved by a simulation. You can also

specify a set of column indexes to load if you don’t want to load all the output data. This is

used as:

• data = ns.Get_Data_Columns(‘OutputData.txt’, [0, 1, …])

There are 2 simple plotting helpers, which are just wrappers for matplotlib code designed to

generate simple plots in just one line of code (which may be enough for many cases):

• ns.Plot_Data(x, y, xlabel, ylabel, title, label, imageFile)

• ns.PlotPolar_Data(r, theta_deg, xlabel, ylabel, title, label, imageFile)

Finally for working with OVF2 files 2 functions are provided, which work with numpy

arrays:

66

• ns.Write_OVF2(filename, vec, nodes, rect_m)

• vec, nodes, rect_m = ns.Read_OVF2(filename)

In the above vec is a numpy 1-dimensional array, nodes = [Nx, Ny, Nz] contains the number

of cells along x, y, z, and rect_m = [sx, sy, sz, ex, ey, ez] contains the mesh rectangle start

and end coordinates in metres.

Look through the examples in Tutorial 0 and run them. Examples of other scripted

simulations will be contained in further tutorials.

Multiple Remote Clients and Advanced Options

You can control a remote Boris instance by specifying the correct IP address and port as:

• ns = NSClient(scriptserverip, scriptserverport, cudaDevice)

The server port can be set in the required Boris instance (serverport command). This allows

controlling multiple Boris instances from the same script, where each instance is assigned a

different server port – for example, if you have multiple GPUs on the same workstation you

can connect in localhost mode, but have each Boris instance configured to listen on a

different port. You can also control multiple Boris instances in a cluster in the same way, or

even multiple instances on a multi-GPU workstation by specifying the required CUDA

device (cudaDevice = 0, 1, 2, …).

NetSocks includes functionality for executing simulations in parallel, which is useful for

parameter-scan type of simulations when multiple GPUs are available. In this case

NSMultiClient should be used instead of NSClient:

• nsm = NSMultiClient(scriptserverports, cudaDevices)

67

NSMultiClient is a wrapper class for a list of NSClient objects, each with a different server

port and cuda device assigned through the scriptserverports and cudaDevices lists

respectively. A simulation method should be defined, which must take an NSClient object as

first argument, followed by any number of arguments. To launch multiple simulations use the

Run method in NSMultiClient, by passing it the simulation method name, followed by a

number of arguments. These arguments must be lists of same length – the length determines

the number of simulations to run – and the number of arguments must match that expected by

the simulation method. As an example see the BorisScriptTemplate_Multiple.py example

script below. Before running it, there must two Boris instances running, each configured to

listen to the respective ports (1000, and 1001), assuming at least 2 GPUs are present.

BorisScriptTemplate_Multiple.py:

from NetSocks import NSMultiClient

nsm = NSMultiClient(scriptserverports = range(1000, 1002), cudaDevices = range(0,

2))

#reset to default, but turn off verbosity so messages do not clash due to parallel

execution

nsm.configure(True, False)

####################################

#example simulation method

#this must take an NSClient object as first argument, and any other number of

arguments after

def simulate(ns, H, Ms, angle):

FM = ns.Ferromagnet([200e-9, 100e-9, 20e-9], [5e-9])

FM.param.Ms = Ms

FM.setfield(H, 90, angle)

FM.addpinneddata('Ha')

ns.cuda(1)

ns.Relax(['iter', 10000])

####################################

#example parameters passed to simulate function

H = [10e3, 20e3, 30e3, 40e3]

Ms = [600e3, 700e3, 800e3, 900e3]

angle = 45

#execute simulations (4 in total) in parallel using all configured devices (2

devices, thus 2 simulations each)

nsm.Run(simulate, H, Ms, [angle]*len(H))

Note that we used nsm.configure(True, False). The second parameter sets the script verbosity

to False, which is recommended with multiple clients to avoid a flood of messages.

68

Server Configuration

You can set the server port using: serverport port

You can set the server password using: serverpassword password

You can set the server response time using: serversleepms time_ms

The server polls the configured network socket for messages every time_ms ms. The default

value is 5 ms. Lower values will increase server responsivity, but will increase the CPU load

which could impact the execution time of some CPU-based simulations.

Embedded Python Scripts

In embedded mode Boris invokes the Python interpreter, thus communication through

network sockets is not required. This presents a number of advantages as shown below. To

run a script in embedded mode, then configure the NSClient object:

• ns = NSClient(embedded = True)

In emebbed mode the output from print commands will appear in the Boris console. Also

matplotlib plots will be redirected to sequentially numbered png files, saved in the simulation

script directory, instead of being shown. This requires the custom matplotlib backend

module, boris_matplotlib_backend.py, to be in the same directory as NetSocks.py.

Embedded Python scripts may still be launched from a Python IDE, or alternatively directly

from a terminal through the Boris executable. Command line arguments are shown in the

table below.

69

Command Line Arguments

Description

-help / -h

Show help. Doesn't launch program.

-version

Show program version. Doesn't launch program.

-p serverport (password)

Set script server port and optional password. Default port 1542, and no

password.

-g gpu_select

Set gpu selection:

gpu_select = -1: automatic gpu selection (default).

gpu_select = 0, 1, 2, ...: select indicated gpu if available.

-s python_script_file

Launch program and execute indicated Python script file (doesn't have

to include termination). If path not given then use the default

Simulations directory. Other flags shown below modify execution

behaviour of Python script.

-sp stride offset1 (offset2, (…))

When running Python script from console (specified with -s flag),

specify how for loops should be executed in parallel.

Embedded Python scripts will attempt to run for loops in parallel, if

they are immediately preceded by the directive: #pragma parallel for

Default behaviour is to split the for loop iterations evenly between all

available GPUs. This setting allows specifying the stride and offsets to

be executed by this program instance, in this order: stride offset1 offset2

…. Stride is the total number of GPUs or machines used to execute the

script, offset is any number between 0 and stride - 1.

Example scenario: 2 machines are available, first with 2 GPUs and a

second with 1 GPU. In this case 2/3rd of the work should be executed

on the first machine, and 1/3rd on the second. Thus use a stride of 3 and

give offsets 0 and 1 to the first machine, and offset 2 to the second.

Thus on the first machine launch: Boris.exe -s script.py -sp 3 0 1, and

on the second machine launch: Boris.exe -s script.py -sp 3 2. Note, the

first machine will automatically split the two offsets between the two

GPUs. Do not assign more offsets than there are GPUs on any machine.

-sd 0/1

After execution of Python script from the command line on a multi-

GPU machine, delete any temporary Python script files generated : 1

(default), or keep: 0.

When running in embedded mode it is possible to inject Python code into the simulation loop.

This takes the form of a Python function which is executed by Boris after every simulation

iteration, and enables the user to implement custom advanced execution algorithms and data

saving. The function, together with any parameters, should be passed to the Run method. The

function must return the input parameters (which can be updated during the function

70

exection), preceded by an integer value 0 or 1. This first return parameter determines if the

simulation will keep iterating (1) or stop (0). As an example see

BorisScriptTemplate_Embedded.py, where the simulation stops as soon as a magnetization

switching event is detected, and saves data when the change in magnetization along the

applied field direction, between data save events, exceeds 5 kA/m. Whilst there is a small

cost to executing the Python function every iteration, this is negligible except for very small

simulations with fast iteration.

BorisScriptTemplate_Embedded.py:

from NetSocks import NSClient

ns = NSClient(embedded = True); ns.configure(True)

####################################

Py = ns.Ferromagnet([160e-9, 80e-9, 10e-9], [5e-9])

def check_switching(Mx_previous, Mx_previous_save):

M = Py.showdata('<M>')

if M[0] * Mx_previous < 0.0:

print("Switching detected. Stopping simulation.")

return 0, M[0], Mx_previous_save

#trigger a save when Mx has changed by more than 5 kA/m

if M[0] - Mx_previous_save > 5e3:

Mx_previous_save = M[0]

ns.savedata()

return 1, M[0], Mx_previous_save

Py.setfield(4e4, 90, 0)

Py.setangle(90, 170)

ns.setsavedata('Mswitching.txt', ['time'], ['<M>', Py])

Mx = Py.showdata('<M>')[0]

ns.Run(check_switching, Mx, Mx)

Another possibility with embedded Python scripts, is to parallelize for loops for multi-GPU

workstations. This is useful for parameter-scan type of simulations. The advantage over the

un-embedded method of multi-GPU parameter scans is the simplicity of use. Only one Boris

instance is required, and a for loop may be parallelized simply by preceding it with the

#pragma parallel for comment. When this is encountered, all available GPUs will be used,

with for loop iterations shared between them. As an example, which is an alternative to the

un-embedded method discussed above, see the BorisScriptTemplate_Multiple_Embedded.py

script.

71

BorisScriptTemplate_Multiple_Embedded.py:

from NetSocks import NSClient

ns = NSClient(embedded = True)

####################################

#example simulation method

def simulate(H, Ms, angle):

ns.configure(True)

FM = ns.Ferromagnet([200e-9, 100e-9, 20e-9], [5e-9])

FM.param.Ms = Ms

FM.setfield(H, 90, angle)

FM.addpinneddata('Ha')

ns.cuda(1)

ns.Relax(['iter', 10000])

####################################

angle = 45

#pragma parallel for

for (H, Ms) in zip([10e3, 20e3, 30e3, 40e3], [600e3, 700e3, 800e3, 900e3]):

simulate(H, Ms, angle)

Multiple for loops may be parallelized in the same script. By default all available GPUs will

be used, however if you want to use only a subset of avbailable GPUs then you should launch

the script from a terminal by specifying the required GPUs using the -sp flag. For example, if

4 GPUs are available, to use only the first 2 GPUs the script should be launched as:

Launching the BorisScriptTemplate_Multiple_Embedded.py script with 2 GPUs only, on

Windows:

Boris.exe –s BorisScriptTemplate_Multiple_Embedded.py –sp 2 0 1

Launching the BorisScriptTemplate_Multiple_Embedded.py script with 2 GPUs only, on

Linux-based OS:

./BorisLin –s BorisScriptTemplate_Multiple_Embedded.py –sp 2 0 1

If all GPUs are required, then you could simply launch the Python script from inside a Python

IDE.

72

Tutorial 10 – Current-Induced Domain Wall Movement

In this tutorial you will learn how to obtain current-induced domain wall velocity curves.

First we will simulate these using only console commands, including processing of output

data, then we will use a Python script to more accurately determine the domain wall velocity.

The simplest available method for enabling spin torques is to use the Zhang-Li spin-transfer-

torque (STT) formulation. In this formulation the calculated charge current density is used to

calculate the following spin torques on the magnetization (included as additive terms in the

normalized LLG equation):

() ( )

[ ]

mum

m

u∇×

−∇

=..

β

STT

T

where

2

1

β

µ

+

=e

M

P

B

S

C

Ju

To enable this use the ode command and select the LLG-STT equation with the RK4

evaluation method. You will also need to enable the transport module (use the modules

command and select the transport module). As before you need to set two electrodes

(setdefaultelectrodes) and enable the moving mesh algorithm (preparemovingmesh).

In the above equation we have two new parameters: P, the charge current spin polarisation,

and β, the STT non-adiabaticity parameter. These can be edited by using the params

command and double-clicking on the respective interactive console objects; the default

values are set for permalloy.

Exercise 10.1

a) Set a 320 × 80 × 10 nm permalloy rectangle with 5x5x5 nm cellsize (3D problem).

Enable the moving mesh algorithm and let the domain wall relax in zero field. Enable

the transport module, set the default electrode configuration and enable the LLG-STT

equation with the RK4 evaluation method.

73

b) Set a simulation schedule to vary the current density from 1011 A/m2 up to 1012 A/m2

in 10 steps, saving the domain wall shift (dwshift), current (I), voltage (V), and charge

current density (Jc) output data. Each current density value should be maintained for 5

ns with a data saving schedule every 10 ps.

Hint: You should use the V_seq simulation schedule stage. This sets a sequence of voltage

values with given start and stop values in a number of steps. Since we have a simple

geometry you can calculate the required voltage values using the formula:

σ

C

Jl

V=

where l is the distance between electrodes (length of the magnetic mesh) and σ is the

electrical conductivity (see the set value using the params command). Solution: use V_seq

with -4.57mV; -45.7mV; 10.

Note: You can also vary the current density using the I_seq schedule stage – this defines a

sequence of current values. Setting current values directly enables the constant current mode,

i.e. the voltage drop between electrodes is adjusted during the simulation to keep the current

constant (the opposite of this is the constant voltage mode).

You can see the current settings using the electrodes command.

You can also set individual values in the console using the setpotential or setcurrent

commands.

c) Obtain and plot the domain wall velocity, v, using the method introduced in Tutorial 5

with dp_linreg, as a function of current density. Convert the current density to spin

drift velocity using the formula:

2

1

β

µ

+

=eM

P

uB

S

C

J

Verify that the following formula holds:

74

α

β

=

u

v

Domain wall velocity curves may also be obtained using a Python script, allowing more

accurate determination of the velocity. The problem with the dp_linreg method, it also

includes in the regression domain wall displacement points at the start of each step. Since the

domain wall requires some time to reach a steady state velocity (the acceleration is not zero at

the start of each step) these initial displacement values should ideally be discarded.

With a Python script you can set a single voltage stage without saving any data (e.g. for 3 ns);

this is followed by another voltage stage (e.g. again for 3 ns) during which data is saved, then

a linear regression is performed to extract the velocity value for the set voltage value. The

script then proceeds to set the next voltage value and so on.

Exercise 10.2

Repeat the simulation in Exercise 10.1 but using a Python script to control the simulation

flow and data output.

(Solution: see the attached Python script)

Note, in general you may need to adjust the stage duration times (1 – to achieve steady state

motion and 2 – to generate enough data to obtain a representative velocity value using linear

regression) for best results, but the suggested 3 ns, 3 ns breakdown will be sufficient for this

exercise.

When setting up the Python script you will need to use the appropriate commands to edit the

stage stopping and saving conditions, as well as the stage value. These commands are:

edistagevalue

editstagestop

editdatasave

Note: an alternative and more elegant higher level method of setting up simulation stages and

configuring data saving in Python scripts is shown in sections Simulation Schedules and

Output Data, which the solution script uses.

75

Tutorial 11 – Oersted Fields

In addition to spin-transfer torques, electrical currents may also interact with magnetization

via the generated Oersted fields. In this tutorial we will set-up a simple bilayer mesh

structure, consisting of a magnetic wire and a non-magnetic metallic capping layer, then

repeat the domain wall velocity simulations from the previous tutorial but also taking into

account the generated Oersted field. Since the bilayer structure has a broken mirror symmetry

in the z direction we might expect the domain wall speed to be different depending on the

current direction.

To enable the Oersted field you need to enable the Oersted module (use the modules

command). The Oersted module is an electric super-mesh module, i.e. it is calculated on the

electric super-mesh with a separately controlled cell-size. After enabling the Oersted and

transport modules bring up the configured meshes using the mesh command. You should

now notice the electric super-mesh rectangle with its cellsize; the electric super-mesh is

simply the smallest rectangle containing all meshes with the transport module enabled.

To set a non-magnetic metallic capping layer you will need to add a new mesh, in particular

an electrical conductor mesh. This is done using the command:

addconductor name rectangle

Exercise 11.1

Set-up a simulation space as for Exercise 10.1. Add a metallic capping layer on top of the

magnetic layer with a 5 nm thickness and enable its transport module (solution: use

addconductor cap 0 0 10nm 320nm 80nm 15nm).

Set the electrodes to contact both meshes (use setdefaultelectrodes after both meshes had

their transport module enabled).

Enable the Oersted field module and set a potential of 10mV (setpotential 10mV).

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Compute a single iteration (computefields) and display the Oersted field (use the display

command and select the Oersted display option on the super-mesh display line).

Figure 11.1 – Computed Oersted field for Exercise 11.1.

Exercise 11.2

Continuing from Exercise 11.1, use a Python script to simulate the domain wall velocity for

both positive and negative currents in the magnitude range 1011 A/m2 to 1012 A/m2 in 10

steps. Plot the two ve

locity curves and compare them to the expected v = (β/α)u relation, as well as the simulations

without an Oersted field. (Solution: see the Python script in the tutorial resources).

Note: when plotting v vs u you need to set the sign of u to be in the direction of electron drift,

i.e. opposite sign to that of the charge current density.

In the above exercise the default electrical conductivity value of the capping layer is the same

as for permalloy (this can be edited using the params command). In this case the current

densities are the same in both layers. If you want to save output data for a particular mesh

(e.g. Jc, the charge current density, which is mesh dependent) then you must focus on the

required mesh first before adding that particular data to the output list.

You can focus on a particular mesh by clicking on the mesh name (bring up the list of meshes

with mesh then click on a mesh name). Alternatively you can double click in the mesh

graphical viewer on the required mesh.

77

Results from Exercise 11.2 are shown in Figure 11.2. The data obtained in Figure 11.2 could

be improved further. The problem with using linear regression directly on the raw dwshift

data, it contains steps due to the mesh discretisation. This is particularly problematic at low

domain wall velocities where only a few steps are contained in the raw data, which can make

the extracted velocity inaccurate.

Figure 11.2 – Simulation results from Exercise 11.2 showing a) domain wall velocity plotted

against spin drift velocity for cases with and without Oersted fields and also compared with

the analytical formula, and b) domain wall speed difference plotted against spin drift speed

for cases with and without Oersted fields.

One possibility is to collect data for a longer time, but this is inefficient. Another possibility

is to assume the domain wall displacement is linear with time (in this exercise this is a valid

assumption). You can then either get rid of the repeated points before using linear regression,

or replace the repeated points using linear interpolation. There is a built-in command for this,

and is covered in a later tutorial on skyrmion movement (dp_replacerepeats).

78

Tutorial 12 – Surface Exchange, Multi-Layered Demagnetization and

CUDA

Surface Exchange

Using the surfexchange module, two or more ferromagnetic meshes can be surface exchange

coupled, allowing simulations of magnetic multilayers with RKKY interaction. The strength

of the surface exchange coupling is controlled using the J1 and J2 material parameters:

negative values result in anti-ferromagnetic coupling, positive values in ferromagnetic

coupling. The J1 parameter controls the strength of bilinear surface exchange, and J2 controls

the strength of biquadratic surface exchange.

Type params and have a look at J1 and J2. For two meshes in surface exchange coupling, it

is the top mesh J1 and J2 values that are used. This allows setting different coupling strength

and types for the bottom and top of a mesh in a multi-layered structure. Boris allows surface

exchange coupling only for xy planes, thus a multi-layered structure should be designed with

the layers stacking along the z direction. Two ferromagnetic meshes will be surface exchange

coupled if they both have the surfexchange module enabled and there’s no other

ferromagnetic mesh with the surfexchange module enabled in between them along the z

direction. The coupling will only be calculated for cells which overlap in the xy plane.

Multi-Layered Demagnetization

To add another ferromagnetic mesh use the addmesh command. The demag module for each

mesh only calculates the demagnetizing field for that ferromagnetic mesh alone. When 2 or

more ferromagnetic meshes are being used, if you want to compute the overall demagnetizing

field you need to use the supermesh sdemag demagnetizing field module. In this case the

individual demag modules are disabled and the overall demagnetizing field is computed for

the collection of individual ferromagnetic meshes, including all stray field contributions.

There are two ways to do this. The default method is called multi-layered convolution, and

you can see the settings for this by using the command:

79

multiconvolution

This is an exact method of computing demagnetizing fields for a collection of computational

meshes, which is able to handle arbitrary spacing and relative positioning between the layers

without sacrificing accuracy or computational performance – see S. Lepadatu, Journal of

Applied Physics 126, 103903 (2019) for details. In many cases you can rely on the default

settings for multi-layered convolution, but for more advanced control you should read the

information below.

Further info:

You can force the demagnetizing field to be computed in a 2D approximation in each mesh

by clicking on the respective interactive console object (see output of the multiconvolution

command, Force 2D : 2D meshes). This allows each mesh to have an arbitrary thickness, and

is appropriate if the meshes are thin enough for the 2D approximation to hold. This option is

turned off by default (Force 2D : Off), resulting in a 2D or 3D convolution in each mesh

depending on their cellsizes. If the 2D approximation for each mesh is not appropriate you

should use this option as long as the z cellsizes are the same in all meshes. If the z cellsizes

differ the computation with Force 2D : Off may not be accurate, and in this case you should

use the 2D layering option (Force 2D : 2D layered). This is similar to the 2D meshes option,

however instead of forcing each mesh to be 2D, it is instead split into multiple layers, with

each layer thickness set by the z cellsize in each respective mesh. This option is thus the most

general case and allows exact computation of demagnetizing fields without any

approximations, however it is in general more computationally expensive and should only be

selected if the other 2 options are not appropriate.

For multiple computational meshes with unequal sizes, the algorithm works by first

transferring the magnetization values to scratch spaces with a common discretization. You

can specify what this discretization should be, but by default it is calculated for you (see

output of the multiconvolution command).

Another method of calculating demagnetizing fields for a collection of computational meshes

is to use the so-called supermesh demagnetization. This is achieved by disabling the multi-

layered convolution algorithm (see output of the multiconvolution command). This method

80

calculates the demagnetizing field on the ferromagnetic supermesh by transferring

magnetization and demagnetizing field values to and from the ferromagnetic supermesh using

a local averaging smoother. Remember the ferromagnetic supermesh is the smallest rectangle

containing all the ferromagnetic meshes and can be viewed using the mesh command.

Computations on the ferromagnetic supermesh are done using its independent cellsize as can

be seen in the console output of the mesh command. This cellsize will need to be carefully

determined in each case to ensure accuracy of results. As a rule you should set the cellsize to

be the minimum value out of the individual mesh cellsize values required to compute the

demagnetizing field accurately separately. In most cases of interest, if full accuracy is

required, this method is much slower than multi-layered convolution. It is also far less

flexible, as it cannot accurately handle spacing between layers which cannot be exactly

discretized.

Another use for the sdemag module without multi-layered convolution, is to calculate the

demagnetizing field in an individual mesh with a different cellsize to that used for the

exchange interaction. The exchange interaction typically requires a smaller cellsize to ensure

accuracy, thus this method can be used to improve computational speed for larger

simulations.

Starting from the default state, add another ferromagnetic mesh with dimensions of 80 nm ×

80 nm × 20 nm, separated from the first mesh by 2 nm along the z direction. Enable the

surfexchange module for both meshes, as well as the sdemag module. Now run the

simulation and observe the result – see Figure 12.1.

Figure 12.1 – Anti-ferromagnetic surface exchange coupled ferromagnetic meshes

81

You can also add a metallic spacer layer in between, using the addconductor command,

however this will not affect magnetic computations directly; it will be needed however if

charge or spin transport computations are also enabled.

Exercise 12.1

a) Set-up a synthetic ferrimagnetic (SyF) Ni80Fe20 bilayer with elliptical shape of 320

nm × 160 nm, thickness values of 20 nm and 10 nm respectively, and with a

separation of 2 nm between layers. Simulate a hysteresis loop for this SyF structure

along 1° to the x-axis direction, plotting the average magnetization for the entire

bilayer against field (you will need to calculate this from the magnetization saved for

the two layers separately – see notes below).

You should use the SDesc with LLGStatic solver (use ode command) and set an mxh

threshold value not higher than 10-5. The field step should not be greater than 1 kA/m.

b) Repeat the same exercise but this time set-up a synthetic antiferromagnetic bilayer

(SAF) with the same overall thickness as above – i.e. 15nm thick layers with 2 nm

spacing.

When adding data to the output list, you can select the mesh for which it applies, if

applicable, e.g. magnetization output. You can do this either by adding data when the

required mesh is in focus, or editing that data entry later to change the applicable mesh name.

For the exercise above you will need to add to the output the magnetization for both meshes

as two separate entries (use data command and follow instructions therein).

You will need to apply the field for the hysteresis loop to both meshes, not just the first

permalloy mesh. Use the stages command and add a Hpolar_seq stage. You will see the field

is set to be applied only to the current mesh in focus, e.g. Hpolar_seq <permalloy>. Instead,

you will need to edit this by double-clicking on the added stage entry, and changing the name

from permalloy to supermesh. After this the entry should read Hpolar_seq <supermesh>.

The field sequence will now be applied to both ferromagnetic meshes.

82

Figure 12.2 – Hysteresis loops obtained for the SyF and SAF bilayers in Exercise 12.1

When working with multiple ferromagnetic meshes, all the commands that affect changes in

a ferromagnetic mesh have an optional parameter which specifies which mesh to use. If only

one ferromagnetic mesh is created the mesh name doesn’t need to be specified explicitly. If

multiple meshes are used, unless the name is specified the settings are applied either to the

current mesh in focus, or to the supermesh, depending on the command.

For example the setangle and setfield commands will make changes to all ferromagnetic

meshes unless a specific name is specified – see the help for these commands (?setangle,

?setfield). On the other hand the loadmaskfile command (remember you can just drag a .png

file to the mesh viewer instead of typing this command) only applies the shape to the current

mesh in focus.

CUDA Computations

If you have a CUDA-enabled graphics card you can enable GPU computations using the

cuda command:

cuda 1

If CUDA computations are not available for your computer, typing the cuda command will

show an N/A status. You can also see how much CPU and GPU-addressable memory you

have by using the memory command.

83

When using CUDA computations, for optimum efficiency you will want to limit the display

update frequency (use the iterupdate command). Boris is designed to limit memory transfers

between GPU and CPU-addressable memory to an absolute minimum, as this is a critical

bottleneck in performance. To display mesh data in the viewer, an average display data is

computed on the GPU, then transferred to CPU-addressable memory so it can be used by

graphics routines. This transfer can slow down computations, especially if the display

viewing coarseness is small (remember you can use the mouse wheel to change viewing

coarseness).

84

Tutorial 13 – Dzyaloshinskii-Moriya Exchange

Dzyaloshinskii-Moriya interaction (DMI) may be included in the simulation by enabling the

DMExchange, or the iDMExchange module. The former is used for bulk DMI, whilst the

latter is used for interfacial DMI. The strength of the DMI interaction is controlled using the

D material parameter (use params command). Néel skyrmions may be generated in the xy

plane using the skyrmion command:

skyrmion core chirality diameter position

In the above command the core parameter sets the z direction of the skyrmion core (-1 or 1),

the chirality parameter sets the radial direction rotation (-1 for away from core, 1 for towards

core). The diameter and position may be specified using metric units, with the position

requiring 2 components. Figure 13.1 shows examples of relaxed Néel (iDMExchange) and

Bloch (DMExchange) skyrmions for both D > 0 and D < 0.

Figure 13.1 – a) Bloch skyrmion for D < 0, b) Bloch skyrmion for D > 0, c) Néel skyrmion

for D < 0, and d) Néel skyrmion for D > 0.

85

For the following exercises, when computing a relaxed magnetization state you should use

the SDesc with LLGStatic solver, remembering to set a low mxh convergence value (at least

10-6).

Exercise 13.1

a) Obtain relaxed Néel skyrmions for both D > 0 and D < 0 in an ultrathin (1 nm) Co

layer with perpendicular magnetization. You will need to use the iDMExchange

module. Use material parameters as Ms = 600 kA/m, A = 10 pJ/m, |D| = 1.5 mJ/m2,

K1 = 380 kJ/m3 for uniaxial anisotropy with easy axis along z direction. To reduce the

skyrmion diameter apply an out-of-plane magnetic field opposing the skyrmion core,

e.g. 15 kA/m along the 0, 0 (polar coordinates) direction.

b) Obtain Bloch skyrmions for both D > 0 and D < 0. You may use the same parameters

as above, but this time set a thickness of 10 nm and use the DMExchange module.

You will need to set a larger out-of-plane magnetic field to control the skyrmion

diameter.

c) For part a) compute the skyrmion diameter and topological charge.

To complete Exercise 13.1 c), you will need to extract a profile along the skyrmion diameter,

then fit it using an analytical model for a skyrmion. Boris provides this functionality through

the dp_fitskyrmion command. The z component of M is fitted using the following formula

(see X.S. Wang et al., Commun. Phys. 1, 31 (2018)):

( )

2/,

4

,

/)sinh

/sinh

arctan2cos)( 2

0

SuSZ MKK

K

D

w

wxx

wR

MxM

µ

π

−==

























−

=

Here MS (saturation magnetization), R (skyrmion radius), x0 (skyrmion center position), and

w are used as fitting parameters, and in particular the Ms and w values after the fit should

match the expected values from the material parameters set.

Thus to obtain the skyrmion radius first use dp_getprofile to extract a profile through the

center of the skyrmion, then use dp_fitskyrmion on the z component of M. The fitting works

86

for both skyrmion topological charges, but the fitted value of MS will change sign depending

on the sign of the topological charge.

To calculate the topological charge, there is a built-in command, dp_topocharge. This

command solves the following formula over the xy plane S:

∫











×=

S

dxdy

dy

d

dx

d

Qmm

m.

4

1

π

Exercise 13.2

For the same parameters as in Exercise 13.1, compute the skyrmion diameter as a function of

out-of-plane field strength from 2 kA/m to 20 kA/m in 1 kA/m steps. You should set-up a

Python script to automate this simulation.

Figure 13.2 – Skyrmion diameter as a function of out-of-plane field strength obtained in

Exercise 13.2.

87

Tutorial 14 – Simulations with non-zero temperature

Landau-Lifshitz-Bloch equation and Curie temperature

Non-zero temperature simulations may be performed by using the Landau-Lifshitz-Bloch

(LLB) equation – use the ode command then select the appropriate equation to solve.

With a non-zero temperature the magnetization length is no longer a constant, but depends on

the applied field strength. This is modelled via a longitudinal susceptibility included in the

LLB equation. Instead, we talk about the equilibrium magnetization, which is the stable

magnetization length at a given field. Thus at zero temperature the equilibrium magnetization

coincides with the saturation magnetization. With a non-zero temperature the equilibrium

magnetization gradually decreases, reaching zero at the Curie temperature. Other parameters

which change with temperature include the exchange stiffness and magnetization damping.

With a non-zero temperature the damping is now divided into two terms: transverse damping

(coincides with the Gilbert damping at zero temperature) and longitudinal damping.

In the simplest case the temperature inside the mesh is uniform, and is controlled using the

temperature command, which sets the mesh base temperature:

temperature value (meshname)

When enabling the LLB equation you will also need to set appropriate temperature

dependences for some material parameters, including Ms, damping, A, and susrel (the relative

longitudinal susceptibility). The longitudinal damping used in the LLB equation is not

available as a separate material parameter, but is automatically calculated based on the

transverse damping parameter (damping). Default temperature dependences for these

parameters may be generated based on the Curie temperature of the material – for details see

S. Lepadatu, Journal of Applied Physics 120, 163908 (2016). You can do this using the

curietemperature command:

curietemperature value (meshname)

88

Setting material parameters temperature dependences

Exercise 14.1

Set a Curie temperature of 870 K (appropriate for Ni80Fe20) and obtain plots of the

temperature dependences of the Ms, damping, A, and susrel material parameters (see below).

Almost all material parameters available in Boris can be assigned a temperature dependence.

This is achieved by specifying a scaling law, t. The value of a parameter at a temperature T is

then obtained as value_at_T_K = value_at_0_K × t(T) – any computational routine in Boris

for a which a parameter is used, obtains an updated value in this way where appropriate,

where T is either the base temperature (uniform temperature mode) or the local temperature

(non-uniform temperature mode). To see the currently set temperature dependences use the

paramstemp command. You can set a temperature dependence by supplying a text equation,

where T is the temperature value (see console output for paramstemp), or by loading an

array using the setparamtemparray command. Further details on using text equations are

given in a dedicated tutorial. Once a temperature dependence array has been set (e.g. after

using the curietemperature command), you can load the set temperature dependence into an

internal data processing array using the dp_dumptdep command:

dp_dumptdep meshname paramname max_temperature dp_index

For example to see the set temperature dependence for Ms use the following:

dp_dumptdep permalloy Ms 870 0

dp_save Ms_scaling 0

In the file Ms_scaling.txt (saved under the current working directory – see data command)

you will see a single column with the scaling coefficients. These are saved in increments of 1

K, from 0 K up to 870 K. Internally the scaling coefficients are obtained from the user loaded

array at 1 K increments, irrespective of how the user specified the temperature dependence –

missing temperature points are filled in using interpolation. During computations the scaling

89

coefficients are obtained by interpolating the nearest 2 temperature scaling points. To reset all

parameters temperature dependences you can use the clearparamstemp command.

Field dependence of material parameters temperature dependences

With non-zero temperature simulations, the equilibrium magnetization also depends on the

strength of the applied magnetic field. This dependence is enabled by setting the strength of

the net atomic moment of the material, specified in Boris as multiples of the Bohr magneton.

This is done using the command:

atomicmoment (value)

If this value is not zero, whenever the applied magnetic field changes, the temperature

dependences of all the parameters affected by the Curie temperature setting (see above) are

recalculated. Moreover, the longitudinal susceptibility is directly proportional to this value so

must be set correctly whenever the LLB equation is used.

Non-zero temperature simulations

Exercise 14.2

Simulate the hysteresis loops in a 160 × 160 × 5 nm permalloy circle at zero temperature

(using the LLG equation), as well as at room temperature (297 K, using the LLB equation)

and compare the two loops. With the LLB equation the time step for numerical stability is

usually lower – you might need to investigate this.

90

Tutorial 15 – Thermal Fields

When non-zero temperature modelling is considered, an additional effect that can be included

is lattice thermal agitation. This gives rise to fluctuations in magnetic moments, and may be

modelled by introducing appropriate stochastic fields and torques. In Boris thermal fields

may be enabled by selecting a stochastic magnetization dynamics equation, e.g. sLLB – use

the ode command then select the appropriate equation to solve. For further information see S.

Lepadatu & M.M. Vopson, Materials 10, 991 (2017).

When solving stochastic equations, the choice of available ODE evaluation methods is more

limited since they must be able to handle the stochasticity introduced. Currently the best fixed

time-step method available in Boris is RK4. Since this method is a fixed time step method

you will need to investigate the time step required for numerical stability. You can use the

default time step as a starting point.

There is also an adaptive time-step method called AHeun which you can use.

Exercise 15.1

Simulate out-of-plane hysteresis loops at room temperature in a 256 nm × 256 nm Co

rectangle with perpendicular magnetization, with 4 nm thickness, and cubic 4 nm cellsize.

Use material parameters as Ms = 600 kA/m, A = 10 pJ/m, and K1 = 380 kJ/m3 for uniaxial

anisotropy with easy axis along z direction. You should simulate hysteresis loops with and

without thermal fields for comparison.

91

Tutorial 16 – Heat Flow Solver and Joule Heating

Heat equation

Non-uniform temperature simulations may be enabled by selecting the heat module. Any

mesh with this module enabled will solve the heat equation as a function of time. If any two

meshes with the heat module enabled are in contact, then heat flow across the interface (also

referred to as a composite media boundary) is automatically calculated based on the

continuity of heat flux and temperature perpendicular to the composite media boundary.

There is a special type of mesh, referred to as an insulator mesh in Boris, which can be used

to model substrates. You can add an insulator mesh using:

addinsulator name rectangle

When the heat module is enabled, the thermal cell discretisation cellsize becomes available in

the mesh descriptions (use the mesh command). This can be controlled independently of the

magnetic and electric cellsize (if enabled), and can also be set independently of other cellsize

values in other meshes.

The heat equation time step may be set using:

setheatdt value

This value shouldn’t be larger than the magnetization dynamics equation time step (setdt),

since during computations the heat equation time is incremented only up to the current

magnetization equation time (the global time, or total time – see the time output data). If this

value is lower, the heat equation will be iterated multiple times until it catches up to the

magnetization equation time.

The mesh temperature may be set as before using the temperature command. This sets a

uniform mesh temperature as a starting point, but depending on the simulation configuration

the mesh temperature can change. This is true particularly if the mesh ambient temperature is

92

different. For the heat equation, boundary conditions for cells not at a composite media

boundary are set based on Newton’s law of cooling – i.e. Robin boundary conditions are

used. These require an ambient temperature (the surrounding temperature) and a heat transfer

coefficient (the Robin coefficient). To adjust these values you may use the ambient

command, then double click on the respective interactive objects to modify their values.

Note, when the temperature command is used, setting a mesh temperature automatically sets

the ambient temperature to the same value too. You may also choose to have thermally

insulating boundary conditions by selecting the appropriate options displayed by the ambient

command.

Parameters for heat transport

A few parameters are used to specify the thermal properties of the material, in particular the

thermK (thermal conductivity) and the shc (specific heat capacity) material parameters – see

these by using the params command. Additionally the density (mass density) parameter also

enters the heat equation.

Note, all material parameters with a temperature dependence enabled will now also vary non-

uniformly throughout the mesh (if heat module enabled), taking on the value set by the local

cell temperature value.

You may obtain the mesh average temperature through the <Temp> output data – use data

command. The mesh temperature may also be displayed (Temp) – use the display command.

Joule heating

If the transport module is also enabled in the same mesh as the heat module, Joule heating is

taken into account. This results in a heat source term in the heat equation due to the charge

current density, J as:

σ

2

),(.

),(

ρJ

r

r+∇∇=

∂

∂tTK

t

tT

C

93

In the next exercise you will investigate the effect of a voltage pulse on a Ni80Fe20 nanowire

placed on a SiO2 substrate, similar to the work in S. Lepadatu, Journal of Applied Physics

120, 163908 (2016). To model a very long wire on a substrate (say the wire is oriented along

the x-axis) you should set the x-axis ends of both the magnetic wire and substrate as

insulating since in this case the heat flux is oriented only along the y and z directions. Similar

considerations apply to the substrate if you want it to be effectively infinite in the x-y plane

and depth – set insulating boundary conditions in the required directions. Note in this latter

case the modelled substrate must still be large enough for the temperature evolution to be

correct for the required duration – for details see S. Lepadatu, Journal of Applied Physics

120, 163908 (2016). The x-axis ends of the magnetic wire should have electrodes so a

uniform current density is achieved – remember you can use the setdefaultelectrodes

command. You can leave the Robin heat transfer coefficient (see ambient command output)

to the default value as this is appropriate for ventilated air surrounding.

The SiO2 substrate may be added by using the addinsulator command and enabling its heat

module. You will also need to enter appropriate values for thermal conductivity, specific heat

capacity and density, and similarly for the Ni80Fe20 magnetic wire.

Note, typically the thermal cellsize can be greater than the magnetic (or electric) cellsize, and

again can be set independently in different meshes (composite media boundary conditions do

not require the discretizations to match on the two contacting meshes, for any computational

routines used in Boris; generic composite media boundary computational routines are used

which are second order accurate in space for all meshes). For the purposes of the next

exercise you can use a simple cubic cellsize with a 10 nm side for all the thermal

discretization lengths (note, if the mesh thickness is 10 nm then the z cellsize will be adjusted

so there are at least 2 computational cells along the z direction – 3D solver used).

Exercise 16.1

Create a Ni80Fe20 nanowire with 160 nm width, 10 nm thickness and 640 nm length, centered

on a SiO2 substrate with 800 nm width, 640 nm length and 150 nm depth.

Enable heat equation computations in both meshes and transport module in the Ni80Fe20

nanowire. By setting appropriate insulating boundary conditions for heat conduction, define

the nanowire to be effectively infinite along the x axis, and the substrate elongated in the x-y

94

plane and depth. Set the ambient temperature (as well as the base temperature – the starting

temperature) to be the room temperature value (297 K). The default thermal conductivity,

specific heat capacity and mass density values are appropriate for Ni80Fe20. For SiO2 you

should edit these as K = 1.4 W/mK (thermK), C = 730 J/kgK (shc), and ρ = 2200 kg/m3.

a) Set a voltage step with 50 ns duration which results in a current density of 1012 A/m2

at T = 297 K. Use a temperature dependence for the electrical conductivity σ such

that:

T

025.

0

1

0

+

=

σ

The above formula represents the default temperature dependence set for electrical

conductivity - see paramstemp command. Obtain the average temperature and

current density as a function of time in the Ni80Fe20 mesh both for the heating cycle

(first 50 ns), as well as the next 50 ns of the cooling cycle when the voltage is set to

zero. (Note, just for this part, to speed up the computations you may want to disable

any magnetic computations in the Ni80Fe20 nanowire by disabling the demag,

exchange, and zeeman modules).

b) Set a transverse domain wall in the center of the nanowire through the

preparemovingmesh command and relax it. Redo the simulation in part a), but this

time also obtain the domain wall displacement as a function of time when using the

LLG-STT equation (use the ode command).

c) Repeat part b) but this time use the LLB-STT equation with a Curie temperature of

870 K. Note, you will need to reduce the time step significantly for the LLB equation

for numerical stability.

95

Figure 16.1 – a) Geometry used for Exercise 16.1, showing a magnetic wire on a SiO2

substrate with Joule heating computations enabled – heat is generated in the magnetic

wire due to an applied charge current density. b) Average temperature in the permalloy

nanowire, also showing the current density during and after the applied voltage pulse. c)

Domain wall displacement simulated using the LLG-STT and LLB-STT equations. For

the latter a Curie temperature of 870 K was set.

a)

b) c)

96

Tutorial 17 – Spin Transport Solver

In addition to the simple Ohm’s law used to obtain the charge current density with the

transport module, Boris also integrates a 3D spin current solver based on the spin drift-

diffusion equations – see S. Lepadatu, Scientific Reports 7, 12937 (2017). This solver allows

for a number of effects to be computed self-consistently in arbitrary multi-layered geometries

and integrated with the magnetization dynamics solver. These include the spin Hall effect

(SHE), inverse SHE, CPP-GMR, spin diffusion and non-local spin transport effects, spin

pumping, as well as bulk and interfacial spin torques calculated from the spin accumulation

and composite media boundary conditions.

The spin transport solver computes both the charge and spin polarisation current densities, JC

and Js respectively, together with the charge potential, V, and spin accumulation, S:

( )

σσ

σ

µ

θ

µ

βσ

BEESmSEJ ×−+×∇+∇+= ne

P

e

P

e

D

e

D

B

eSHA

B

eDC 2

2



() ( )

( )

mmEz

mme

EεSmEJ

yx

B

iii

B

SHA

e

B

S

n

ee

e

DP

e

∂×∂⊗×+∂×⊗+

+∇−⊗−=

∑

3

2

σµσµ

σ

µ

θσ

µ







where:

( )

mmm .

ii

E∂×= 

σ

( )

mmmzB .

yx

∂×∂=

σ

Here Eσ and Bσ are the directions of the emergent electric field due to charge pumping, and

emergent magnetic field due to topological Hall effect respectively. Js is a rank-2 tensor such

that JSij signifies the flow of the j component of spin polarisation in the direction i. The

electric field is given by E = -∇V and S satisfies the equation of motion:

()











××

+

×

+−−∇=

∂

222

.

ϕ

λ

λλ

mSmmSS

J

S

Jsf

eS D

t

97

In the above equations we have a number of material constants which can be controlled via

the params command:

• De is the electron diffusion constant

• βD is the diffusion spin polarisation - this term leads to CPP-GMR

• P is the charge current spin polarisation – this terms leads to Zhang-Li spin transfer

torques, among other effects

• θSHA is the spin Hall angle (unitless) – the term in the equation for Js leads to SHE,

whilst the term in the equation for JC leads to the inverse SHE; Note there are two

related parameters available in Boris: SHA and iSHA. These represent the spin Hall

angle, but may be set to different values, allowing the SHE or inverse SHE to be

turned on or off in the computations by setting one or the other to zero.

• λsf is the spin flip length

• λJ and λϕ are the exchange rotation and spin dephasing lengths respectively,

describing the absorption of transverse spin components (transverse to m, the

magnetization direction) within a ferromagnetic material

• n is the carrier density (m-3)

• Charge pumping and topological Hall effect may be turned on or off by setting the

pump_eff and the_eff values to 1 or 0 (disabled by default).

Bulk spin torques are included in the computations as:

( )

SmmSmTS××−×−= 22

ϕ

λλ

e

J

eDD

This term is included in the implicit LLG (or LLB) equation as (in practice this term results

in an effective field which is added to Heff):

S

T

m

mHm

m

S

eff

Mtt

1

+

∂

×+×−=

∂

αγ

There is a parameter in Boris, called ts_eff (see params): This is a unitless constant which

multiplies TS and is termed the spin torque efficiency, allowing bulk spin torques to be turned

off (ts_eff = 0) or fully on (ts_eff = 1).

98

There are two possibilities for treating composite media boundaries. The simplest approach is

to assume continuity of a flux and potential – for the spin transport solver these are JC and V

for charge transport and Js and S for spin transport. The continuity conditions are used when

modelling interfaces between two normal metals (N) or two ferromagnets (F); they may also

be used to model interfaces between a normal metal and ferromagnet (N/F) but in this case

typically the second approach is more appropriate, based on interface spin conductances:

() ( )

{ }

( )

{ }

[ ]

( )

[]

mmmVn

J

Vm

Vmmn

Jn

J

m

VnJ

nJ

VGGGG

e

GG

e

GGVGG

S

B

F

S

B

F

S

N

S

F

C

N

C

∆−+

∆+−

=

∆

×+∆××=

−

∆−+∆+−

=

↓↑↓↑

↑↓↑↓

↓↑↓↑

..

ImRe

2

..

...

µ

In the above equations G↑, G↓ are interface conductances for the majority and minority spin

carriers respectively, and G↑↓ is the complex spin mixing conductance. Also ∆V is the

potential drop across the N/F interface (∆V = VF – VN) and ∆VS is the spin chemical potential

drop, where

( )( )

SV

BeS

eD

µσ

//=

. These interface conditions describe the absorption of

transverse spin components at the interface, giving rise to interfacial spin torques:

{ }

( )

{ }

[ ]

SS

h

B

erface

SGG

ed

gVmVmmT ∆×+∆××= ↑↓↑↓ ImRe

int

µ

There is also an associated interfacial spin torque efficiency constant – tsi_eff. The above

term is also included in the magnetization dynamics equation, much in the same way as TS is.

The main difference is this torque is only included in the computational cells at the interface,

where dh is the cellsize normal to the interface – this allows correct computation of interfacial

spin torques for a ferromagnetic layer of given thickness t, independent of its computational

discretization, since the effect on magnetization of the interfacial spin torque is averaged over

its thickness.

Spin pumping is generated at an N/F interface as:

99

{}{ }













∂

+

∂

×=

↑↓

t

g

t

g

B

pump

S

m

JImRe

2

π

µ

Here g↑↓ = (h/e2)G↑↓ and the pumped spin current is used in the calculation of composite

media boundary conditions by including it on the normal metal side of the equations. As with

the spin torques, there’s an associated spin pumping efficiency parameter – pump_eff – which

allows spin pumping to be turned on or off in the computations.

When modelling N/F interfaces you may need to have different interface conductances on

different sides of a ferromagnetic layer (e.g. a Pt/Co/Ta multilayer, where the Pt/Co and

Co/Ta interfaces may need different spin mixing conductances). For this reason, the interface

conductances (G↑, G↓, and G↑↓) are not associated just with a ferromagnetic mesh, but also

appear in the list of parameters for normal metal meshes (conductor meshes -

addconductor). When an N/F interface is defined by the contact of two meshes, the interface

conductances stored in the upper mesh are used – e.g. if the meshes are arranged in a

multilayer structure along the z direction, the upper mesh is that with a higher z coordinate.

To turn off the interface conductance approach to modelling composite media boundaries you

need to set the G↑↓ values to zero for the appropriate mesh. In this case the computations

revert to using the continuity approach described above.

To enable the spin transport solver you need to have the transport module active in the mesh

you want spin transport computations and you must also select a magnetization dynamics

equation with spin accumulation (e.g. LLG-SA, LLB-SA, etc.) – see the ode command.

Using the display command you can select to display a number of associated quantities in the

mesh viewer, including S, bulk and interfacial spin torques, x, y, and z directions for the spin

current.

With the spin transport solver enabled you will need to pay attention to the convergence

constant for the spin accumulation solver. Similarly to the charge potential solver, which

solves a Poisson equation to obtain V within the set convergence constant, the spin

accumulation S is obtained by solving a vector Poisson equation – this equation is obtained

from the equation of motion for S in the “steady state”, i.e. when ∂S/∂t = 0. The response

time-scales of m and S are separated typically by 3 orders of magnitude (ps vs fs time-scales

respectively) thus we only require to obtain the “steady state” values for S for a given

100

magnetization configuration. The vector Poisson equation also uses a convergence constant

and a timeout for the maximum number of allowed sequential iterations, and these values

may be changed by using the tsolverconfig command.

Further info:

Whilst each iteration taken for the Poisson equations for V and S is relatively cheap, typical

problems may require a large number of iterations to reach convergence, which significantly

slows down computations. This is especially true in the initialization stage when the timeout

number of iterations may be reached for the first few iterations; after this, small steps in m

should result in relatively few steps in the solution of S (and where appropriate V). A

recommended general approach is to solve for the steady state V and S values with all spin

torques turned off and for a relaxed starting magnetization configuration. After this, save the

simulation (which also saves the computed V and S), and re-enable the spin torques as

required. From this point initialization should be quicker, with any further iterations in V and

S triggered by changes in m (changing set electrode potential values can also trigger the

Poisson solvers).

Setting the convergence factors too low may result in very slow simulations as the solvers

will require a large number of iterations. You will need to determine the best compromise

between computational speed and accuracy. The default normalized convergence values of

10-6 for V and 10-5 for S Poisson equations are set on the side of accuracy, having been found

to give accurate results in all test cases, but you should still verify this for your particular

simulation.

101

Tutorial 18 – Spin Hall Effect

Exercise 18.1

Consider a single Pt mesh with dimensions 320 nm x 320 nm and 40 nm thickness. Compute

the spin accumulation and z-direction spin current density in response to a set potential of 10

mV with electrodes placed at the x-axis ends. Verify that S obeys the right-hand-rule with

respect to the charge current direction.

For Pt you may use σ = 7×106 S/m, λsf = 1.4 nm and θSHA = 0.1. You may use a cubic cellsize

with 5 nm side.

Plot the y components of the z-direction spin current density and spin accumulation along the

z-axis, through the center of the Pt slab (remember the dp_getprofile command). Verify that

the following relation holds, using the plotted value of the spin current at the center of the Pt

slab:

eJ

J

B

cx

ysz

SHA

µ

θ

/

,

−=

Figure 18.1 – Computed spin accumulation for Exercise 18.1, where the charge current

density is along the negative x direction.

102

Exercise 18.2

a) Continuing from Exercise 18.1, now add a Ni80Fe20 layer with 20 nm thickness on top

of the Pt layer. Make sure to reset to default electrodes (setdefaultelectrodes) so a

uniform charge current density is obtained in each layer. Plot the y components of the

z-direction spin current density along the z axis for both the continuous and spin-

mixing conductance interface models for a) magnetization direction along the injected

spin current, i.e. along the y axis, and b) magnetization direction transverse to the

injected spin current, i.e. along the x axis. Explain the differences between these

cases.

Note, for this exercise you will have to use a smaller cellsize along the z direction. This is due

to the large gradients involved, and is normally the case when N / F multilayers are used. You

should use a cellsize of (5 nm, 5 nm, 1 nm).

Remember you can use the computefields command for this exercise, instead of using run,

since you don’t want to relax the magnetization configuration. It helps to monitor the

transport solver number of iterations and convergence error (in the data box display the

following using the data command and righ-clicking on the respective interactive objects:

v_iter, s_iter, ts_err).

b) Does the relation in Exercise 18.1 hold at the N/F interface, and why not? Investigate

this again with a spin flip length in Pt of 8 nm, checking the relation both at the center

of the Pt layer and at the interface.

103

Figure 18.2 – Spin current density in the z direction for a Pt/Ni80Fe20 bilayer, where the

magnetization in the permalloy mesh is along the y axis (longitudinal).

Figure 18.3 – Spin current density in the z direction for a Pt/Ni80Fe20 bilayer, where the

magnetization in the permalloy mesh is along the x axis (transverse).

104

Tutorial 19 – Spin Pumping and Inverse Spin Hall Effect

In this tutorial you will set-up a ferromagnetic resonance in a magnetic dot, then investigate

the generated spin Hall voltage in a Pt underlayer. Due to the motion of magnetic moments in

the ferromagnetic layer a spin current is pumped in the Pt underlayer, where an electrical

current is generated due to the inverse SHE. This leads to charge accumulation at opposing

sides of the Pt underlayer, and thus an electrical potential is generated.

Exercise 19.1

Setup a ferromagnetic resonance (FMR) at 20 GHz excitation frequency in a Ni80Fe20 circle

with 80 nm diameter and 10 nm thickness, with a bias field applied in the plane of the circle

along the y axis.

First, find the demagnetizing factor in the plane of the circle and set the demag_N module

with demagnetizing factors Nx = Ny = N. Remember you can calculate the demagnetizing

energy for uniform magnetization (e_demag), and the demagnetizing factor is then related to

it by:

2

0

2S

demag NM

µ

ε

=

For this exercise, since you are effectively using the Stoner-Wohlfarth model you can turn off

the exchange module. Calculate the FMR bias field required for resonance at an r.f.

frequency of 20 GHz. You can use Kittel’s formula applicable for elliptical shapes for a bias

field H0 along the y direction:

))()()((

2

00

0syzsyx

e

MNNHMNNHf −+−+=

π

γµ

Using the Hfmr stage type, apply the excitation r.f. field (together with the calculated

orthogonal bias field) in the plane of the circle for a number of cycles, and record the average

magnetization. An r.f. field amplitude of 100 A/m is normally sufficient. If the r.f. field is

applied for a sufficient number of cycles, the magnetization will achieve a steady state

precession at resonance. Determine the number of cycles required by examining the output

105

average magnetization data, then reset and save the simulation – the next time you load the

simulation the FMR precession will start directly in the steady state.

The Hfmr stage consists of the following parameters: H0x, H0y, H0z; Hrfx, Hrfy, Hrfz; r.f. steps;

r.f. cycles.

The bias field (H0) and r.f. field amplitude (Hrf) are specified using Cartesian coordinates.

The r.f. steps is the number of discretisation steps in each r.f. cycle, and the r.f. cycles is the

number of sinusoidal oscillations the r.f. field will be applied for. To set the required 20 GHz

frequency you will need to set the correct combination of r.f. steps and time stopping

condition for each step. For example, since at 20 GHz each period takes 50 ps, if you use 20

r.f. steps per cycle, the time stopping condition for each step should be 2.5 ps (the default

time stopping condition of 50 ps results in a 1 GHz frequency with 20 r.f. steps per cycle, so

you will need to edit this).

Exercise 19.2

Using the prepared simulation from Exercise 19.1, add a Pt underlayer with dimensions 160

nm × 160 nm with the magnetic dot centered, and 20 nm depth.

Enable spin pumping (pump_eff = 1 in the permalloy mesh), and inverse SHE (SHA = iSHA =

0.1) in the Pt mesh. Do not set any electrodes but make sure the transport module is enabled

in both the permalloy and Pt meshes, and the LLG-SA equation is selected so the spin

transport solver is enabled. You will need to refine the electric cellsize in both meshes along

the z direction to 1 nm. You should also relax the transport solver convergence criteria to 10-4

for both the charge and spin solvers.

Obtain the induced spin Hall voltage at the opposing y-axis sides of the Pt mesh and plot

them as a function of time for a few FMR precessions. Note, in the output data (data) you

will have to add <V> (the average calculated voltage) for the Pt mesh two times, editing the

respective rectangles to correspond to the required two sides of the Pt mesh.

106

Figure 19.1 – Inverse spin-Hall effect voltage in a Pt underlayer generated through spin

pumping from a ferromagnetic dot at ferromagnetic resonance.

Figure 19.2 – a) Magnetization precession at ferromagnetic resonance with a 20 GHz r.f.

field, b) inverse spin-Hall effect voltage at resonance on opposing sides of the Pt mesh – see

Figure 19.1.

a) b)

107

Tutorial 20 – Ferromagnetic Resonance

In this tutorial you will learn how to simulate ferromagnetic resonance (FMR), and re-

produce input material parameters. Following this, ino the next tutorial you will investigate

how the spin torques due to spin pumping and the SHE affect the effective damping

observed. An introduction to FMR simulations was given in the preceding Tutorial, and you

must complete it before proceeding. Here we will investigate both field-swept FMR (fixed

excitation frequency) and frequency-swept FMR (fixed bias field).

Field-Swept FMR

In Boris, FMR simulations are best done using a Python script. As you will note from the

previous tutorial, applying an r.f. field excitation requires a number of cycles for the

magnetization precession to reach steady state. For a field-swept FMR simulation, after

changing the bias field you must ensure the magnetization precession is stable before

obtaining output data. The simulation procedure is as follows:

1) Set bias field value and run the simulation for a fixed number of r.f. cycles (the

“chunk” – e.g. 20 or more), but do not save any output data.

2) After the chunk has completed, run the simulation for a single r.f. cycle and save the

output data (<M>).

3) From the saved data obtain the magnetization oscillation amplitude along the r.f. field

direction.

4) Compare the oscillation amplitude against the previous oscillation amplitude (which

is zero if this is the first chunk). If the change exceeds a set threshold (e.g. 0.1%) then

repeat from step 1), otherwise proceed.

5) Record the oscillation amplitude and bias field. Increase bias field value and start

again from step 1) until the field sweep range is completed.

A general-purpose FMR simulation Python script has been prepared and saved in the

examples folder for this Tutorial.

108

Exercise 20.1

Using a Ni80Fe20 square of 80 nm side and 10 nm thickness simulate an FMR peak around the

resonance bias field and plot the resulting magnetization oscillation amplitude against bias

field data. Set the bias field along the –y direction, i.e. at 270° azimuthal angle. Use a Python

script to simulate this as described above. (You may use Nx = Ny = 0.12, with the predicted

resonance field of H0

≅

367 kA/m; aim for at least 50 kA/m either side of resonance).

From the simulated oscillation amplitude versus bias field data, you will need to obtain a

quantity proportional to the absorbed FMR power. The simplest way to do this is to square

the oscillation amplitude data. The FMR power absorption peak is described by a Lorentz

peak function, and you will need to fit this to your squared amplitude data.

Boris has built-in data processing command to help with processing FMR simulation data.

You will need the following commands:

First load bias field and oscillation amplitude from the raw output data file (e.g. named

‘fmr_fieldsweepFMR_data.txt’):

dp_load fmr_fieldsweepFMR_data 0 1 0 1

Next square the magnetization oscillation amplitude data:

dp_muldp 1 1 1

Finally fit a Lorentz peak function to the data:

dp_fitlorentz 0 1

The Lorentz peak function is given as:

22

0

0)(4

)( wxx

w

Syxf +−

+=

109

In the above equation w is the full-width half-maximum (FWHM), and x0 is the peak center.

You can obtain these values from the dp_fitlorentz command, including fitting uncertainties

(Boris has a built-in generic Levenberg-Marquardt algorithm for curve fitting).

The magnetization damping value is related to the full-width half-maximum (∆H) by:

f

H

e

∆

=

π

γµ

α

4

0

Exercise 20.1 continued

Process the output FMR data and verify the damping obtained from the FWHM matches the

set damping value (damping = 0.02).

Figure 20.1 – Simulated FMR peak with Lorentz peak function fit for Exercise 20.1.

Frequency-Swept FMR

Frequency-swept FMR simulations are also possible and are typically much more efficient to

compute than field-swept FMR. This type of simulation is closely related to the method used

to investigate spin-wave dispersion. Here we apply a sinc pulse in the time domain and

capture the magnetization response, which we then transform to the frequency domain using

110

a Fourier transform. The reason for using a sinc pulse is its Fourier transform is a symmetric

hat function with a defined frequency cut-off. Thus in order to capture the required resonance

(and also any required higher resonance modes) we simply need to set the cut-off to a large

enough value.

The sinc pulse excitation is given as:

( )

( ) ( )

( )

00

2

/2sin)

(tt

ft

tfH

tH

cce

−

−=

π

Here fc (Hz) is the cut-off frequency, He is the excitation amplitude, with H(t) taking on the

value He at t = t0, the sinc pulse centre. We also need a fixed bias field, H0, which must be

orthogonal to the excitation field, and in general we must capture the average magnetization

(for now we’ll assume the magnetization is uniform) and use the magnetization component

along the excitation field. We need to capture the magnetization at a time step set by the

Nyquist criterion:

)(

2

1s

f

t

C

S

=

Capturing magnetization data at other time steps is sub-optimal and will result in lost

information if larger than the above formula, or increased noise in the Fourier transform

spectrum if smaller than the above formula. It is recommended the sinc pulse is simulated for

a total time of 2t0. If you want to increase the spectrum resolution (number of points) then

you must increase the t0 value, but still simulate for a total time of 2t0, saving data at a time

step of ts.

We can set this type of excitation using a text equation stage, namely Hequation. To set the

above equation, edit the stage value for the added Hequation stage to: He*sinc(2*PI*fc*(t-

t0)), H0, 0. This will set the excitation field along the x axis, and bias field along the y axis,

with zero z axis field. A dedicated chapter in the manual is given for details on using text

equations. In the above equation t is a reserved parameter, namely the stage time, and the

equation provided is evaluated internally every iteration. The remaining parameters, He, fc,

t0, H0 must be defined in order for the equation evaluation to function. Equation constants

may be given as:

111

equationconstants name value,

e.g. equationconstants fc 200e9.

Other related commands are delequationconstant, and clearequationconstants.

For frequency-swept FMR after taking the Fourier transform, similarly to field-swept FMR,

the output data must be squared. The resulting FMR peak in the frequency domain is also

described by a Lorentz peak, where the FWHM, Δf, is related to the resonance frequency f0,

and Gilbert damping as:

0

2f

f∆

=

α

By capturing a set of frequency-domain FMR peaks as a function of bias field H0 the

following Kittel formula can be verified:

( )( )

SzxSzy

eMkNNHMkNNHf +−++−+= 0

0

2

π

γ

µ

Here Nx, Ny, and Nz are demagnetizing factors such that the bias field is applied along the z

direction. The above formula also includes uniaxial anisotropy, with easy axis along the z

direction, where:

2

0

1

2

S

M

K

k

µ

=

For the following exercise you’ll simulate a thin film Co material interfaced with Pt (this is

stored in the materials database and can be loaded as setmaterial Co/Pt – more on the

materials database in a dedicated chapter. You will also need to simulate a thin film, thus

must set periodic boundary conditions in the xy plane as: pbc x 10, pbc y 10 – more on

periodic boundary conditions in a dedicated tutorial.

112

Exercise 20.2 (Advanced)

Simulate the frequency-swept FMR response of a thin-film Co/Pt material with 2 nm

thickness using the method described above.

You can use a cut-off frequency of 200 GHz, with excitation field amplitude of 1000 A/m,

and vary the bias field between 100 kA/m and 1 MA/m. Simulate both out-of-plane FMR

(anisotropy easy axis and bias field out of plane), and in-plane FMR (anisotropy easy axis

and bias field in the plane, e.g. both along the y direction).

In both cases use the Kittel relation and damping formula to verify the input simulation

parameters using fitting procedures (damping and magneto-crystalline anisotropy).

113

Tutorial 21 – Ferromagnetic Resonance with Spin Torques

Continuing from the previous tutorial, you will now investigate the effect of spin torques due

to the spin-Hall effect on the magnetization damping using a Pt/Ni80Fe20 bilayer. The spin

current generated in the Pt underlayer is absorbed by the Ni80Fe20 layer, resulting in a

combination of damping-like and field-like torques. Depending on the current direction a

decrease or increase of the effective damping is obtained. First the full spin transport solver is

used, and following this a simpler method using an analytical form for the spin-orbit torques

is introduced. Using the full spin transport solver we can also consider the effect of spin

pumping on the effective damping, and this is investigated at the end of this tutorial.

The interfacial spin orbit torque added to the implicit LLG equation, as explained in Tutorial

17, is given by:

{}

( )

{ }

[]

SS

h

B

erface

S

GG

ed

gV

mVm

mT ∆

×+

∆×

×=

↑↓

ImRe

int

µ

For an N/F interface in the x-y plane with uniform current densities we can obtain an

analytical expression for this interfacial torque as (see S. Lepadatu, Scientific Reports 7,

12937 (2017)):

( )

[ ]

pmpmmT ×+××=

G

h

c

B

effSHA

erface

S

r

d

J

e

µ

θ

,

int

Here p = z × eJc, where eJc is the charge current direction, and:

{ } { } { }

{ }( ) { }

,

~~

Im

~

Re

/

1

122

22

GImGReN

GReNGG

)cosh(d N

sfN

SHASHAeff ++

+−











−=

λ

θθ

22

}

~

Im{}

~

Re{}

~

Re{

}

~

Im{

}

~

Re{2}

~

Im{

GGGN

GG

GN

r

G

−

+

=

λ

114

In the above,

( )

N

sf

N

sfN

dtanhN

λ

/

=

,

( )

F

sf

F

sfF

dtanhF

λλ

λ

//=

, and

N

G

σ

/2

~

↑↓

=

. Thus the

interfacial torque has a damping-like and a field-like component. In the limit of abrupt

interface (λφ → 0 or equivalently Re{G↑↓} → ∞) the field-like component tends to zero and

we obtain the following expression for the torque:

()

pmm

T××

=

h

c

B

effSHASOT d

J

e

µ

θ

,

and











−= )

cosh(d N

sfN

SHAeffSHA

λ

θθ

/

1

,

This approximation can also be used when the damping-like torque is much larger than the

field-like torque. This expression is commonly used in the literature to model the spin-orbit

torque resulting from the spin-Hall effect. Note however the spin-Hall angle in this

expression is not the bulk (or intrinsic) spin Hall angle, but an effective spin Hall angle,

scaled by transport parameters; if further the N layer thickness is many times larger than its

spin flip length, we can use the approximation

SHAeffSHA

θθ

≅

,

. In many cases this may not be

true, and moreover the abrupt interface approximation may not be good either, thus to model

the effect of the damping-like torque with the analytical form of the spin-orbit torque, in the

expression for TSOT you should use the full expression for the effective spin-Hall angle given

above.

In Boris you can include this analytical spin-orbit torque using the SOTField module.

Enabling this module in a ferromagnetic mesh introduces an additional effective field into the

LLG equation which results in the TSOT torque given above (as it appears in the implicit LLG

equation). To use it you still need to have the transport module enabled in order to calculate

the charge current density, but instead of selecting LLG-SA (enabling the full spin transport

solver) you should select just the LLG equation (ode command). In the material parameters

for the ferromagnetic mesh (params command) you need to enter the correct effective spin-

Hall angle (SHA) to use with the SOTField module.

115

Exercise 21.1

Using the Ni80Fe20 layer from the previous tutorial, now add a Pt underlayer with the same

dimensions (80 nm × 80 nm × 10 nm), using the Pt parameters from Tutorial 18. Set default

electrodes (resulting in current flow along the x direction), enabling the spin transport solver

both in the Ni80Fe20 and Pt meshes (add transport modules and set the ode solver to LLG-

SA). Make sure to disable spin pumping (pump_eff = 0) and the inverse SHE (iSHA = 0). You

should also disable bulk spin torques (ts_eff = 0), only leaving interfacial spin torques

enabled (tsi_eff = 1). As before you may need to decrease the z-direction electrical cellsize to

ensure accuracy (and numerical convergence!).

Obtain FMR peaks for charge current densities in the Pt layer of Jc = ±1012 A/m2. How does

the damping change with current density direction?

In this case, even though you are using the Stoner-Wohlfarth model (demag_N module), you

should still enabled the exchange module. The reason for this, the spin torques may not be

perfectly uniform (e.g. if the permalloy and Pt layers have the same width, the spin torques

will not be uniform since the spin accumulation has gradients at the sample edges), thus you

do need to take the exchange interaction into consideration.

The change in damping due to a damping-like spin-orbit torque may be roughly

approximated by:

FS

c

B

effSHASHE

dfM

J

e

π

µ

θα

2

,

≅∆

Verify the change in damping obtained from simulations with the above formula.

116

Exercise 21.2

Repeat Exercise 21.1 but this time without the spin transport solver, only using the analytical

form for the spin-orbit torque (SOTField module). You should delete the Pt mesh and reset

the electrodes and potential to give you the correct current density. Calculate an appropriate

effective spin-Hall angle to use. Compare the results with the previous exercise.

Exercise 21.3

Repeat Exercise 21.1, using the full spin-transport solver, but now enable spin pumping (set

pump_eff = 1). What is the increase in damping?

Figure 21.1 – FMR simulations with spin orbit torques for both the full spin transport solver

(ST Solver) and effective field obtained from the analytical spin-orbit torque (SOTField).

117

Tutorial 22 – CPP-GMR

The spin transport solver is also able to reproduce the spin torques in a current-perpendicular

to plane (CPP) giant magneto-resistance (GMR) spin valve, in addition to its magneto-

resistance. Here we will investigate the current-induced switching in a simple generic spin

valve between the parallel and anti-parallel states, see Figure 22.1, and plot the resistance

during these switching events.

A spin valve, in its simplest form, consists of a fixed magnetic layer, a free magnetic layer

which can be switched between an anti-parallel and parallel orientation with respect to the

fixed layer, and a thin metallic spacer layer. The spacer layer thickness can be adjusted to

give either a ferromagnetic or anti-ferromagnetic surface exchange coupling between the two

magnetic layers. In the following simulation we will also add two metallic contacts, top and

bottom.

Figure 22.1 – CPP-GMR spin valve showing the spin accumulation in the spacer layer, top,

and bottom contacts, and the magnetization in the elliptically shaped fixed and free layers for

a) anti-parallel state, and b) parallel state.

118

Exercise 22.1

Setup a generic spin valve structure (i.e. just use the default mesh parameters unless indicated

otherwise) similar to that shown in Figure 22.1. This consists of:

• Bottom and top contacts (addconductor) with dimensions 160 nm × 80 nm × 20 nm.

Disable spin-Hall effects in both (SHA = iSHA = 0).

• Spacer layer with dimensions 160 nm × 80 nm × 2 nm and set it to an elliptical shape

(drag a .png file with a circle shape to the mesh viewer when the spacer layer mesh is

in focus). Disable spin-Hall effects.

• Fixed layer (addmesh) with dimensions 160 nm × 80 nm × 10 nm and elliptical

shape. Disable spin torques and spin pumping in this mesh (ts_eff = tsi_eff = ts_pump

= 0). You should also disable magnetization dynamics in this mesh so the

magnetization is fixed. You can do this by setting the relative gyromagnetic factor to

zero (grel = 0 in params).

• Free layer with dimensions 160 nm × 80 nm × 5 nm and elliptical shape. Disable spin

pumping and bulk spin torques only, in this mesh (i.e. keep tsi_eff = 1).

You will need to add the following modules:

• super-mesh multi-layered demagnetization (sdemag).

• surfexchange modules in both magnetic meshes. Edit the J1 (bilinear surface

exchange energy density) value in the free layer to give you a weak ferromagnetic

coupling; set J1 = 0.1 mJ/m2.

• transport modules in all meshes. Set the electrical cellsize to 5 nm × 5 nm × 1 nm

everywhere except in the spacer layer where you should set it to 5 nm × 5 nm × 0.5

nm.

For the ode solver you should set the LLG-SA equation (thus enabling the spin-transport

solver) with RKF45 evaluation. For output data you should have time, R (resistance), and

<M> (average magnetization) in the free layer. Set electrodes top and bottom

(addelectrode), designating the bottom electrode to be the ground electrode (electrodes).

You need to simulate switching starting from the anti-parallel state (see Figure 22.1) using a

+15 mV pulse for 3 ns, then back to this state with a further -15 mV pulse for 3 ns. Save data

119

every 10 ps for both stages. Before starting the simulation you should relax the starting state

as follows:

1) Set all spin torques to zero and insert a Relax stage with nostop condition at the start.

2) First relax the magnetization in the anti-parallel state without the spin-transport solver

(set ode to LLG).

3) Next enable the spin-transport solver and relax it (run it until the solver no longer

iterates, monitoring v_iter and s_iter data).

4) Re-enable the appropriate spin torques (tsi_eff = 1 in the free layer only), reset, delete

the Relax stage, then save the simulation (savesim).

Explain the resistance change observed by comparing it with the magnetization in the free

layer as a function of time – see Figure 22.2 for expected results.

Figure 22.2 – Change in resistance for the CPP-GMR spin valve of Exercise 22.1, together

with the magnetization along the longitudinal direction in the free layer.

120

Tutorial 23 – Skyrmion Movement with Spin Currents

Skyrmions may be displaced using charge and spin currents. To study their movement a

skyrmion tracking window can be used in Boris. There are two methods available for

tracking a skyrmion, discussed below, both available as data outputs: skyshift and skypos (use

data command).

The skyshift entry needs a rectangle defined, which should be set around the initial position of

a skyrmion, making sure to fully contain it, but don’t leave excessive space around it; the

thickness of this rectangle should be set to the thickness of the ferromagnetic mesh containing

the skyrmion. During a simulation a x-y shift is recorded and saved in the output data file.

This shift is determined by comparing the average magnetization magnitude in the 4

quadrants of the skyrmion tracking window – e.g. if the skyrmion shifts to the right, the

average magnetization magnitude in the 2 right-hand-side quadrants will decrease compared

to the left, thus a right single-cell shift is recorded. Multiple skyshift entries can be defined,

with different rectangles, to track multiple skyrmions. Note, the skyshift entry only works

with data file output, and not in the data box or with the showdata command.

The raw output skyshift data will contain staircase steps due to mesh discretisation. It is

possible to obtain a more natural skyrmion movement path by assuming linear displacement

in between the staircase steps. To remove the stair steps and replace them using linear

interpolation you can use the dp_replacerepeats command on both the x and y skyshift data

columns. The x, y data can then be plotted directly in Cartesian coordinates. This path

normally doesn’t start from (0, 0). To display the skyrmion displacement path relative to its

starting position you should remove this offset using the dp_removeoffset command on both

the x and y data. Finally, if you want to plot this path using polar coordinates you can use the

dp_cartesiantopolar command, included in Boris for convenience. Note, especially when

converting to polar coordinates you should check the processed data correctly represents the

raw data. Problems may occur due to blips in the raw data, especially if the tracking window

was not defined well or the starting state is not sufficiently relaxed, thus the results from this

procedure must be carefully compared with the raw data.

121

The second method uses the skypos data output. Again this needs an initial rectangle defined

around the skyrmion as for the skyshift data output. Skypos uses a slightly more

computationally expensive algorithm to track the skyrmion, but is able to obtain the exact

skyrmion center position, as well as skyrmion diameters along the x and y axes, without

being affected by staircase discretisation artifacts. It is also able to adjust the tracking

window size if the skyrmion diameter changes significantly. The algorithm works by fitting

the skyrmion using an analytical function (the same one used for the dp_fitskyrmion

command) in three steps: 1) Skyrmion is fitted along the x axis through the center of the

skyrmion tracking window in order to find the center position. The tracking window x

position is adjusted to center it. 2) Skyrmion is fitted along the y axis through the center of

the skyrmion tracking window in order to find the center position. The tracking window y

position is adjusted to center it. The y diameter is also recorded at this step. 3) The skyrmion

is again fitted along the x axis to obtain its x diameter.

For the following exercises you should first use the skyshift method of tracking the skyrmion,

the repeat them using the skypos method. Note, you should not use both methods

simultaneously on the same skyrmion.

Exercise 23.1

a) Setup a Pt/Co bilayer with a skyrmion relaxed at the center of the Co layer under a 15

kA/m out-of-plane magnetic field as shown in Figure 23.1. The Pt layer should be a

320 nm × 320 nm × 3 nm rectangle, whilst the Co layer should be a 320 nm diameter

disk with a 1 nm thickness. Relax this magnetization configuration.

For Pt you should use σ = 7×106 S/m, λsf = 1.4 nm and θSHA = 0.19. Use a

discretisation cellsize of (4 nm, 4 nm, 0.5 nm). Set iSHA to zero.

For Co you should use σ = 5×106 S/m, λsf = 38 nm, λJ = 2 nm, λϕ = 4 nm, Gmix = 1.5

PS/m2, grel = 1.3, α = 0.03, MS = 600 kA/m, A = 10 pJ/m, D = -1.5 mJ/m2, K1 = 380

kJ/m3 with uniaxial anisotropy perpendicular to the plane. You should also enable the

interfacial DM exchange module. Use a discretisation cellsize of (4 nm, 4 nm, 1 nm)

for magnetic computations and (4 nm, 4 nm, 0.25 nm) for spin transport

122

computations. In the Co mesh only enable the interfacial spin torques, not the bulk

spin torques or spin pumping.

Figure 23.1 – Skyrmion in a Co disk on a Pt underlayer.

b) Enable the spin transport solver in both meshes and set electrodes at the x-axis ends of

the Pt mesh only. Set a -20 mV potential for 3 ns and save the time and skyshift (or

skypos) data every 10 ps. (for skyshift and skypos define a rectangle around the initial

position of the skyrmion). Simulate the skyrmion movement path with SHE enabled

(SHA = 0.19 in the Pt mesh), as well as without SHE (SHA = 0 in the Pt mesh), and

plot them in polar coordinates.

123

c) Simulate the skyrmion movement path without the spin transport solver but with the

SOTfield module enabled. Still keep the transport module enabled to calculate the

charge current density. For the SOTfield module set a suitable effective spin Hall

angle in the Co mesh (SHA). You can use the effective spin Hall angle formula from

Tutorial 21, but note this is only strictly applicable for uniform magnetization and

spin currents. You can use this as a starting point, but will need to adjust the effective

spin Hall angle.

Plot the skyrmion path in polar coordinates and compare it with the path obtained as

the difference between the SHE and no SHE simulations above – see Figure 23.2 for

expected results.

Figure 23.2 – Skyrmion movement paths obtained in Exercise 23.1.

124

Tutorial 24 – Roughness and Staircase Corrections

Staircase Corrections

With finite difference discretisation, errors can arise due to a staircase effect when

discretising curved boundaries. In micromagnetics the largest errors arise in the

demagnetizing field and may be reduced by decreasing the discretisation cellsize. This

method is inefficient however since for most problems the results converge when the

discretisation cellsize is close to the exchange length of the material – thus to further reduce

this everywhere just to improve the discretisation accuracy at a boundary is very inefficient.

Note, for materials where the demagnetizing energy dominates the exchange length may be

defined as:

2

0

2

S

ex M

A

l

µ

=

For systems where the anisotropy energy dominates (

2

/

2

0S

u

M

K

µ

>

), the exchange length

may be defined as:

u

ex

K

A

l=

Instead of refining this, a good approximation may be achieved by computing a correction

field using a finely discretised demagnetization kernel, but applying it at run-time to the

coarsely discretised mesh, as described in S. Lepadatu, Journal of Applied Physics 118,

243908 (2015). This correction field is typically similar to an uniaxial anisotropy field when

averaged.

To enable staircase corrections in a particular magnetic mesh you must enable its Roughness

module. When applying a mask shape to the mesh, staircase corrections will now

automatically be taken into account. You must enable the Roughness module and reset the

125

mesh shape before applying the mask to correctly enable staircase corrections. You must also

set the required refinement using the refineroughness command:

refineroughness mx my mz

When calculating the correction field factors, the shape is first discretised on a fine mesh as

set by the refineroughness parameters. Thus if the coarse mesh has cellsize (hx, hy, hz), the

fine mesh used for correction field initialization has cellsize (hx / mx, hy / my, hz / mz).

Typically the improvement in accuracy is small above m > 10, so the refinement set should

not be excessive. Since a demagnetizing kernel must be computed for the fine mesh, the

initialisation time may become very long, and the available memory may be exceeded if the

m factors are set too large. You must also set them before applying the mask shape.

With the Roughness module enabled, setting a mask shape may result in a slightly different

shape than without. This is because a fine shape is internally obtained first, then the coarse

mesh shape is calculated to be the smallest shape which includes the fine shape on the coarse

mesh – this is a requirement of the corrections calculation method. To clear the staircase

corrections, effectively setting the fine mesh shape to the coarse mesh shape you can use:

clearroughness

There’s an energy density term associated with the correction fields, and this is available as a

data parameter: e_rough. The demagnetizing energy density, e_demag, still corresponds to

the coarse mesh shape; the sum of e_rough and e_demag is the approximated demagnetizing

energy for the fine mesh shape.

Exercise 24.1

Set a 80 nm diameter Ni80Fe20 disk with 10 nm thickness. Calculate the demagnetizing

energy density as a function of in-plane uniform magnetization orientation from 0° through

360° by saturating in a strong magnetic field (106 A/m). Repeat this computation but now set

the shape with the Roughness module enabled and a refinement of (10, 10, 1) –

126

refineroughness. Compare the demagnetizing energies for the coarse mesh and the

approximated demagnetizing energy for the fine mesh (e_rough + e_demag).

For the above exercise, in theory the demagnetizing energy should be constant for a circle as

the field rotates. In practice, due to discretisation errors a shape anisotropy effect is observed

(the magnetization is not fully saturated even at 106 A/m, so some non-uniformity persists).

With staircase corrections enabled this anisotropy should be significantly reduced, thus closer

to the ideal uniform demagnetizing energy – see Figure 24.1. The refinement can be

increased but further improvement is small.

Figure 24.1 – Demagnetizing energy computed for a circle with and without staircase

corrections.

Edge and Surface Roughness

The same model used to reduce staircase corrections may be applied to compute the effect of

topological roughness with variations below the exchange length of the material. As before,

coefficients for a roughness field are computed at initialisation depending on the shape of a

finely discretised mesh (the mesh with topological roughness applied), and that of the coarse

mesh (the actual mesh used in computations but without roughness).

127

A roughness profile may be applied using a built-in algorithm, or alternatively a mask may be

used. See Figures 24.2 – 4 for examples of real surface scans, processed into a grayscale

image suitable for use as masks. These may be found in the Examples folder for this tutorial.

Figure 24.2 – Granular surface roughness profile

Figure 24.3 – Maze-like surface roughness profile.

Figure 24.4 – Elongated defects, or stripes, surface roughness profile.

To apply a roughness profile using a mask, instead of simply dragging the file to the mesh

viewer (as you would do when applying a shape), you should also specify the depth to which

128

you want to apply the profile. For example, with the mask shown in Figure 24.4, to apply it to

a 4.5 nm depth (the coarse discretisation cellsize is 5 nm so this keeps the coarse mesh shape

intact) you need to use:

loadmaskfile 4.5nm (directory\)Stripes

This will apply a surface roughness profile on the top face up to 4.5 nm depth – black results

in 0 depth cut, whilst white results in full depth cut (i.e. 4.5 nm); values on the greyscale in

between are correlated linearly with the depth cut. For the actual surface roughness profile

obtained see Figure 24.5.

In this example the starting mesh has dimensions of 320 nm × 320 nm × 10 nm, and the

roughness refinement was set to (4, 4, 10) – refineroughness. To view the set roughness,

under display select the Roughness option for the respective mesh. To apply the surface

roughness to the bottom face, negative values need to be set for the depth value – see help for

loadmaskfile command.

Figure 24.5 – Applied surface roughness using the mask in Figure 24.4.

You can also apply edge and surface roughness using a built-in console command. Currently

two methods are available: roughenmesh, surfroughenjagged. The roughenmesh

command applies a completely random roughness profile to one of the 6 faces as indicated –

129

see help for this command. The surfroughenjagged command applies a jagged profile to

either the top, bottom, or both, faces – see help for this command.

Exercise 24.2

Set a 160 nm × 160 nm × 10 nm permalloy rectangle and apply the Stripes roughness profile

from Figure 24.4 (use file in Examples folder) to a depth of 4.5 nm. Use a roughness

refinement of (4, 4, 10). Simulate the hysteresis loops along the x and y directions and

compare them. (You should apply the field at a slight angle to the x and y directions to avoid

artifacts associated with a finite geometry – in particular for the easy axis you want to avoid

the “U” shape configuration at zero field which can happen if the field is perfectly along the

x axis.)

Since permalloy does not have a magneto-crystalline anisotropy, without roughness it is

expected the two hysteresis loops will be identical. With roughness applied an effective

anisotropy is observed, due to the orientation of the surface roughness stripes as seen in

Figure 24.5.

Figure 24.6 – Hysteresis loops for Exercise 24.2, showing a roughness-induced anisotropy

effect.

130

Tutorial 25 – Defects and Impurities

Material parameters in Boris may also be assigned a spatial variation, in addition to a

temperature dependence. This spatial variation will be taken into account in all routines

where the material parameters appear, allowing inclusion of material defects and impurities

in simulations as appropriate.

To see the currently set parameters spatial variation use the command:

paramsvar

You can use a pre-defined method of generating defects by following the instructions

displayed after using the paramsvar command. Currently these include: random, jagged,

defects, faults. To see the spatial variation generated, under display select the ParamVar

option, making sure to select the required parameter under the paramvar list. The generated

spatial variation is stored as an array of coefficients, multiplying the base parameter value.

For examples of these profiles see Figures 25.1 – 4.

Figure 25.1 – Random parameter variation between 0.9 and 1.1 with generator seed 1.

131

Figure 25.2 – Jagged parameter variation between 0.9 and 1.1 with 30 nm average spacing

and generator seed 1.

Figure 25.3 – Defects parameter variation between 0.9 and 1.1 with diameters in the range 20

nm to 50 nm, and 40 nm average spacing with generator seed 1.

132

Figure 25.4 – Faults parameter variation between 0.9 and 1.1 with 20 nm to 50 nm fault

length, -30° to 30° fault orientation and 50 nm average spacing with generator seed 1.

Exercise 25.1

Generate MS defects in a 640 nm × 640 nm × 10 nm mesh as shown in Figures 25.1 – 4.

You can also set a custom parameter variation using an image as a mask file, although this is

now a legacy option and documented in a previous version (v2.4). Instead if you want to set

an arbitrary parameter variation you should use the ovf2 file option, which can be

programmatically generated – a routine is included in the NetSocks module, allowing for

easy generation of parameter spatial variation. This is covered in a separate chapter in the

manual on working with ovf2 files. You can also set parameter variation using a text

equation, which simultaneously allows for both spatial and temporal dependence. This is

covered in a separate chapter in the manual on working with text equations.

133

Tutorial 26 – Polycrystalline and Granular Films

Boris includes a Voronoi tessellation generator, both 2D and 3D, which can be used to

generate polycrystalline and granular films.

Figure 26.1 – Polycrystalline film showing K1 parameter variation generated using vor2D

generator between 0.9 and 1.1 with 40 nm spacing and generator seed 1.

Figure 26.2 – Polycrystalline film showing easy axis (ea1) parameter variation generated

using vorrot2D generator with polar angle range 70° to 110°, azimuthal angle range -90° to

90°, with 40 nm spacing and generator seed 1.

Polycrystalline films may be simulated by generating parameter variations using one of the

Voronoi tessellation generators under the paramsvar command. These include:

134

vor2d min, max; spacing; seed – Used for 2d crystallites in the xy plane.

vor3d min, max; spacing; seed – Used for 3d crystallites.

vorbnd2d min, max; spacing; seed – Used for 2d crystallites in the xy plane, but parameter

variation generated randomly only at Voronoi cell boundaries.

vorbnd3d min, max; spacing; seed – Used for 3d crystallites, but parameter variation

generated randomly only at Voronoi cell boundaries.

vorrot2d min_polar, max_polar; min_azimuthal, max_azimuthal, spacing; seed – Used for

2d crystallites in the xy plane, specifically magneto-crystalline anisotropy easy axes.

vorrot3d min_polar, max_polar; min_azimuthal, max_azimuthal, spacing; seed – Used for

3d crystallites, specifically magneto-crystalline anisotropy easy axes.

Both 2D and 3D crystallites may be generated using the generators listed above. For an

example of a 2D polycrystalline film with K1 (magneto-crystalline anisotropy) variation see

Figure 26.1. In order to generate crystallites with a varying magneto-crystalline anisotropy

easy axis orientation you can use either the vorrot2d or vorrot3d generator – for example

see Figure 26.2 for the ea1 parameter having the same polycrystalline structure as in Figure

26.1. Figure 26.2 shows the rotation to be applied, as a vector quantity. For an easy axis base

value set along the x-axis this coincides with the resulting easy axis orientation.

You can also generate a parameter variation at the Voronoi cell boundaries rather than in the

cells themselves. This can be done using the vorbnd2d and vorbnd3d generators. This could

be useful for example to modify the electrical conductivity at the grain boundaries only.

In order to generate a granular film with non-magnetic phase separation you can use one of

the following commands:

generate2dgrains spacing (seed)

generate3dgrains spacing (seed)

These commands generate granular films directly on the magnetization mesh – see for

example Figure 26.3. You can also combine this with parameter variations generated using

the same generator seed and sizes (e.g. grains with varying MS values).

135

When generating grains for the magnetic mesh you can choose to generate grains for the

electrical conductivity mesh also. To carry the grain structure over you should generate the

grains first without the Transport module enabled. After enabling the Transport module the

granular structure is also applied to the electrical conductivity mesh. This could be useful for

example in a multi-layered structure. If you apply the grain structure with the Transport

module already enabled, the grains are not generated for the electrical conductivity also. You

could combine this with a Voronoi generator for the elC material parameter however (e.g.

vorbnd2d or vorbnd3d, or even vor2d or vor3d) as mentioned above.

Figure 26.3 – Granular film (80 nm thick) with non-magnetic phase separation, generated

using generate3dgrains command with 50 nm spacing and generator seed 1. The image

shows a magnetization configuration at zero field.

136

Tutorial 27 – Periodic Boundary Conditions

Periodic boundary conditions (PBC) may be applied when computing demagnetizing fields.

Enabling this setting also affects exchange coupling (e.g. Exchange, DMExchange,

iDMExchange), resulting in a wrap-around effect. PBCs are useful to simulate periodic

arrays, or to approximate effectively infinite thin films or tracks using only a finite simulation

window. PBCs are applicable to both single mesh demagnetization (Demag), as well as

supermesh demagnetization (SDemag), including multi-layered demagnetization. Moreover,

when enabling the Roughness module, PBCs are also taken into account when calculating an

effective roughness field.

To enable PBCs use the command:

pbc

The configuration for all meshes, or the supermesh if applicable, will be shown. You can

enable PBCs in any direction by setting a number of images using the interactive console

objects (10 images are set by default when enabled, which can be edited if required; the

default setting is for no PBCs). If the SDemag module is enabled you can edit the PBC

settings on the supermesh, otherwise you need to edit them for the individual meshes.

When using PBCs with multi-layered convolution you need to ensure the problem is

physically meaningful. For example the recommended approach is to use a z direction

stacking of layers, then either x or y PBCs are fine, but z PBCs will not lead to physically

meaningful results in this case.

Exercise 27.1

Repeat Exercise 24.2, but this time set default PBCs along both x and y. Compare the

hysteresis loops.

When using PBCs with the Roughness module enabled, you should be careful about setting a

combination of a large number of PBC images and fine roughness refinement

137

(refineroughness). This can result in excessive initialization time due to the large demag

kernel calculated by the Roughness module.

Figure 27.1 – Hysteresis loops obtained for Exercise 27.1. The small peaks in the hard axis

loop are artifacts resulting from the finite simulation mesh size, and could be decreased by

increasing the simulation mesh rectangle.

138

Tutorial 28 – Ultrafast Demagnetization

Ultrafast demagnetization processes may be studied in Boris using the LLB equation (or

sLLB) coupled to a two-temperature model. The default heat equation doesn’t differentiate

between lattice and electron temperature, i.e. it is a 1-temperature model. With the two-

temperature model, two coupled equations for the electron and lattice temperatures are given

as:

)(

),(

ρ

)(),(.

),(

ρ

lee

l

leee

e

TTG

t

tT

C

STTGtTK

t

tT

C

−=

∂

+−−∇∇=

∂

r

Here Ce and Cl are the electron and lattice specific heat capacities,

ρ

is the mass density, K is

the thermal conductivity, and Ge is the electron-lattice coupling constant, typically of the

order 1018 W/m3K.

To change the temperature model use the following command (or use the interactive console

output from tmodel command):

tmodel num_temperatures (meshname)

e.g. tmodel 2 sets the two-temperature model for the currently focused mesh (nnot applicable

to insulator meshes). The material parameters in the above equations are available as usual

under the params command console output (also see list of parameters under the Material

Parameters section in this manual).

In the above equation S is a heat source; a heat source may be specified in simulation using

one of the following stage types: Q, Q_seq, Qequation, Qfile. These stages specify a heat

source (W/m3) in the heat equation: Q sets a constant value, Q_seq sets a sequence of values

(similar to Hxyz_seq for external fields), Qequation sets a heat source using a text equation,

thus allowing both spatial and temperature dependence, and finally Qfile sets a spatially

uniform heat source, but with arbitrary time dependence as specified using a text file (see

139

description of stage in console help). Moreover a spatial variation may be assigned to the

parameter Q under the paramsvar command.

A typical heat source from a focused laser pulse is given as:

)/

(

)2ln(4/

)(

exp

)2

ln(

4

/

||

exp

3

2

0

20

0

mW

t

d

PS

R











−−











−

−

=r

r

Here d and tR are full-width at half-maximum (FWHM) values (pulse diameter and duration

respectively) and P0 is the maximum power density. This heat pulse may be simulated using a

Qequation stage set to (pulse centre coincides with simulation mesh centre):

Q0 * exp(-sqrt((x/Lx - 0.5)^2 + (y/Ly - 0.5)^2) / ((d0/Lx)^2/(4*ln(2)))) * exp(-(t-

2*tau)^2/(tau^2/(4*ln(2))))

For this equation we need to define the user constants (equationconstants): i) Q0, ii) d0, iii)

tau.

Exercise 28.1

In this exercise you will simulate the effect of a single laser pulse on a Co/Pt/SiO2 trilayer,

similar to that used in [Phys. Rev. B 102, 094402 (2020)], when the electron temperature

rises above the Curie temperature, and observe creation of Néel skyrmions, plotting the

variation in topological charge magnitude as a function of time. The structure to simulate is

shown in Figure 28.1. The topological charge is computed as:

dxdy

yx

Q

A













∂

×

∂

=∫mm

m.

4

1

π

The topological charge takes on unit values for skyrmions, and may be computed in Boris

using the dp_topocharge command.

140

Figure 28.1 – Trilayer structure similar to that used for Exercise 28.1, consisting of Co (2

nm) / Pt (8 nm) / SiO2 (40 nm), showing (a) temperature during a Gaussian profile laser

pulse, and (b) typical ultrafast laser pulse and temperature time dependence in the three

layers: Co layer maximum electron and lattice temperatures, Pt layer average electron

temperature and SiO2 average temperature. Reproduced from [Phys. Rev. B 102, 094402

(2020)].

See the ufsky_creation.py script in the Examples/Tutorial 0 folder. This script sets up the

simulation for this exercise from scratch. Study the script to understand what all the

commands are used for (if needed look up the command help in the console or manual). Run

this script several times and build a probability distribution of number of skyrmions created

in each run. Is the computed probability distribution described by a Poisson counting

distribution? What is the mean number of skyrmions created?

Hint: to obtain a reasonable probability distribution you will need to run the script at least

50 times – this skyrmion counting process can be automated, and will require up to 1h or 2h

of simulation time depending on your workstation.

141

Further hint: the computed Q value with dp_topocharge will not be an integer for two

reasons: i) stochasticity, ii) cellsize could be too large. However, rounding the Q value will

result in a correct integer Q value. The other possibility is to turn off the stochasticity at the

end of the simulation and allow the magnetization to quickly relax before obtaining the Q

value with dp_topocharge. With an in-plane cellsize of 1 nm this will result in a value very

close to an integer (e.g. -3.98 for 4 skyrmions etc.).

Figure 28.2 – Topological charge magnitude as a function of time, computed for a run of

Exercise 28.1. The resulting skyrmion state after 800 ps is shown in the inset, showing 6

skyrmions created (Q ≅ -6).

142

Tutorial 29 – Magneto-Optical Effect

With circularly polarised pulses the polarisation can be clockwise or anti-clockwise. Due to

the circular polarisation of the laser pulse a strong perpendicular magneto-optical field is

present, given by

( )

zr ˆ

,

0

tfHH

MOMOMO ±

=

σ

. Here fMO gives the spatial and temporal

dependence of the laser pulse, HMO (A/m) gives the strength of the magneto-optical field, and

σ± = ±1 its helicity. Thus for the Gaussian laser pulse from the previous Tutorial, the spatial

and temporal dependence of the magneto-optical field is given by:

( )











−−











−−

=)2ln(4/

)(

exp

)2ln(4/

||

exp,2

2

0

20

R

MO t

tt

d

tf rr

r

In Boris you can enable the magneto-optical effect with the moptical module (addmodule

meshname moptical).

The HMO parameter appears as a material parameter (see params), named Hmo. Thus you

can set the strength of the magneto-optical effect by setting the parameter value. You can also

set the fMO dependence as above by setting a material parameter variation for Hmo

(paramsvar). Thus for the above example you need to set the spatial variation using a text

equation set to:

exp(-sqrt((x/Lx - 0.5)^2 + (y/Ly - 0.5)^2) / ((d0/Lx)^2/(4*ln(2)))) * exp(-(t-

2*tau)^2/(tau^2/(4*ln(2))))

Exercise 29.1

Based on the simulation file from Exercise 28.1, study the effect of a train of circularly

polarised pulses with positive or negative helicities on skyrmion creation. Use a reduced laser

power density which does not raise the temperature above the Curie temperature (set Q0 =

9e20 W/m3). Write a Python script to apply 20 laser pulses with a 6 ps repetition period and

strength of 40 MA/m. Compare the results after 30 pulses for the two helicities.

143

Tutorial 30 – Spin-Wave Dispersion

Similarly to the method of simulating frequency-swept FMR (see tutorial on FMR first), we

can simulate spin-wave dispersion by applying a sinc pulse which has not only a time

dependence, but also a spatial dependence. In the xy plane the excitation field has the form:

( )( ) ( )( ) ( )( )

000

2)( ttfsincyyksincxxksincHtH

ccce

−−−=

π

As before fc is the frequency cut-off (Hz) with t0 the temporal sinc pulse centre. To excite

spin-waves with non-zero wave-vector we also need a spatial dependence for the sinc pulse.

Here kc is the wave-vector cut-off (rad/m), and (x0, y0) is the spatial sinc pulse centre. When

obtaining the spin-wave dispersion, we need to consider the direction of the wave-vector, k.

Instead of sampling the average magnetization, we need to obtain a magentisation profile

along a given direction. The direction in which we sample is the wave-vector direction k.

You can obtain a magentisation profile using the dp_getexactprofile command. Also we

need to sample the magnetisaiton at fixed steps, determined by the Nyquist criterion (as for

temporal samping):

)(m

kC

S

π

=∆

We only need to sample one component of the magnetization, and the spin wave dispersion is

obtained by performing a 2D Fourier transform on the spatial-temporal 2D data, transforming

it to wave-vector-frequency 2D space.

Consider a spin-wave waveguide as an elongated magnetic track. There are three possible

configurations, determined by the direction of the bias field.

Bias field along length – this is called the backward volume configuration.

Bias field along thickness – this is called the forward volume configuration.

Bias field along width – this is called the surface spin wave configuration.

144

In all three cases the excitation and analysed magnetization component need to be

perpendicular to the bias field; for simplicity the analysed magnetization component can be

chosen to be along the excitation direction. When simulating the spin-wave dispersion it is

important to choose the cell-size correctly, as the computed spin-wave dispersion will be

inaccurate at larger wave-vector values if the cellsize is not small enough.

Exercise 30.1

Read through the reference IEEE Trans. Mag. 49, 524 (2013). Simulate the spin-wave

dispersion as described in this reference for the three spin-wave configurations described

above. Use periodic boundary conditions along the length only.

You should make the following modifications to the proposed problem:

Cellsize along length should be 1 nm; the specified 2 nm cellsize results in a large

discrepancy between computations and analytical formulas at larger k values.

You should simulate for a total time of 2×t0 as explained in the tutorial on FMR. For this

exercise it is suggested you use a t0 value of 200 ps, although 100 ps is still fine. A value of

50 ps results in a coarse frequency spectrum.

The spatial sampling interval should be 4 nm due to the Nyquist criterion.

Hint: use an Hequation stage set to 'H0, He * sinc(kc*(x-Lx/2))*sinc(kc*(y-

Ly/2))*sinc(2*PI*fc*(t-t0)), 0' for the backward volume configuration.

When analysing the spin-wave dispersion you can use the following formula:

)/(

2

0

sradk

M

A

www

S

Mn

µ

+=

Here wn is the resonance frequency (rad/s) for the nth spin-wave mode at k = 0, and may be

extracted from the computed spectrum at k = 0, or alternatively you can use an analytical

formula to predict it (not given here). The value wM is given as γMS, where γ = µ0|γe|

(2.212761569×105 mA/s).

145

Figure 30.1 – Spin-wave dispersion computed in Exercise 30.1 together with analytical

predictions (even spin-wave modes excited n = 0, 2, 4, 6, 8, 10, 12). A damping value of 0.01

was used.

146

Tutorial 31 – Two-Sublattice Model

In a micromagnetics formulation we can study antiferromagnetic, ferrimagnetic, as well as

binary ferromagnetic alloys using a two-sublattice model, where we consider magnetic orders

of two sub-lattices A, B, and couple them using inter-lattice exchange stiffness terms. The

two-sublattice stochastic LLB equation (sLLB) is given below:

()( )

( )

),(.

~

,||,

||,

,,

,

BAi

M

t

ithi

ii

i

ithieff

i

ieffii

i

=++

+××

−×−=

∂

⊥

ηM

HM

HH

MM

HM

M

α

γ

α

γ

The reduced gyromagnetic ratio is given by

( )

2,

1/

~

iii ⊥

+=

αγγ

, and the reduced transverse and

longitudinal damping parameters by

i

ii

m/

~

(||),

(||), ⊥⊥

=

α

, where

0,

/

)()( iSii MT

MTm =

, with

0,iS

M

denoting the zero-temperature saturation magnetization, and Mi ≡ |Mi|. The damping

parameters are continuous at TN – the phase transition temperature – and given by:

( )

N

ii

N

Niejeiji

ii

N

Niejeiji

ii

TT

T

TT

Tmm

T

TT

Tmm

T

≥==

<













+

=

<













+

−=

⊥

,

3

2

,

~

/3

2

,

~

/3

1

||,,

,,

||,

,,

,

αα

ττ

αα

ττ

αα

We denote

N

T

~

the re-normalized transition temperature, given by:

( )

BA

ABBABA

N

T

τττ

4

2

~

2

+−++

=

The micromagnetic parameters

τ

i and

τ

ij ∈ [0, 1], are coupling parameters between exchange

integrals and the phase transition temperature, such that

τ

A +

τ

B = 1 and |J| = 3

τ

kBTN. Here J

is the exchange integral for intra-lattice (i = A,B) and inter-lattice (i,j = A,B, i ≠ j) coupling

respectively. For a simple antiferromagnet we have

τ

A =

τ

B =

τ

AB =

τ

BA = 0.5. The normalized

equilibrium magnetization functions me,i are obtained from the Curie-Weiss law as:

147

( )

[ ]

Tk

H

TT

mmBm Bext

i

Nij

jeiieie /

/

~

30

,,,

µ

µτ

τ

++=

,

where

x

xx

B/

1

)coth(

)( −=

, and µi is the atomic magnetic moment. The magnetization

length is not constant, and can differ from the equilibrium magnetization length, giving rise to

a longitudinal relaxation field which includes both intra-lattice and inter-lattice contributions:

( )

()

Ni

ji

ie

je

i

j

i

NBij

i

N

i

je

j

i

ie

j

e

ie

i

j

i

NBij

ie

i

TT

m

Tk

TT

m

Tk

m

>































−+−

=

<











































−−











−+











−

=

,

ˆ

.

ˆ

~

3

~

1

,

1

ˆ

.

ˆ

1

~

2

3

1

~

2

1

,

||,

0||,0

||,

2,

2

,

2,

2

||,

0

2,

2

||,

0

||,

mmmH

χ

µµ

τ

χµ

χ

µµ

τ

χ

µ

Here

iii m/

ˆmm =

, and the relative longitudinal susceptibility is

0,0||,||, /

~iSii M

µχχ

=

, where:

( )

( )( ) ( )

2

||, /

~

3/

~

31/

~

31

/

~

3/

~

31

~

TTBBT

BTTBT

TBBTTBTB

Tk

NjijiijjNjiNi

jiNij

jjNjii

iB ′′

−

′

−

′

−

′′

+

′

−

′

=

ττττ

τµτµ

χ

and

( )

[ ]

TTmmBB

Nijjeiiemi ie

/

~

3

,,

,

ττ

+

′

≡

′

.

The direct exchange term includes the usual intra-lattice contribution, as well as

homogeneous and non-homogeneous inter-lattice contributions, and is given by:

j

je

ie

inh

j

jeie

ih

i

2ie

i

iex

MM

A

MM

A

M

AMMMH

2

,,

0

,

,,0

,

2

,0

,

4

2∇

++∇=

µµµ

The intra-lattice exchange stiffness Ai has the temperature dependence

2,

0ieii mAA =

, whilst the

inter-lattice exchange stiffnesses have the temperature dependences

jeieinhhinhh mmA

A,,

0),(),(=

.

Finally, the terms Hth,i and ηth,i are stochastic quantities with zero spatial, vector components,

and inter-lattice correlations, and whose components follow Gaussian distributions with zero

mean and standard deviations given respectively by:

148

( )

tV

M

Tk

tVM

Tk

H

iSii

B

stdi

th

i

Si

i

iB

i

stdith

∆

=

∆

−

=⊥

⊥

0

0,||,

.

,

0,

0

||,,

,

.

,

2

1

µ

γα

η

µ

γ

αα

α

Here V is the stochastic computational cellsize volume, and Δt is the stochastic time-step.

In Boris a mesh with the two-sublattice model may be added using the command:

addafmesh meshname rectangle

Default temperature dependences may be generated as for a ferromagnetic mesh with the

command (same command as for a ferromagnetic mesh, hence the naming Curie):

curietemperature value (meshname)

Control of two-sublattice model meshes is exactly the same as for a ferromagnetic mesh, but

the list of parameters and available modules is different (see modules and params). For a

description of how the modules handle the two-sublattice model meshes see the chapter on

Modules in the manual.

Some important parameters you need to control are:

1) Micromagnetic

τ

coupling factors. Set these using the tau command as:

tau tau11 tau22 tau12 tau21

2) Atomic moments µA, µB. Set these using the atomicmoment command as:

atomicmoment mu1 mu2 meshname

Here mu1 and mu2 are the atomic moment values in units of the Bohr magneton, and

meshname must be the name of an antiferromagnetic mesh. Note this command is

also used to set the atomic moment for the LLB equation for a ferromagnetic mesh,

149

which only takes a single mu value – this is the default behaviour, so giving the

meshname parameter is important here.

3) Most magnetic parameters now have 2-sublattice values, so you can control them

separately if needed. The two inter-lattice exchange stiffness coupling terms, Ah and

Anh, only appear in two-sublattice model meshes.

When using the sLLB equation, by default the stochastic field generation time-step is the

same as the time-step using for the equation evaluation. You may need to control this

separately (set it to a larger value), and this is possible using the setdtstoch command – see

the output of the stochastic command for an interactive display.

You can also set the stochastic cellsize value to be different than the magnetic cellsize value

using the scellsize command.

Exercise 31.1 (Advanced)

The two-sublattice sLLB equation produces a distribution of magnetization lengths on the

two sub-lattices, mA, mB, which is expected to obey the following bi-variate Boltzmann

probability distribution:

( )

()

)

,,,

(

3

~

2

~

3

4

exp

,

2

,

2,

2

||,

,

2

2,

2

,

0,

2

jiB

Aji

mT

k

m

mmTk

m

T

km

V

M

mmmP

iNB

ij

je

jej

i

jNB

iji

ie

i

ei

Bi

ei

i

S

iBA

i

≠

=































−

+

−

−∝

τ

χ

χτµ

µ

Write a general-purpose Python script which tests the above equation against computed bi-

variate probability distributions for any possible combination of T, TN,

τ

i,

τ

ij, µi, and

0,iS

M

parameters. Test it for particular values (e.g. antiferromagnetic case).

Hint: there is a useful command built into Boris which computes a histogram for the two-

sublattice model, namely dp_histogram2 – see help for this command (there’s also a

dp_histogram command which works for ferromagnetic meshes). See Figure 31.1 for a

typical output you should obtain from your script.

150

Figure 31.1 – Computed two-sublattice normalized magnetization length probability

distribution at T/TN = 0.99 (colored surface) for an antiferromagnet in Exercise 31.1,

compared with the bi-variate Boltzmann probability distribution prediction (wire-frame).

Exercise 31.2

Using the default antiferromagnetic mesh (addafmesh) in Boris, verify the Kittel formula for

antiferromagnetic resonance is reproduced (see F. Keefer and C. Kittel, Phys. Rev. 85, 329

(1952)):

( )

AEA

HHHH

w++= 2

0

γ

Here HA is the uniaxial anisotropy field along the bias field, HA = 2K1/µ0MS, where MS is the

magnetization length on a sub-lattice, and HE is the Weiss exchange field, given by HE =

4Ah/µ0MS.

Hint: use the LLG equation, and adapt a previous exercise on frequency-swept FMR to

simulate a frequency-swept FMR peak and obtain the resonance frequency. You should either

use H0 set to zero, or small bias field values as the above formula becomes inaccurate at

large bias field values.

151

For the above exercise you should fit a Lorentz peak function with both symmetric and

asymmetric components for a more accurate result:

2

0

0)

(4

)

(

)( wxx

xx

Aw

Syxf +

−

+

+=

Such a fitting procedure has already been built into Boris and can be accessed using the

dp_fitlorentz2 command.

Figure 31.2 – Antiferromagnetic resonance peak with fitted symmetric and asymmetric

Lorentz peak function, verifying Kittel’s formula in Exercise 31.2.

152

Tutorial 32 – Exchange Bias

The exchange bias field on a ferromagnetic layer from an antiferromagnet is given as:

j

FS

ex

tM

JmH

0

µ

=

Here MS and tF are the saturation magnetization and thickness of the ferromagnetic layer, and

mj is the exchange bias field direction from the antiferromagnet. This effect may be modelled

in Boris using the surfexchange module, since the bilinear surface exchange field is given as:

j

FS

i

t

M

JmH

0

1

µ

=

Here mj is the magnetization direction on sub-lattice A of an interfacing antiferromagnetic

mesh. Thus in order to model exchange bias in Boris you need an antiferromagnetic mesh

(addafmesh), and a ferromagnetic mesh (addmesh) in contact with it, and they both need the

surfexchange module enabled, remembering it is the top mesh (in order of z axis direction)

which sets the J1 value.

Exercise 32.1

Simulate the exchange bias effect in a Fe 2nm thin film using a generic antiferromagnetic

material, 10 nm thick (use the addafmesh command to create a default antiferromagnetic

mesh). Enable in-plane uniaxial anisotropy for the antiferromagnetic material (x axis), and set

the antiferromagnetic sub-lattice A magnetization direction to result in a bias effect towards

the +ve side.

You can add the Fe material from the materials database, and you should enable cubic

anisotropy for it. You can use periodic boundary conditions in the xy plane, simulating an

area of 320 × 320 nm2. Set the J1 surface exchange constant to 0.2 mJ/m2. Note: avoid using

SDesc for this problem as it can be problematic, instead use LLGStatic with RKF45 and mxh

of 3

×

10-5.

153

Tutorial 33 – Magneto-Elastic Effect and Elastodynamics Solver

The magneto-elastic effect may be simulated in Boris using the melastic module. The

magneto-elastic effect can be included for a cubic crystal using a strain tensor. The strain

tensor is given as:













=

zzyzxz

yz

yy

xy

xzxyxx

εεε

S

.

Here we define the diagonal strain vector as Sd = (εxx, εyy, εzz), and off-diagonal strain vector

as Sod = (εyz, εxz, εxy). The strain tensor can have a spatial dependence, and currently needs to

either be loaded from ovf2 files (strain computed with an external package), or alternatively a

displacement vector field can be loaded (using ovf2 files, computed externally), and the strain

tensor computed as:













∂

+

∂

+

∂

+

∂

=













∂

=

x

u

y

u

x

u

z

u

y

u

z

u

z

u

y

u

x

u

y

x

z

x

z

y

od

z

y

x

d

,,

2

1

,,

S

In the simplest case a uniform stress may be applied which results in a constant strain with

zero off-diagonal terms. From the strain tensor, for a cubic crystal with orthogonal axes e1, e2,

e3, and magneto-elastic constants B1, B2, we have the following diagonal and off-diagonal

energy density terms:

[ ]

).)(.)(.().)(.)(.().)(.)(.(2

).().().().().().(

1322313212,

3

2

32

2

21

2

11,

eSememeSememeSemem

eSemeSemeSem

dodododmel

ddddmel

B

++=

ε

To set a uniform stress use the command (similar to the setfield command):

setstress magnitude polar azimuthal (meshname)

154

A uniform stress may also be set using the Sunif stage. When applying a uniform stress, T,

the strain tensor is generated based on the material Young’s modules and Poisson ratio as:

T

S













−

=

1

ν

m

Y

Young’s moduls and Poisson ratio are available as material parameters as Ym and Pr

respectively. The magneto-elastic constants are available as material parameters as MEc (2-

component parameter for B1 and B2 respectively). The orthogonal axes e1, e2, e3 are set by the

magento-crystalline anisotropy axes (ea1 is e1, ea2 is e2 and e3 = e1 × e2.

When applying a uniform stress the mesh origin (0, 0, 0) is a fixed point, and no shear strain

or physical displacement is allowed. This means a positive stress value along the x axis

results in elongation, whilst a negative stress value along the x axis results in compression.

You can run computations with a non-uniform strain, but in the current version this must be

computed externally. There are two ways of setting a non-uniform strain. The simplest

method is to compute the displacement vector map u externally and save it into a ovf2 file.

This can then be loaded into the currently focused mesh using the command (must have

melastic module enabled):

loadovf2disp filename

After the displacement map is loaded the strain tensor used for computations is obtained

using the equations above.

You can also load the strain tensor directly, but this requires saving the diagonal and off-

diagonal components in two separate ovf2 files in vector data format. The diagonal file will

then contain vector data with strain components xx, yy, zz, and the off-diagonal file will

contain vector data with strain components yz, xz, xy (in this order). The ovf2 files can then

be loaded as:

loadovf2strain filename_diag filename_odiag

155

With the melastic module enabled, there is a separate cellsize for the strain tensor, controlled

using the mcellsize command, and this should be set before loading the externally computed

strain or displacement.

Exercise 33.1

Simulate hysteresis loops for a 5 nm thick Fe thin film (found in materials database) with

cubic anisotropy and melastic module enabled, along the x axis. Use a magneto-elastic

coefficient B1 = 8 MJ/m3. Simulate three cases: i) no strain, ii) compressive stress along the x

axis of 0.7 GPa, iii) tensile stress along the x axis of 0.7 GPa. Explain the differences

between the 3 curves.

Figure 33.1 – Hysteresis loops computed in Exercise 33.1 for a Fe thin film, with and

without mechanical stress.

It is also possible to compute strains in response to 1) external forces, 2) magnetostriction, 3)

thermoelastic effect, as well as use of elastodynamics solver. In this case at least one fixed

surface should be defined, where the mechanical displacement is zero:

surfacefix (meshname) rect_or_face

The easiest way is to specify an entire fixed face for a given mesh (typical use case), using

string literals -x, x, -y, y, -z, z. For example surfacefix -z designates the entire bottom face as

156

fixed. Alternatively a rectangle may be specified in absolute coordinates, but this should

intersect with at least one mesh face. Multiple fixed surfaces may be specified.

An elastodynamics solver time-step also needs to be specified using the seteldt command.

For stability this should satisfy the Courant, Friedrichs, Lewy (CFL) condition, and should

also not exceed the magnetisation solver time-step:

()

22

2

p

1/

−

∆+∆+∆<∆ zyxvt

Here vp is the elastic compressional wave velocity (vp is typically 4000 to 6000 m/s), and ∆x,

∆y, ∆z are the mechanical cell-size dimensions. Setting the elastodynamics solver time-step to

zero disables it and keeps the currently calculated strain fixed. Elastic stiffness coefficients

(cC material parameter) and mass density (density) should also be specified. There’s also a

mechanical damping coefficient, which results in mechanical energy dissipation (mdamping

material parameter).

In addition to magneto-elastic coefficient, MEc, there are also corresponding coefficients for

magnetostriction effects (mMEc) – in theory these are identical to MEc, but can be set to zero

in order to disable magnetostriction. A linear thermoelastic expansion coefficient may also be

specified (thalpha), which should be used in conjuction with the heat flow solver. For full

details see the melastic module description in Modules.

Depending on the use case, it may also be necessary to reset the elastodynamics solver to

initial conditions, and this can be achieved using the resetelsolver command.

Finally, external forces may be specified using the surfacestress command:

surfacestress (meshname) rect_or_face vector_equation

The same format as for the surfacefix command is used here, but additionally the external

stimulus (units of N/m2, i.e. a pressure) is specified using a vector text equation (3

components). The equation depends on x and y spatial coordinates, which are relative to the

157

defined surface rectangle, and also on time t. See the User-Defined Text Equations section for

details on text equations. Any number of external stimuli may be specified.

Exercise 33.2

Setup a surface acoustic wave (SAW) with 200 nm wavelength, in a Fe track, 2 µm long, 50

nm wide, and 40 nm thick. Assume the Fe track is on a stiff substrate, and generate the SAW

using i) an external pressure of 1 GPa amplitude on the -x face, and ii) using interdigitated

electrodes on the top surface with a 0.5 GPa shear pressure amplitude. Assume the

compressional wave velocity is vp = 4500 m/s for the default elastic coefficients (c11 = 300

GPa, c12 = 200 GPa, c44 = 50 GPa). Obtain the mechanical displacement profile after 5 ns.

You may disable magnetostriction.

Notes: the SAW condition is

λ

= vp / f, thus the external stimulus should have the form

)/

2

sin(

||0

λ

π

t

v

F

. For interdigitated electrodes the electrodes spacing and width should be

λ

/4

with a π radians phase difference between them. It is also necessary to allow for energy

dissipation otherwise waves will be reflected from the opposite end of the stimulus. This may

be achieved by setting the mechanical damping parameter with an absorbing boundary spatial

variation: use the abl_tanh generator so the mechanical damping reaches a value of 1016

kg/m3s at one end, but has a relatively small value elsewhere (e.g. 1010 kg/m3s), so SAW

attenuation is confined to one end only.

Figure 33.2 – Longitudinal mechanical displacement profile for the 200 nm SAW generated

in Exercise 33.2. The SAW is absorbed at the far end to prevent reflections.

158

Shapes and Regions

There are a number of built-in elementary shapes (rectangle, disk, triangle, ellipsoid,

torus, pyramid, tetrahedron, cone) which can be combined to form composite shapes. These

shapes can then be used as 1) physical objects in meshes, 2) as masks to set magnetization

directions, and 3) as regions to set material parameter values. The shapes also have

transformation functions, including scaling, translation, rotation, array generation.

Elementary shapes can be generated using the console commands: shape_rect,

shape_disk, shape_triangle, shape_tetrahedron, shape_pyramid, shape_cone,

shape_ellipsoid, shape_torus. These functions use modifiers, including rotation, array

generation, and method (add, subtract), set using the commands: shape_rotation,

shape_repetitions, shape_displacement, shape_method.

A more powerful approach to working with shapes consists in using Python scripted

simulations. The NetSocks.py module defines a class called Shape. This class can be used to

define the above elementary shapes, which can then be combined using arithmetic operators

(+,-) to form composite shapes. The shapes can then be physically set in a mesh, used to set a

material parameter spatial variation by effectively defining a region, or used as a mask to set

magnetization directions. They can also be transformed using methods defined in the Shape

class, as detailed in the tables below.

Table SR.I – Elementary shapes methods with given dimensions (m) and center position (m).

Overloaded operators are + and -, and the result is a composite shape.

Shape class: elementary shapes methods

disk(dimensions, position)

pyramid(dimensions, position)

rect(dimensions, position)

cone(dimensions, position)

triangle(dimensions, position)

ellipsoid(dimensions, position)

tetrahedron(dimensions, position)

torus(dimensions, position)

159

Table SR.II – Shape transformation methods. Rotations are specified using YXZ Tait-Bryan

angles in degrees.

Shape class: transformation methods for both composite and elementary shapes

setdimensions(dimensions)

Set dimensions of first elementary shape contained, and

scale everything else in proportion.

scale(scalefactors)

Scale dimensions of shape.

setposition(position)

Set position of shape, defined by the position of the first

elementary shape contained.

move(positionshift)

Translate position of shape.

setrotation(rotation)

Set rotation of shape, around shape position as defined by

the first elementary shape contained.

rotate(rotation)

Rotate shape around shape position as defined by the first

elementary shape contained.

setrepetitions(repetitions,

displacement)

Set number of repetitions and displacement of shape for

generating array.

Once a shape has been defined, we can use 1) shape_set command to set a physical shape in

the focused mesh, 2) shape_setangle command to set magnetization angle in the defined

shape, and 3) shape_setparam command to set the material parameter spatial variation

scaling in the region defined by the shape.

The following contains examples, with Python scripts included in the Examples/Shapes and

Regions directory.

from NetSocks import NSClient, Shape

ns = NSClient(); ns.configure(True)

########################################

l, w, t = 800e-9, 800e-9, 100e-9

FM = ns.Ferromagnet([l, w, t], [5e-9])

ns.delrect()

tor = Shape.torus([l, w, t])

disk = Shape.disk([l/2, w/2, t])

#set disk centred

FM.shape_set(disk.move([l/2, w/2, t/2]))

160

#set torus centred

FM.shape_set(tor.move([l/2, w/2, t/2]))

#set magnetization angles of defined shapes

FM.shape_setangle(disk, [180, 0])

FM.shape_setangle(tor, [90, 90])

The above example generates a torus and a disk, shown in Figure SR.1.

Figure SR.1 – Physical shapes generated in a magnetic mesh.

The following sets the MS material parameter value in the region defined by the torus and

disk shapes, with the result shown in Figure SR.2.

#set material parameter values of regions defined by shapes

Ms = FM.param.Ms.setparam()

FM.param.Ms.shape_setparam(disk, 600e3 / Ms)

FM.param.Ms.shape_setparam(tor, 1200e3 / Ms)

Figure SR.2 – Material parameter (MS) spatial variation set in defined regions.

161

The following exemplifies the use of composite shapes and transformation functions, with the

result shown in Figure SR.3.

from NetSocks import NSClient, Shape

ns = NSClient(); ns.configure(True)

########################################

l, w, t = 800e-9, 800e-9, 200e-9

FM = ns.Ferromagnet([l, w, t], [4e-9])

ns.delrect()

#define hollow half-torus

tor1 = Shape.torus([l, w, t])

tor2 = Shape.torus([l - t/2 + t*0.75/2, w - t/2 + t*0.75/2, t*0.75])

htor = tor1 - tor2 - Shape.rect([l, w/2, t]).move([0, -w/4, 0])

#define masks for left and right sides of half-torus

left = htor - Shape.rect([l/2, w/2, t]).move([l/4, w/4, 0])

right = htor - Shape.rect([l/2, w/2, t]).move([-l/4, w/4, 0])

#set shape and magnetization directions for left and right sides

FM.shape_set(htor.move([l/2, w/2, t/2]))

FM.shape_setangle(left.move([l/2, w/2, t/2]), [90, 90])

FM.shape_setangle(right.move([l/2, w/2, t/2]), [90, 270])

#rectangular base

base = Shape.rect([l * 0.6, w*0.5, 10e-9])

FM.shape_set(base.move([l/2, w/4, t/3]))

#repeated tetrahedra

d = 40e-9

tetra = Shape.tetrahedron([d, d, d])

tetra.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

FM.shape_set(tetra.move([l * 0.2 + d, d, t/3 + d/2]))

FM.shape_setangle(tetra, [90, 0])

#repeated pyramids

pyramid = Shape.pyramid([d, d, d])

pyramid.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

FM.shape_set(pyramid.move([l * 0.2 + 3*d, d, t/3 + d/2]))

FM.shape_setangle(pyramid, [0, 0])

#repeated rotated triangles

triangle = Shape.triangle([d, d, d])

triangle.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

triangle.rotate([0, 90, 45])

FM.shape_set(triangle.move([l * 0.2 + 5*d, d, t/3 + d/2]))

FM.shape_setangle(triangle, [180, 0])

#repeated rotated excentric tubes

tube = Shape.disk([d, d, d]) - Shape.disk([d/2, d/2, d]).move([d/5, 0, 0])

tube.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

tube.rotate([-30, 45, 45])

FM.shape_set(tube.move([l * 0.2 + 7*d, d, t/3 + d/2]))

FM.shape_setangle(tube, [90, 90])

162

#repeated ellipsoids

ellipsoid = Shape.ellipsoid([d/2, d*1.5, d*0.75])

ellipsoid.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

FM.shape_set(ellipsoid.move([l * 0.2 + 9*d, d, t/3 + d/2]))

FM.shape_setangle(ellipsoid, [90, 270])

#repeated cones

cone = Shape.cone([d, d, d])

cone.setrepetitions([1, w / (4*d), 1], [0, d * 2, 0])

FM.shape_set(cone.move([l * 0.2 + 11*d, d, t/3 + d/2]))

FM.shape_setangle(cone, [45, 45])

ns.displaydetail(2e-9)

Figure SR.3 – Examples of elementary and composite physical shapes generated in a

magnetic mesh.

As a final example we generate a 3D array of a composite, rotated and scaled shape, with the

result shown in Figure SR.4.

from NetSocks import NSClient, Shape

ns = NSClient(); ns.configure(True)

########################################

l, w, t = 800e-9, 800e-9, 800e-9

FM = ns.Ferromagnet([l, w, t], [5e-9])

ns.delrect()

disk = Shape.disk([300e-9, 300e-9, 300e-9])

triangle = Shape.triangle([150e-9, 150e-9, 300e-9])

shape = disk - triangle.move([60e-9, 60e-9, 0])

163

shape.rotate([10, 10, 10])

shape.scale([0.3, 0.3, 0.3])

shape.setrepetitions([4, 4, 4], [200e-9, 200e-9, 200e-9])

FM.shape_set(shape.setposition([100e-9, 100e-9, 100e-9]))

Figure SR.4 – Example 3D array of composite and rotated physical shapes.

164

User-Defined Text Equations

Arbitrary text equations may be supplied by the user using fundamental functions, a number

of special functions, and mathematical operators as detailed below. These equations are

evaluated efficiently at run-time every iteration. Text equations use a number of reserved

variables depending on the context, including x, y, z to define spatial variation, t to define

temporal dependence, T to define temperature dependence, and also use a number of reserved

constants as listed below. Both scalar and vector text equations may be supplied as

appropriate. The contexts in which text equations may be used are:

1. Setting stage values

The available stage types are: i) Hequation (set external field using a vector text equation), ii)

Vequation (set electrode potential drop using a scalar text equation), iii) Iequation (set ground

electrode current using a scalar text equation), iv) Tequation (set mesh base temperature

using a scalar text equation – applicable when heat module disabled), v) Qequation (set heat

source in heat equation using a scalar equation).

Reserved variables:

x, y, z (spatial coordinates in meters, relative to mesh where used), t (stage time in seconds).

Defined constants:

Lx, Ly, Lz (mesh dimensions in meters), Tb (mesh base temperature), Ss (stage step).

Example: 0, 0, He * sinc(kc*(x-Lx/2))*sinc(kc*(y-Ly/2))*sinc(2*PI*fc*(t-t0)),

The above example is used with Hequation (vector equation), and applies a spatial and

temporal sinc pulse at the centre of the mesh with time centre at t0, for the z field component.

The constants He, kc, fc, t0 are defined by the user – see below.

165

2. Parameter temperature dependence

Any material parameter for which a temperature dependence is allowed (see output of

paramstemp command), may be assigned a temperature dependence defined using a text

equation. To set this either use the interactive console, or use the following command:

setparamtempequation (meshname) paramname text_equation

In Python scripts you can use ns.setparamtempequation(paramname, text_equation), or for a

specific mesh as ns.setparamtempequation(meshname, paramname, text_equation).

Reserved variables:

T (temperature, either mesh base temperature if heat module not enabled, or the local

temperature if heat module is enabled – in the latter case the temperature can be non-uniform

and the parameter value is evaluated according to the local computational cell temperature).

Defined constants:

Tb (mesh base temperature), Tc (set mesh Curie (or Néel) temperature).

Example: me(T/Tc)^2

The above example applies a squared Curie-Weiss scaling relation (me), for temperatures

ranging from 0 to Tc.

3. Parameter spatial variation

Any material parameter for which a spatial variation dependence is allowed (see output of

paramsvar command), may be assigned a spatial and temporal dependence defined using a

text equation. To set this either use the interactive console, or use the following command:

setparamvar (meshname) paramname equation text_equation

Note in the above command the field “equation” specifies the type of generator used to

generate a spatial variation and must be inputted literally as above.

166

Reserved variables:

x, y, z (spatial coordinates in meters, relative to mesh where used), t (stage time in seconds).

Defined constants:

Lx, Ly, Lz (mesh dimensions in meters).

Example: exp(-(x-Lx/2)^2/Sx) * exp(-(y-Ly/2)^2/Sy) / (exp(0)^2)

The above example applies a Gaussian spatial variation scaling in the xy plane at the centre

of the mesh.

Functions

Allowed functions in text equations are given below.

Function Name

Description

sin

Fundamental sin function.

sinc

sinc(x) = sin(x) / x

cos

Fundamental cos function.

tan

tan(x) = sin(x) / cos(x)

sinh

sinh(x) = [exp(x) - exp(-x)] / 2

cosh

cosh(x) = [exp(x) + exp(-x)] / 2

tanh

tanh(x) = sinh(x) / cosh(x)

sqrt

Square root.

exp

Fundamental exp function.

asin

Inverse sin function.

acos

Inverse cos function.

atan

Inverse tan function.

asinh

Inverse hyperbolic sin function.

acosh

Inverse hyperbolic cos function.

atanh

Inverse hyperbolic tan function.

ln

Natural logarithm.

log

Logarithm base 10.

abs

Modulus function.

sgn

Sign function.

floor

Floor function (round down).

ceil

Ceil function (round up).

167

round

Round function (round to nearest integer).

step

Step function: step(x) = 0 for x < 0, = 1 for x>= 0.

swav

Square wave function with period 2π, s.t. swav(0+) = +1, swav(π+) = -1.

twav

Triangular wave function with period 2π, s.t. twav(0) = +1, twav(π) = -1.

me

Normalized Curie-Weiss law.

chi

Normalized relative longitudinal susceptibility.

me1

Normalized Curie-Weiss law for sub-lattice A.

me2

Normalized Curie-Weiss law for sub-lattice B.

chi1

Normalized relative longitudinal susceptibility for sub-lattice A.

chi2

Normalized relative longitudinal susceptibility for sub-lattice B.

alpha1

Transverse damping temperature dependence.

alpha2

Longitudinal damping temperature dependence.

Any nested combination of functions is allowed.

Special equation expanders

sum

This is the sum function, with format given as: sum(i;low;high;func(<i>)).

Thus func(i) is summed for i ranging from integers low to high inclusive; note the format i

must appear in func, namely <i>. The variable named i can be changed to a different string

literal but must not clash with any equation variables, or reserved names. For example the

expression:

sum(j;0;10;sin(2*PI*<j>*x)), evaluates to:

( )

∑

=

10

0

2sin

j

jx

π

Nested sum functions are also allowed, and may be combined with any of the reserved

function names in the table above. Note that using excessive limits (e.g. integer range over

1000) in the sum function will result in very slow evaluation times and should be avoided,

especially if the evaluation is to be performed in every computational mesh cell.

168

Allowed mathematical operators

Operator Symbol (in precedence order)

Description

^

Exponentiation operator.

/

Division operator.

*

Multiplication operator.

-

Subtraction operator.

+

Addition operator.

All functions, variables, and constants must be linked by an operator, assumed multiplication

is not allowed, e.g. 2*sin(x) is a valid equation, 2sin(x) is not a valid equation in the current

program version.

Bracketing

Only round brackets are allowed, (, ), which must appear in equal numbers.

Numerical constants

Numbers may be specified in floating point or scientific format.

Reserved constants

A number of pre-defined numerical constants are available. Since the names in the table

below are reserved, they must not clash with any user defined constants.

Symbol

Description

PI

3.1415926535897932384626433833

mu0

Free space permeability: 4*PI*10-7 (N/A2)

muB

Bohr magneton: 9.27400968e-24 (Am2)

ec

Electron charge: 1.60217662e-19 (C)

hbar

Reduced Planck constant: 1.054571817e-34 (m2kg/s)

kB

Boltzmann constant: 1.3806488e-23 (m2kg/s2K)

gamma

Gyromagnetic ratio modulus: 2.212761569e5 (m/As)

169

User defined constants

Any number of user defined constants, named using alphanumeric strings, may be given, with

the restriction they must not clash with any of the reserved names in the tables above, or

equation variables. Equation constants may be given as:

equationconstants name value

Other related commands are delequationconstant, and clearequationconstants.

Vector and scalar equations

Scalar equation have a single component which must conform to the above rules. Vector

equations with 3 components are allowed where appropriate, and must be specified using

comma separators as: component1, component2, component3. Here the three components are

scalar equations.

170

Working with OVF2 Files

Data may be loaded and saved using the OVF2 file format introduced in OOMMF. This

allows setting magnetic shapes programmatically, as well as importing and exporting data to

other software which support this file format.

OVF2 files may also be used to save mesh data numerically, not only for magnetization, but

for any mesh quantity which may be displayed on screen, including material parameter

spatial variation. The latter may also be set programmatically using OVF2 files, via the ovf2

spatial variation generator (see below).

OVF2 files allow scalar or vector formats, and both are handled in Boris. Data may be saved

in natural text format (text), single precision binary (bin4), or double precision binary (bin8).

Commands which handle OVF2 files

loadovf2mag (meshname) (renormalize_value) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing magnetization data, into the currently

focused mesh or given meshname (which must be ferromagnetic), mapping the data to the

current mesh dimensions. By default the loaded data will not be renormalized:

renormalize_value = 0. If a value is specified for renormalize_value, the loaded data will be

renormalized to it (e.g. this would be an Ms value).

saveovf2mag (meshname) (n) (data_type) (directory/)filename

Save an OOMMF-style OVF 2.0 file containing magnetization data from the currently

focused mesh or given meshname (which must be ferromagnetic). You can normalize the data

to Ms0 value by specifying the n flag (e.g. saveovf2mag n filename) - by default the data is

not normalized. You can specify the data type as data_type = bin4 (single precision 4 bytes

per float), data_type = bin8 (double precision 8 bytes per float), or data_type = text. By

default bin8 is used.

171

saveovf2 (meshname) (data_type) (directory/)filename

Save an OOMMF-style OVF 2.0 file containing data from the currently focused mesh or

given meshname. You can specify the data type as data_type = bin4 (single precision 4 bytes

per float), data_type = bin8 (double precision 8 bytes per float), or data_type = text. By

default bin8 is used.

loadovf2temp (meshname) (directory/)filename

Load a temperature distribution into the currently focused mesh or given meshname, which

must have the heat module enabled. You can disable the heat solver by setting setheatdt 0.

loadovf2curr (meshname) (directory/)filename

Load a current density distribution into the currently focused mesh or given meshname,

which must have the transport module enabled. This should normally be used with transport

solver iteration disabled (but transport module enabled) so the loaded current distribution

remains constant: disabletransportsolver 1.

loadovf2disp (meshname) (directory/)filename

Load mechanical displacement data into the currently focused mesh or given meshname,

which should have the melastic module enabled. A strain will be computed internally.

loadovf2strain (meshname) (directory/)filename_diag filename_odiag

Load strain tensor data into the currently focused mesh or given meshname, which should

have the melastic module enabled. Two vector files are required, one for the diagonal strain

tensor components, and the other for the off-diagonal components, where a symmetric strain

tensor for a cubic crystal is assumed.

172

loadovf2field (meshname) (directory/)filename

Load magnetic field into the currently focused mesh or given meshname, which should have

the Zeeman module enabled. Alternatively. if meshname is supermesh, load a global field

with absolute rectangle coordinates and its own discretization.

setparamvar (meshname) paramname ovf2 filename

Set the named parameter, paramname, spatial dependence for the named mesh, meshname,

using an ovf2 file with filename, located in current working directory. The type of data in the

OVF2 file must match the expected format of the parameter (scalar or vector). Once you’ve

loaded the ovf2 file you can display the set spatial variation by selecting the ParamVar

option under display, and clicking on the required parameter under paramsvar.

The provided NetSocks.py module used for Python scripts, provides a convenient method to

handle OVF2 files: Write_OVF2, which allows writing a Python list into an OVF2 file. An

example of programmatically setting a magnetic shape using an OVF2 file generated in a

Python script is given in Tutorial 0.

saveovf2param (meshname) (data_type) paramname (directory/)filename

Save an OOMMF-style OVF 2.0 file containing the named parameter spatial variation data

from the named mesh (currently focused mesh if not specified). You can specify the data type

as data_type = bin4 (single precision 4 bytes per float), data_type = bin8 (double precision 8

bytes per float), or data_type = text. By default bin8 is used.

173

Materials Database

Material definitions can be saved in a materials database. This includes base material

parameter values. The default database is called BorisMDB.txt. You can see this using the

command:

materialsdatabase

The default materials database can be updated from a shared database stored on a server. The

shared database can be seen at: https://boris-spintronics.uk/online-materials-database. To

update your local BorisMDB database using the latest material parameter definitions, use the

command:

updatemdb

You can also switch to an alternative custom database using the materialsdatabase

command. To add a new computational mesh with given material parameters you can use the

addmaterial command. The type of material will determine the type of computational mesh

generated. For example the ferromagnetic type will generate a computational mesh with

LLG/LLB solvers enabled. The conductor mesh type will generate a computational mesh

with only the transport and heat solvers enabled, while the insulator mesh will only have the

heat solver enabled (e.g. a substrate material).

Users can also send in entries to be added to the centrally stored database. This can be done

using the requestmdbsync command. Before sending entries, you must properly format the

material entry. The procedure is described as follows.

1. Add a new entry to your local BorisMDB file.

Suppose you want to enter a new ferromagnetic material. First create a ferromagnetic

mesh in Boris (addmesh), and set as many parameter values as possible. Next, add

the entry to your local BorisMDB file using:

addmdbentry meshname (materialname)

174

Here meshname is the name of the mesh as it appears in Boris (the one you created

using addmesh), and materialname is the name of the material, or entry, you want to

create, if different.

2. Edit the material description fields in BorisMDB.txt

There are 5 description fields for the entry:

Name, Formula, Type, Description, Contributor

Name is the name of your new material entry, which should not already be in the

shared online database. This should already be filled.

Formula is the symbolic formula for the material. Make sure to fill this.

Type is already filled for you, and is the type of computational mesh for which the

material applies.

Description should have a very brief description for the material entry, with any

useful information. Make sure to fill this.

Contributor is the name of the entry contributor; leave as N/A if you don’t want to

specify this.

There is another field called State. This specifies if the entry was taken from the

online database (SHARED) or if it’s a new user-created entry (LOCAL). You don’t

need to change this.

175

3. Set references for the parameters

After each parameter there is a column called DOI. This must hold a DOI reference

for where the material parameter value was taken from, or derived. You can leave it

as N/A only if not applicable, but in most cases should be properly referenced.

4. Unspecified material parameters

If you cannot reasonably give an entry for a material parameter value then override it

as “N/A”.

5. Send in the entry

After properly formatting the entry, you can upload it to a holding database using the

requestmdbsync command:

requestmdbsync materialname (email)

Materialname is the name of the material you’ve just created. If you specify an email

address you will receive feedback about whether the entry was added to the shared

database or not – the entry will be verified for validity before being added to the

online materials database. Once entered there, it will be visible at https://boris-

spintronics.uk/online-materials-database, and other users can update their databases

with it.

176

Meshes

Boris uses cell-centred finite difference discretization. In the simplest case a single magnetic

mesh may be used, which occupies a rectangular space. Object shapes may be defined inside

the mesh, however these must always be contained within the reserved mesh rectangular

space. The size of the mesh rectangle is set using the meshrect command as:

meshrect (meshname) sx sy sz ex ey ez

The coordinates sx, sy, sz, ex, ey, ez are the Cartesian mesh rectangle start and end

coordinates (units m). In the simplest case where a single mesh is used, only the end

coordinates need be provided, as the start is assumed to be (0, 0, 0):

meshrect ex ey ez

Boris also allows multiple meshes to be used, each with its own independent discretization

and independent mesh rectangle. In this case meshes are addressed using their individual

names, i.e. meshname. This is a common pattern with most commands: for single-mesh

simulations the meshname parameter doesn’t need to be provided as it’s optional; but should

be specified for multi-mesh simulations.

The magnetic discretization, or cellsize, can be set using:

cellsize (meshname) cx cy cz

Here cx, cy, cz are cellsize values (unit m). The mesh rectangle must be subdivided by the

cellsize into an integer number of cells. If the set cellsize does not conform to this

requirement it will be automatically adjusted to the nearest values which result in a correct

discretization.

There are several types of meshes: ferromagnetic meshes (FM), 2-sublattice magnetic

meshes, e.g. ferrimagnetic, antiferromagnetic, or binary alloys (AFM), atomistic spin cubic

meshes (ASC), electrical conductor meshes (C), electrical insulator meshes (I), and fixed

magnetic dipole meshes (D).

Micromagnetic meshes (FM, AFM) have a set micromagnetic dynamics equation – see

Differential Equations. Atomistic meshes (ASC) have a set atomistic dynamics equation –

177

see Differential Equations. Other types of meshes do not have a magnetic dynamics equations

set. Electrical conductor meshes (C) allow electron charge and spin transport computations

(transport module), and heat flow computations (heat module), while electrical insulator

meshes (I) only allow heat flow computations. Magnetic dipole meshes (D) define a fixed

magnetic dipole for the purposes of generating a magnetic stray field. Advanced simulations

allows any combinations of these mesh types, e.g. multilayered devices which might have

ferromagnetic, antiferromagnetic, non-magnetic spacer layers, and substrate materials

included.

The different mesh types also differ in their available computational modules – see Modules.

When multiple meshes are defined, they are contained in a supermesh (S). The supermesh has

its own modules which work across multiple meshes – see Modules. For example these are

the sdemag (supermesh magnetostatic field module, which allows computing the

demagnetizing field from any number of meshes), strayfield (computes the stray field from

dipole meshes, to be included in magnetic meshes), and Oersted modules (computes the

Oersted field from all meshes with a current density defined, and includes it in all magnetic

meshes). The supermesh rectangle is automatically set as the smallest rectangle which

encompasses all magnetic meshes.

Ferromagnetic meshes (FM):

To erase all meshes and set a single FM mesh:

setmesh name rectangle

To add another FM mesh:

addmesh name rectangle

Two-sublattice meshes (AFM):

To erase all meshes and set a single AFM mesh:

setafmesh name rectangle

To add another AFM mesh:

addafmesh name rectangle

178

Atomistic simple-cubic meshes (ASC):

To erase all meshes and set a single ASC mesh:

setameshcubic name rectangle

To add another ASC mesh:

addameshcubic name rectangle

Electrical conductor meshes (C):

To add a C mesh:

addconductor name rectangle

Electrical insulator meshes (I):

To add a I mesh:

addinsulator name rectangle

Magnetic dipole meshes (D):

To add a D mesh:

adddipole name rectangle

Materials

A material may be added from the materials database, which contains a number of defined

materials parameters, and also defines the type of mesh to generate. See Materials Database

for further details.

To erase all meshes and set a single material from the materials database (name must match a

material name in the materials database):

addmaterial name rectangle

To add another mesh from the materials database (name must match a material name in the

materials database):

setmaterial name rectangle

179

Cellsizes

There are different mesh discretizations for each of their physical properties (magnetic,

electric, thermal, mechanical), even within the same mesh.

Magnetic

cellsize (meshname) cx cy cz

This affects the discretization of all magnetic quantities.

Electric

ecellsize (meshname) cx cy cz

This affects the discretization of quantities related to charge and spin transport in the

transport module.

Thermal

tcellsize (meshname) cx cy cz

This affects the discretization of quantities related to heat flow in the heat module.

Elastic

mcellsize (meshname) cx cy cz

This affects the discretization of quantities related to mechanical elastic properties in the

melastic module.

Supermesh magnetic cellsize

fmscellsize cx cy cz

This is used by the sdemag module with multiconvolution 0 (not recommended in general).

Supermesh electric cellsize

escellsize cx cy cz

This is used by the Oersted module – when computing the Oersted field the current density

from individual meshes is transferred to the electric supermesh at this discretization. The

Oersted field is then also computed at this discretization, and finally transferred back to

individual magnetic meshes.

180

Mesh Objects in Python Scripts

A higher level, and recommend, approach to working with meshes in Python scripts, is to

create them as objects using respective classes. Any commands which take a meshname as a

parameter can then be accessed directly from the mesh object created. Also see Material

Parameters section on accessing parameters through a mesh object, and Output Data section

on using mesh objects for setting output data.

When creating a mesh object, a rectangle and a primary quantity cellsize must be given. For

magnetic meshes the primary cellsize is for magnetic properties discretization, for conductor

meshes the primary cellsize is for electrical conduction discretization, and for insulator

meshes the primary cellsize is for temperature discretization.

The examples below show how different types of meshes may be created in a Python script.

You can create any number of mesh objects in this way, and the first one in a code context

erases all existing meshes (e.g. erases the existing default permalloy mesh).

Ferromagnetic meshes (FM):

FM = ns.Ferromagnet(rect, cellsize)

Use the created object to access commands which take meshname as a parameter, e.g. to set

the electrical cellsize also:

FM.ecellsize(ecellsize)

Two-sublattice meshes (AFM):

AFM = ns.AntiFerromagnet(rect, cellsize)

Atomistic simple-cubic meshes (ASC):

ASC = ns.Atomistic(rect, cellsize)

Electrical conductor meshes (C):

NM = ns.Conductor(rect, cellsize)

181

Electrical insulator meshes (I):

I = ns.Insulator(rect, cellsize)

Magnetic dipole meshes (D):

Dipole = ns.Dipole(rect, cellsize)

Materials

Depending on the mesh object type, different options will be visible (e.g. material

parameters, output data, display options, etc.). You can also use the materials database to

create a mesh, and the mesh object type will be determined directly by the type of material

created.

#This creates a Ferromagnet mesh with material definitions of Co

Co1 = ns.Material(‘Co’, rect, cellsize)

#This creates a Conductor mesh with material definitions of Pt

Pt1 = ns.Material(‘Pt’, rect, cellsize)

182

Differential Equations

This section outlines the magnetization dynamics equations solved – to see a list in the

console use the ode command.

To set an equation to solve and evaluation method use the setode command as setode

equation evaluation, e.g. setode LLG-STT RK4:

setode equation evaluation

For fixed time step evaluation methods you can set the time step as setdt value. For adaptive

time step evaluation methods this sets the starting value:

setdt value

To fine tune the adaptive time step controller the astepctrl command may be used, but should

not normally be necessary.

For a list of available equations and evaluation methods see below. Parameters entering the

equations are:

A number of evaluation methods are available for the magnetization dynamics equations.

These are fixed step methods Euler (Euler – 1st order), trapezoidal Euler (TEuler – 2nd order)

and Runge-Kuta (RK4 – 4th order). Adaptive time-step methods are the adaptive Heun

(AHeun – 2nd order), the multi-step Adams-Bashforth-Moulton (ABM – 2nd order), Runge-

Kutta-Bogacki-Shampine (RK23 – 3rd order with embedded 2nd order error estimator), Runge-

Kutta-Fehlberg (RKF45 – 4th order with embedded 5th order error estimator), Runge-Kutta-

Cash-Karp (RKCK45 – 4th order with embedded 5th order error estimator), Runge-Kutta-

Dormand-Prince (RKDP54 – 5th order with embedded 4th order error estimator), and Runge-

Kutta-Fehlberg (RKF56 – 5th order with embedded 6th order error estimator) For static

problems a steepest descent solver is available, SDesc, using Barzilai-Borwein stepsize

selection formulas. For further details see [S. Lepadatu, IEEE Trans. Mag. 58, 1 (2022)].

Symbol

(MM units)

(A units)

g

rel

α

M

s

(A/m)

P

β

g

rel,i

α

i

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

grel

(1)

damping

(1)

Ms

(1)

P

(1)

beta

(1)

grel_AFM

(2)

damping_AFM

(2)

Ms_AFM

(2)

mu_s

(1)

183

LLG – Landau-Lifshitz-Gilbert (FM)

The normalized LLG equation (m = M / MS) in implicit form is given by:

tt ∂

∂

×+×−=

∂

∂m

mHm

m

αγ

Here

erel

g

γµγ

0

=

, where



/

Be g

µ

γ

−=

is the electron gyromagnetic ratio and grel is a

relative gyromagnetic factor (grel = 1 by default giving

γ

= 2.212761569×105 m/As).

Damping constant is

α

. In explicit form the normalized LLG equation is given by:

()

Hm

m

Hm

m×

×

+

−

×

+

−=

∂

2

11

α

αγ

α

γ

t

LLG (AFM)

In antiferromagnetic (or ferrimagnetic, or binary ferromagnetic alloys) meshes, a two-

sublattice model is used, where the two sub-lattices are labelled A, B, and each have an LLG

equation set. Coupling between the equations is done through the effective fields, in

particular the exchange field. Each sub-lattice has its own set of simulation parameters. In

implicit and normalized form this is given by:

),(, BAi

tt

i

iiiii

i

=

∂

×+×−=

∂

∂m

mHm

m

αγ

LLG (ASC)

Atomistic equation for magnetic dipole spin S with magnetic moment µS is given in

normalized form:

tt S∂

∂

×+×−=

∂

∂S

SHS

S

µ

α

γ

184

LLG-STT – Landau-Lifshitz-Gilbert with Spin-Transfer Torques (FM)

The LLG equation can be complemented by Zhang-Li spin-transfer torques. In implicit form

this becomes:

() ( )

mu

mm

u

m

mHm

m∇

×

−∇

+

∂

×+×−=

∂

∂.

β.

αγ t

t

The spin-drift velocity u is given by:

2

β1

1

2

μ

+

=

s

B

eM

Pg

Ju

The LLG-STT equation in explicit form is given by:

( )

()( ) () ( ) ( ) ( )( )

[ ]

m

mumm

umm

u

Hmm

Hm

m

∇

−−∇

×−

−∇+

+

+××

+

−×

+

−=

∂

..

β.β

.β1

1

11

1

γ

2

22

α

ααα

α

αγ

α

t

LLG-STT (AFM)

In implicit and normalized form this is given by:

( ) ( )

)

,(,

.β

.α

γ

ii

BAi

t

ii

iii

i

ii

i

=∇

×−∇

+

∂

×+

×−=

∂

∂m

umm

u

m

mH

m

Here:

),(,

β

1

2

μ

2

,

BAi

eM

Pg

is

B

i

=

+

=Ju

LLG-STT (ASC)

In implicit and normalized form this is given by (magnetic dipole spin S with direction

S

ˆ

):

( ) ( )

SuSSu

S

SHS

Sˆ

.

ˆ

β

ˆ

.

ˆ

ˆˆ

ˆ∇×−∇+

∂

×+×−

=

∂

tt

αγ

185

The spin-drift velocity u is given by (µS is the magnetic moment, and V the lattice unit cell

volume):

2

β

1

2

μ

+

=

S

B

e

VPg

µ

Ju

LLB – Landau-Lifshitz-Bloch (FM)

For non-zero temperature simulations the LLB equation should be used and in implicit form

is given by (un-normalized):

( )

MHM

M

HM

M.

α

~

γ

α

~

γ

||

+

∂

×+×

−=

∂

⊥

tt

Here for T < TC (TC is the Curie temperature)

( )

C

TT 3/1−=

⊥

αα

,

C

TT 3

/2

||

αα

=

and

m/

~⊥⊥ =

αα

,

m

/

~

||||

αα

=

, where m is the magnetization length normalized to its zero

temperature value, i.e.

0

/||

S

Mm M=

. For T > TC

C

T

T3/

2

||

=

⊥

αα

.

The effective field H must be complemented by a longitudinal susceptibility field given by

(which due to vector cross products only affects the longitudinal torque term):











>













−

+−

≤











−

=

C

e

l

TT

mT

TT

m

,

)(5

3

1

~

,

~

2

1

2

||0

2

χµ

m

H

The field and temperature-dependent equilibrium magnetization, me, is given by:











+= Tk

H

T

m

BTm

B

extC

ee 0

3

)(

µµ

, where B(x) = coth(x) – 1/x is the Langevin function, µ is the

atomic moment.

The relative longitudinal susceptibility is given by (units 1/T), where

||

χ

is the longitudinal

susceptibility (unitless):

0

0||||

/)(

)/3)((1

)(

~

S

CB

MT

TTxB

xB

Tk

T

µχ

µ

χ

=

′

−

′

=

, with

TTmx

Ce

/3=

.

186

Further, we have the temperature dependence

)()(

0

TmMTM

eS

S

=

. Temperature dependences

for other magnetic parameters may also be set. To use the LLB equation a Curie temperature

needs to be set, which automatically also calculates the above default temperature

dependences: curietemperature Tc.

In explicit form the LLB equation becomes:

( ) ( )

MHM

M

HMM

M

HM

M.

α

~

γ

1

α

~

1

γα

~

α

~

1

γ

||

22

+××

+

−×

+

−=

∂

⊥

t

LLB (AFM)

In explicit and un-normalized form this is given by:

( ) ( )

),(,

.

~~

~~ ||,

||,

,

,BAi

MMt iii

i

iieffii

i

iieffii

i=+××

−×−=

∂

∂⊥MHMHMM

HM

M

α

γ

α

γγ

Here

( )

2,

α

~

1/

~iii ⊥

+=

γγ

and Mi ≡ |Mi|. As before

iii m/

~(||),(||), ⊥⊥ =

αα

, where

0,

/

)

()(

iSii

M

TM

Tm =

with

0,iS

M

denoting the zero-temperature saturation magnetization. The

damping parameters are continuous at TN – the phase transition temperature also set using the

curietemperature command – and given by:

( )

N

ii

N

Niejeiji

ii

N

Niejeiji

ii

TT

T

TT

Tmm

T

TT

Tmm

T

≥==

<













+

=

<













+

−=

⊥

,

3

2

,

~

/3

2

,

~

/3

1

||,,

,,

||,

,,

,

αα

ττ

αα

ττ

αα

Here we denote

N

T

~

the re-normalized transition temperature, given by:

( )

BAABBABA

N

T

ττττττ

4

2

~

2+−++

=

The micromagnetic parameters

τ

i and

τ

ij ∈ [0, 1], are coupling parameters between exchange

constants and the phase transition temperature, such that

τ

A +

τ

B = 1 and |J| = 3

τ

kBTN. Here J

187

is the exchange constant for intra-lattice (i = A,B) and inter-lattice (i,j = A,B, i ≠ j) coupling

respectively. These are set using the tau command.

The normalized equilibrium magnetization functions me,i are:

( )

[ ]

TkHTTmmBm BextiNijj

eiieie //

~

30,,,

µµττ

++=

The longitudinal relaxation field includes both intra-lattice and inter-lattice contributions as:

( )

()

N

i

j

i

ie

je

i

j

i

N

Bij

i

N

i

j

e

j

i

ie

j

e

i

e

i

j

i

N

Bij

ie

i

T

m

Tk

T

m

Tk

m

>































−

+−

=

<











































−

−











−+











−

=

,

ˆ

.

ˆ

~

3

~

1

,

1

ˆ

.

ˆ

1

~

2

3

1

~

2

1

,

||,

0||,

0

||,

2,

2

,

2,

2

||,

0

2,

2

||,0

||,

mmmH

χ

µµ

τ

χµ

χ

µµ

τ

χµ

Here

iii m/

ˆmm =

, and the relative longitudinal susceptibility is

0,0||,||, /

~iSii M

µχχ

=

, where:

( )

( )( ) ( )

2

||,

/

~

3/

~

31/

~

3

1

/

~

3

/

~

3

1

~

T

TB

BT

BTT

BT

T

BB

TT

B

TB

T

k

N

ji

jiij

jN

ji

Ni

j

iN

ij

j

jN

ji

i

B

′

−

′

−

′

−

′′

+

′

−

′

=

τ

ττ

τ

τµ

χ

,

and

( )

[ ]

TTmmBB

Nijjeiiemi

ie

/

~

3

,,

,

ττ

+

′

≡

′

.

188

LLB-STT – Landau-Lifshitz-Bloch with Spin-Transfer Torques (FM)

In implicit form we have the LLB-STT equation as:

( ) ( ) ( )

MuM

M

MuMHM

M

HM

M∇×−∇++

∂

×+×

−=

∂

⊥

.

β

..

α

~

γ

α

~

γ

||

tt

In this case the spin-drift velocity is given by:

2

β1

1

2

μ

+

=

S

B

eM

Pg

Ju

In explicit form the LLB-STT equation becomes:

( ) ( )

()

( )( ) ( ) ()()()( )











∇

−

∇×

−

−∇

+

+××

+

−×

+

−=

∂

⊥

⊥⊥

⊥

M

Mu

M

Mu

M

Mu

M

HM

M

HMM

M

H

M

.

α

~

βα

~

.

α

~

β

.βα

~

1

α

~

1

.

α

~

γ

1

α

~

1

γα

~

α

~

1

γ

22

||

22

t

LLB-STT (AFM)

In explicit and un-normalized form this is given by:

( ) ( )

( )

()

( )

()

()( )











∇

−

∇×

−

−∇

+

++

××

+

−

×

+

−

=

∂

⊥

⊥⊥

⊥

i

iii

i

ii

i

ii

i

ii

i

ii

i

ii

i

ii

t

MMuM

M

MuM

M

Mu

MH

M

H

MM

M

HM

M

..

α

~

βα

~

.

α

~

β

.βα

~

1

α

~

1

.

α

~

γ

1

α

~

1

γα

~

α

~

1

γ

2

,,,

i,

2,

i||,i

2,

i,

2,

Here:

),(,

β1

1

2

μ

2

,

BAi

eM

Pg

iS

B

i=

+

=Ju

189

sLLG – Stochastic Landau-Lifshitz-Gilbert (FM)

The explicit and normalized sLLG equation is given as:

))((

1

)(

1

22 thermalthermal

tHHmmHHm

m+××

+

−+×

+

−=

∂

α

αγ

α

γ

Each vector component of the thermal field follows a Gaussian distribution with zero mean

and standard deviation given by:

tVM

Tk

H

s

B

∆

=

0

2

γµ

α

σ

V is the volume of the stochastic computational cell, and ∆t is the time step used to update the

stochastic field. The stochastic field has zero spatial and vector component correlations. By

default V is the mesh cellsize, and ∆t is the evaluation method time-step, however different

values may be set using scellsize and setdtstoch commands.

sLLG (AFM)

In explicit and normalized form this is given by:

( ) ( )

),(,

1

1,

2

,

2BAi

tith

iii

i

ii

ithi

i

ii =+××

+

−+×

+

−=

∂

∂HHmm

HHm

m

α

γα

α

γ

As with the sLLG equation the thermal field standard deviation is given by:

tVM

Tk

H

iS

i

Bi

stdith ∆

=0,0

.

,

2

µγ

α

sLLG (ASC)

tt S

thermal ∂

∂

×++×−=

∂

∂S

SHHS

S

µ

α

γ

)(

Here

t

Tk

H

SB

B

∆

=

µµγµ

α

σ

0

2

.

190

sLLB – Stochastic Landau-Lifshitz-Bloch (FM)

For the stochastic LLB equation we have both a thermal field and thermal torque, and is

given by:

()( ) ( )

thermal

tηMHM

M

H

HM

M

HM

M+++×

×

+

−

×

+

−=

∂

⊥

.

α

~

γ

1

α

~

1

γα

~

α

~

1

γ

||

2

The components of the thermal field and torque follow Gaussian distributions with no

correlations, zero mean and standard deviation given respectively by:

tV

M

Tk

H

s

B

∆

−

=

⊥

⊥0

0

||

)(2

1

γµ

αα

α

σ

tV

MTk

sB

∆

=

0

||

2

µ

γα

η

σ

sLLB (AFM)

This is given by:

( )( ) ( )

),(.

~~

~~ ,||,

||,

,,

,

,BAi

MMt ithiii

i

iithieffii

i

iieffii

i=+

++××−×

−=

∂

∂⊥ηMHMHH

MMHM

M

α

γ

α

γγ

As with sLLB we have (i = A, B):

( )

tV

MTk

tVM

Tk

H

iSiiB

stdith

iS

i

iiB

i

stdith

∆

=

∆

−

=

⊥

0

0,||,

.

,

0,0

||,,

,

.

,

2

1

µ

γα

η

µγ

αα

α

191

sLLG-STT – Stochastic Landau-Lifshitz-Gilbert with Spin-Transfer Torques (FM, AFM,

ASC)

This is similar to the LLG-STT equation, but also has the thermal field from the sLLG

equation added.

sLLB-STT – Stochastic Landau-Lifshitz-Bloch with Spin-Transfer Torques (FM, AFM)

This is similar to the LLB-STT equation, but also has the thermal field from the sLLB

equation added to the damping torque term, as well as the additional thermal torque term.

LLG-SA, sLLG-SA, LLB-SA, sLLB-SA – Equations with Spin Accumulation (FM)

The LLG, LLB, sLLG, and sLLB equations also appear in the forms LLG-SA, LLB-SA,

sLLG-SA, and sLLB-SA. When using these equations the spin transport solver is enabled and

a spin accumulation S is calculated. This gives rise to bulk and interfacial torques which are

added to the respective equation. For example for the LLG equation we obtain the LLG-SA

equation as:

S

tt T

M

HM

M+

∂

×

+×−=

∂

α

γ

The bulk spin-accumulation torque is given by:

( )

SmmSmT

S

××−×−=

22

ϕ

λλ

e

J

e

DD

This is included as an additional effective field in the explicit forms of the equations:











×

+

=

22

||

ϕ

λλ

γ

SmS

M

H

S

J

e

D

Note there are no SA version for the STT equations. This is because the Zhang-Li STTs

result from the bulk TS torque as a special case (see e.g. S. Lepadatu, Scientific Reports 7,

12937 (2017)).

192

Interfacial spin-accumulation torques are also present when N/F interfaces are used:

{}

()

{ }

[ ]

SS

h

B

erface

S

GG

ed

gVmV

mm

T∆×+

∆×

×

=

↑↓↑↓

ImRe

int

µ

,

where ∆VS = VS,F – VS,N and

( )( )

SV

BeS

eD

µσ

//=

.

This is included as an additional effective field in the explicit forms of the equations:

{}{ }

( )

S

h

B

GG

ed

gV

Vm

M

H

S

∆

+∆

×

−

=

↑↓↑↓

ImRe

|

1

µ

γ

Currently antiferromagnetic meshes do not have a drift-diffusion model included, so the SA

versions are not active with two-sublattice equations. This will be included in a future

version.

LLGStatic – Static Landau-Lifshitz-Gilbert

This equation is used for static problems, i.e. where only the relaxed magnetization state is

required. It is the explicit LLG equation without the precession term, and with the damping

factor set to 1. This is given by:

Hm

m

m××−

=

∂

2

γ

t

193

Modules

Modules typically correspond to an additive field in the total effective field H appearing in

the equations shown in the Differential Equations section:

...

21

++== HHHH

eff

To enable a module you need to add it using the addmodule command. To disable a module

you need to delete it using the delmodule command. Most modules also have an energy

density term associated with their effective field contributions, available as an output data

parameter. All contributions are evaluated on a cell-centered uniform finite difference mesh,

with all differential operators evaluated to second order accuracy.

To see a list of available modules in the console use the modules command. To add a module

to simulations use the addmodule command as addmodule modulename, e.g. addmodule

iDMexchange. For multi-mesh simulations you can specify the mesh the module should be

added to, e.g. addmodule meshname iDMexchange.

In Python scripts you can use ns.addmodule(‘modulename’), or for a specific mesh

ns.addmodule(‘meshname’, ‘modulename’). To add a supermesh module you need to use

meshname as supermesh, e.g. addmodule supermesh sdemag. To remove a module from

simulations use the delmodule command, with the same parameters as addmodule. For a list

of available module names see below.

When adding a new mesh, or resetting to default, only the demag, exchange, and Zeeman

modules are enabled by default. In Python scripts you can also specify directly which

modules should be used in a single command, after mesh creation (recommened approach),

e.g.:

FM = ns.Ferromagnet(rect, cellsize)

FM.modules([‘Zeeman’, ‘demag’, ‘exchange’, ‘aniuni’])

Applicable meshes are also indicated for each module below: ferromagnetic (FM), 2 sub-

lattice (AFM), atomistic simple cubic (ASC), conductor (C), insulator (I), dipole (D),

supermesh (S).

194

Modules Table of Contents:

anibi ....................................................................................................................................... 195

anitens ................................................................................................................................... 196

aniuni ..................................................................................................................................... 197

anicubi ................................................................................................................................... 198

demag_N ............................................................................................................................... 199

demag .................................................................................................................................... 200

dipoledipole ........................................................................................................................... 201

DMExchange ........................................................................................................................ 202

exchange ................................................................................................................................ 203

heat ........................................................................................................................................ 204

iDMExchange ....................................................................................................................... 205

melastic.................................................................................................................................. 207

moptical ................................................................................................................................. 212

mstrayfield ............................................................................................................................ 212

Oersted .................................................................................................................................. 213

roughness .............................................................................................................................. 214

sdemag ................................................................................................................................... 215

SOTfield ................................................................................................................................ 217

STfield ................................................................................................................................... 218

strayfield ............................................................................................................................... 220

surfexchange ......................................................................................................................... 221

transport ............................................................................................................................... 225

viDMExchange ..................................................................................................................... 231

Zeeman .................................................................................................................................. 233

195

anibi – Biaxial Magneto-Crystalline Anisotropy (FM, AFM, ASC)

addmodule anibi

FM:

]

).

()

.

()

.)(.[(

2

)

.(

2

22

2

11

2

21

0

2

0

1bbbbbb

S

A

SM

K

M

Ke

e

me

meememe

e

mH +−

=

µµ

Here m = M / MS.

Output data parameter e_anis:

2

12

2

1

).

().(

]).

(1[

b

bA

KK em

eme

m+−=

ε

AFM:

),(],).

().().)(.[(

2

).(

2

11

2

21

,0

,2

,0

,1

BAi

M

K

M

K

bbibibbibi

iS

i

AAi

iS

i

=+−= ee

memeemem

eemH

µµ

Here mi = Mi / MS,i.

Output data parameter e_anis:

2/

)(

B

A

εε

ε

+=

, where

),(,).().(]).(1[ 2

2

1,2

2

,1 BAiKK bibiiAiii =+−= ememem

ε

ASC:

]).

ˆ

)(.

ˆ

().

ˆ

)(.

ˆ

[(

2

).

ˆ

(

2

2211

2

21

0

2

0

1bbbbbb

SB

AA

SB

KK eeSeSeeSeSeeSH ++=

µµµµ

µµ

.

Here

)(

ˆBS

µµ

SS =

.

Output data parameter e_anis:

2

12

2

1).

ˆ

().

ˆ

(]).

ˆ

(1[ bbA KK eSeSeS +−=

ε

Symbol

(MM units)

(A units)

K

1

(J/m3)

(J)

K

2

(J/m3)

(J)

e

A

e

b1

e

b2

M

S

(A/m)

K

1,i

(J/m3)

K

2,i

(J/m3)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

K1

(1)

K2

(1)

ea1

(3)

ea2

(3)

ea3

(3)

Ms

(1)

K1_AFM

(2)

K2_AFM

(2)

Ms_AFM

(2)

mu_s

(1)

196

anitens – ‘Tensorial’ Magneto-Crystalline Anisotropy (FM, AFM, ASC)

addmodule anitens

This module allows any magneto-crystalline anisotropy to be specified by giving the

coefficients in the energy expansion:

∑ ∑

≥=

++ ≥

=

00

,,

)(

oo

k

ji kj

i

kjio

ijk

d

γβαε

Here

α

= m.e1,

β

= m.e2, and

γ

= m.e3. Also m = M / MS.

To specify which energy terms are to be included in computations, use the setktens

command as follows: setktens dxn1yn2zn3 …

Here d is the coefficient for the term, n1, n2, n3 are respective integer powers for the

direction cosines, and x, y, z are string literals. Powers of 0 can simply be omitted, powers of

1 don’t have to be written explicitly. All other modules (aniuni, anibi, anicubi) are special

cases, so for example cubic anisotropy may be implemented in anitens to 6th order as:

setktens x2y2 x2z2 y2z2 x2y2z2

By convention the existing material parameters K1, K2, K3 (and K1_AFM, K2_AFM,

K3_AFM) now apply to 2nd, 4th, and 6th order terms in the energy expansion. For any other

orders you have to specify the energy density value directly in the string set through setktens,

as a prefactor for the respective term.

Symbol

(MM units)

(A units)

K

1

(J/m3)

(J)

K

2

(J/m3)

(J)

K

3

(J/m3)

(J)

e

1

e

2

e

3

K

1,i

(J/m3)

K

2,i

(J/m3)

K

3,i

(J/m3)

Param.

(dim.)

K1

(1)

K2

(1)

K3

(1)

ea1

(3)

ea2

(3)

ea3

(3)

K1_AFM

(2)

K2_AFM

(2)

K3_AFM

(2)

197

aniuni – Uniaxial Magneto-Crystalline Anisotropy (FM, AFM, ASC)

addmodule aniuni

FM:

AAA

S

AA

SM

K

M

KeememeemH ).]().(1[

4

).(

22

0

2

0

1−+=

µµ

Here m = M / MS.

Output data parameter e_anis:

22

2

1]).(1[]).(1[ AA KK emem −+−=

ε

AFM:

),(

,)

.

]()

.

(1[

4

)

.

(

2

,

0

,2

,

0

,1

BA

i

M

K

M

K

AA

iA

i

S

i

A

Ai

i

S

i

=−

+= ee

m

eme

e

mH

µ

.

Here mi = Mi / MS,i.

Output data parameter e_anis:

2/)(

BA

εεε

+=

, where

),(,]).(1[]).(1[ 22

,2

2

,1 BAiKK AiiAiii =−+−= emem

ε

ASC:

A

AA

S

B

A

SB

KK ee

SeSe

eSH ).

ˆ

]().

ˆ

(1[

4

).

ˆ

(

2

0

2

0

1

−+=

µµ

µµµ

µ

.

Here

)(

ˆB

S

µ

SS =

.

Output data parameter e_anis:

22

2

1]

).

ˆ

(1[]

).

ˆ

(1[ AA K

KeSeS −+−=

ε

Symbol

(MM units)

(A units)

K

1

(J/m3)

(J)

K

2

(J/m3)

(J)

e

A

M

S

(A/m)

K

1,i

(J/m3)

K

2,i

(J/m3)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

K1

(1)

K2

(1)

ea1

(3)

Ms

(1)

K1_AFM

(2)

K2_AFM

(2)

Ms_AFM

(2)

mu_s

(1)

198

anicubi – Cubic Magneto-Crystalline Anisotropy (FM, AFM, ASC)

addmodule anicubi

FM:

][

2

)]()()(

[

2

22

3

2

22

1

0

2

22

3

22

2

22

1

0

1

γβα

βγα

γ

αβ

µ

βαγγαβγβα

µ

ee

eeee

H++

−+++++−

=

SS

M

K

M

K

Here

α

= m.e1,

β

= m.e2, and

γ

= m.e3. Also m = M / MS.

Output data parameter e_anis:

222

2

2222

22

1

][

γβαγβγαβαε

KK +++=

AFM:

),(],[

2

)]()()([

2

22

3

22

2

22

1

,0

,2

22

3

22

2

22

1

,0

,1

BAi

M

K

M

K

iiiiiiiii

iS

i

iiiiiiiii

iS

i

=++−

+++++−=

γβαγβαγβα

µ

βαγγαβγβα

µ

eee

eeeH

Here

α

i = mi.e1,

β

i = mi.e2, and

γ

i = mi.e3. Also mi = Mi / MS,i.

Output data parameter e_anis:

2/)(

BA

εεε

+=

, where

),(,]

[222

,2

2222

22

,

1BAiK

Kiiiiii

iii

iii =++

+=

γβαγ

βγαβαε

ASC:

][

2

)]()()([

2

22

3

22

2

22

1

0

2

22

3

22

2

22

1

0

1

γβαβγαγαβ

µµµ

βαγγαβγβα

µµµ

eeeeeeH ++−+++++−=

SBSB

KK

Here

1

.

ˆeS=

α

,

2

.

ˆeS=

β

and

3

.

ˆeS=

γ

. Also

)(

ˆBS

µµ

SS =

.

Output data parameter e_anis:

222

2

222222

1

][

γβαγβγαβαε

KK +++=

Symbol

(MM units)

(A units)

K

1

(J/m3)

(J)

K

2

(J/m3)

(J)

e

1

e

2

e

3

M

S

(A/m)

K

1,i

(J/m3)

K

2,i

(J/m3)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

K1

(1)

K2

(1)

ea1

(3)

ea2

(3)

ea3

(3)

Ms

(1)

K1_AFM

(2)

K2_AFM

(2)

Ms_AFM

(2)

mu_s

(1)

199

demag_N – Stoner-Wohlfarth Magnetostatic Interaction (FM, AFM, ASC)

addmodule demag_N

FM:

),

,( z

yxiMN

Hi

i

i=−

=

Here Nz = 1 - Nx - Ny.

Output data parameter e_demag:

H

M.

2

0

µ

ε

−

=

AFM:

),,

(2

/)

(

,,

),(

,

zy

xi

M

MNH

B

iA

iiB

A

i

=

+−=

Output data parameter e_demag:

2

/

).(

2

0BA

MM

H+

−=

µ

ε

ASC:

)

,,(

/

3

z

yxi

aS

NH

Biii

=

−=

µ

Here Nz = 1 - Nx - Ny, and a is the unit cell size.

Output data parameter e_demag:

HS.

23

0aB

µ

ε

−=

Symbol

(MM units)

(A units)

M

S

(A/m)

M

S,i

(A/m)

µ

S

(µB)

N

x

, N

y

Param.

(dim.)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

Nxy

(2)

200

demag - Magnetostatic Interaction (FM, AFM, ASC)

addmodule demag

FM:

∫

∈

−

−=

V

d

r

rrMr

rN

r

H)()

()

(00

Here N is a rank-2 tensor with the following symmetry:













=

zzyzxz

yzyyxy

xzxyxx

NNN

N

N is computed using the formulas in A.J. Newell et al., “A Generalization of the

Demagnetizing Tensor for Nonuniform Magnetization” J. Geophys. Res. 98, 9551 (1993).

Output data parameter e_demag:

HM.

2

0

µ

ε

−=

AFM:

∫

∈

+

−−=

V

BA

BA d

r

rMrM

rrNrH 2

)()(

)()( 00),(

Output data parameter e_demag:

2/).(

2

0BA

MMH +−=

µ

ε

ASC:

Same formulas as for FM, where

∑

=

ii

B

hSM 3

µ

. Here the sum runs over all spins in a

macrocell with size h. The macrocell is set using dmcellsize command, and must be a

multiple of the unit cell in all directions.

Symbol

(MM units)

(A units)

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

201

dipoledipole – Dipole-Dipole Interaction (ASC)

addmodule dipoledipole

ASC:

Interaction field at spin i given by:

∑

≠

−

=

j

iij

jij

ijj

B

i

r

3

ˆ

)

ˆ

.(3

4

S

rrS

H

π

µ

Here

ij

rr

rˆ

=

is the distance vector from spin i to spin j.

Output data parameter e_demag, averaged over entire mesh from individual unit cell

contributions, where a is the unit cell size:

ii

B

i

aH

S.

2

3

0

µµ

ε

−

=

The dpin-dipole field may also be calculated using macrocells, where

∑

=

jji SM

is the total

moment of the macrocell (units of

µ

B). Here the sum runs over all spins in a macrocell i with

size h. The macrocell is set using dmcellsize command, and must be a multiple of the unit

cell in all directions. Then we have:

3

33

ˆ

)

ˆ

.(

3

4hr

i

B

ji ij

j

ijijj

B

i

M

MrrM

H

µ

π

µ

−

=

∑

≠

The output data parameter e_demag is not computed in macrocell mode.

Symbol

(A units)

µ

S

(µB)

Param.

(dim.)

mu_s

(1)

202

DMExchange – Dzyaloshinskii-Moriya Bulk Exchange Interaction (FM, AFM, ASC)

addmodule DMExchange

FM:

M

H×∇

−=

2

0

2

S

M

D

µ

Non-homogeneous Neumann boundary conditions are used to evaluate the curl operator:

Mn

n

M×=

∂

A

D

2

, where n is the surface normal.

The DM exchange field adds to the direct exchange field (see exchange module).

Output data parameter e_exch:

()

exch

H

M+

−= .

2

0

µ

ε

AFM:

);,,(

2

,,0

2,0

jiBAji

MM

D

M

D

jh

B

SAS

h

ii

iS

i

i≠=×+×∇−= MdMH

µ

η

µ

, where

1+

=

A

η

,

1−=

B

η

.

Output data parameter e_exch:

2/

)(

B

A

εεε

+=

, where

( )

iexchiii ,

0

.

2HHM +−=

µ

ε

ASC:

j

Nj ij

SB

DSrH ˆ

ˆ

0

×= ∑

∈

µµµ

, where sum runs over all neighbors j, such that

ij

r

ˆ

is the unit vector

from this spin (i) to neighbour spin j. Here

)(

ˆBS

µµ

SS =

.

Output data parameter e_exch:

( )

exch

B

aHHS +−= .

23

0

µµ

ε

, where a is the unit cell size.

Symbol

(MM units)

(A units)

D

(J/m2)

(J)

A

(J/m)

D

i

(J/m2)

D

h

(J/m3)

d

h

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

D

(1)

A

(1)

D_AFM

(2)

Dh

(1)

dh_dir

(3)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

203

exchange – Direct Exchange Interaction (FM, AFM, ASC)

addmodule exchange

FM:











∇−

∇= )|(|

||2

2

22

2

0

M

MM

A

S

M

MH

µ

Homogeneous Neumann boundary conditions are used to evaluate the Laplacian operator.

Output data parameter e_exch:

HM.

2

0

µ

ε

−=

This is equivalent to:























∂

+













∂

+













∂

=

2

zyxM

A

S

MMM

ε

AFM:

),,,(,

)(

4

2

,,

0

,

2

,,0

,

2

,0

ji

B

Aji

MM

A

MM

M

A

M

A

j

BSA

S

i

nh

i

jii

BS

AS

i

h

i

2iS

i

i≠=∇+

××

−∇= M

MMM

MH

µµµ

Output data parameter e_exch:

2/)(

BA

εεε

+=

, where

iii

HM .

2

0

µ

ε

−=

ASC:

∑

∈

=

Nj j

SB

JSH ˆ

0

µµµ

, where sum runs over all neighbors j.

Here

)(

ˆBS

µµ

SS =

.

Output data parameter e_exch:

HS.

2

3

0

a

B

µµ

ε

−=

, where a is the unit cell size.

Symbol

(MM units)

(A units)

A

(J/m)

A

i

(J/m)

J

(J)

A

h,i

(J/m3)

A

nh,i

(J/m)

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

A

(1)

A_AFM

(2)

J

(1)

Ah

(2)

Anh

(2)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

204

heat – Heat Equation Solver (FM, AFM, ASC, C, I, D)

addmodule heat

The heat equation in the 1-temperature model with Joule heating and any other additional

heat sources (S) is given by:

)

(

)

,

(

)

,(

)

,(

)(

.

)

,

(

)

(

ρ)

(

2

r

rJ

rr

r

σ

t

tQ

t

TK

t

T

C+

+

∇∇

=

∂

Robin boundary conditions are used to evaluate the differential operators.

The heat equation is evaluated using the simple forward-time centered-space method. The

heat equation time-step required is normally (though not always) comparable to the

magnetization equation time-step thus a more time-efficient method (e.g. Crank-Nicolson) is

not normally required.

In the two-temperature model, the heat equation is given as:

[ ]

[]

),()

,()

(

)

,(

)(ρ)(

),(),()

,()

(),()(.

),(

)

(ρ)(

tTtT

G

t

T

C

tQtT

tTGtTK

t

tT

C

lee

l

le

ee

e

rrr

r

rr

rrrr

r

−

=

∂

+−−∇∇

=

∂

Here Ce and Cl are the electron and lattice specific heat capacities,

ρ

is the mass density, K is

the thermal conductivity, and Ge is the electron-lattice coupling constant, typically of the

order 1018 W/m3K.

Symbol

ρ

(kg/m

3

)

C, C

l

(J/kgK)

C

e

(J/kgK)

K

(W/mK)

G

e

(W/m

3

K)

Q

(W/m

3

)

σ

(S/m)

Param.

(dim.)

density

(1)

shc

(1)

shc_e

(1)

thermK

(1)

G_e

(1)

Q

(1)

elC

(1)

205

iDMExchange – Dzyaloshinskii-Moriya Interfacial Exchange Interaction (FM, AFM, ASC)

addmodule iDMExchange

FM:

)

ˆ

)

.((

2

0

z

S

M

D∇−

∇

=zM

H

µ

Non-homogeneous Neumann boundary conditions are used to evaluate the differential

operators:

Mnz

n

M××=

∂

∂)

ˆ

(

2A

D

, where n is the surface normal.

The DM exchange field adds to the direct exchange field (see exchange module).

Output data parameter e_exch:

H

M.

2

0

µ

ε

−=

AFM:

),(),

ˆ

).((

2

,

2,0

BAiM

M

D

izi

iS

i

=∇−∇= zMH

µ

Non-homogeneous Neumann boundary conditions are used to evaluate the differential

operators:

),,,()],()

ˆ

[(

)1(2

2

jiBAjic

cA

D

jii

ii

≠=−××

−

=

∂

∂MMnz

n

M

,

where ci = Anh,i / 2Ai and n is the surface normal.

Output data parameter e_exch:

2/)(

BA

εεε

+=

, where

iii

HM .

2

0

µ

ε

−=

Symbol

(MM units)

(A units)

D

(J/m2)

(J)

A

(J/m)

A

i

(J/m)

D

i

(J/m2)

A

nh,i

(J/m)

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

D

(1)

A

(1)

A_AFM

(2)

D_AFM

(2)

Anh

(2)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

206

ASC:

j

Nj ij

SB

DSzrH ˆ

)

ˆ

(

0

××=

∑

∈

µµµ

, where sum runs over all neighbors j, such that

ij

r

ˆ

is the unit

vector from this spin (i) to neighbour spin j. Here

)(

ˆBS

µµ

SS =

.

Output data parameter e_exch:

( )

exch

B

aHHS +−= .

2

3

0

µ

ε

, where a is the unit cell size.

207

melastic – Magneto-elastic effect (FM, AFM, C, I)

addmodule melastic

Uniform Stress or Imported Files Mode

The magneto-elastic effect can be included for a cubic crystal using a strain tensor. The strain

tensor is given as:













=

zzyzxz

yzyyxy

xzxyxx

εεε

ε

.

Here we define the diagonal strain vector as Sd = (εxx, εyy, εzz), and off-diagonal, or shear,

strain vector as Sod = (εyz, εxz, εxy). The strain tensor can have a spatial dependence, and can

be loaded from two ovf2 files (diagonal and shear strain components computed with an

external package – loadovf2strain), or alternatively a displacement vector field can be

loaded (using ovf2 files, computed externally – loadovf2disp). From the displacement u =

(ux, uy, uz), the strain tensor is computed as:

),,,(,

2

1zyxqp

p

u

q

uqp

pq =













∂

+

∂

=

ε

In the simplest case a uniform stress may be applied which results in a constant strain with

zero off-diagonal terms – setstress. This is given below for uniform external pressure F = (Fx,

Fy, Fz), where Y is Young’s modulus and

ν

is Poisson’s ratio:

























−−

=

z

y

x

d

F

Y1

1

νν

S

Symbol

ρ

(kg/m

3

)

B

1

, B

2

(J/m

3

)

B

1

, B

2

(J/m

3

)

e

1

e

2

e

3

Y

(Pa)

ν

c

11

, c

12

, c

44

(N/m

2

)

η

(kg/m

3

s)

α

T

(K

-1

)

Param.

(dim.)

density

(1)

MEc

(2)

mMEc

(2)

ea1

(3)

ea2

(3)

ea3

(3)

Ym

(1)

Pr

(1)

cC

(3)

mdampin

g

(1)

thalpha

(1)

208

From the strain tensor, for a cubic crystal with orthogonal axes e1, e2, e3, and magneto-elastic

constants B1, B2, we have the following diagonal and off-diagonal energy density terms

(e_mel output data is the sum of the two contributions):

[ ]

).)(.)(.().)(.)(.().)(.)(.(2

).().().().().().(

1322313212,

3

2

32

2

21

2

11,

eSememeSememeSemem

eSemeSemeSem

dodododmel

ddddmel

B

++=

ε

For example, for axes coninciding with the x-y-z axes (default setting), this gives the total

energy density:

][2][

2

222

1yzzyxzzxxyyxzzzyyyxxxmel

mmmmmmBmmmB

εεεεεεε

+++++=

Here m = M / MS.

The effective field can be computed using the usual formula:

m

H∂

∂

−= mel

S

mel M

ε

µ

0

1

.

Elastodynamics Solver

The elastodynamics equations are solved in the velocity-stress representation, included in a

magneto-thermo-elastic model, based on three coupled dynamics equations for

magnetization, temperature, and elasticity properties:

),,(,

,,

2

zyxpv

qt

v

ρ

QTK

t

T

Cρ

tt

p

zyxq

pqp

eff

=−

∂

=

∂

+∇=

∂

×+×

−=

∂

∑

=

η

σ

α

γ

m

mHm

m

The first equation is the LLG equation, although any of the implemented magnetization

dynamics equations may be used, and the second expression is the heat equation (see heat

module description). The third expression is the elastodynamics equation, written in the

209

velocity-stress representation, where v is the velocity,

σ

is the symmetric stress tensor, and

η

is a mechanical damping factor resulting in energy dissipation. This is solved using the finite-

difference time-domain (FDTD) scheme. The stress and strain for a cubic crystal are related

using:

),,,,,(,)()2()(2 2

1012111211 rqpzyxrqpmBTTcccc pTrrqqpppp ≠≠=−−+−++=

αεεεσ

),,

,,

(,

2

244

qpzy

xq

pmmBc

q

ppqpq

≠=

−=

εσ

Here c11, c12, c44 are elastic stiffness coefficients,

α

T is the linear thermal expansion

coefficient, and T0 is the ambient temperature (ambient command). The terms involving B1

and B2 give the magnetostriction contribution. Note, the inverse thermo-elastic effect (change

in temperature due to lattice strains) is not currently included in the solver.

Taking the time derivative of strain and stress expressions allows replacing the

displacement with velocity, and we obtain a system of first-order differential equations for

elastodynamics, written in the velocity-stress representation as:

),

,(

,

z

yxp

v

q

t

v

ρp

z

yxq

pqp =

−

∂

=

∂

∂∑

=

η

σ

),,,,,

(,2

)2(

11211

1211

r

qpzyxrqp

t

m

B

t

T

cc

r

v

q

v

c

p

v

c

t

p

pT

r

q

ppp

≠

≠=

∂

−

∂

+−













∂

+

∂

+

∂

=

∂

α

σ

),,,,(,2

2

44

qpzyxqp

t

m

t

m

mB

p

v

q

v

c

t

p

q

p

qppq

≠=













∂

+

∂

−













∂

+

∂

=

∂

σ

Time derivatives of temperature and magnetization are evaluated directly in the dynamical

magneto-thermo-elastic model. The equations are solved using the FDTD scheme, with

staggered velocity and stress components as indicated in the figure below for a given

computational cell:

210

Staggered velocity and stress components – Velocity, stress, magnetization and temperature

components for finite difference discretization of the magneto-thermo-elastic model. Velocity

components are edge-centred, diagonal stress components are at vertices, shear stress components are

face-centred, and magnetization and temperature values are cell-centred.

The temperature and magnetization values are cell-centred; their values and that of

their time derivative at vertices, edge and face centres, are obtained to second order accuracy

in space using interpolation. Spatial derivatives at inner points are evaluated using standard

finite differences to second order accuracy. There must be at least one fixed surface where the

mechanical displacement is zero, since no translational motion is allowed, and thus the

velocity is also zero (add/delete a fixed surface using the surfacefix and delsurfacefix

commands). Velocity components on a fixed surface are set directly as zero, whilst

derivatives of velocity components perpendicular to a fixed surface are evaluated using

standard expressions with Dirichlet boundary condition of zero. As an example, for a fixed xz

surface, vx = vz = 0, whilst for

yv

y

∂∂ /

at the boundary the Dirichlet condition vy = 0 applies.

For all other surfaces (free surfaces) boundary stress values are prescribed from external

forces (add/delete an external force surface using the surfacestress and delsurfacestress

commands) as (

δ

is the Kronecker delta):

),,,(,

,,

z

yxrpFr

zyxq rqpq ==

∑

=

δσ

Here p is normal to the free surface. The boundary stress values are either set directly for

components on free surfaces, or else are used as Dirichlet boundary conditions to evaluate

211

derivatives of stress perpendicular to a free surface. As an example, for a fixed xz surface,

yyy

F

=

σ

is set directly, whilst

x

xy

F=

σ

and

z

yz

F

=

σ

are used as Dirichlet boundary

conditions to evaluate

y

xy ∂∂ /

σ

and

y

yz ∂∂ /

σ

respectively, at the boundary. Finally, required

derivatives of velocity perpendicular to a free surface are obtained as:

)

,,,,,(,

2

21

1

11

1

11

12

11

12

11

rqpzyxrqp

t

m

c

B

t

T

c

r

v

q

v

c

t

F

c

p

v

p

pT

r

qpp

≠≠=

∂

+

∂











++













∂

+

∂

−

∂

=

∂

α

For multi-layered structures, composite media boundary (CMB) conditions are also required.

These consist of enforcing continuity of velocity and stress components located on the CMB

interface, achieved by interpolating respective values either side of the interface. These

values are set after the elastodynamics equation is iterated in all layers. It is also necessary to

specify initial values for stress and velocity. The velocity initial value is zero throughout,

whilst the initial values of stress components, which are not on a free surface, are computed

by taking the initial strain to be zero. To reset the solver to initial conditions use the

resetelsolver command.

The time-step for the elastodynamics equation (set using seteldt command) must

satisfy the Courant, Friedrichs, Lewy condition,

()

2

22

p

1/ −

−− ∆

+∆+∆<

∆zyxv

t

, where vp is

the elastic compressional wave velocity (typically vp = 6000 m/s), and ∆x, ∆y, ∆z are the cell-

size dimensions. The mechanical cellsize is set using the mcellsize command.

212

moptical – Magneto-optical effect (FM, AFM, ASC)

addmodule moptical

This module applies a z-axis field as:

( )

z

rˆ

,

0

tfHH

MOMOMO ±

=

σ

Here

0

MO

H

±

σ

is set using the Hmo parameter, and fMO is its spatial (and temporal) variation.

Energy density term (output data parameter: e_mo):

HM.

0

µε

−=

For two-sublattice models the same field is applied to both sublattices.

mstrayfield – Stray field from fixed magnetic dipoles (FM, AFM, ASC)

addmodule mstrayfield

This is similar to the strayfield module, but instead of computing stray field on the

supermesh and then transferring them to an individual mesh using interpolation, the stray

field is computed directly in the given mesh with its magnetization discretization.

213

Oersted – Oersted Field (S)

addmodule supermesh Oersted

Effective field contribution:

∫

∈

−=

V

Cd

r

rrJ

rr

KrH )()

()( 00

Here K is a rank-2 tensor with the following symmetry:













−−

−

=

0

yzxz

yz

xy

xzxy

KK

K

K is computed using the formulas in B. Krüger, “Current-Driven Magnetization Dynamics:

Analytical Modeling and Numerical Simulation”, PhD Dissertation, University of Hamburg

(2011) – Appendix D, page 118. Also JC is the current density, and the Oersted field is

computed from all meshes with transport module enabled (thus on the electric supermesh).

For two-sublattice models the Oersted field is applied equally to both sublattices.

214

roughness – Roughness Field and Staircase Magnetostatic Corrections (FM, AFM)

addmodule roughness

Effective field contribution computed on the coarse mesh (i.e. the actual mesh discretisation

used at run-time with NV number of discretisation cells):

)(

)

()

,

()

()(

00

0

00

VG

V

∈











−−=

∑

∈

rr

M

rr

rrNrH

r

Here N is the demagnetizing tensor computed on the fine mesh with NVr number of

discretisation cells, and:











−∈∨−

∈∧−

=

R

Vr

V

VV

V

N

G

0

01

1

),( rr

rr

V is the smooth body without roughness and VR is the mesh with roughness, and we require

VR ⊆ V. If the coarse cellsize has dimensions (hx, hy, hz), the fine cellsize must have

dimensions (hx / mx, hy / my, hz / mz), where the m factors are integers. The function

)

,

()

(0

0r

rr

r

N

r

G

V

∑

∈

−

is computed at initialisation on the finely discretised mesh then averaged

up to the coarse mesh (each coarse cellsize value is obtained as an average of its contained

fine cellsize values).

Energy density term (output data parameter: e_rough):

H

M.

20

µ

ε

−=

Details can be found in: S. Lepadatu, “Effective field model of roughness in magnetic nano-

structures” J. Appl. Phys. 118, 243908 (2015).

For two-sublattice models the roughness field is computed using the average magnetization,

and applied equally to both sublattices.

215

sdemag – Supermesh Magnetostatic Interaction (S)

addmodule supermesh sdemag

I. Multi-Layered convolution (multiconvolution 1). This is the default. Can handle multiple

meshes which do not fit onto a regular supermesh grid.

A generalisation of the single layer convolution algorithm is used here. We can write the

convolution sum as:

kkl

Vni ijikijklkl Vnk

iij

∈

′

=−

′

−=

′

∑

∈

=

rrMhhrrNrH

r

;,...,1,)(),,()(

,...,1

In the demagnetizing tensor for the equation above we explicitly specify the cellsize, h, of the

two computational meshes the tensor relates. Since we have n terms of the form appearing in

the single layer convolution sum, we can again apply the convolution theorem. This time for

each output mesh (H) we have n input meshes (M), together with n kernels. Thus to calculate

the outputs in all n meshes we require a total of n2 sets of kernel multiplications in the

transform space. This is illustrated in the figure below.

Multi-Layered convolution algorithm for n computational meshes. The magnetization input

of each mesh is transformed separately using a FFT algorithm, either directly (dotted line), or

by first transferring to a scratch space with a common discretisation cellsize, using a

weighted average smoother (solid lines). In the transform space the inputs are multiplied with

pre-computed kernels for a total of n2 sets of point-by-point multiplications. Finally the

output demagnetizing fields are obtained using an inverse FFT algorithm, which are set

directly in the output meshes (dotted line), or transferred using a weighted average smoother

if the discretisation cellsizes differ (solid lines).

216

II. Supermesh demagnetization (multiconvolution 0). Only use if multiple meshes fit onto a

regular supermesh grid. Not recommended otherwise.

The same formulas as for the demag module are used when computing demagnetizing fields

on the uniformly discretised super-mesh. The ferromagnetic super-mesh may have a cellsize

which differs from that of the individual ferromagnetic meshes. In this case a weighted

average smoother is used to transfer magnetization to the super-mesh and demagnetizing field

values back from the super-mesh.

Consider a discrete distribution of magnetization values M at points V = {ri; i∈P}. Let h be

the cellsize of the input mesh, with the set of cells {ci; i∈P} centered around the points ri. To

obtain the magnetization value at a point r′ in a cell c with dimensions h′ we introduce the

definitions di = |r′−ri|, dV = |h′+h|/2, and

iVi

ddd −=

~

. The weighted average is given as:

∑

∈

=

′

Pi ii

w)

(

)( r

Mr

M

where

∑

∈

=





∅≠

∩

=

Pi iiT

i

T

ii

i

d

otherwise

cc

d

w

δ

~~ ,

0

,1

~

217

SOTfield – Spin-Orbit Torque Field (FM, AFM)

addmodule SOTfield

The spin-orbit torque is given by:

( )

)(

,pmpmmT ×+××= G

h

c

B

effSHASOT r

d

J

e

µ

θ

Here p = d × eJc, where d is the spin current direction incident on the interface (default z

corresponding to an HM/FM bilayer stacked along z axis), and eJc is the current density

direction. dh is the FM layer thickness. The transport module needs to be enabled and

configured so a current density is available.

FM:

( )

ppm

HG

h

c

B

effSHA

S

SOT r

d

J

eM +×

−=

µ

θ

γ

,

1

Here

Re g||

0

γµ

γ

=

, where

e

γ

is the electron gyromagnetic ratio and gR is a relative

gyromagnetic factor. Also m = M / MS.

AFM:

( )

),

(

1,

,

,BAir

d

J

eM Gi

h

c

B

effSHA

iS

i

iSOT =+

×−

=pp

mH

µ

θ

γ

Here

iR

ei g,0 ||

γµγ

=

, where

e

γ

is the electron gyromagnetic ratio and gR,i is a relative

gyromagnetic factor. Also mi = Mi / MS,i.

Symbol

M

s

(A/m)

M

S,i

(A/m)

θ

SHA,eff

r

G

g

R

g

R,i

d

Param.

(dim.)

Ms

(1)

Ms_AFM

(2)

SHA

(1)

flST

(1)

grel

(1)

grel_AFM

(2)

STp

(3)

218

STfield – Slonczewski Spin Torques Field (FM)

addmodule STfield

The spin torque is given by (including damping-like and field-like components):

()

[]

p

mp

mm

T×+

××=

G

zB

ST r

d

J

e

η

µ

).

(

).

(p

m

pm BA

q

B

A

q

−

+

=−

+

η

Here p is the spin polarization from the fixed layer, and Jz is the current density in the z

direction (this module is applicable for layers stacked along the z direction, i.e. CPP-GMR

geometry), and d is the thickness of the free layer. The angular dependence is specified

through the

η

function, which is parmetrized by q+, q-, A, and B. The field-like coefficient is

rG. The transport module needs to be enabled and configured so a current density is

available.

This results in an effective field in the magnetization dynamics equation given by:

( ) ( )

p

pmH

G

zB

S

ST

r

d

J

eM +×−=

µ

θη

γ

1

Here

Re g

|

0

γ

µγ

=

, where

e

γ

is the electron gyromagnetic ratio and gR is a relative

gyromagnetic factor. Also m = M / MS. This effective field is applied equally to all cells

along the z direction.

Non-constant polarization mode

In the above approach, the polarization p was set to a fixed direction using the the STp

parameter. In a multi-layered simulation, it is possible to use the polarization value directly

from neighboring magnetic layers in a stack along the z direction. This mode is selected

simply by setting STp to be a zero vector. Then, the program will search for neighboring

Symbol

M

s

(A/m)

q

+

, q

-

A, B

r

G

g

R

p

q

+

, q

-

(top)

A, B

(top)

r

G

(top)

Param.

(dim.)

Ms

(1)

STq

(2)

STa

(2)

flST

(1)

grel

(1)

STp

(3)

STq2

(2)

STa2

(2)

flST2

(1)

219

magnetic layers, top and bottom along the z direction, and in each cell next to the interface

use the p value (normalized magnetization direction) obtained from the neighboring layer.

Thus, e.g. for a bilayer F / Fbot, the effective field used for F at interface cells is obtained as:

( ) ( )

ppmH

G

zB

S

ST

r

h

J

e

M+×−=

µ

θ

η

γ

1

,

).()

.( pmpm BA

q

B

A

q

−

+

=−+

η

Here h is the z-direction cellsize of the F layer. Thus, unlike in the fixed polarization mode,

here the effective field is included at interface cells only. Over the entire thickness the total

spin torque still averages to a 1 / d dependence, but this approach results in a more realistic

description for thicker layers, since typically the spin torque from an adjacent layer is much

stronger close to the interface.

Aditionally, the non-constant polarization mode allows inclusion of spin torques from both

top and bottom layers independently, e.g. as in a trilayer Ftop / F / Fbot. The spin torque from

Ftop on F is still calculated using the above formula, however the spin torque parameters can

be set independently using the STq2, STa2, and flST2 parameters, i.e. STq2 = (q+, q-), STa2 =

(A, B), an flST2 = rG for cells at the Ftop / F interface, with p taken from Ftop for each interface

cell. In general these values may be obtained by fitting the angular dependence of spin

torques computed self-consistently using the spin transport solver for any particular

geometry.

220

strayfield – Stray field from fixed magnetic dipoles (S)

addmodule supermesh strayfield

If Md is the magnetization of a uniformly magnetized prism with dimensions d, then the

magnetic field (stray field) at a distance r from the centre of the prism is given by:

d

dM

dr

N

H)

,(

−

=

Here Nd is a rank-2 tensor with the following symmetry:













=

zzyzxz

yzyyxy

xzxyxx

d

NN

N

NN

N

Nd is computed using the formulas in A. Andreev et al., “Universal Method for the

Calculation of Magnetic Microelectronic Components: the Saturated Ferromagnetic

Rectangular Prism and the Rectangular Coil.” ICSE2000 Proceedings, Nov. 2000, 187. Note

these formulas are equivalent to the Newell formulas used to compute the demagnetizing

tensor.

This module computes the stray field resulting from all dipole meshes and includes it in all

magnetic meshes. For two-sublattice models the field is applied equally to both sublattices.

221

surfexchange – Surface Exchange Interaction (FM, AFM, ASC)

addmodule surfexchange

This module allows surface exchange coupling between different meshes along the z

direction. In order for 2 meshes to be exchange coupled they both need to have the

surfexchange module enabled, and must be stacked adjacently along the z direction. The

coupling effective fields are computed between pairs of cells either side of the

interface/separation. For 2 coupled meshes it is the top mesh which sets the material

parameters to be used.

The different types of exchange coupling currently possible are given below.

RKKY (FM to FM coupling):

Let mi and mj be the normalized magnetization values of two cells (normalized to MS), i and

j, which are surface exchange coupled across a gap between two ferromagnetic meshes. Let ∆

be the thickness of the ferromagnetic layer for which the surface exchange field is computed.

The surface exchange field at cell i, from cell j, is given as:

jji

S

j

S

ij

M

J

M

JmmmmH ).(

2

0

2

0

1

∆

+

∆

=

µµ

The effective field is applied to all cells along the z direction if a 3D simulation mesh is used

(surface exchange field applicable for thin films).

Output data parameter e_surfexch:

2

21

).

(.

jiji

ij

JJ mm

mm ∆

−

∆

−

=

ε

Symbol

(MM units)

(A units)

J

1

, J

A

(J/m2)

J

2

, J

B

(J/m2)

J

S

(J)

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

Param.

(dim.)

J1

(1)

J2

(1)

Js

(1)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

222

RKKY (ASC to ASC coupling):

The surface exchange field at cell i, from cell j, is given as:

j

S

B

S

ij JS

Hˆ

0

µ

µµ

=

The effective field is applied only to cells at the interface.

Output data parameter e_surfexch:

ji

S

ij

a

JSS ˆ

.

ˆ

3

−=

ε

, where a is the unit cell size.

RKKY (ASC to FM coupling):

Micromagnetic and atomistic meshes may also be coupled using RKKY. This is a

combination of the above equations, where magnetization values are obtained by averaging

over atomistic spins and dividing by cell-size volume. Thus to compute effective field in FM,

the FM to FM coupling formula is used, but magnetization on the ASC side is obtained by

averaging over atomistic spins. To compute the effective field in ASC, the ASC to ASC

coupling formula is used, but the magnetization direction in FM sets the spin direction.

Exchange Bias (AFM to FM coupling):

Let mi be the normalized magnetization (normalized to MS) of cell i in the ferromagnetic

mesh, and mjA, mjB be the normalized magnetization (normalized to MS,A respectively MS,B)

values in cell j for sub-lattices A and B of the 2-sublattice mesh (antiferromagnet). Let ∆ be

the thickness of the ferromagnetic layer for which the surface exchange field is computed.

The surface exchange field at cell i, from cell j, is given as:

Bj

S

B

Aj

S

A

ij

M

J

M

J

,

0

,

0

mm

H∆

+

∆

=

µµ

The effective field is applied to all cells along the z direction if a 3D simulation mesh is used

(surface exchange field applicable for thin films).

223

Output data parameter e_surfexch:

B

ji

B

A

j

i

A

ij

J

,

.. m

m

m∆

−

∆

−

=

ε

Exchange Bias (FM to AFM coupling):

Let mj be the normalized magnetization (normalized to MS) of cell j in the ferromagnetic

mesh, and miA, miB be the normalized magnetization (normalized to MS,A respectively MS,B)

values in cell i for sub-lattices A and B of the 2-sublattice mesh (antiferromagnet). Let ∆ be

the thickness of the 2-sublattice mesh for which the surface exchange field is computed. The

surface exchange field at cell i, from cell j, is given as:

j

AS

A

ij

M

JmH ∆

=

,

0

,

µ

,

j

BS

B

Bij

M

JmH ∆

=

,0

,

µ

The effective field is applied to all cells along the z direction if a 3D simulation mesh is used

(surface exchange field applicable for thin films).

Output data parameter e_surfexch:

j

Bi

B

jAi

A

ij

JJ mm

mm .

2

.

2

,

∆

−

∆

−=

ε

Exchange Coupling (AFM to AFM coupling):

Let mj,A, mj,B, miA, miB be the normalized magnetization (normalized to MS,A respectively

MS,B) of cells i, j for sub-lattices A and B of the 2-sublattice meshes. Let ∆ be the thickness of

the 2-sublattice mesh for which the surface exchange field is computed. The surface

exchange field at cell i, from cell j, is given as:

Aj

AS

A

Aij M

J

,

,0

,mH ∆

=

µ

,

Bj

BS

B

Bij M

J

,

,0

,mH ∆

=

µ

The effective field is applied to all cells along the z direction if a 3D simulation mesh is used

(surface exchange field applicable for thin films).

224

Output data parameter e_surfexch:

AjBi

B

AjAi

A

ij

JJ

,,,, .

2

.

2mmmm ∆

−

∆

−=

ε

Exchange Bias (ASC to AFM coupling):

Micromagnetic and atomistic meshes may also be coupled using exchange bias. This is a

combination of the above equations, where magnetization values are obtained by averaging

over atomistic spins and dividing by cell-size volume – see description of RKKY (ASC to

FM coupling).

225

transport – Charge and Spin-Transport Solver (FM, AFM, C, ASC)

addmodule transport

Charge Transport (FM, AFM, C, ASC)

When solving only for the charge current density, a Poisson-type equation for V is solved as:

()( )

σ

∇∇

−

=∇ .

2

V

For V Dirichlet boundary conditions are used at boundaries containing a fixed potential

electrode, otherwise Neumann boundary conditions are used. The conductivity may have an

AMR contribution (AMR given as a percentage value) calculated as (FM only):

( )

,

100/1 2

0

dAMR+

=

σ

where

||||

.

MJ

C

d=

.

From V the charge current density is obtained as Jc = -σ∇V, and is used for Joule heating

computations and to obtain spin torques (Zhang-Li STT and SOT).

The Poisson equation is evaluated using the successive over-relaxation (SOR) algorithm with

black-red ordering for parallelization.

Charge and Spin Transport (FM, C, ASC)

Charge and spin current densities are given as (see S. Lepadatu, “Unified treatment of spin

torques using a coupled magnetization dynamics and three-dimensional spin current solver”

Scientific Reports 7, 12937 (2017) for details):

Symbol

(MM units)

σ

(S/m)

AMR

(%)

Param.

(dim.)

elC

(1)

amr

(1)

Symbol

(MM units)

P

D

e

(m2/s)

n

(m-3)

β

D

λ

sf

(m)

λ

J

(m)

λ

ϕ

(m)

↓

↑GG ,

(S/m

2

)

↑↓

G

(S/m2)

Param.

(dim.)

P

(1)

De

(1)

n

(1)

betaD

(1)

l_sf

(1)

l_sf

(1)

l_phi

(1)

Gi

(2)

Gmix

(2)

226

( )

σ

µ

θ

µ

βσ

B

EE

S

mS

EJ ×

−+

×

∇+

∇+= ne

P

e

P

e

D

e

D

B

e

SHA

B

eDC 2

2



( ) ( )

( )

mmEzmmeEεSmEJ yx

B

iii

BB

SHAe

B

Sneee

DP

e∂×∂⊗×+∂×⊗++∇−⊗−= ∑3

2

σµσµ

σ

µ

θσ

µ







Here:

()

mmm .

ii

E∂×= 

σ

()

m

mm

zB .

yx ∂×

∂=

σ

Here Eσ and Bσ are the directions of the emergent electric field due to charge pumping, and

emergent magnetic field due to topological Hall effect respectively.

With the full spin transport solver enabled both V and S are computed using Poisson-type

equations as:

()()( )

( )

[]

() ( )

[ ]

V

y

n

e

Pe

P

e

D

V

xy

x

yx

yxy

y

xy

yx

x

B

e

D

∇∂

×

∂+

∂

×∂

−∂

×

∂+

∂×

∂

−

∂

+∂×

+∂

×∂

+

∂×

∂+

∇∇

+

∇

−

=

∇

..

.

2

.

2

22

2

m

mm

m

mm

x

m

mmm

m

mm

mS









σ

µσ

β

σ

and

( )

[]

( ) ( )

[]

( )

22

2

2222

3

2

22

2

22

2

..

ϕ

λλλ

σµ

σ

µθ

σµσµ

m

Smm

SS

mm

m

mm

mmm

m

εE

mmES

×

+

×

++

∂×

∂+

∂

×∂

−∂×

∂+∂

×

∂+

∂

×+∂

×∂

+∂

×∂

+

∇

+∇+∇−=

∇

Jsf

xyxyxy

yxyxyx

e

B

y

xyyxx

e

B

e

SHA

e

B

e

B

EE

nD

e

De

eD

V

D

P

eD

P

e







For boundaries containing an electrode with a fixed potential, differential operators applied to

V use a Dirichlet boundary condition. For other external boundaries the following non-

homogeneous Neumann boundary conditions are used:

227

( )

( ) ()

nεEn

S

n

S

n

..

.

B

SHA

B

SHA

e

D

e

D

V

e

µ

σ

θ

µ

σ

θ

=∇

×

∇

=

∇

At N/F composite media boundaries the following conditions are applied:

( ) ( )

{ }

( )

{ }

[ ]

( )

[ ]

mmmVnJ

VmVmm

nJnJ

mVnJnJ

VGGGG

e

GG

e

GG

V

GG

S

B

F

S

SS

B

F

S

N

S

F

C

N

C

∆−+∆+

−=

∆×+∆××

=−

∆−

+∆

+

−==

↓↑

↓

↑

↑↓↑↓

↓↑↓↑

..

ImRe

2

..

...

µ

Spin pumping is included on the N side of the above equations as:

{ } { }













∂

+

∂

×= ↑↓↑↓

t

G

t

G

eB

pump

Smm

mJ ImRe

2



µ

At N/F interfaces, interfacial spin torques are obtained as (hF is the discretisation cellsize of

the F layer in the direction normal to the composite media boundary) :

{ }

( )

{ }

[ ]

SS

F

B

SGG

eh

gVmVmmT ∆×+∆××= ↑↓↑↓ ImRe

µ

From S, bulk spin torques are obtained as:

()

Sm

mS

m

T×

×

−×

−=

22

ϕ

λ

e

J

e

S

DD

Both Poisson equations are evaluated using the SOR algorithm with black-red ordering for

parallelization. Note, whilst the SOR algorithm is robust in evaluating the spin transport

equations in arbitrary multi-layers with composite media boundary conditions, it does suffer

from slow convergence in particular for lower target solver errors. This algorithm is due to be

replaced with a more efficient method in a future version.

Finally, different contributions in the spin transport solver may be turned on or off using the

parameters pump_eff (spin pumping), cpump_eff (charge pumping), the_eff (topological Hall

effect), ts_eff (bulk spin torques), tsi_eff (interfacial spin torques).

228

TMR (I)

In tunnel barriers separating two ferromagnetic layers, such that cos

θ

is the dot product of

pairs of magnetization unit vectors either side of the barrier, we calculate the conductivity

from the angular-dependent resistance, R(θ), obtained from Slonczewski’s formula:

θ

cos

1

)

(

0

p

ap

p

ap

R

RR

R

+

−

+

=

Here

)/(2

0pappap RRRRR +=

is the resistance for perpendicular orientation of ferromagnetic

layers’ magnetizations, with Rap and Rp being resistances for antiparallel and parallel

orientations respectively. We use the definition

ppap RRR

TMR /)( −

=

. The tunnel barrier’s

resistance area (RA) product is obtained for the parallel orientation of magnetization vectors,

i.e.

A

RRA

p

=

, and it is used as a parameter in the model. The conductivity is then expressed

as

ARd

I

)(/

θσ

=

, where dI is the insulator layer thickness and A is the barrier area.

Thus, within this model the insulator layer is treated as a poor conductor, and a charge current

density is computed throughout a simulated structure by taking the electrical potential V

(where

V−∇

=E

), to be continuous across all interfaces. Further, across the tunnel barrier we

impose the requirement of net spin current conservation, which gives the following equations

within the tunnel barrier:

( )( )

σ

∇∇

−=∇ .

2

V

0

2=∇ S

Within this model spin currents are conserved across the tunnel barrier, but discontinuities in

the spin accumulation can arise between the two sides of the barrier.

Symbol

(MM units)

R

p

A

(Ωm2)

R

ap

A

(Ωm2)

σ

(S/m)

D

e

(m2/s)

λ

sf

(m)

↓

↑

GG ,

(S/m

2

)

↑↓

G

(S/m2)

Param.

(dim.)

RAtmr_p

(1)

RAtmr_ap

(1)

elC

(1)

De

(1)

l_sf

(1)

Gi

(2)

Gmix

(2)

229

The

)(

θ

R

formula is set using the tmrtype command, where the default setting of 1 is the

Slonczewski formula above. The other possibility is setting of 0, which sets the formula:

2

/

)cos

1

)(()(

θ

−−+= papp RRRR

.

Metallic conduction channels may be introduced by setting the elC material parameter to

have a spatial variation (non-zero values of elC will be treated using metallic conduction,

using the parameters De,

λ

sf,

↓↑

GG ,

, and

↑↓

G

).

TAMR (FM, ASC)

Tunelling anisotropic magnetoresistance (TAMR) may be included in magnetic meshes,

which gives rise to a dependence of the conductivity depending on the set material

parameters and TAMR formula.

By default the following TAMR formula is used:

( )

)1(100/1

2

1

0

dTAMR −+

=

σ

. Here

11

.em=d

, where m is the magnetization direction.

It is possible to specify a custom formula for TAMR, which takes into account the tamr

parameter, as well as the three axes ea1, ea2, ea3. This is done using the tamrequation

command (see Commands). This command takes a user text equation (see User-Defined Text

Equations), where the text equation uses parameters TAMR, d1, d2, d3. Here TAMR = tamr /

100 is the ratio, where tamr is the material parameter set as a percentage value. Also

)3

,2,1(,. == i

d

ii

em

.

Symbol

(MM units)

σ

0

(S/m)

TAMR

(%)

e

1

e

2

e

3

Param.

(dim.)

elC

(1)

tamr

(1)

ea1

(3)

ea2

(3)

ea3

(3)

230

Thermoelectric Effect (FM, C, ASC)

The thermoelectric effect may be included by enabling both the transport and heat modules,

and setting a non-zero Seebeck coefficient (SC – this is the S material parameter). In this case

any temperature gradients will generate a thermoelectric effect.

The electric field is now obtained as

TSV

C

∇−−∇=E

. From

0. =∇ J

, where

EJ

σ

=

, we

now obtain the following contribution to the Poisson equation for V, which the solver uses for

computations:

TSV

C22

∇−=∇

There are two computation modes: 1) open potential mode, and 2) externally set potential

mode (this is the default mode). In the default mode, where the potential on electrodes is

fixed, then no net thermoelectric current can be generated. In the open potential mode, an

external resistance is defined (e.g. this would be the external resistance across defined

electrodes, to which the device interfaces), R0, which allows a net thermoelectric current to be

generated by setting the potential drop across electrodes as V0 = R0 IT. Here IT is the total

thermoelectric current flowing into the ground electrode, where the corresponding

thermoelectric current density is

TS

CC

∇=

σ

J

.

The thermoelectric effect computation mode is set using the openpotentialresistance

command.

Set R0 to zero to disable open potential mode:

openpotentialresistance 0

Set a non-zero R0 value to enable net thermoelectric current generation (it is recommended R0

is set to a matched value, i.e. set to the base device resistance between electrodes, R), e.g.:

openpotentialresistance 100

Symbol

(MM units)

σ

0

(S/m)

S

C

(V/K)

Param.

(dim.)

elC

(1)

S

(1)

231

viDMExchange – Vectorial Dzyaloshinskii-Moriya interfacial exchange interaction (FM,

AFM, ASC)

addmodule viDMExchange

FM:

)

ˆ

).

((

2

0

x

S

x

M

D∇−∇

=x

MH

µ

,

)

ˆ

).((

2

0

y

S

y

M

D∇−∇= yMH

µ

,

)

ˆ

).

((

2

0

z

S

zM

M

D∇−

∇= zM

H

µ

Then:

zyx

H

HHH

γβα

+

+=

, where

),,(

γβα

=

DMI

d

is a unit vector.

Non-homogeneous Neumann boundary conditions are used to evaluate the differential

operators:

M

nz

ny

n

x

n

M××

+×

=

∂

∂)]

ˆ

()

ˆ

(

)

ˆ

(

[

2

γ

βα

A

D

, where n is the surface normal.

The DM exchange field adds to the direct exchange field (see exchange module).

Output data parameter e_exch:

H

M.

2

0

µ

ε

−

=

AFM:

),(),

ˆ

).((

2

,

2,0

,BAiM

M

D

iyi

iS

i

ix =∇−∇= xMH

µ

),(),

ˆ

).((

2

,

2,0

,BAiM

M

D

iyi

iS

i

iy =∇−∇= yMH

µ

),(),

ˆ

).((

2

,

2,0

,

BAiM

M

D

izi

iS

i

iz

=∇−∇= zMH

µ

Then:

),

(,

,,

,

BAi

izi

yixi

=++= HH

HH

γβα

, where

),,(

γβα

=

DMI

d

is a unit vector.

Non-homogeneous Neumann boundary conditions are used to evaluate the differential

operators:

),,

,(),()]

ˆ

()

ˆ

()

ˆ

([

)1(

2

jiB

Ajic

c

A

D

ji

i

ii

i

≠=−××+×+×

−

=

∂

∂MMnzn

ynx

n

M

γβα

,

where ci = Anh,i / 2Ai, and n is the surface normal.

Symbol

(MM units)

(A units)

D

(J/m2)

(J)

A

(J/m)

A

i

(J/m)

D

i

(J/m2)

A

nh,i

(J/m)

M

s

(A/m)

M

S,i

(A/m)

µ

S

(µB)

d

DMI

Param.

(dim.)

D

(1)

A

(1)

A_AFM

(2)

D_AFM

(2)

Anh

(2)

Ms

(1)

Ms_AFM

(2)

mu_s

(1)

Ddir

(3)

232

Output data parameter e_exch:

2/)( BA

εεε

+=

, where

iii

HM .

2

0

µ

ε

−=

ASC:

j

Nj ij

SB

x

DSx

rH ˆ

)

ˆ

(

0

××=

∑

∈

µµµ

,

j

Nj ij

SB

y

DSyrH ˆ

)

ˆ

(

0

××=

∑

∈

µµµ

,

j

Nj ij

S

B

z

DS

zr

Hˆ

)

ˆ

(

0

×

×=

∑

∈

µ

Then:

z

yx

HHHH

γ

βα

++=

, where

),,(

γβα

=

DMI

d

is a unit vector.

Here the sums run over all neighbors j, such that

ij

r

ˆ

is the unit vector from this spin (i) to

neighbour spin j. Here

)(

ˆ

BS

µµ

SS =

.

Output data parameter e_exch:

( )

exch

B

aH

HS +

−= .

2

3

0

µ

ε

, where a is the unit cell size.

233

Zeeman – Applied Magnetic Field (FM, AFM, ASC)

addmodule Zeeman

Effective field contribution:

)(rHH Hext c=

Output data parameter e_zee:

HM.

0

µε

−=

For two-sublattice models the field is applied equally to both sublattices.

Symbol

c

H

Param.

(dim.)

cHa

(1)

234

Material Parameters

To set a material parameter value use the setparam command as setparam meshname

paramname value, e.g. setparam permalloy A 1e-11, or even setparam permalloy A 10pJ/m.

setparam paramname value

To set the parameter in a specific mesh:

setparam meshname paramname value

In Python scripts you can set a parameter value as ns.setparam(‘paramname’, value), or for a

specific mesh as ns.setparam(‘meshname’, ‘paramname’, value). A higher level approach to

working with material parameters in Python is through a mesh object, e.g.:

#create a ferromagnetic mesh with given rectangle and discretization

FM = ns.Ferromagnet(rect, cellsize)

#now access material parameters through the param class in FM

#e.g. the line below sets the exchange stiffness value

FM.param.A = 1e-11

#to read a parameter value:

A = FM.param.A.setparam()

For list of available parameter names see below. To see a list of available parameters in the

console use the params command.

Format:

paramname: name in equations (units)

Description.

A: A (J/m)

Exchange stiffness.

A_AFM: Ai (J/m)

Exchange stiffness.

Ah: Ah,i (J/m3)

Homogeneous inter-lattice exchange coupling per lattice constant.

235

Anh: Anh,i (J/m3)

Non-homogeneous inter-lattice exchange stiffness.

amr: AMR (%)

Anisotropic magneto-resistance as a percentage of base resistance.

beta:

β

(unitless)

Spin-transfer torque non-adiabaticity parameter.

betaD:

β

D (unitless)

Diffusion spin polarisation.

cC: (c11, c12, c44) (N/m2)

Stiffness constants for a cubic crystal.

cHA: cHA (unitless)

Applied field spatial variation parameter, which multiplies the applied field value.

cpump_eff: cpump_eff (unitless)

Charge-pumping efficiency.

cT: cT (unitless)

Set temperature spatial variation parameter, which multiplies the set temperature value. To

enable spatial variation of temperature you need to have the heat module enabled. If you want

the set temperature to remain constant you need to disable the heat equation by using

setheatdt 0.

D: D (J/m2)

Dzyaloshinskii-Moriya exchange constant.

D: D (J)

Atomistic Dzyaloshinskii-Moriya exchange constant.

236

Ddir: dD (unitless)

Interfacial DMI symmetry axis.

D_AFM: Di (J/m2)

Dzyaloshinskii-Moriya exchange constant.

damping:

α

(unitless)

Gilbert magnetization damping.

damping_AFM:

α

i (unitless)

Gilbert magnetization damping.

Ddir: dDMI (unitless)

Interfacial Dzyaloshinskii-Moriya interaction direction unit vector.

De: De (m2/s)

Electron diffusion constant.

density:

ρ

(kg/m^3)

Mass density.

Dh: Dh (J/m3)

Homogeneous Dzyaloshinskii-Moriya exchange constant for 2-sublattice model.

dh_dir: dh (unitless)

Homogeneous Dzyaloshinskii-Moriya interaction unit vector for 2-sublattice model.

ea1: ea1 (unit vector)

Uniaxial magneto-crystalline anisotropy symmetry axis, or first cubic magneto-crystalline

anisotropy symmetry axis.

ea2: ea2 (unit vector)

Second cubic magneto-crystalline anisotropy symmetry axis.

237

ea3: ea3 (unit vector)

Third cubic magneto-crystalline anisotropy symmetry axis.

elC:

σ

(S/m)

Base electrical conductivity.

flST: rG (unitless)

Field-like spin orbit torque coefficient.

G_e: Ge (W/m3K)

Electron-lattice coupling constant (two temperature model).

Gi: G

↑

, G

↓

(S/m^2)

Interface spin-dependent conductivity (for majority and minority carriers). The top contacting

mesh sets the interface value, thus Gi is available in both magnetic and non-magnetic meshes.

Gmix: G

↑↓

= Re{G

↑↓

} + i Im{G

↑↓

} (S/m^2)

Interface spin-mixing conductivity (real and imaginary parts). The top contacting mesh sets

the interface value, thus Gmix is available in both magnetic and non-magnetic meshes.

grel: grel (unitless)

Relative electron gyromagnetic ratio.

grel_AFM: grel,i (unitless)

Relative electron gyromagnetic ratio.

Hmo: Hmo (A/m)

Magneto-optical field strength.

iSHA:

θ

SHA (unitless)

Spin Hall angle used for the inverse spin Hall effect.

238

J: J (J)

Atomistic direct exchange constant.

J1: J1 (J/m2)

Bilinear surface exchange coupling. For coupled meshes it is the top mesh that sets the J

values.

J2: J2 (J/m2)

Biquadratic surface exchange coupling. For coupled meshes it is the top mesh that sets the J

values.

joule_eff:

η

J (unitless)

Joule heating effect efficiency.

Js: JS (J)

Atomistic surface exchange constant.

K1: K1 (J/m3)

Magneto-crystalline anisotropy energy density.

K1: K1 (J)

Atomistic magneto-crystalline anisotropy energy.

K1_AFM: K1,i (J/m3)

Magneto-crystalline anisotropy energy density.

K2: K2 (J/m3)

Magneto-crystalline anisotropy energy density, higher order.

K2: K2 (J)

Atomistic magneto-crystalline anisotropy energy, higher order.

239

K2_AFM: K2,i (J/m3)

Magneto-crystalline anisotropy energy density, higher order.

K3: K3 (J/m3)

Magneto-crystalline anisotropy energy density, higher order.

K3: K3 (J)

Atomistic magneto-crystalline anisotropy energy, higher order.

K3_AFM: K3,i (J/m3)

Magneto-crystalline anisotropy energy density, higher order.

l_J:

λ

J (m)

Spin exchange rotation length.

l_phi:

λϕ

(m)

Spin dephasing length.

l_sf:

λ

sf (m)

Spin-flip length.

MEc: B1, B2 (J/m3)

Magneto-elastic coefficients.

mMEc: B1, B2 (J/m3)

Magneto-elastic coefficients used for magnetostriction (should be same as MEc).

Ms: Ms (A/m)

Saturation magnetization.

Ms_AFM: Msi (A/m)

Saturation magnetization.

240

mdamping:

η

(kg/m3s)

Mechanical damping.

mu_s: µS (µB)

Atomistic magnetic moment (in units of µB).

n: n (m-3)

Conduction electrons density.

Nxy: Nx, Ny (unitless)

In-plane demagnetizing factors (used by demag_N module)

P: P or

βσ

(unitless)

Charge current spin polarization.

Pr:

ν

(unitless)

Poisson’s ratio.

pump_eff:

η

pump (unitless)

Spin pumping efficiency.

Q: Q (W/m3)

Heat source added to the heat equation. Can be non-uniform by setting a spatial variation.

RAtmr_ap: Rap×A (Ωm2)

TMR RA product for antiparallel state.

RAtmr_p: Rp×A (Ωm2)

TMR RA product for parallel state.

SHA:

θ

SHA (unitless)

Spin Hall angle used for the spin Hall effect.

241

S: S (V/K)

Seebeck coefficient.

shc: C, Cl (J/kgK).

Specific heat capacity (total – one temperature model, lattice – two temperature model).

shc_e: Ce (J/kgK).

Electronic specific heat capacity (two temperature model).

STa: A, B (unitless).

Parameters in Slonczewski spin torques angular dependence (denominator constants).

STq: q+, q- (unitless).

Parameters in Slonczewski spin torques angular dependence (symmetric and asymmetric

proportional components).

STp: p (unitless).

Spin polarization from fixed layer in Slonczewski spin torques (unit vector), or spin current

direction for SOT.

s_eff:

η

s (unitless)

Stochasticity efficiency parameter.

susrel: χ|| (As2/kg)

Longitudinal (parallel) susceptibility divided by µ0MS.

susrel_AFM: χ||,i (As2/kg)

Longitudinal (parallel) susceptibility divided by µ0MS,i.

tamr: tamr (%)

Tunnelling anisotropic magnetoresistance.

242

thalpha:

α

T (K-1)

Linear coefficient of thermal expansion.

the_eff:

η

the (unitless)

Topological Hall effect efficiency.

thermK: K (W/mK)

Thermal conductivity.

ts_eff:

η

ts (unitless)

Spin accumulation torque efficiency in the bulk.

tsi_eff:

η

tsi (unitless)

Spin accumulation torque efficiency at interfaces.

Ym: Y (Pa)

Young’s modulus.

243

Parameters Temperature Dependence

Material parameters may be assigned a temperature dependence where appropriate. There are

two available methods: using an array, or using a text equation. In all cases the temperature

dependence is the scaling to be applied to the base parameter value. Thus if A0 is the set

material parameter base value, then a(T) is its set temperature dependence, such that the

material parameter as a function of temperature is A(T) = A0 a(T).

1) To set a temperature dependence using a text equation:

setparamtempequation (meshname) paramname text_equation

For details see User-Defined Text Equations. When using a text equation both a temperature

and time dependence may be set.

2) To set a temperature dependence using an array loaded from a file:

setparamtemparray (meshname) paramname filename

The file must contain 2 tab-spaced columns. The first gives the temperature value, and the

second gives the scaling value at the respective temperature.

To remove a material parameter temperature dependence use:

clearparamstemp (meshname) (paramname)

The above commands may be used in Python scripts through the param class in mesh objects,

e.g.:

#create a ferromagnetic mesh with given rectangle and discretization

FM = ns.Ferromagnet(rect, cellsize)

#Now access temperature dependences commands through the param class

#e.g. the lines below work for the exchange stiffness

FM.param.A.setparamtemparray(filename)

FM.param.A.setparamtempequation(equation_string)

FM.param.A.clearparamstemp()

244

Parameters Spatial Variation

Material parameters may be assigned a spatial variation where appropriate. There are four

types of methods used to set a spatial variation: using files, shapes, text equation, or a built-in

generator.

For scalar material parameters the spatial variation is the scaling to be applied to the base

parameter value. Thus if A0 is the set material parameter base value, then a(x,y,z) is its set

spatial variation, such that the material parameter as a function of position is A(x,y,z) = A0

a(x,y,z).

For vector material parameters (e.g. anisotropy axes) the spatial variation also requires 3

components, and rotates the base unit vector value. Thus if e0 = (e0x, e0y, e0z) is the base unit

vector, and n = (nx, ny, nz) is the set spatial variation, then the resultant vector used in

computations is:

























−

−−

=













z

y

x

xyz

xyzyxyxy

xyzxxyyx

z

y

x

e

nn

nnnnnn

e

0

//

Here nxy = (nx2 + ny2)0.5. Thus if the base unit vector is set to [1, 0, 0], the spatial variation

gives directly the resultant vector direction.

1) To set a spatial variation using a text equation:

setparamvar (meshname) paramname equation text_equation

For details see User-Defined Text Equations. When using a text equation both a spatial

variation and time dependence may be set.

2) To set a spatial variation using a file you can use either a mask file (png file) or an ovf2

file:

setparamvar (meshname) paramname mask filename

or

setparamvar (meshname) paramname ovf2 filename

245

The mask file method uses a grayscale png image file where black means scaling value of 0

and white means scaling value of 1. This method only assigns an x, y dependence.

The ovf2 file method specifies the scaling coefficients directly. For details see Working with

OVF2 Files.

3) To set a spatial variation using a shape:

setparamvar (meshname) paramname shape shape_definition value

This method is mainly intended to be used from Python scripts since shape definitions are

very long to type in manually, and can comprise single or composite shapes. Here a single

multiplier value is set for the given shape. For details see Shapes and Regions.

4) To set a spatial variation using a generator:

setparamvar (meshname) paramname generator_name (arguments…)

This method uses built-in generators, where generator_name is one of the generators listed

below. Each has a different set of arguments. If no arguments are given then default values

are set.

random

setparamvar (meshname) paramname random min max seed

Uniform random variation between min and max with generator seed (≥ 0).

jagged

setparamvar (meshname) paramname jagged min max spacing seed

A set of uniform random values between min and max are generated on a square grid in the xy

plane with given spacing (units m). A generator seed can also be given (≥ 0). The remaining

points are obtained using bi-linear interpolation. This is useful for generating random jagged

variations with a given in-plane correlation length (the spacing).

246

defects

setparamvar (meshname) paramname defects min max min_dia max_dia spacing seed

Circular defects are generated in the xy plane, at an average spacing (units m), and with

diameters (units m) in the range min_dia to max_dia. A generator seed can also be given (≥

0). The circular defects take on a value at the centre randomly between min and max. Within

the defect the value changes linearly along the radius from value of 1 at the extremity to the

centre value.

faults

setparamvar (meshname) paramname faults min_val max_val min_len max_len min_deg

max_deg spacing seed

Fault lines are generated in the xy plane, at an average spacing (units m), and with length

(units m) in the range min_len to max_len. The fault lines orientation can vary in the plane

between min_deg and max_deg (azimuthal angle in degrees). A generator seed can also be

given (≥ 0).

vor2D

setparamvar (meshname) paramname vor2D min max spacing seed

Voronoi 2D tessellation in the xy plane, with a Voronoi cell size given by spacing (units m).

Each Voronoi cell takes on a fixed value randomly between min and max. A generator seed

can also be given (≥ 0).

vor3D

setparamvar (meshname) paramname vor3D min max spacing seed

Voronoi 3D tessellation, with a Voronoi cell size given by spacing (units m). Each Voronoi

cell takes on a fixed value randomly between min and max. A generator seed can also be

given (≥ 0).

247

vorbnd2D

setparamvar (meshname) paramname vorbnd2D min max spacing seed

Voronoi 2D tessellation in the xy plane, with a Voronoi cell size given by spacing (units m).

Values vary randomly between min and max only at the boundaries of the Voronoi cells. A

generator seed can also be given (≥ 0).

vorbnd3D

setparamvar (meshname) paramname vorbnd3D min max spacing seed

Voronoi 3D tessellation, with a Voronoi cell size given by spacing (units m). Values vary

randomly between min and max only at the boundaries of the Voronoi cells. A generator seed

can also be given (≥ 0).

vorrot2D

setparamvar (meshname) paramname vorrot2D minpol_deg maxpol_deg minazim_deg

maxazim_deg spacing seed

Voronoi 2D tessellation in the xy plane, with a Voronoi cell size given by spacing (units m).

This is intended to be used for vectorial material parameters only (e.g. anisotropy axes), and

in each Voronoi cell a fixed rotation is generated using polar and azimuthal angles, randomly

in the range minpol_deg to maxpol_deg for polar angle in degrees, and minazim_deg to

maxazim_deg for azimuthal angle in degrees. A generator seed can also be given (≥ 0).

vorrot3D

setparamvar (meshname) paramname vorrot3D minpol_deg maxpol_deg minazim_deg

maxazim_deg spacing seed

Voronoi 3D tessellation, with a Voronoi cell size given by spacing (units m). This is intended

to be used for vectorial material parameters only (e.g. anisotropy axes), and in each Voronoi

cell a fixed rotation is generated using polar and azimuthal angles, randomly in the range

minpol_deg to maxpol_deg for polar angle in degrees, and minazim_deg to maxazim_deg for

azimuthal angle in degrees. A generator seed can also be given (≥ 0).

248

abl_pol

setparamvar (meshname) paramname abl_pol ratio_-x ratio_+x ratio_-y ratio_+y

ratio_-z ratio_+z min max exponent

Absorbing boundary layer with polynomial slopes of given exponent. The slopes vary

between max value at the mesh exterior boundary and min value in the interior. The slopes

may be applied to –x, +x, –y, +y, –z, +z boundaries as specified by the ratio values. These are

ratios of the mesh dimensions along the respective axes, with a value of zero meaning no

slope is applied to the respective boundary. For example ratio_-x value of 0.1 means apply

slope to the left for the first 10% of the mesh length L, so max value is taken at x = 0, and min

value is taken at x = l = 0.1 × L. Specifically, the following formula is used:

l

xmx

s

lxml

x

l

m

M

xs

n

≥=

≤≤+−

−

=

,)(

0,)

(

)(

)

(

.

Here M is max, m is min, l = ratio_-x × L, n = exponent, etc.

abl_tanh

setparamvar (meshname) paramname abl_tanh ratio_-x ratio_+x ratio_-y ratio_+y

ratio_-z ratio_+z min max sigma_nm

Absorbing boundary layer with tanh slopes of given width specified by sigma_nm. The slopes

vary between max value at the mesh exterior boundary and min value in the interior. The

slopes may be applied to –x, +x, –y, +y, –z, +z boundaries as specified by the ratio values.

These are ratios of the mesh dimensions along the respective axes, with a value of zero

meaning no slope is applied to the respective boundary. For example ratio_-x value of 0.1

means apply slope to the left for the first 10% of the mesh length L, so max value is taken at x

= 0, and min value is taken at x = l = 0.1 × L. Specifically, the following formula is used:

lxmxs

lxm

llx

l

mM

xs

≥=

≤≤+























+











−

−

=

,)(

0,

2

tanh

2/

tanh

)2/tanh(2

)(

σσσ

.

Here M is max, m is min, l = ratio_-x × L,

σ

= sigma_nm × 10-9, etc.

249

abl_exp

setparamvar (meshname) paramname abl_exp ratio_-x ratio_+x ratio_-y ratio_+y

ratio_-z ratio_+z min max sigma_nm

Absorbing boundary layer with exponential slopes of given width specified by sigma_nm.

The slopes vary between max value at the mesh exterior boundary and min value in the

interior. The slopes may be applied to –x, +x, –y, +y, –z, +z boundaries as specified by the

ratio values. These are ratios of the mesh dimensions along the respective axes, with a value

of zero meaning no slope is applied to the respective boundary. For example ratio_-x value of

0.1 means apply slope to the left for the first 10% of the mesh length L, so max value is taken

at x = 0, and min value is taken at x = l = 0.1 × L. Specifically, the following formula is used:

lxmxs

lxm

lx

l

mM

xs

≥=

≤≤+











−











−

−

=

,)(

0,1exp

1)/exp(

)(

σσ

.

Here M is max, m is min, l = ratio_-x × L,

σ

= sigma_nm × 10-9, etc.

To remove a material parameter spatial variation use:

clearparamsvar (meshname) (paramname)

The above commands may be used in Python scripts through the param class in mesh objects,

e.g.:

#create a ferromagnetic mesh with given rectangle and discretization

FM = ns.Ferromagnet(rect, cellsize)

#Now access spatial variation commands through the param class

#e.g. the lines below work for the exchange stiffness

FM.param.A.setparamvar(generator_name, parameters_list)

FM.param.A.clearparamsvar()

250

Simulation Schedules

A simulation may be started using the run command:

run

This will execute the currently configured simulation schedule. A schedule is defined by a

number of stages. There are a number of different stages available, and each has a set

stopping condition, and a set data saving condition. Stages are numbered sequentially starting

from 0, and there must always be at least one stage set.

To set the stopping condition for a stage:

editstagestop index stoptype (stopvalue)

Here index is the stage number, stoptype is one of the available stopping conditions (see

below) and stopvalue is the associated numerical value for that condition.

Available Stopping Conditions

nostop: stage will never finish.

iter: stage will finish when the number of iterations specified through stopvalue have

completed.

mxh: stage will finish when the normalized maximum torque, calculated as |M×H|/MS2, falls

below that specified through stopvalue (for simulations with stochasticity the average

normalized torque is used instead).

dmdt: stage will finish when the normalized maximum torque, calculated as |dM/dt|/MS, falls

below that specified through stopvalue (for simulations with stochasticity the average

normalized torque is used instead).

time: stage will finish when the set simulation time specified through stopvalue has elapsed.

mxh_iter: combination of mxh and iter conditions – stop when either condition is satisfied.

251

dmdt_iter: combination of dmdt and iter conditions – stop when either condition is satisfied.

Stage Types

There are several different types of stages (see below), and some may be associated with a

particular mesh. Some stages also accept a parameter.

To set a single stage and erase all others:

setstage (meshname) stagetype

To append a stage to the simulation schedule:

addstage (meshname) stagetype

To delete a stage:

delstage index

When a new stage is defined, which requires some value to be set (e.g. a stage which sets a

magnetic field), then a default value is first assumed. This can then be edited as:

editstagevalue index value

Relax: this is the default starting stage, is not associated with any mesh and does not set any

values.

Hxyz: set a magnetic field in the named mesh using Cartesian coordinates (or focused mesh if

name not given).

For example the following defines a stage with a 1 kA/m field set along the x direction:

setstage Hxyz

editstagevalue 0 1e3 0 0

In multi-mesh simulations the field may be applied to all meshes by specifying the meshname

as supermesh:

setstage supermesh Hxyz

252

Hxyz_seq: set a field sequence from a start to an end magnetic field value, given in Cartesian

coordinates, in a number of steps.

For example the following defines a field sequence from -1 kA/m to +1 kA/m in 10 steps (so

field step is 2 kA/m / 10):

setstage Hxyz_seq

editstagevalue 0 -1e3 0 0 1e3 0 0 10

For such multi-step stages, the stopping condition applies to each step.

Hpolar_seq: set a field sequence from a start to an end magnetic field value, given in polar

coordinates with angles in degrees, in a number of steps.

For example the following defines a field sequence from -1 kA/m to +1 kA/m in 10 steps

along the 30 degrees xy-plane azimuthal angle:

setstage Hpolar_seq

editstagevalue 0 -1e3 90 30 1e3 90 30 10

Hfmr: set a bias field and r.f. field for FMR simulations. Note: this is deprecated now, and

whilst it may still be used the recommended approach is to use the Hequation stage instead.

The r.f. field is discretized in a number of steps, sd, and a number of r.f. cycles may be set.

This stage type should be used with a time stopping condition. Since this is a multi-step stage,

the stopping condition applies to each step. Thus if an r.f. frequency f is required, then the

time stopping condition should be: tS = (1 / f) / sd.

For example the following defines a bias field of 1 MA/m along the y axis, with an r.f.

excitation of 100 A/m amplitude along the x axis, lasting for 1000 cycles with a frequency of

10 GHz, and each cycle discretized in 20 steps:

setstage Hfmr

editstagevalue 0 0 1e6 0 100 0 0 20 1000

editstagestop 0 time 5e-12

Hequation: field specified using a vector text equation.

For example the FMR example above would be specified as:

setstage Hequation

editstagevalue 0 100*sin(2*PI*10e9*t), 1e6, 0

editstagestop 0 time 100e-9

253

Hfile: field specified using an input text file. The file must contain tab-separated columns as

1) first column specifies time values in seconds, increasing order, 2) following columns

specify values to set, so requires 3 columns for the 3 field components. This stage should use

a time stopping condition. The number of steps in this stage is determined by the maximum

time value in the file (so last row entry) divided by the time stopping condition – i.e. each

step lasts for the set time stopping condition which determines the resolution. Every new step

sets a new field value; if the time values specified in the file are coarser, then linear

interpolation is used.

V, V_seq, Vequation, Vfile: as for H but sets electrode potential difference. Mesh name not

applicable.

I, I_seq, Iequation, Ifile: as for H but sets electrode electrical current. Mesh name not

applicable.

T, T_seq, Tequation, Tfile: as for H sets mesh base temperature. As for H can use supermesh

to set base temperature for all meshes.

Q, Q_seq, Qequation, Qfile: as for T but sets heat source.

Sunif: set uniform stress.

MonteCarlo: run a Mone-Carlo computation, either in an atomistic mesh, or micromagnetic

mesh – for details see Monte Carlo Algorithms.

Finally, stages may also be configured to save data – see Output Data for details.

At any point to stop the simulation from the console use the stop command. Another useful

command is reset, which sets the stage-step (sstep) counter to zero, iterations (iter) counter to

zero, and also the time (time).

254

Stages in Python Scripts

Rather than using separate commands to set, add, and edit stage stop and data saving

conditions, stage objects are available which make the Python scripts simpler. These are

shown below. When called, these stages are executed immediately. In the general form below

round brackets indicate optional arguments.

Relax:

General form:

ns.Relax([(stop_condition, (stop_value)), (save_condition, (save_value))])

Example:

ns.Relax([‘mxh_iter’, [1e-5, 10000], ‘iter’, 100])

Hxyz:

General form:

ns.Hxyz([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Example:

FM = ns.Ferromagnet(rectangle, cellsize)

ns.Hxyz([FM, [1e4, 1e4, 0], ‘mxh_iter’, [1e-5, 10000], ‘iter’, 100])

or:

ns.Hxyz([‘supermesh’, [1e4, 1e4, 0], ‘mxh_iter’, [1e-5, 10000], ‘iter’, 100])

Hxyz_seq:

General form:

ns.Hxyz_seq([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Hpolar_seq:

General form:

ns.Hpolar_seq([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Hfmr:

General form:

ns.Hfmr([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

255

Hequation:

General form:

ns.Hequation([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Hfile:

General form:

ns.Hfile([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

V, V_seq, Vequation, Vfile:

General form:

ns.V([stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

I, I_seq, Iequation, Ifile:

General form:

ns.I([stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

T, T_seq, Tequation, Tfile:

General form:

ns.T([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Q, Q_seq, Qequation, Qfile:

General form:

ns.Q([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

Sunif:

General form:

ns.Sunif([Mesh, stage_value, (stop_condition, (stop_value)), (save_condition,

(save_value))])

MonteCarlo:.

General form:

ns.MonteCarlo([acceptance_rate, (stop_condition, (stop_value)), (save_condition,

(save_value))])

256

Monte Carlo Algorithms

Boris uses parallelized Monte Carlo algorithms in atomistic meshes (see [S. Lepadatu

et al., JMMM 540, 168460 (2021)]), and a micromagnetic Monte Carlo algorithm

implemented for FM and AFM meshes [S. Lepadatu, J. Appl. Phys. 130, 163902 (2021)].

Note, the demagnetizing interaction and dipole-dipole interactions can be included in the

parallelized Monte Carlo algorithms. The Monte Carlo algorithms may also be used at 0 K as

energy minimizers.

Atomistic Monte Carlo

To use Monte Carlo then set a MonteCarlo stage. For micromagnetic meshes this

requires a Curie temperature to be set (curietemperature), and in all cases a non-zero

temperature should be set (temperature).

There are serial and parallel versions for the atomistic meshes, selected using the

mcserial command. By default the parallel version is enabled, and the serial version should

only be used for testing. To switch between the classical and constrained version in atomistic

meshes use the mcconstrain command in the required mesh (different meshes can use

different types of Monte Carlo algorithm, including with different constraining directions):

To set the classical Monte Carlo algorithm (the default):

mcconstrain (meshname) 0

To set the constrained Monte Carlo algorithm specify the constraining direction as a unit

vector, e.g. along the x axis:

mcconstrain (meshname) 1 0 0

You can also disable the Monte Carlo algorithm in some meshes if needed (status = 1):

mcdisable (meshname) status

When running a MonteCarlo stage, iteration of the dynamics equation is disable so effective

field are no longer computed. This means energy density output data is no longer available. If

you want to have these data computed then use (status = 1):

257

mccomputefields status

When enabled this slows down the iteration; when using micromagnetic meshes with the

demag module set this is automatically enabled.

The stage value for MonteCarlo is the target acceptance rate, set to 0.5 by default. This can

be adjusted, e.g. the following sets a 0.6 target acceptance rate:

setstage MonteCarlo

editstagevalue 0 0.6

During iteration the trial move cone angle is adjusted in order to reach the set target

acceptance rate. The minimum and maximum allowed cone angles can be controlled as:

mcconeangle min_angle_deg max_angle_deg

Thus in order to set a fixed cone angle, set the minimum and maximum allowed values to be

equal.

The set cone angle and actual acceptance rate are available in the MCparams output data –

see Output Data.

Micromagnetic Monte Carlo

In FM and AFM meshes, the MonteCarlo stage will use the micromagnetic Monte Carlo

algorithm introduced in [S. Lepadatu, J. Appl. Phys. 130, 163902 (2021)]. This differs from

the usual atomistic Monte Carlo algorithm, in that in addition to a rotation trial move, a

magnetization length compound trial move is also introduced. This is based on the Maxwell-

Boltzmann distribution of the micromagnetic free energy F:

()

Z

TkF

mf

B

//)

(exp)

(

2

mm −=

, where

( )

∑−=

iBii TkFmZ /)(exp

2m

Here

0

/S

MMm =

. In particular the magnetization length probability distribution is given by

(Zl is a renormalization factor):

( )











>

























−

+

−

<











−−

=

C

B

S

Ce

Be

S

l

T

TT

mT

Tk

mVM

TTmm

Tkm

VM

Z

m

mf

,

10

3

1

~

2

exp

,

~

8

exp

)(

2

||

2

0

2

22

2

||

0

2

χ

258

For symbol defitions see the LLB equation in Differential Equations section.

This leads to a modified trial move acceptance formula for FM as (see the publication for

details):

()











∆

−

=

→T

kF

m

min

B

A

P

B

A

B

accept

/exp

,1

)

(

4

, where

)()(

AB

FFF mm −=∆

.

For AFM meshes, trial moves are considered on the two sub-lattices in pairs, with acceptance

formulas and procedure detailed below.

1) Move

A

M

to

A

M

~

:

( )

BABA

A

FF

FMMMM ,,

~−=

∆

2) Accept/reject:

( )











∆−= TkF

M

minPBA

A

Aaccept /exp

~

,1 4

4

,

3) Move

B

M

to

B

M

~

:

( )

BABAB

FFF MMMM ,

~

,−=∆

4) Accept/reject:

( )











∆−= T

kF

M

minPB

B

Baccept /exp

~

,1 4

4

,

Here the micromagnetic free energy is a function of both

A

M

and

B

M

, and in particular the

magnetization length probability distribution is proportional to:

( )

),,,(

3

~

2

~

3

4

exp,2

,

2,

2

||,

,

2

2,

2

,

0,

2

jiBAji

mTk

m

mmTk

m

mm

Tkm

V

M

mmmP iNBij

je

jej

i

jNBiji

ie

iei

Biei

iS

iBAi

≠=































−

+

−

−∝

τ

χ

χτµ

µ

For symbol defitions see the two-sublattice LLB equation in Differential Equations section.

259

Simulation Algorithms

Domain Wall Movement

The domain wall movement algorithm has been described in detail in Tutorial 5 – Domain

Wall Movement and Data Processing section.

Dipole Shifting

Dipole meshes may be shifted during a simulation, which allows computations with a shifting

stray field pattern. The supermesh strayfield module should be enabled, or the mstrayfield

module enabled in each mesh where a stray field is required.

Dipole shifting can be achieved efficiently at runtime using 1 of 2 methods (for details see

description of respective commands in the Commands section).

1) shiftdipole command

This allows a single Dipole mesh to be shifted by the required x, y, z amount. For

efficiency this method should only be used with embedded Python scripts.

2) dipolevelocity command

It is possible to assign velocities for Dipole meshes at the set-up stage, which will

allow them to be shifted accordingly during a simulation.

Global Field Shifting

A global magnetic field may be set using the loadovf2field command (set on the supermesh).

This may later be cleared using the clearglobalfield command.

The loaded global field pattern may be shifted using the shiftglobalfield command (for

efficiency, this should only be used with embedded Python scripts).

260

Data Extraction Algorithms

Skyshift Algorithm

This is available through the skyshift output data (see Output Data section). This algorithm

allows tracking of a skyrmion in a given mesh, by defining a rectangle relative to that mesh,

such that the skyrmion is roughly in the centre of this rectangle. The output of this algorithm

are the x and y shifts of the skyrmion, relative to the starting position.

Any number of skyrmions may be tracked in this way, by adding multiple skyshift output

data. This requires the different skyshift output data to have different initial rectangles.

The algorithm works by dividing the defined rectangle into quadrants. The z direction

average magnetization is computed separately in these quadrants, and the rectangle position

is adjusted in order to keep these values the same (which occurs when the skyrmion is

centred). Thus the rectangle shift becomes the skyrmion shift output data.

Skypos Algorithm

This is an alternative skyrmion tracking algorithm, which additionally allows obtaining

skyrmion diameters, as well as tracking skyrmions with changing size. As with the skyshift

data, the skypos output data (see Output Data section) requires a given mesh, and an initial

rectangle relative to that mesh, such that the skyrmion is roughly in the centre of this

rectangle. Any number of skyrmions may be tracked in this way, by adding multiple skypos

output data. This requires the different skypos output data to have different initial rectangles.

The output data are the skyrmion x and y positions relative to the containing mesh, as well as

the skyrmion x and y diameters.

The algorithm works by analysing horizontal (x) and vertical (y) profiles through the centre of

the defined rectangle. Magnetization z component crossing points are detected, which allows

extraction of skyrmion centre positions along x and y directions, as well as respective

261

diameters. The tracking rectangle position is adjusted in order to keep the skyrmion in the

centre of the rectangle. The tracking rectangle size is also adjusted depending on the size of

the skyrmion. This is determined by a multiplicative factor, set using the skyposdmul

command. The default value is 2.0, which means the tracking rectangle sizes along x and y

are set to twice the skyrmion diameters along x and y respectively. If multiple densly packed

skyrmions are being tracked, it may be necessary to reduce this value, but it is recommended

this should exceed a multiplier of 1.2 otherwise loss of tracking can result.

It is recommended the skypos algorithm be used only ay 0 K, since thermal fluctuations will

result in loss of skyrmion tracking due to multiple crossing points. If thermal fluctuations are

included, either the skyshift algorithm should be used, or a custom skypos-type algorithm may

be implemented which includes averaging over many iterations. Such an algorithm may best

be implemented using embedded Python scripts.

Dwpos algorithm

It is also possible to track individual domain walls using the dwpos_x, dwpos_y, and dwpos_z

output data (see Output Data section), which obtain the domain wall centre position and

width.

A rectangle is required, and a domain wall profile will be extracted through the rectangle

centre (along x for dwpos_x, along y for dwpos_y, and along z for dwpos_z), where each

profile point is obtained as the average in the transverse rectangle cross-section. The

appropriate magnetization component in the extracted profile is then fitted using a tanh

function, namely

)/)(tanh(

0

∆−− xx

π

, where

0

x

is the domain wall centre position and

∆

its

width.

It is possible to specify which magnetization component is used for tanh fitting, using the

dwposcomponent command. By default the tanh component is automatically detected, but it

is also possible to specify directly the x, y, or z components.

262

Demagnetizing Field Polynomial Extrapolation

In order to speedup dynamical simulations when using higher order explicit

evaluation methods, it is possible to approximate the demagnetizing fields using polynomial

extrapolation at sub-steps of the evaluation method. For details see [S. Lepadatu, IEEE Trans.

Mag. 58, 1 (2022)]. Thus, the demagnetizing field need only be computed once per iteration,

and with the correct polynomial approximation no, or negligible, solution error should arise.

This approach is disabled by default, with full computation of the demagnetizing field

at all sub-steps of the evaluation method. To enable it use the following command to set the

required approximating polynomial order (in general the polynomial order should match the

evaluation method order – see [S. Lepadatu, IEEE Trans. Mag. 58, 1 (2022)]):

evalspeedup level

level 0: no speedup (default)

level 1: step, polynomial order 0.

level 2: linear, polynomial order 1.

level 3: quadratic, polynomial order 2.

level 4: cubic, polynomial order 3.

level 5: quartic, polynomial order 4.

level 6: quintic, polynomial order 5.

It’s also possible, although not recommended in general, to manually control the time step at

which the demagnetizing field is computed (setdtspeedup command). In general this is

linked to the differential equation evaluation method time step (linkdtspeedup command).

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Output Data

Output data may be obtained in the following ways. 1) The simulation schedule may

be configured to extract output data automatically during a simulation. 2) Output data may

also be obtained in interactive mode from the console. 3) Python scripts may be used to issue

commands which output data to files, or as return parameters.

1) Saving data during a simulation schedule.

Data saving conditions may be set for stages (see Simulation Schedules) as:

editdatasave index savetype (savevalue)

During simulations a data save is triggered when the set condition is met. There are a few

data save types available (savetype), detailed below:

none: no data save set.

stage: save at the end of every stage.

step: save at the end of every step in this stage.

iter: save every given number of iterations. For example the following saves every 100

iterations in the first stage:

editdatasave 0 iter 100

time: save at given time interval. For example the following saves every 100ps in the first

stage:

editdatasave 0 time 100ps

When data saving is triggered an output data file is used (out_data.txt by default), which is

set using:

savedatafile (directory/)filename

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The data to be saved is contained in a list, which may be configured as follows.

To set a single output data entry and erase all others:

setdata (meshname) dataname (rectangle)

Here, dataname is the name of a possible output data field. Possible names are given below.

The different data types may be applicable to a given mesh, and have a rectangle set (e.g.

averaging operations).

To append a new output data entry:

adddata (meshname) dataname (rectangle)

To delete an existing output data entry:

deldata index

Here, output data entries are contained sequentially with index number starting from zero.

The default configuration contains, in this order, 0: sstep, 1: iter, 2: time, 3: Ha for the default

permalloy mesh, 4: <M> for the default permalloy mesh with the entire mesh rectangle.

During a data saving event, the configured list is saved to the set output data file in a tab-

spaced row. Different output data types have different number of component, e.g. iter is the

iteration number and has just one component, sstep is the stage-step value and has 2

components, <M> is the average magnetization in the configured rectangle relative to the

given mesh and has 3 components.

The different output data types are listed below.

sstep: (2 components); stage-step.

time: (1 component); simulation time.

stime: (1 component); stage time.

iter: (1 component); simulation iteration number.

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siter: (1 component); stage iteration number.

dt: (1 component); evaluation time-step.

heat_dT: (1 component); heat equation time-step.

mxh: (1 component); normalized torque as |M×H|/MS2.

dmdt: (1 component); normalized torque as |dM/dt|/MS.

Ha: (3 components, needs mesh name); applied field.

<M>: (3 components, needs mesh name and rectangle relative to mesh); average

magnetization in given rectangle.

<M>th: (3 components, needs mesh name and rectangle relative to mesh); thermodynamic

average magnetization in given rectangle. (experimental, do not use)

<M2>: (3 components, needs mesh name and rectangle relative to mesh); average

magnetization in given rectangle, from sub-lattice B for 2-sublattice meshes.

|M|mm: (2 components, needs mesh name and rectangle relative to mesh); minimum-

maximum magnetization length in given rectangle.

Mx_mm: (2 components, needs mesh name and rectangle relative to mesh); minimum-

maximum magnetization x component in given rectangle.

My_mm: (2 components, needs mesh name and rectangle relative to mesh); minimum-

maximum magnetization y component in given rectangle.

Mz_mm: (2 components, needs mesh name and rectangle relative to mesh); minimum-

maximum magnetization z component in given rectangle.

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MCparams: (2 components, needs mesh name); Monte Carlo algorithm cone angle and

acceptance rate.

<Jc>: (3 components, needs mesh name and rectangle relative to mesh); average current

density in given rectangle.

<Jsx>: (3 components, needs mesh name and rectangle relative to mesh); average x-direction

spin current in given rectangle.

<Jsy>: (3 components, needs mesh name and rectangle relative to mesh); average y-direction

spin current in given rectangle.

<Jsz>: (3 components, needs mesh name and rectangle relative to mesh); average z-direction

spin current in given rectangle.

<mxdmdt>: (3 components, needs mesh name and rectangle relative to mesh); average m ×

dm/dt, where m = M / Ms, in given rectangle.

<dmdt>: (3 components, needs mesh name and rectangle relative to mesh); average dm/dt,

where m = M / Ms, in given rectangle.

<mxdmdt2>: (3 components, needs mesh name and rectangle relative to mesh); average mB ×

dmB/dt, where mB = MB / Ms,B, in given rectangle for sub-lattice B when using 2-sublattice

mesh.

<dmdt2>: (3 components, needs mesh name and rectangle relative to mesh); average dmB/dt,

where mB = MB / Ms,B, in given rectangle for sub-lattice B when using 2-sublattice mesh.

<mxdm2dt>: (3 components, needs mesh name and rectangle relative to mesh); average mA ×

dmB/dt, where mA,B = MA,B / Ms,A,B, in given rectangle for sub-lattice B when using 2-

sublattice mesh.

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<m2xdmdt>: (3 components, needs mesh name and rectangle relative to mesh); average mB ×

dmA/dt, where mA,B = MA,B / Ms,A,B, in given rectangle for sub-lattice B when using 2-

sublattice mesh.

<V>: (1 component, needs mesh name and rectangle relative to mesh); average charge

potential in given rectangle.

<S>: (3 components, needs mesh name and rectangle relative to mesh); average spin

accumulation in given rectangle.

<elC>: (1 component, needs mesh name and rectangle relative to mesh); average electrical

conductivity in given rectangle.

<T>: (1 component, needs mesh name and rectangle relative to mesh); average temperature

in given rectangle.

<T_l>: (1 component, needs mesh name and rectangle relative to mesh); average lattice

temperature in given rectangle, when using 2-temperature model.

V: (1 component); electrodes potential difference.

I: (1 component); ground electrode charge current.

R: (1 component); electrical resistance.

<u>: (3 components, needs mesh name and rectangle relative to mesh); average mechanical

displacement in given rectangle.

<Sd>: (3 components, needs mesh name and rectangle relative to mesh); average linear strain

in given rectangle (components as

ε

xx,

ε

yy,

ε

zz).

<Sod>: (3 components, needs mesh name and rectangle relative to mesh); average shear

strain in given rectangle (components as

ε

yz,

ε

xz,

ε

xy).

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e_demag: (1 component, needs mesh name and rectangle relative to mesh); average

demagnetizing energy density in given rectangle.

e_exch: (1 component, needs mesh name and rectangle relative to mesh); average exchange

energy density in given rectangle.

t_exch: (3 components, needs mesh name and rectangle relative to mesh); average exchange

interaction torque in given rectangle, to be used in atomistic meshes only.

e_surfexch: (1 component, needs mesh name and rectangle relative to mesh); average surface

exchange energy density in given rectangle.

t_surfexch: (3 components, needs mesh name and rectangle relative to mesh); average surface

exchange interaction torque in given rectangle, to be used in atomistic meshes only.

e_zee: (1 component, needs mesh name and rectangle relative to mesh); average Zeeman

energy density in given rectangle.

t_zee: (3 components, needs mesh name and rectangle relative to mesh); average applied field

torque in given rectangle, to be used in atomistic meshes only.

e_stray: (1 component, needs mesh name and rectangle relative to mesh); average dipole

stray field energy density in given rectangle.

t_stray: (3 components, needs mesh name and rectangle relative to mesh); average dipole

stray field torque in given rectangle, to be used in atomistic meshes only.

e_mo: (1 component, needs mesh name and rectangle relative to mesh); average magneto-

optical energy density in given rectangle.

e_mel: (1 component, needs mesh name and rectangle relative to mesh); average magneto-

elastic energy density in given rectangle.

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e_anis: (1 component, needs mesh name and rectangle relative to mesh); average magneto-

crystalline anisotropy energy density in given rectangle.

t_anis: (3 components, needs mesh name and rectangle relative to mesh); average magneto-

crystalline anisotropy interaction torque in given rectangle, to be used in atomistic meshes

only.

e_rough: (1 component, needs mesh name and rectangle relative to mesh); average roughness

module energy density in given rectangle.

e_total: (1 component); total energy density.

dwshift: (1 component); domain wall shift when using domain wall movement algorithm (see

Tutorial 5 – Domain Wall Movement and Data Processing).

dwpos_x: (2 components, needs mesh name and rectangle relative to mesh); Domain wall

position and width if applicable. A profile is extracted through the set rectangle along the x

direction, then a tanh fitting function is used to obtain the domain wall position and width.

The magnetization component fitted ia controlled using the dwposcomponent command.

dwpos_y, dwpos_z: As for dwpos_x, but the profile is extracted along the y, respectively z

direction.

skyshift: (2 components, needs mesh name and rectangle relative to mesh); Skyrmion xy

position is extracted using the skyshift algorithm in the given starting relative rectangle. The

rectangle tracks the skyrmion, thus multiple skyrmions may be tracked by assigning multiple

skyshift output data with different initial rectangles. For further details see Tutorial 23 –

Skyrmion Movement with Spin Currents. Note: this is considered obsolete and you should

use skypos instead.

skypos: (4 components, needs mesh name and rectangle relative to mesh); Skyrmion xy

position and xy diameters are extracted using the skypos algorithm in the given starting

relative rectangle. The rectangle tracks the skyrmion, thus multiple skyrmions may be tracked

by assigning multiple skypos output data with different initial rectangles. For further details

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see Tutorial 23 – Skyrmion Movement with Spin Currents. To fine-tune this algorithm

particularly for tracking multiple skyrmions which may be close together see the skyposdmul

command.

Q_topo: (1 component, needs mesh name and rectangle relative to mesh); Topological charge

computed in given rectangle. Also see the related dp_topocharge command.

v_iter: (1 component); Spin-transport solver, number of iterations for V convergence

(potential).

s_iter: (1 component); Spin-transport solver, number of iterations for S convergence (spin

accumulation).

ts_err: (1 component); Spin-transport solver achieved convergence error.

TMR: (1 component, needs mesh name and rectangle relative to mesh); Tunneling magneto-

resistance in an insulator mesh separating two ferromagnetic meshes.

commbuf: (0 components); This is a special entry which doesn’t save any data, but triggers

executation of the command buffer which allows more complex and/or pre-processed data to

be saved during a simulation schedule, e.g. profiles, ovf files, etc. – for details see Command

Buffering.

It’s also possible to configure the simulation schedule to save image files, e.g. for creating

videos. You can set an image file base using the saveimagefile command. To enable

automatic image saving then set the image save flag using saveimageflag command.

2) Interactive data output

With any of the data types above, the output may be obtained at any time using the

showdata command:

showdata dataname (meshname, (rectangle))

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There are a number of commands which extract output data. These include (see command

descriptions for details): savemeshimage, saveovf2mag, saveovf2, dp_getexactprofile,

averagemeshrect, dp_topocharge, dp_calctopochargedensity, dp_histogram,

dp_histogram2, dp_anghistogram.

3) Data output in Python scripts

Any of the commands above may be using from a Python script. Some have return

parameters which may be read (see command descriptions). For example with showdata:

data = ns.showdata(dataname, meshname, rectangle)

As a specific example the following obtains the average magnetization in a 20 nm cube:

data = ns.showdata('<M>', [0, 0, 0, 20e-9, 20e-9, 20e-9])

There is also an easy method of configuring output data during a simulation using the

setsavedata method in NSClient:

ns.setsavedata(filename, [data_name, (Mesh, (rectangle))], ...)

For example the following configures to save the time and average magnetization in part of a

mesh:

FM = ns.Ferromagnet(rectangle, cellsize)

ns.setsavedata(fileName, [‘time’], [‘<M>’, FM, sub_rectangle])

It is also possible to extract data from physical mesh quantities using a number of built-in

commands. For details see the Mesh Display section, Mesh quantities output in Python

scripts subsection.

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Command Buffering

NOTE: The command buffering technique is considered largely obsolete since v3.8. Whilst it

is still available, the recommended approach is to use embedded Python scripts when more

complex data types need to be extracted during a simulation. See Tutorial 9 – Scripting using

Python for help with embedded Python scripts.

When using Python scripts to control simulations, sometimes more complex data

types need to be extracted, which is not normally possible with a simulation schedule. For

example you may want to obtain magnetization profiles at fixed time intervals, ∆t, or save

the entire magnetization configuration to ovf files periodically. This is possible with a Python

script by setting a single stage which lasts for time ∆t, saving any required data from the

script, then looping until finished. The problem with this approach, if ∆t is small so data

needs to be saved often, it can be slow due to communication time between the script and

Boris. If the same set of commands needs to be executed each time data is saved, then it’s

possible to buffer these commands in Boris, so whenever a data save is triggered the

command buffer is executed.

When using NSClient in a Python script, all commands have a parameter called

bufferCommand. By default this is False, which means the command is executed straight

away. If bufferCommand = True, then instead of executing the command, it is appended to

the execution buffer (from the console this is the buffercommand command). The command

buffer may be executed at any point by calling the runcommbuffer command. Alternatively

the commbuf output data may be set so the command buffer is executed during a simulation

schedule whenever a data save is triggered. The buffer may be cleared by using the

clearcommbuffer command.

As an example the following script saves the entire magnetization configuration to ovf files

every 10 iterations using the commbuf output data, as well as the iteration, time, and average

magnetization to an output text file:

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from NetSocks import NSClient

ns = NSClient(); ns.configure(True)

ns.setdata('iter')

ns.adddata('time')

ns.adddata('<M>')

ns.adddata('commbuf')

ns.savedatafile('simdata.txt')

ns.editstagestop(0, 'time', 10e-9)

ns.editdatasave(0, 'iter', 10)

ns.saveovf2mag('mag_%iter%.ovf', bufferCommand = True)

ns.Run()

Another thing to notice here is the name of the output ovf file : ‘mag_%iter%.ovf’. Here

‘%iter%’ means replace ‘%iter%’ by the actual value of the iter data. Thus the above

command will save the magnetization to files numbered sequentially as mag_0.ovf,

mag_10.ovf, mag_20.ovf, etc. This works in general, and ‘%dataname%’ may be used with

most commands which work with input/output files, where dataname is any of the output

data names as listed in Output Data.

As a further example see the solutions to Tutorial 30 – Spin-Wave Dispersion, where

command buffering is used to extract magnetization profiles and append them to an output

file in a single simulation schedule run.

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Effective Fields and Energies Spatially Resolved Output Data

It is possible to obtain the energy density and effective fields from individual modules

with full spatial resolution, rather than just average values. Each module typically computes a

different contribution to the total effective field. The total effective field is available under

the Heff display (see Mesh Display). Once displayed it can be saved for later analysis using

the saveovf2 command, or profiles extracted from the display (dp_getexactprofile), etc.

In order to obtain information from invididual modules this needs to be configured using:

displaymodule (meshname) modulename

Here modulename is the name of the required module (see Modules). Once configured, then

the module effective field will be displayed when Heff is selected for display (display

(meshname) Heff). At this point data may be extracted as explained above. In order to get the

energy density with spatial resolution then Ed needs to be displayed instead (display

(meshname) Ed).

Once a module is selected in this way, the module data needs to be recomputed so the

effective fields are refreshed:

computefields

Only one module may be configured in this way at any one time, thus if the effective field or

energy density is required from multiple modules this procedure needs to be applied to each

in turn.

The module effective field display may be turned off by using:

displaymodule (meshname) none

For example the following saves the demagnetizing field and demagnetizing energy density

to ovf files:

displaymodule demag

computefields

display Heff

saveovf2 Hdemag.ovf

display Ed

saveovf2 Edemag.ovf

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Mesh Display

The mesh viewer may be configured using the display command as:

display (meshname) name

This sets the physical quantity to be displayed in the mesh viewer for the given mesh

(different meshes can be configured to display different things). The possible entries that can

be specified with name are:

Nothing: don’t display anything for this mesh.

M: magnetization.

M2: magnetization of sub-lattice B for 2-sublattice meshes.

M12: magnetization, showing simultaneously sub-lattices A and B for 2-sublattice meshes.

mu: magnetic moments for atomistic meshes.

Heff: total or module effective field (see Effective Fields and Energies Spatially Resolved

Output Data)

Heff2: total or module effective field of sub-lattice B for 2-sublattice meshes (see Effective

Fields and Energies Spatially Resolved Output Data)

Heff12: total or module effective field showing simultaneously sub-lattices A and B for 2-

sublattice meshes (see Effective Fields and Energies Spatially Resolved Output Data)

Ed: module energy density spatial variation (see Effective Fields and Energies Spatially

Resolved Output Data)

Ed2: module energy density spatial variation of sub-lattice B for 2-sublattice meshes (see

Effective Fields and Energies Spatially Resolved Output Data)

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Jc: current density.

V: charge potential.

elC: electrical conductivity.

S: spin accumulation.

Jsx: x-direction spin current.

Jsy: y-direction spin current.

Jsz: z-direction spin current.

Ts: bulk spin torque obtained when running the spin-transport solver.

Tsi: interfacial spin torque obtained when running the spin-transport solver.

Temp: temperature.

u: mechanical displacement.

S_d: strain tensor, diagonal components.

S_od: strain tensor, off-diagonal components.

ParamVar: parameter spatial variation. To select which parameter spatial variation is to be

displayed use the setdisplayedparamsvar command.

Roughness: applied mesh roughness (see Tutorial 24 – Roughness and Staircase Corrections).

Cust_V: specialcustom display of vectorial quantities, used with some commands.

Cust_S: special display of scalar quantities, used with some commands.

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The mesh display handles both scalar and vectorial quantites. For vectorial quantities it is

possible to display only one of the components. To do this use the vecrep command:

vecrep (meshname) vecreptype

vecreptype = 0, full.

vecreptype = 1, x component only.

vecreptype = 2, y component only.

vecreptype = 3, z component only.

vecreptype = 4, vector direction only.

vecreptype = 5, vector magnitude only.

It’s also possible to display quantities computed on the supermesh by setting meshname to

supermesh in the display command. The possible entries that can be specified with name on

the supermesh are:

Nothing: don’t display anything on supermesh.

Hdemag: demagnetizing field from sdemag module when multiconvolution 0.

HOe: Oersted field.

Hstray: stray field from dipole meshes.

Finally, the mesh viewer rotation, scaling etc., possible by using the mouse, may also be

controlled using the commands: rotcamaboutorigin, rotcamaboutaxis, adjustcamdistance,

shiftcamorigin.

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Mesh quantities output in Python scripts

The above physical quantities are associated with meshes, and they can be accessed using the

quant class in mesh objects. There are a number of commands available for each to allow

data extraction (see Commands section for details on usage):

averagemeshrect, dp_getexactprofile, getvalue, saveovf2, shape_get.

As an example, the following saves the effective field to an ovf2 file:

FM = ns.Ferromagnet(rectangle, cellsize)

FM.quant.Heff.saveovf2(filename)

All mesh display quantities listed above may be accessed in this way, when available for the

given mesh type.

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Commands

2dmulticonvolution status

Switch to multi-layered convolution and force it to 2D layering in each mesh (2), or 2D convolution for each

mesh (1), or allow 3D (0).

addafmesh name rectangle

Add antiferromagnetic mesh with given name and rectangle (m). The rectangle can be specified as: sx sy sz ex

ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

addameshcubic name rectangle

Add an atomistic mesh with simple cubic structure, with given name and rectangle (m). The rectangle can be

specified as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start

point as the origin.

addconductor name rectangle

Add a normal metal mesh with given name and rectangle (m). The rectangle can be specified as: sx sy sz ex ey

ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

adddata (meshname) dataname (rectangle)

Add dataname to list of output data. If applicable specify meshname and rectangle (m) in mesh. If not specified

and required, focused mesh is used with entire mesh rectangle.

adddipole name rectangle

Add a rectangular dipole with given name and rectangle (m). The rectangle can be specified as: sx sy sz ex ey ez

for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

addelectrode electrode_rect

Add an electrode in given rectangle (m).

addinsulator name rectangle

Add an insulator mesh with given name and rectangle (m). The rectangle can be specified as: sx sy sz ex ey ez

for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

addmaterial name rectangle

Add a new mesh with material parameters loaded from the materials database. The name is the material name as

found in the mdb file (see materialsdatabase command); this also determines the type of mesh to create, as well

as the created mesh name. The rectangle (m) can be specified as: sx sy sz ex ey ez for the start and end points in

Cartesian coordinates, or as: ex ey ez with the start point as the origin.

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Script return values: meshname - return name of mesh just added (can differ from the material name).

addmdbentry meshname (materialname)

Add new entry in the local materials database from parameters in the given mesh. The name of the new entry is

set to materialname if specified, else set to meshname. For a complete entry you should then edit the mdb file

manually with all the appropriate fields shown there.

addmesh name rectangle

Add a ferromagnetic mesh with given name and rectangle (m). The rectangle can be specified as: sx sy sz ex ey

ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

addmodule (meshname) handle

Add module with given handle to named mesh. If mesh name not specified, this is set for all applicable meshes.

addpinneddata (meshname) dataname (rectangle))

Add new entry in data box (at the end) with given dataname and meshname if applicable. A rectangle may also

be specified if applicable, however this will not be shown in the data box.

addrect (meshname) rectangle

Fill rectangle (m) within given mesh (focused mesh if not specified). The rectangle coordinates are relative to

mesh.

addstage (meshname) stagetype

Add a generic stage type to the simulation schedule with name stagetype, specifying a meshname if needed (if

not specified and required, focused mesh is used).

adjustcamdistance dZ

Adjust camera distance from origin using dZ. This is the same as right mouse hold and up-down drag on mesh

viewer.

ambient (meshname) ambient_temperature

Set mesh ambient temperature (all meshes if meshname not given) for Robin boundary conditions : flux normal

= alpha * (T_boundary - T_ambient).

Script return values: ambient_temperature - ambient temperature for named mesh.

astepctrl err_fail dT_incr dT_min dT_max

Set parameters for adaptive time step control: err_fail - repeat step above this (the tolerance), dT_incr - limit

multiplicative increase in dT using this, dT_min, dT_max - dT bounds.

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Script return values: err_fail dT_incr dT_min dT_max

atomicmoment (meshname) ub_multiple

Set atomic moment as a multiple of Bohr magnetons for given mesh (all ferromagnetic meshes if not specified).

This affects the temperature dependence of 'me' (see curietemperature command). A non-zero value will result

in me(T) being dependent on the applied field.

Script return values: ub_multiple - atomic moment multiple of Bohr magneton for named mesh.

averagemeshrect (meshname (quantity)) (rectangle (dp_index))

Calculate the average value depending on currently displayed quantities, or the named quantity if given (see

output of display command for possible quantities). The rectangle is specified in relative coordinates to the

named mesh (focused mesh if not specified); if not specified average the entire focused mesh. If dp_index is

specified then also append value to dp array.

Script return values: value

benchtime

Show the last simulation duration time in ms, between start and stop; used for performance becnhmarking.

Script return values: value

blochpreparemovingmesh (meshname)

Setup the named mesh (or focused mesh) for moving Bloch domain wall simulations: 1) set movingmesh

trigger, 2) set domain wall structure, 3) set dipoles left and right to remove end magnetic charges, 4) enable

strayfield module.

bprint message

Print message in console.

buffercommand command (params...)

Append command with parameters to command buffer.

cellsize (meshname) value

Change cellsize of meshname (m). The cellsize can be specified as: hx hy hz, or as: hxyz. If meshname not

given use the focused mesh.

Script return values: cellsize - return cellsize of focused mesh, or of meshname if given.

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center

Center mesh view and scale to fit window size.

chdir directory

Change working directory.

Script return values: directory

checkupdates

Connect to boris-spintronics.uk to check if updates to program or materials database are available.

clearcommbuffer

Clear commands stored in command buffer.

clearelectrodes

Delete all currently set electrodes.

clearequationconstants

Clear all user-defined constants for text equations.

clearglobalfield

Clear a set global field (previously loaded with loadovf2field on supermesh).

clearmovingmesh

Clear moving mesh settings made by a prepare command.

clearparamstemp (meshname) (paramname)

Clear material parameter temperature dependence in given mesh. If meshname not given clear temperature

dependences in all meshes. If paramname not given clear all parameters temperature dependences.

clearparamsvar (meshname) (paramname)

Clear parameter spatial dependence in given mesh. If meshname not given clear spatial dependence in all

meshes. If paramname not given clear all parameters spatial dependence.

clearroughness (meshname)

Clear roughness in given mesh (focused mesh if not specified) by setting the fine shape same as the coarse M

shape.

clearscreen

Clear all console text.

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clearstrainequations

Clear strain equations from all meshes.

computefields

Run simulation from current state for a single iteration without advancing the simulation time.

copymeshdata meshname_from meshname_to (...)

Copy all primary mesh data (e.g. magnetization values and shape) from first mesh to all other meshes given - all

meshes must be of same type.

copyparams meshname_from meshname_to (...)

Copy all mesh parameters from first mesh to all other meshes given - all meshes must be of same type.

coupletodipoles status

Set/unset coupling to dipoles : if magnetic meshes touch a dipole mesh then interface magnetic cells are coupled

to the dipole magnetization direction.

crosstie (meshname) (direction (radius thickness centre))

Set magnetization in a crosstie state with given direction (-1 or +1) starting at given centre, over given radius

and thickness. If not specified then entire mesh is used, centred. If mesh name not specified, the focused mesh is

used.

cuda status

Switch CUDA GPU computations on/off.

Script return values: status

curietemperature (meshname) curie_temperature

Set Curie temperature for ferromagnetic mesh (all ferromagnetic meshes if not specified). This will set default

temperature dependencies as: Ms = Ms0*me, A = Ah*me^2, D = D0*me^2, K = K0*me^3 (K1 and K2),

damping = damping0*(1-T/3Tc) T < Tc, damping = damping0*2T/3Tc T >= Tc, susrel = dme/d(mu0Hext).

Setting the Curie temperature to zero will disable temperature dependence for these parameters.

Script return values: curie_temperature - Curie temperature for named mesh.

data

Shows list of currently set output data and available data.

Script return values: number of set output data fields

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dataprecision precision

Set number of significant figures used for saving data to file as well as data display. The default (and minimum)

is 6.

Script return values: precision

default

Reset program to default state.

deldata index

Delete data from list of output data at index number. If index number is -1 then delete all data fields, leaving just

a default time data field - there must always be at least 1 output data field.

delelectrode index

Delete electrode with given index.

delequationconstant name

Delete named user constant used in text equations.

delmdbentry materialname

Delete entry in the local materials database (see materialsdatabase for current selection).

delmesh meshname

Delete mesh with given name.

delmodule (meshname) handle

Delete module with given handle from named mesh (focused mesh if not specified).

delpinneddata index

Delete entry in data box at given index (index in order of appearance in data box from 0 up).

delrect ((meshname) (rectangle))

Void rectangle (m) within given mesh (focused mesh if not specified). The rectangle coordinates are relative to

mesh. If rectangle not given void entire mesh.

delstage index

Delete stage from simulation schedule at index number. If index number is -1 then delete all stages, leaving just

a default Relax stage - there must always be at least 1 stage set.

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delsurfacefix index

Delete a fixed surface with given index (previously added with surfacefix). Use index -1 to delete all fixed

surfaces.

delsurfacestress index

Delete an external stress surface with given index (previously added with surfacestress). Use index -1 to delete

all external stress surfaces.

designateground electrode_index

Change ground designation for electrode with given index.

dipolevelocity meshname velocity (clipping)

Set velocity for dipole shifting algorithm (x, y, z, components). Optionally specify a clipping distance (x, y, z

components) - i.e. dipole shift clipped. Default is 0.5 nm; to disable clipping set to zero.

Script return values: velocity clipping

disabletransportsolver status

Disable iteration of transport solver so any set current density remains constant.

Script return values: status

diskbufferlines lines

Set output disk buffer number of lines - default is 100. You shouldn't need to adjust this.

Script return values: bufferlines

display (meshname) name

Change quantity to display for given mesh (focused mesh if not specified).

displaybackground (meshname) name

Change background quantity to display for given mesh (focused mesh if not specified).

displaydetail size

Change displayed detail level to given displayed cell size (m).

Script return values: size

displaymodule (meshname) modulename

Select module for effective field and energy density display (focused mesh if not specified). If modulename is

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none then display total effective field.

displayrenderthresholds thresh1 thresh2 thresh3

Set display render thresholds of number of displayed cells for faster rendering as: 1) switch to using simpler

elements (vectors only), 2) don't display surrounded cells, 3) display on checkerboard pattern even if not

surrounded (vectors only). Set to zero to disable.

Script return values: thresh1 thresh2 thresh3

displaythresholds minimum maximum

Set thresholds for foreground mesh display : magnitude values outside this range are not rendered. If both set to

0 then thresholds are ignored.

displaythresholdtrigger trigtype

For vector quantities, set component to trigger thresholds on. trigtype = 1 (x component), trigtype = 2 (y

component), trigtype = 3 (z component), trigtype = 5 (magnitude only)

displaytransparency foreground background

Set alpha transparency for display. Values range from 0 (fully transparent) to 1 (opaque). This is applicable in

dual display mode when we have a background and foreground for the same mesh.

dmcellsize (meshname) value

Change demagnetizing field macrocell size of meshname, for atomistic meshes (m). The cellsize can be

specified as: hx hy hz, or as: hxyz. If meshname not given use the focused mesh.

Script return values: cellsize - return demagnetizing field macrocell size of focused mesh, or of meshname if

given.

dp_add dp_source value (dp_dest)

Add value to dp array and place it in destination (or at same position if destination not specified).

dp_adddp dp_x1 dp_x2 dp_dest

Add dp arrays : dp_dest = dp_x1 + dp_x2

dp_anghistogram (meshname) dp_x dp_y (cx cy cz (nx ny nz (numbins min max)))

Calculate an angular deviation histogram with given number of bins, minimum and maximum bin values, from

the magnetization of the given mesh (must be magnetic; focused mesh if not specified). Save histogram in dp

arrays at dp_x, dp_y. If cx cy cz values given, then first average in macrocells containing (cx, cy, cz) individual

cells. If unit vector direction nx ny nz not given, then angular deviation is calculated from the average

magnetization direction. If histogram parameters not given use 100 bins with minimum and maximum

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magnetization magnitude values.

dp_append dp_original dp_new

Append data from dp_new to the end of dp_original.

dp_calcsot hm_mesh fm_mesh

For the given heavy metal and ferromagnetic meshes calculate the expected effective spin Hall angle and field-

like torque coefficient according to analytical equations (see manual).

Script return values: SHAeff, flST.

dp_calctopochargedensity (meshname)

Calculate topological charge density spatial dependence for the given mesh (must be magnetic; focused mesh if

not specified). Output available in Cust_S.

dp_cartesiantopolar dp_in_x dp_in_y (dp_out_r dp_out_theta)

Convert from Cartesian coordinates (x,y) to polar (r, theta).

dp_chunkedstd dp_index chunk

Obtain stadard deviation from every chunk number of elements in dp array, then average all the chunk std value

to obtain final std. In the special case where chunk is the number of elements in the dp array (set chunk = 0 for

this case), the chunked std is the same as the usual std on the mean.

Script return values: std.

dp_clear indexes...

Clear dp arrays with specified indexes.

dp_clearall

Clear all dp arrays.

dp_coercivity dp_index_x dp_index_y

Obtain coercivity from x-y data: find first crossings of x axis in the two possible directions, with uncertainty

obtained from step size.

Script return values: Hc_up Hc_up_err- Hc_up_err+ Hc_dn Hc_dn_err- Hc_dn_err+.

dp_completehysteresis dp_index_x dp_index_y

For a hysteresis loop with only one branch continue it by constructing the other direction branch (invert both x

and y data and add it in continuation) - use only with hysteresis loops which are expected to be symmetric.

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dp_countskyrmions (meshname) (x y radius)

Calculate the number of skyrmions for the given mesh (must be magnetic; focused mesh if not specified),

optionally in the given circle with radius and centered at x y (relative values). Use Qmag = Integral(|m.(dm/dx x

dm/dy)| dxdy) / 4PI.

Script return values: Q - the calculated topological charge.

dp_crossingsfrequency dp_in_x dp_in_y dp_level dp_freq_up dp_freq_dn (steps)

From input x-y data build a histogram of average frequency the x-y data crosses a given line (up and down,

separated). The line varies between minimum and maximum of y data in given number of steps (100 by default).

Output the line values in dp_level with corresponding crossings frequencies in dp_freq_up and dp_freq_dn.

dp_crossingshistogram dp_in_x dp_in_y dp_level dp_counts (steps)

From input x-y data build a histogram of number of times x-y data crosses a given line (up or down). The line

varies between minimum and maximum of y data in given number of steps (100 by default). Output the line

values in dp_level with corresponding number of crossings in dp_counts.

dp_div dp_source value (dp_dest)

Divide dp array by value and place it in destination (or at same position if destination not specified).

dp_divdp dp_x1 dp_x2 dp_dest

Divide dp arrays : dp_dest = dp_x1 / dp_x2

dp_dotprod dp_vector ux uy uz dp_out

Take dot product of (ux, uy, uz) with vectors in dp arrays dp_vector, dp_vector + 1, dp_vector + 2 and place

result in dp_out.

dp_dotproddp dp_x1 dp_x2

Take dot product of dp arrays : value = dp_x1.dp_x2

dp_dumptdep meshname paramname max_temperature dp_index

Get temperature dependence of named parameter from named mesh up to max_temperature, at dp_index -

temperature scaling values obtained.

dp_erase dp_index start_index length

From dp_index array erase a number of points - length - starting at start_index.

dp_extract dp_in dp_out start_index (length)

From dp_in array extract a number of points - length - starting at start_index, and place them in dp_out.

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dp_fitadiabatic (meshname) (abs_err Rsq T_ratio (stencil))

Fit the computed self-consistent spin torque (see below) using Zhang-Li STT with fitting parameters P and beta

(non-adiabaticity) using a given square in-plane stencil (default size 3) in order to extract the spatial variation of

P. Cut-off values for absolute fitting error (default 0.1), Rsq measure (default 0.9), and normalized torque

magnitude (default 0.1) can be set - value of zero disables cutoff. The focused mesh must be ferromagnetic,

have the transport module set with spin solver enabled, and we also require Jc and either Ts or Tsi to have been

computed. The fitting is done on Ts, Tsi, or on their sum depending if they’ve been enabled or not. Output

available in Cust_S.

dp_fitdw dp_x dp_y

Fit domain wall tanh component to obtain width and center position : M(x) = A * tanh(PI * (x - x0) / D).

Script return values: D, x0, A, std_D, std_x0, std_A.

dp_fitlorentz dp_x dp_y

Fit Lorentz peak function to x y data : f(x) = y0 + S dH / (4(x-H0)^2 + dH^2).

Script return values: S, H0, dH, y0, std_S, std_H0, std_dH, std_y0.

dp_fitlorentz2 dp_x dp_y

Fit Lorentz peak function with both symmetric and asymmetric parts to x y data : f(x) = y0 + S (dH + A * (x -

H0)) / (4(x-H0)^2 + dH^2).

Script return values: S, A, H0, dH, y0, std_S, std_A, std_H0, std_dH, std_y0.

dp_fitnonadiabatic (meshname) (abs_err Rsq T_ratio (stencil))

Fit the computed self-consistent spin torque (see below) using Zhang-Li STT with fitting parameters P and beta

(non-adiabaticity) using a given square in-plane stencil (default size 3) in order to extract the spatial variation of

beta. Cut-off values for absolute fitting error (default 0.1), Rsq measure (default 0.9), and normalized torque

magnitude (default 0.1) can be set - value of zero disables cutoff. The focused mesh must be ferromagnetic,

have the transport module set with spin solver enabled, and we also require Jc and either Ts or Tsi to have been

computed. The fitting is done on Ts, Tsi, or on their sum depending if they’ve been enabled or not. Output

available in Cust_S.

dp_fitskyrmion dp_x dp_y

Fit skyrmion z component to obtain radius and center position : Mz(x) = Ms * cos(2*arctan(sinh(R/w)/sinh((x-

x0)/w))).

Script return values: R, x0, Ms, w, std_R, std_x0, std_Ms, std_w.

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dp_fitsot (meshname) (hm_mesh) (rectangle)

Fit the computed self-consistent interfacial spin torque using SOT with fitting parameters SHAeff and flST

(field-like torque coefficient). hm_mesh specifies the heavy metal mesh from which to obtain the current density

- if not specified, then the current density in meshname is used instead. The fitting is done inside the specified

rectangle for the given mesh (focused mesh if not specified), with the rectangle specified using relative

coordinates as sx sy sz ex ey ez (entire mesh if not specified). This mesh must be ferromagnetic, have the

transport module set with spin solver enabled, and we also require Jc and Tsi to have been computed.

Script return values: SHAeff, flST, std_SHAeff, std_flST, Rsq.

dp_fitsotstt (meshname) (hm_mesh) (rectangle)

Fit the computed self-consistent spin torque (see below) using Zhang-Li STT with fitting parameters P and beta

(non-adiabaticity), and simultaneously also using SOT with fitting parameters SHAeff and flST (field-like

torque coefficient). hm_mesh specifies the heavy metal mesh from which to obtain the current density for SOT.

The fitting is done inside the specified rectangle for the given mesh (focused mesh if not specified), with the

rectangle specified using relative coordinates as sx sy sz ex ey ez (entire mesh if not specified). The focused

mesh must be ferromagnetic, have the transport module set with spin solver enabled, and we also require Jc and

either Ts or Tsi to have been computed to have been computed. The fitting is done on Ts, Tsi, or on their sum

depending if they’ve been enabled or not.

Script return values: SHAeff, flST, P, beta, std_SHAeff, std_flST, std_P, std_beta, Rsq.

dp_fitstt (meshname) (rectangle)

Fit the computed self-consistent spin torque (see below) using Zhang-Li STT with fitting parameters P and beta

(non-adiabaticity). The fitting is done inside the specified rectangle for the given mesh (focused mesh if not

specified), with the rectangle specified using relative coordinates as sx sy sz ex ey ez (entire mesh if not

specified). The focused mesh must be ferromagnetic, have the transport module set with spin solver enabled,

and we also require Jc and either Ts or Tsi to have been computed. The fitting is done on Ts, Tsi, or on their

sum depending if they’ve been enabled or not.

Script return values: P, beta, std_P, std_beta, Rsq.

dp_get dp_arr index

Show value in dp_arr at given index - the index must be within the dp_arr size.

Script return values: value

dp_getampli dp_source pointsPeriod

Obtain maximum amplitude obtained every pointsPeriod points.

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Script return values: amplitude.

dp_getaveragedprofile (meshname) start end step dp_index

This is used in conjuction with dp_getexactprofile. Get in dp arays starting at dp_index the averaged profile

built with dp_getexactprofile. start, end and step are the same parameters used by dp_getexactprofile. Calling

this command causes the average counter to be reset to zero. Thus a sequence consists of 1) dp_getexactprofile

executed as many times as needed (e.g. to average stochasticity), called with dp_index = -1, 2)

dp_getaveragedprofile to read out the profile and reset averaging counter, ready for next batch of

dp_getexactprofile calls.

Script return values: number of averages

dp_getexactprofile (meshname (quantity)) start end step dp_index (stencil)

Extract profile of physical quantity displayed on screen, or the named quantity if given (see output of display

command for possible quantities), for named mesh (focused mesh if not specified), directly from the mesh (so

using the exact mesh resolution not the displayed resolution), along the line specified with given start and end

relative cartesian coordinates (m) and with the given step size (m). If stencil specified - as x y z (m) - then obtain

profile values using weighted averaging with stencil centered on profile point. If dp_index >= 0 then place

profile in given dp arrays: up to 4 consecutive dp arrays are used, first for distance along line, the next 3 for

physical quantity components (e.g. Mx, My, Mz) so allow space for these starting at dp_index. If dp_index = -1

then just average profile in internal memory, to be read out with dp_getaveragedprofile. Profile will wrap-

around if exceeding mesh size.

dp_getprofile start end dp_index

Extract profile of physical quantities displayed on screen, at the current display resolution, along the line

specified with given start and end cartesian absolute coordinates (m). Place profile in given dp arrays: up to 4

consecutive dp arrays are used, first for distance along line, the next 3 for physical quantity components (e.g.

Mx, My, Mz) so allow space for these starting at dp_index. NOTE : if you only want to extract the profile from

a single mesh, then use dp_getexactprofile instead; this command should only be used to extract profiles across

multiple meshes.

dp_histogram (meshname) dp_x dp_y (cx cy cz (numbins min max))

Calculate a histogram with given number of bins, minimum and maximum bin values, from the magnetization

magnitude of the given mesh (must be magnetic; focused mesh if not specified). Save histogram in dp arrays at

dp_x, dp_y.If cx cy cz values given, then first average in macrocells containing (cx, cy, cz) individual cells. If

histogram parameters not given use 100 bins with minimum and maximum magnetization magnitude values.

dp_histogram2 (meshname) dp_x dp_y (numbins min max M2 deltaM2)

Calculate a histogram for the given 2-sublattice mesh (focused mesh if not specified) with given number of bins,

minimum and maximum bin values for sub-lattice A, if the corresponding magnetization magnitude in sub-

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lattice B equals M2 within the given deltaM2. Save histogram in dp arrays at dp_x, dp_y. If histogram

parameters not given use 100 bins with minimum and maximum magnetization magnitude values, with M2 set

to MeB and deltaM2 set 0.01*MeB respectively.

dp_linreg dp_index_x dp_index_y (dp_index_z dp_index_out)

Fit using linear regression to obtain gradient and intercept with their uncertainties. If dp_index_z is specified

multiple linear regressions are performed on adjacent data points with same z value; output in 5 dp arrays

starting at dp_index_out as: z g g_err c c_err.

Script return values: g g_err c c_err.

dp_load (directory/)filename file_indexes... dp_indexes...

Load data columns from filename into dp arrays. file_indexes are the column indexes in filename (.txt

termination by default), dp_indexes are used for the dp arrays; count from 0. If directory not specified, the

default one is used.

dp_mean dp_index (exclusion_ratio)

Obtain mean value with standard deviation. If exclusion_ratio (>0) included, then exclude any points which are

greater than exclusion_ratio normalised distance away from the mean.

Script return values: mean stdev.

dp_minmax dp_index

Obtain absolute minimum and maximum values, together with their index position.

Script return values: min_value min_index max_value max_index.

dp_monotonic dp_in_x dp_in_y dp_out_x dp_out_y

From input x-y data extract monotonic sequence and place it in output x y arrays.

dp_mul dp_source value (dp_dest)

Multiply dp array with value and place it in destination (or at same position if destination not specified).

dp_muldp dp_x1 dp_x2 dp_dest

Multiply dp arrays : dp_dest = dp_x1 * dp_x2

dp_newfile (directory/)filename

Make new file, erasing any existing file with given name. If directory not specified, the default one is used.

dp_peaksfrequency dp_in_x dp_in_y dp_level dp_freq (steps)

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From input x-y data build a histogram of average frequency of peaks in the x-y data in bands given by the

number of steps. The bands vary between minimum and maximum of y data in given number of steps (100 by

default). Output the line values in dp_level with corresponding peak frequencies in dp_freq.

dp_pow dp_source exponent (dp_dest)

Exponentiate source aray values and place it in destination (or at same position if destination not specified).

dp_rarefy dp_in dp_out (skip)

Pick elements from dp_in using the skip value (1 by default) and set them in dp_out; e.g. with skip = 2 every 3rd

data point is picked. The default skip = 1 picks every other point.

dp_remanence dp_index_x dp_index_y

Obtain remanence from x-y data: find values at zero field in the two possible directions.

Script return values: Mr_up Mr_dn.

dp_removeoffset dp_index (dp_index_out)

Subtract the first point (the offset) from all the points in dp_index. If dp_index_out not specified then processed

data overwrites dp_index.

dp_replacerepeats dp_index (dp_index_out)

Replace repeated points from data in dp_index using linear interpolation: if two adjacent sets of repeated points

found, replace repeats between the mid-points of the sets. If dp_index_out not specified then processed data

overwrites dp_index.

dp_save (directory/)filename dp_indexes...

Save specified dp arrays in filename (.txt termination by default). If directory not specified, the default one is

used. dp_indexes are used for the dp arrays; count from 0.

dp_saveappend (directory/)filename dp_indexes...

Save specified dp arrays in filename (.txt termination by default) by appending at the end. If directory not

specified, the default one is used. dp_indexes are used for the dp arrays; count from 0.

dp_saveappendasrow (directory/)filename dp_index

Save specified dp array in filename (.txt termination by default) as a single row with tab-spaced values,

appending to end of file. If directory not specified, the default one is used.

dp_saveasrow (directory/)filename dp_index

Save specified dp array in filename (.txt termination by default) as a single row with tab-spaced values. If

directory not specified, the default one is used.

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dp_sequence dp_index start_value increment points

Generate a sequence of data points in dp_index from start_value using increment.

dp_set dp_arr index value

Set value in dp_arr at given index - the index must be within the dp_arr size.

dp_showsizes (dp_arr)

List sizes of all non-empty dp arrays, unless a specific dp_arr index is specified, in which case only show the

size of dp_arr.

Script return values: dp_arr size if specified

dp_smooth dp_in dp_out window_size

Smooth data in dp_in using nearest-neighbor averaging with given window size, and place result in dp_out

(must be different).

dp_sub dp_source value (dp_dest)

Subtract value from dp array and place it in destination (or at same position if destination not specified).

dp_subdp dp_x1 dp_x2 dp_dest

Subtract dp arrays : dp_dest = dp_x1 - dp_x2

dp_sum dp_index

Obtain sum of all elements in dp array.

Script return values: sum.

dp_thavanghistogram (meshname) dp_x dp_y cx cy cz (nx ny nz (numbins min max))

Calculate an angular deviation histogram with given number of bins, using thermal averaging in each macrocell

containing (cx, cy, cz) individual cells, minimum and maximum bin values, from the magnetization of the given

mesh (must be magnetic; focused mesh if not specified). Save histogram in dp arrays at dp_x, dp_y. If unit

vector direction nx ny nz not given, then angular deviation is calculated from the average magnetization

direction. If histogram parameters not given use 100 bins with minimum and maximum magnetization

magnitude values.

dp_thavhistogram (meshname) dp_x dp_y cx cy cz (numbins min max)

Calculate a histogram with given number of bins, using thermal averaging in each macrocell containing (cx, cy,

cz) individual cells, minimum and maximum bin values, from the magnetization magnitude of the focused mesh

(must be magnetic). Save histogram in dp arrays at dp_x, dp_y. If histogram parameters not given use 100 bins

with minimum and maximum magnetization magnitude values.

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dp_topocharge (meshname) (x y radius)

Calculate the topological charge for the given mesh (must be magnetic; focused mesh if not specified),

optionally in the given circle with radius and centered at x y (relative values). Q = Integral(m.(dm/dx x dm/dy)

dxdy) / 4PI.

Script return values: Q - the calculated topological charge.

dwall (meshname) longitudinal transverse width position

Create an idealised domain wall (tanh profile for longitudinal component, 1/cosh profile for transverse

component) along the x-axis direction in the given mesh (focused mesh if not specified). For longitudinal and

transverse specify the components of magnetization as x, -x, y, -y, z, -z, i.e. specify using these std::string

literals. For width and position use metric units.

dwposcomponent value

Set vector component used to extract dwpos_x, dwpos_y, dwpos_z data: this should be the component with a

tanh profile. -1: automatic (detect tanh component), 0: x, 1: y, 2: z. Default is automatic, but you might want to

specify component as the automatic detection can fail when very noisy (e.g. close to Tc for atomistic

simulations).

Script return values: value

ecellsize (meshname) value

Change cellsize of meshname for electrical conduction (m). The cellsize can be specified as: hx hy hz, or as:

hxyz. If meshname not given use the focused mesh.

Script return values: cellsize - return electrical conduction cellsize of focused mesh, or of meshname if given.

editdata index (meshname) dataname (rectangle)

Edit entry in list of output data at given index in list. If applicable specify meshname and rectangle (m) in mesh.

If not specified and required, focused mesh is used with entire mesh rectangle.

editdatasave index savetype (savevalue)

Edit data saving condition in simulation schedule. Use index < 0 to set condition for all stages.

editstage index (meshname) stagetype

Edit stage type from simulation schedule at index number.

editstagestop index stoptype (stopvalue)

Edit stage/step stopping condition in simulation schedule. Use index < 0 to set condition for all stages.

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editstagevalue index value

Edit stage setting value in simulation schedule. The value type depends on the stage type.

editsurfacefix index rect

Edit rectangle of fixed surface with given index.

editsurfacestress index rect

Edit rectangle of stress surface with given index.

editsurfacestressequation index equation

Edit equation of stress surface with given index.

electrodes

Show currently configured electrodes.

equationconstants name value

Create or edit user constant to be used in text equations.

errorlog status

Set error log status.

escellsize value

Change cellsize for electric super-mesh (m). The cellsize can be specified as: hx hy hz, or as: hxyz

Script return values: cellsize - return cellsize for electric super-mesh.

evalspeedup level

Enable/disable evaluation speedup by extrapolating demag field at evaluation substeps from previous field

updates. Status levels: 0 (no speedup), 1 (step), 2 (linear), 3 (quadratic), 4 (cubic), 5 (quartic), 6 (quintic). If

enabling speedup strongly recommended to always use quadratic type.

Script return values: level

exchangecoupledmeshes (meshname) status

Set/unset direct exchange coupling to neighboring meshes : if neighboring ferromagnetic meshes touch the

named mesh (focused mesh if not specified) then interface magnetic cells are direct exchange coupled to them.

Script return values: status

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excludemulticonvdemag (meshname) status

Set exclusion status (0 or 1) of named mesh from multi-layered demag convolution (focused mesh if not

specified).

flower (meshname) (direction (radius thickness centre))

Set magnetization in a flower state with given direction (-1 or +1) starting at given centre, over given radius and

thickness. If not specified then entire mesh is used, centred. If mesh name not specified, the focused mesh is

used.

flusherrorlog

Clear error log.

fmscellsize value

Change cellsize for ferromagnetic super-mesh (m). The cellsize can be specified as: hx hy hz, or as: hxyz

Script return values: cellsize - return cellsize for ferromagnetic super-mesh.

generate2dgrains (meshname) spacing (seed)

Generate 2D Voronoi cells in the xy plane at given average spacing in given mesh (focused mesh if not

specified). The seed is used for the pseudo-random number generator, 1 by default.

generate3dgrains (meshname) spacing (seed)

Generate 3D Voronoi cells at given average spacing in given mesh (focused mesh if not specified). The seed is

used for the pseudo-random number generator, 1 by default.

getmeshtype meshname

Show mesh type of named mesh.

Script return values: meshtype.

getvalue (meshname (quantity)) position

Read value at position depending on currently displayed quantites, or the named quantity if given (see output of

display command for possible quantities), in named mesh (focused mesh if not specified).

Script return values: value

gpukernels status

When in CUDA mode calculate demagnetization kernels initialization on the GPU (1) or on the CPU (0).

Script return values: status

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imagecropping left bottom right top

Set cropping of saved mesh images using normalized left, bottom, right, top values: 0, 0 point is left, bottom of

mesh window and 1, 1 is right, top of mesh window.

individualshape status

When changing the shape inside a mesh set this flag to true so the shape is applied only to the primary displayed

physical quantity. If set to false then all relevant physical quantities are shaped.

Script return values: status

insulatingside (meshname) side_literal status

Set temperature insulation (Neumann boundary condition) for named mesh side (focused mesh if not specified).

side_literal : x, -x, y, -y, z, -z.

Script return values: status_x status_-x status_y status_-y status_z status_-z - insulating sides status for named

mesh.

invertmag (meshname) (components)

Invert magnetization direction. If mesh name not specified, the focused mesh is used. You can choose to invert

just one or two components instead of the entire vector: specify components as x y z, e.g. invertmag x

isrunning

Checks if the simulation is running and sends state value to the calling script.

Script return values: state - return simulation running state.

iterupdate iterations

Update mesh display every given number of iterations during a simulation.

Script return values: iterations - return number of iterations for display update.

linkdtelastic flag

Links elastodynamics solver time-step to magnetic ODE time-step if set, else elastodynamics time-step is

independently controlled.

linkdtspeedup flag

Links speedup time-step to ODE time-step if set, else speedup time-step is independently controlled.

linkdtstochastic flag

Links stochastic time-step to ODE time-step if set, else stochastic time-step is independently controlled.

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linkstochastic (meshname) flag

Links stochastic cellsize to magnetic cellsize if flag set to 1 for given mesh, else stochastic cellsize is

independently controlled. If meshname not given set for all meshes.

loadmaskfile (meshname) (z_depth) (directory/)filename

Apply .png mask file to magnetization in given mesh (focused mesh if not specified); i.e. transfer shape from

.png file to mesh - white means empty cells. If image is in grayscale then void cells up to given depth top down

(z_depth > 0) or down up (z_depth < 0). If z-depth = 0 then void top down up to all z cells.

loadovf2curr (meshname) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing current density data, into the given mesh (which must have

transport module enabled; focused mesh if not specified), mapping the data to the current mesh dimensions.

loadovf2disp (meshname) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing mechanical displacement data, into the given mesh (which must

be ferromagnetic and have the melastic module enabled; focused mesh if not specified), mapping the data to the

current mesh dimensions. From the mechanical displacement the strain tensor is calculated.

loadovf2field (meshname) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing applied field data, into the given mesh (which must be

magnetic; focused mesh if not specified), mapping the data to the current mesh dimensions. Alternatively if

meshname is supermesh, then load a global field with absolute rectangle coordinates and its own discretization.

loadovf2mag (meshname) (renormalize_value) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing magnetization data, into the given mesh (which must be

magnetic; focused mesh if not specified), mapping the data to the current mesh dimensions. By default the

loaded data will not be renormalized: renormalize_value = 0. If a value is specified for renormalize_value, the

loaded data will be renormalized to it (e.g. this would be an Ms value).

loadovf2strain (meshname) (directory/)filename_diag filename_odiag

Load an OOMMF-style OVF 2.0 file containing strain tensor data, into the given mesh (which must be

ferromagnetic and have the melastic module enabled; focused mesh if not specified), mapping the data to the

current mesh dimensions. The symmetric strain tensor is applicable for a cubic crystal, and has 3 diagonal

component (specified in filename_diag with vector data as xx, yy, zz), and 3 off-diagonal components (specified

in filename_odiag with vector data as yz, xz, xy).

loadovf2temp (meshname) (directory/)filename

Load an OOMMF-style OVF 2.0 file containing temperature data, into the given mesh (which must have heat

module enabled; focused mesh if not specified), mapping the data to the current mesh dimensions.

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loadsim (directory/)filename

Load simulation with given name.

makevideo (directory/)filebase fps quality

Make a video from .png files sharing the common filebase name. Make video at given fps and quality (0 to 5

worst to best).

manual

Opens Boris manual for current version.

matcurietemperature (meshname) curie_temperature

Set indicative material Curie temperature for ferromagnetic mesh (focused mesh if not specified). This is not

used in calculations, but serves as an indicative value - set the actual Tc value with the curietemperature

command.

Script return values: curie_temperature - Indicative material Curie temperature for named mesh.

materialsdatabase (mdbname)

Switch materials database in use. This setting is not saved by savesim, so using loadsim doesn't affect this

setting; default mdb set on program start.

mccomputefields status

Enable/disable modules update during Monte Carlo stages. Disabled by default, but if you want to save energy

density terms you need to enabled this - equivalent to running ComputeFields after every Monte Carlo step.

Script return values: status

mcconeangle min_angle max_angle

Set Monte Carlo cone angle limits (1 to 180 degrees). Setting min and max values the same results in a fixed

cone angle Monte Carlo algorithm.

Script return values: min_angle max_angle

mcconstrain (meshname) value

Set value 0 to revert to classic Monte-Carlo Metropolis for ASD. Set a unit vector direction value (x y z) to

switch to constrained Monte Carlo as described in PRB 82, 054415 (2010). If meshname not specified setting is

applied to all atomistic meshes.

mcdisable (meshname) status

Disable or enable Monte Carlo algorithm for given mesh (focused mesh if not specified).

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mcellsize (meshname) value

Change cellsize of meshname for mechanical solver (m). The cellsize can be specified as: hx hy hz, or as: hxyz.

If meshname not given use the focused mesh.

Script return values: cellsize - return mechanical cellsize of focused mesh, or of meshname if given.

mcserial (meshname) value

Change Monte-Carlo algorithm type. 0: parallel (default); red-black ordering parallelization 1: serial (testing

only); spins picked in random order. If meshname not specified setting is applied to all atomistic meshes.

memory

Show CPU and GPU-addressable memory information (total and free).

mesh

Display information for all meshes.

meshfocus meshname

Change mesh focus to given mesh name.

Script return values: meshname - return name of focused mesh.

meshfocus2 meshname

Change mesh focus to given mesh name but do not change camera orientation.

Script return values: meshname - return name of focused mesh.

meshrect (meshname) rectangle

Change rectangle of meshname (m). The rectangle can be specified as: sx sy sz ex ey ez for the start and end

points in Cartesian coordinates, or as: ex ey ez with the start point as the origin. If meshname not given use the

focused mesh.

Script return values: rectangle - return rectangle of focused mesh, or of meshname if given.

mirrormag (meshname) axis

Mirror magnetization in a given axis, specified as x, y, z, e.g. mirrormag x. If mesh name not specified, the

focused mesh is used.

modules (meshname (modules...))

Show interactive list of available and currently set modules. If meshname specified then return all set modules

names, and set given modules if specified.

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Script return values: modulenames

movingmesh status_or_meshname

Set/unset trigger for movingmesh algorithm. If status_or_meshname = 0 then turn off, if status_or_meshname =

1 then turn on with trigger set on first ferromagnetic mesh, else status_or_meshname should specify the mesh

name to use as trigger.

movingmeshasym status

Change symmetry type for moving mesh algorithm: 1 for antisymmetric (domain walls), 0 for symmetric

(skyrmions).

Script return values: status

movingmeshthresh value

Set threshold used to trigger a mesh shift for moving mesh algorithm - normalised value between 0 and 1.

Script return values: threshold

multiconvolution status

Switch between multi-layered convolution (true) and supermesh convolution (false).

ncommon sizes

Switch to multi-layered convolution and force it to user-defined discretisation, specifying sizes as nx ny nz.

ncommonstatus status

Switch to multi-layered convolution and force it to user-defined discretisation (status = true), or default

discretisation (status = false).

neelpreparemovingmesh (meshname)

Setup the named mesh (or focused mesh) for moving Neel domain wall simulations: 1) set movingmesh trigger,

2) set domain wall structure, 3) set dipoles left and right to remove end magnetic charges, 4) enable strayfield

module.

newinstance port (cudaDevice (password))

Start a new local Boris instance with given server port, and optionally cuda device number (0/1/2/3/...); a value

of -1 means automatically determine cuda device. If password not blank this should be specified otherwise

server will not start.

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nextstage

Increment stage value by one.

ode

Show interactive list of available and currently set ODEs and evaluation methods.

onion (meshname) (direction (radius1 radius2 thickness centre))

Set magnetization in an onion state with given direction (-1 clockwise, +1 anti-clockwise) starting at given

centre, from radius1 to radius2 and thickness. If not specified then entire mesh is used, centred. If mesh name

not specified, the focused mesh is used.

openpotentialresistance resistance

Set open potential mode for electrodes, by setting a resistance value - set zero to disable open potential mode. In

this mode the electrode potential is determined during the simulation, e.g. if thermoelectric effect is enabled,

then setting electrodes in open potential mode allows a thermoelectric current to flow through the electrodes,

with potential determined as thermoelectric current generated times resistance set.

Script return values: resistance

params (meshname)

List all material parameters. If meshname not given use the focused mesh.

paramstemp (meshname)

List all material parameters temperature dependence. If meshname not given use the focused mesh.

paramsvar (meshname)

List all material parameters spatial variation. If meshname not given use the focused mesh.

pbc (meshname) flag images

Set periodic boundary conditions for magnetization in given mesh (must be magnetic, or supermesh;

supermesh/all meshes if name not given). Flags specify types of perodic boundary conditions: x, y, or z; images

specify the number of mesh images to use either side for the given direction when calculating the demagnetising

kernel - a value of zero disables pbc. e.g. pbc x 10 sets x periodic boundary conditions with 10 images either

side for all meshes; pbc x 0 clears pbc for the x axis.

preparemovingmesh (meshname)

Setup the named mesh (or focused mesh) for moving transverse (or vortex) domain wall simulations: 1) set

movingmesh trigger, 2) set domain wall structure, 3) set dipoles left and right to remove end magnetic charges,

4) enable strayfield module.

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prngseed (meshname) seed

Set PRNG seed, for computations with stochasticity, in given mesh (all meshes if name not given). If seed = 0

then system tick count is used as seed (default behavior), otherwise the fixed set value is used. NOTE: seed

values > 0 only take effect for computations with cuda 1.

Script return values: seed

raapbiasequation meshname text_equation

Set equation for bias dependence of TMR anti-parallel RA value in given mesh (must be insulator with tmr

module added) - the text equation uses V for the bias value. Call with empty equation string to clear any

currently set equation.

random (meshname (seed))

Set random magnetization distribution in mesh, with pseud-random number generator seed (default 1 if not

specified). If mesh name not specified, the focused mesh is used.

randomxy (meshname (seed))

Set random magnetization distribution in the xy plane in mesh, with pseud-random number generator seed

(default 1 if not specified). If mesh name not specified, the focused mesh is used.

rapbiasequation meshname text_equation

Set equation for bias dependence of TMR parallel RA value in given mesh (must be insulator with tmr module

added) - the text equation variables are RAp (parallel TMR RA), RAap (antiparallel TMR RA), and V (bias

across insulator mesh). Call with empty equation string to clear any currently set equation.

refineroughness (meshname) value

Set roughness refinement cellsize divider in given mesh (focused mesh if not specified), i.e. cellsize used for

roughness initialization is the magnetic cellsize divided by value (3 components, so divide component by

component).

Script return values: value - roughness refinement.

refreshmdb

Reload the local materials database (see materialsdatabase for current selection). This is useful if you modify the

values in the materials database file externally.

refreshscreen

Refreshes entire screen.

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renamemesh (old_name) new_name

Rename mesh. If old_name not specified then the focused mesh is renamed.

requestmdbsync materialname (email)

Request the given entry in the local materials database is added to the online shared materials database. This

must be a completed entry - see manual for instructions. The entry will be checked before being made available

to all users through the online materials database. If you want to receive an update about the status of this

request include an email address.

reset

Reset simulation state to the starting state.

resetelsolver

Reset the elastodynamics solver to unstrained state.

resetmesh (meshname)

Reset to constant magnetization in given mesh (focused mesh if not specified).

robinalpha (meshname) robin_alpha

Set alpha coefficient (all meshes if meshname not given) for Robin boundary conditions : flux normal = alpha *

(T_boundary - T_ambient).

Script return values: robin_alpha - Robin alpha value for named mesh.

rotcamaboutaxis dAngle

Rotate camera about its axis using an angle change. This is the same as right mouse hold and left-right drag on

mesh viewer.

rotcamaboutorigin dAzim dPolar

Rotate camera about origin using a delta azimuthal and delta polar angle change. This is the same as middle

mouse hold and drag on mesh viewer.

roughenmesh (meshname) depth (side (seed))

Roughen given mesh (focused mesh if not specified) to given depth (m) on named side (use side = x, y, z, -x, -y,

-z as literal, z by default). The seed is used for the pseudo-random number generator, 1 by default.

run

Run simulation from current state.

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runcommbuffer

Execute commands stored in command buffer.

runscript (directory/)filename

Execute a Python-scripted simulation (filename must be a Python script). If directory not specified then default

directory is used.

runscriptnewinstance (directory/)filename

Execute a Python-scripted simulation (filename must be a Python script) in a new program instance. If directory

not specified then default directory is used.

runstage stage

Run given simulation stage only.

savecomment (directory/)filename comment

Save comment in given file by appending to it.

savedata (append_option)

Save data (as currently configured in output data - see setdata/adddata commands) to output file (see

savedatafile command). append_option = -1 : use default append behaviour (i.e. new file with header created for

first save, then append). append_option = 0 : make new file then save. append_option = 1 : append to file.

savedatafile (directory/)filename

Change output data file (and working directory if specified).

Script return values: filename

savedataflag status

Set data saving flag status.

Script return values: status

saveimage ((directory/)filename)

Save currently displayed mesh image, with a centered view of displayed quantity only, to given file (as .png). If

directory not specified then default directory is used. If filename not specified then default image save file name

is used.

saveimagefile (directory/)filename

Change image file base (and working directory if specified).

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Script return values: filename

saveimageflag status

Set image saving flag status.

Script return values: status

savemeshimage ((directory/)filename)

Save currently displayed mesh image to given file (as .png). If directory not specified then default directory is

used. If filename not specified then default image save file name is used.

saveovf2 (meshname (quantity)) (data_type) (directory/)filename

Save an OOMMF-style OVF 2.0 file containing data from the given mesh (focused mesh if not specified),

depending on currently displayed quantites, or the named quantity if given (see output of display command for

possible quantities). You can specify the data type as data_type = bin4 (single precision 4 bytes per float),

data_type = bin8 (double precision 8 bytes per float), or data_type = text. By default bin8 is used.

saveovf2mag (meshname) (n) (data_type) (directory/)filename

Save an OOMMF-style OVF 2.0 file containing magnetization data from the given mesh (which must be

magnetic; focused mesh if not specified). You can normalize the data to Ms0 value by specifying the n flag (e.g.

saveovf2mag n filename) - by default the data is not normalized. You can specify the data type as data_type =

bin4 (single precision 4 bytes per float), data_type = bin8 (double precision 8 bytes per float), or data_type =

text. By default bin8 is used.

saveovf2param (meshname) (data_type) paramname (directory/)filename

Save an OOMMF-style OVF 2.0 file containing the named parameter spatial variation data from the given mesh

(focused mesh if not specified). You can specify the data type as data_type = bin4 (single precision 4 bytes per

float), data_type = bin8 (double precision 8 bytes per float), or data_type = text. By default bin8 is used.

savesim (directory/)filename

Save simulation with given name. If no name given, the last saved/loaded file name will be used.

scalemeshrects status

When changing a mesh rectangle scale and shift all other mesh rectangles in proportion if status set.

Script return values: status

scellsize (meshname) value

Change cellsize of meshname for stochastic properties (m). The cellsize can be specified as: hx hy hz, or as:

hxyz. If meshname not given use the focused mesh.

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Script return values: cellsize - return stochastic properties cellsize of focused mesh, or of meshname if given.

scriptserver status

Enable or disable the script communication server. When enabled the program will listen for commands

received using network sockets on port 1542.

selectcudadevice number

Select CUDA device to use from available devices. The device CUDA Compute version must match Boris

CUDA version. Devices numbered from 0 up, default selection at startup is device 0.

Script return values: number

serverpassword password

Set/change script server password - this is used to authenticate remote client messages. By default no password

is set (blank).

serverport port

Set script server port.

Script return values: port

serversleepms time_ms

Set script server thread sleep time in ms: lower value makes server more responsive, but increases CPU load.

Script return values: time_ms

setafmesh name rectangle

Set a single antiferromagnetic mesh (deleting all other meshes) with given name and rectangle (m). The

rectangle can be specified as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey

ez with the start point as the origin.

setameshcubic name rectangle

Set a single atomistic mesh (deleting all other meshes) with simple cubic structure, with given name and

rectangle (m). The rectangle can be specified as: sx sy sz ex ey ez for the start and end points in Cartesian

coordinates, or as: ex ey ez with the start point as the origin.

setangle (meshname) polar azimuthal

Set magnetization angle (deg.) in mesh uniformly using polar coordinates (all magnetic meshes if name not

given).

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setatomode equation evaluation

Set differential equation to solve in atomistic meshes, and method used to solve it (same method is applied to

micromagnetic and atomistic meshes).

setconductor name rectangle

Set a single metal mesh (deleting all other meshes) with given name and rectangle (m). The rectangle can be

specified as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start

point as the origin.

setcurrent current

Set a constant current source with given value. The potential will be adjusted to keep this constant current.

Script return values: current

setcurrentdensity (meshname) Jx Jy Jz

Set a constant current density vector with given components in given mesh (focused mesh if not specified).

Must have transport module added, but transport solver iteration will be disabled.

setdata (meshname) dataname (rectangle)

Delete all currently set output data and set dataname to list of output data. If applicable specify meshname and

rectangle (m) in mesh. If not specified and required, focused mesh is used with entire mesh rectangle.

setdefaultelectrodes (sides)

Set electrodes at the x-axis ends of the given mesh, both set at 0V. If sides specified (literal x, y, or z) then set at

respective axis ends instead of default x-axis ends. Set the left-side electrode as the ground. Delete all other

electrodes.

setdipole name rectangle

Set a single dipole (deleting all other meshes) with given name and rectangle (m). The rectangle can be specified

as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as

the origin.

setdisplayedparamsvar (meshname) paramname

Set param to display for given mesh (focused mesh if not specified) when ParamVar display is enabled (to show

spatial variation if any).

setdt value

Set differential equation time-step (only applicable to fixed time-step methods).

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Script return values: dT

setdtspeedup value

Set time step for evaluation speedup.

Script return values: dTspeedup

setdtstoch value

Set time step for stochastic field generation.

Script return values: dTstoch

seteldt value

Set elastodynamics equation solver time step. Zero value disables solver.

Script return values: value - elastodynamics equation time step.

setelectrodepotential electrode_index potential

Set potential on electrode with given index.

Script return values: potential

setelectroderect electrode_index electrode_rect

Edit rectangle (m) for electrode with given index.

setfield (meshname) magnitude polar azimuthal

Set uniform magnetic field (A/m) using polar coordinates. If mesh name not specified, this is set for all

magnetic meshes - must have Zeeman module added.

Script return values: <Ha_x, Ha_y, Ha_z> - applied field in Cartesian coordinates for, or of meshname if given.

setheatdt value

Set heat equation solver time step. Zero value disables solver.

Script return values: value - heat equation time step.

setinsulator name rectangle

Set a single insulator mesh (deleting all other meshes) with given name and rectangle (m). The rectangle can be

specified as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start

point as the origin.

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setktens term1 (term2 ...)

Set the anisotropy energy terms for tensorial anisotropy. Each term must be of the form dxn1yn2zn3, where d is

a number, n1, n2, n3 are integers >= 0. x, y, z are string literals.

setmaterial name rectangle

Set a single mesh with material parameters loaded from the materials database (deleting all other meshes). The

name is the material name as found in the mdb file (see materialsdatabase command); this also determines the

type of mesh to create, as well as the created mesh name. The rectangle (m) can be specified as: sx sy sz ex ey

ez for the start and end points in Cartesian coordinates, or as: ex ey ez with the start point as the origin.

Script return values: meshname - return name of mesh just added (can differ from the material name).

setmesh name rectangle

Set a single ferromagnetic mesh (deleting all other meshes) with given name and rectangle (m). The rectangle

can be specified as: sx sy sz ex ey ez for the start and end points in Cartesian coordinates, or as: ex ey ez with

the start point as the origin.

setobjectangle (meshname) polar azimuthal position

Set magnetization angle in solid object only containing given relative position (x y z) uniformly using polar

coordinates. If mesh name not specified, the focused mesh is used.

setode equation evaluation

Set differential equation to solve in both micromagnetic and atomistic meshes, and method used to solve it

(same method is applied to micromagnetic and atomistic meshes).

setodeeval evaluation

Set differential equation method used to solve it (same method is applied to micromagnetic and atomistic

meshes).

setparam (meshname) paramname (value)

Set the named parameter to given value. If meshname not given use the focused mesh.

Script return values: value - return value of named parameter in named mesh (focused mesh if not specified).

setparamtemparray (meshname) paramname filename

Set the named parameter temperature dependence using an array in the given mesh (all applicable meshes if not

given). This must contain temperature values and scaling coefficients. Load directly from a file (tab spaced).

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setparamtempequation (meshname) paramname text_equation

Set the named parameter temperature dependence equation for the named mesh (all applicable meshes if not

given).

setparamvar (meshname) paramname generatorname (arguments...)

Set the named parameter spatial dependence for the named mesh (all applicable meshes if not specified) using

the given generator (including any required arguments for the generator - if not given, default values are used).

setpotential potential

Set a symmetric potential drop : -potential/2 for ground electrode, +potential/2 on all other electrodes.

Script return values: potential

setrect (meshname) polar azimuthal rectangle

Set magnetization angle in given rectangle of mesh (relative coordinates) uniformly using polar coordinates. If

mesh name not specified, the focused mesh is used.

setschedulestage stage

Set stage value, but do not run simulation. Must be a valid stage number.

Script return values: stage

setsordamping damping_v damping_s

Set fixed damping values for SOR algorithm used to solve the Poisson equation for V (electrical potential) and S

(spin accumulation) respectively.

Script return values: damping_v damping_s

setstage (meshname) stagetype

Delete all currently set stages, and set a new generic stage type to the simulation schedule with name stagetype,

specifying a meshname if needed (if not specified and required, focused mesh is used).

setstagevalue index

Set configured value for given stage index.

setstress (meshname) magnitude polar azimuthal

Set uniform mechanical stress (Pa) using polar coordinates. If mesh name not specified, this is set for all

magnetic meshes - must have MElastic module added. Setting a uniform stress disables the elastodynamics

solver.

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Script return values: <Tsig_x, Tsig_y, Tsig_z> - applied mechanical stress in Cartesian coordinates for focused

mesh, or of meshname if given.

shape_cone (meshname) len_x len_y len_z cpos_x cpos_y cpos_z

Set a conical shape in given mesh (focused mesh if not specified) with given lengths (x, y, z) at given centre

position coordinates (x, y, z).

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shape_disk (meshname) dia_x dia_y cpos_x cpos_y (z_start z_end)

Set a disk shape in given mesh (focused mesh if not specified) with given x and y diameters at given centre

position with x and y coordinates. Optionally specify z start and end coordinates, otherwise entire mesh

thickness used.

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z

shape_displacement x y z

Set modifier for shape generator commands: displacements along x, y, z to use with shape repetitions

(shape_repetitions).

Script return values: value

shape_ellipsoid (meshname) dia_x dia_y dia_z cpos_x cpos_y cpos_z

Set a prolate ellipsoid shape in given mesh (focused mesh if not specified) with given diameters (x, y, z) at

given centre position coordinates (x, y, z).

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shape_get (meshname (quantity)) name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_psi rot_theta rot_phi

repeat_x repeat_y repeat_z disp_x disp_y disp_z method ...

Get average value from a shape in given mesh (focused mesh if not specified), depending on currently displayed

quantites, or the named quantity if given (see output of display command for possible quantities), where shape

definition is shown in shape_set command help. NOTE: this command does not work yet with CUDA enabled,

will be finished in a future version.

shape_method method

Set modifier for shape generator commands: method to use when applying shape as string literal: add or sub.

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Script return values: name

shape_pyramid (meshname) len_x len_y len_z cpos_x cpos_y cpos_z

Set a pyramid shape in given mesh (focused mesh if not specified) with given lengths (x, y, z) at given centre

position coordinates (x, y, z).

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shape_rect (meshname) len_x len_y cpos_x cpos_y (z_start z_end)

Set a rectangle shape in given mesh (focused mesh if not specified) with given x and y lengths at given centre

position with x and y coordinates. Optionally specify z start and end coordinates, otherwise entire mesh

thickness used.

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z

shape_repetitions x y z

Set modifier for shape generator commands: number of shape repetitions along x, y, z.

Script return values: value

shape_rotation psi theta phi

Set modifier for shape generator commands: rotation using psi (around y), theta (around x) and phi (around z) in

degrees.

shape_set (meshname) name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_psi rot_theta rot_phi repeat_x

repeat_y repeat_z disp_x disp_y disp_z method ...

General command for setting a shape: meant to be used with scripted simulations, not directly from the console.

Can be used with the return data from shape_... commands, and can take multiple elementary shapes to form a

composite object. Set a named shape in given mesh (focused mesh if not specified) with given dimensions (x, y,

z) at given centre position coordinates (x, y, z).

shape_setangle (meshname) name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_psi rot_theta rot_phi repeat_x

repeat_y repeat_z disp_x disp_y disp_z method ... theta polar

Set magnetization angle (theta - polar, phi - azimuthal) in degrees for given shape identifier (or composite

shape) in given mesh (focused mesh if not specified). The shape identifier is returned by one of the shape_...

commands, also same parameters taken by the shape_set command. As with shape_set this command is intended

for scripted simulations only.

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shape_setparam (meshname) paramname name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_psi rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method ... scaling_value

Set material parameter scaling value in given shape only, in given mesh (focused mesh if not specified). If you

want to set effective parameter value directly, not as a scaling value, then set base parameter value (setparam) to

1 and the real parameter value through this command.

shape_tetrahedron (meshname) len_x len_y len_z cpos_x cpos_y cpos_z

Set a tetrahedron shape in given mesh (focused mesh if not specified) with given lengths (x, y, z) at given centre

position coordinates (x, y, z).

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shape_torus (meshname) len_x len_y len_z cpos_x cpos_y cpos_z

Set a torus shape in given mesh (focused mesh if not specified) with given lengths (x, y, z) at given centre

position coordinates (x, y, z).

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shape_triangle (meshname) len_x len_y cpos_x cpos_y (z_start z_end)

Set a triangle shape in given mesh (focused mesh if not specified) with given x and y lengths at given centre

position with x and y coordinates. Optionally specify z start and end coordinates, otherwise entire mesh

thickness used.

Script return values: shape object definition as name dim_x dim_y dim_z cpos_x cpos_y cpos_z rot_theta

rot_phi repeat_x repeat_y repeat_z disp_x disp_y disp_z method

shearstrainequation (meshname) text_equation

Set equation for shear strain components, using a vector text equation (3 equations separated by commas) in

given meshname (focused mesh if not specified). This will disable the elastodynamics solver in all meshes.

shiftcamorigin dX dY

Shift camera origin using dX dY. This is the same as left mouse hold and drag on mesh viewer.

meshrect meshname shift

Shift the dipole mesh by the given amount (will only take effect when called on a dipole mesh). shift is specified

using x y z coordinates.

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shiftglobalfield shift

Shift rectangle of global field (previously loaded with loadovf2field on supermesh).

Script return values: rect (global field rectangle)

showa (meshname)

Show predicted exchange stiffness (J/m) value for current focused mesh (must be atomistic; focused mesh if not

specified), using formula A = J*n/2a, where n is the number of atomic moments per unit cell, and a is the atomic

cell size.

Script return values: A

showdata (meshname) dataname (rectangle)

Show value(s) for dataname. If applicable specify meshname and rectangle (m) in mesh. If not specified and

required, focused mesh is used with entire mesh rectangle.

Script return values: varies

showk (meshname)

Show predicted uniaxial anisotropy (J/m^3) constant value for current focused mesh (must be atomistic; focused

mesh if not specified), using formula K = k*n/a^3, where n is the number of atomic moments per unit cell, and a

is the atomic cell size.

Script return values: A

showlengths (meshname)

Calculate a number of critical lengths for the focused mesh (must be ferromagnetic; focused mesh if not

specified) to inform magnetization cellsize selection. lex = sqrt(2 A / mu0 Ms^2) : exchange length, l_Bloch =

sqrt(A / K1) : Bloch wall width, l_sky = PI D / 4 K1 : Neel skyrmion wall width.

showmcells (meshname)

Show number of discretisation cells for magnetization for focused mesh (must be ferromagnetic; focused mesh

if not specified).

Script return values: n

showms (meshname)

Show predicted saturation magnetization (A/m) value for current focused mesh (must be atomistic; focused

mesh if not specified), using formula Ms = mu_s*n/a^3, where n is the number of atomic moments per unit cell,

and a is the atomic cell size.

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Script return values: Ms

showtc (meshname)

Show predicted Tc value (K) for current focused mesh (must be atomistic; focused mesh if not specified), using

formula Tc = J*e*z/3kB, where e is the spin-wave correction factor, and z is the coordination number.

Script return values: Tc

skyposdmul (meshname) multiplier

Set skyrmion diameter multiplier for given meshname (focused mesh if not specified) which determines skypos

tracking rectangle size. Default is 2.0, i.e. tracking rectangle side is twice the diameter along each axis. Reduce

if skyrmion density is too large, but at the risk of losing skyrmion tracking (recommended to keep above 1.2).

Script return values: multiplier

skyrmion (meshname) core chirality diameter position

Create an idealised Neel-type skyrmion with given diameter and centre position in the x-y plane (2 relative

coordinates needed only) of the given mesh (focused mesh if not specified). Core specifies the skyrmion core

direction: -1 for down, 1 for up. Chirality specifies the radial direction rotation: 1 for towards core, -1 away

from core. For diameter and position use metric units.

skyrmionbloch (meshname) core chirality diameter position

Create an idealised Bloch-type skyrmion with given diameter and centre position in the x-y plane (2 relative

coordinates needed only) of the given mesh (focused mesh if not specified). Core specifies the skyrmion core

direction: -1 for down, 1 for up. Chirality specifies the radial direction rotation: 1 for clockwise, -1 for anti-

clockwise. For diameter and position use metric units.

skyrmionpreparemovingmesh (meshname)

Setup the named mesh (or focused mesh) for moving skyrmion simulations: 1) set movingmesh trigger, 2) set

domain wall structure, 3) set dipoles left and right to remove end magnetic charges, 4) enable strayfield module.

ssolverconfig s_convergence_error (s_iters_timeout)

Set spin-transport solver convergence error and iterations for timeout (if given, else use default).

Script return values: s_convergence_error s_iters_timeout

stages

Shows list of currently set simulation stages and available stage types.

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Script return values: number of set stages

startupscriptserver status

Set startup script server flag.

startupupdatecheck status

Set startup update check flag.

statictransportsolver status

If static transport solver is set, the transport solver is only iterated at the end of a stage or step. You should set a

high iterations timeout if using this mode.

Script return values: status

stochastic

Shows stochasticity settings : stochastic cellsize for each mesh and related settings.

stop

Stop simulation without resetting it.

stopscript

Stop any currently running Python script.

strainequation (meshname) text_equation

Set equation for diagonal strain components, using a vector text equation (3 equations separated by commas) in

given meshname (focused mesh if not specified). This will disable the elastodynamics solver in all meshes.

surfacefix (meshname) rect_or_face

Add a fixed surface for the elastodynamics solver. A plane rectangle may be specified in absolute coordinates,

which should intersect at least one mesh face, which will then be set as fixed. Alternatively an entire mesh face

may be specified using a string literal: -x, x, -y, y, -z, or z. In this case a named mesh will also be required

(focused mesh if not given).

surfacestress (meshname) rect_or_face vector_equation

Add an external stress surface for the elastodynamics solver. A plane rectangle may be specified in absolute

coordinates, which should intersect at least one mesh face, which will then be set as fixed. Alternatively an

entire mesh face may be specified using a string literal: -x, x, -y, y, -z, or z. In this case a named mesh will also

be required (focused mesh if not given). The external stress is specified using a vector equation, with variables

x, y, t. Coordinates x, y are relative to specified rectangle, and t is the time.

319

surfroughenjagged (meshname) depth spacing (seed (sides))

Roughen given mesh (focused mesh if not specified) surfaces using a jagged pattern to given depth (m) and

peak spacing (m). Roughen both sides by default, unless sides is specified as -z or z (string literal). The seed is

used for the pseudo-random number generator, 1 by default.

tamrequation meshname text_equation

Set conductivity equation for TAMR effect (must be a mesh with transport module added) - the text equation

uses parameters TAMR, d1, d2, d3. Here TAMR = tamr / 100 is the ratio, where tamr is the material parameter

set as a percentage value. di = eai.m, i = 1, 2, 3, where eai is the crystalline axis and m is the magnetization

direction. Call with empty equation string to clear any currently set equation and use default conductivity

formula: cond / elC = 1 / (1 + TAMR * (1 - d1*d1)), where elC is the base conductivity (material parameter).

Note, the formula set always multiplies elC to obtain conductivity due to TAMR.

tau (meshname) tau_11 tau_22 (tau_12 tau_21)

Set ratio of exchange parameters to critical temperature (Neel) for antiferromagnetic mesh (all antiferromagnetic

meshes if not specified). tau_11 and tau_22 are the intra-lattice contributions, tau_12 and tau_21 are the inter-

lattice contributions.

Script return values: tau_11 tau_22 tau_12 tau_21

tcellsize (meshname) value

Change cellsize of meshname for thermal conduction (m). The cellsize can be specified as: hx hy hz, or as:

hxyz. If meshname not given use the focused mesh.

Script return values: cellsize - return thermal conduction cellsize of focused mesh, or of meshname if given.

temperature (meshname) value

Set mesh base temperature (all meshes if meshname not given) and reset temperature. Also set ambient

temperature if Heat module added. If the base temperature setting has a spatial dependence specified through

cT, this command will take it into account but only if the Heat module is added. If you want the temperature to

remain fixed you can still have the Heat module enabled but disable the heat equation by setting the heat dT to

zero (setheatdt 0).

Script return values: value - temperature value for focused mesh.

threads number

Set number of threads to use for cuda 0 computations. Value of zero means set maximum available.

Script return values: num_threads

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tmodel (meshname) num_temperatures

Set temperature model (determined by number of temperatures) in given meshname (focused mesh if not

specified). Note insulating meshes only allow a 1-temperature model.

tmrtype (meshname) setting

Set formula used to calculate TMR angle dependence in named mesh (all meshes if not given). setting = 0: RA

= (RAp + (RAap - RAp) * (1 - cos(theta))/2). setting = 1 (Slonczewski form, default): RA = 2*RAp/[(1 +

RAp/RAap) + (1 - RAp/RAap)*cos(theta)]

Script return values: setting

tsolverconfig convergence_error (iters_timeout)

Set transport solver convergence error and iterations for timeout (if given, else use default).

Script return values: convergence_error iters_timeout

updatemdb

Switch to, and update the local materials database from the online shared materials database.

updatescreen

Updates all displayed values on screen and also refreshes.

vecrep (meshname) vecreptype

Set representation type for vectorial quantities in named mesh, or supermesh (focused mesh if not specified).

vecreptype = 0 (full), vecreptype = 1 (x component), vecreptype = 2 (y component), vecreptype = 3 (z

component), vecreptype = 4 (direction only), vecreptype = 5 (magnitude only).

versionupdate action (target_version)

Incremental version update command. action = check : find if incremental update is available for current

program version. action = download : download available incremental updates if any (or up to target version if

specified). action = install : install incremental updates if any (or up to target version if specified), downloading

if needed. action = rollback : roll back to specified target version if possible.

vortex (meshname) longitudinal rotation core (rectangle)

Create a vortex domain wall with settings: longitudinal (-1: tail-to-tail, 1: head-to-head), rotation (-1: clockwise,

1: counter-clockwise), core (-1: down, 1: up). The vortex may be set in the given rectangle (entire mesh if not

given), in the given mesh (focused mesh if not specified).

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References

There are a number of articles which cover various parts of the software.

General

• S. Lepadatu, “Boris computational spintronics — High performance multi-mesh

magnetic and spin transport modeling software”, Journal of Applied Physics 128,

243902 (2020)

Differential equation solvers

• S. Lepadatu “Speeding Up Explicit Numerical Evaluation Methods for

Micromagnetic Simulations Using Demagnetizing Field Polynomial Extrapolation”

IEEE Transactions on Magnetics 58, 1 (2022)

Multilayered convolution

• S. Lepadatu, “Efficient computation of demagnetizing fields for magnetic multilayers

using multilayered convolution” Journal of Applied Physics 126, 103903 (2019)

Parallel Monte Carlo algorithm

• S. Lepadatu, G. Mckenzie, T. Mercer, C.R. MacKinnon, P.R. Bissell, “Computation

of magnetization, exchange stiffness, anisotropy, and susceptibilities in large-scale

systems using GPU-accelerated atomistic parallel Monte Carlo algorithms” Journal of

Magnetism and Magnetic Materials 540, 168460 (2021)

Micromagnetic Monte Carlo algorithm (with demagnetizing field parallelization)

• S. Lepadatu “Micromagnetic Monte Carlo method with variable magnetization length

based on the Landau–Lifshitz–Bloch equation for computation of large-scale

thermodynamic equilibrium states” Journal of Applied Physics 130, 163902 (2021)

322

Roughness effective field

• S. Lepadatu, “Effective field model of roughness in magnetic nano-structures” Journal

of Applied Physics 118, 243908 (2015)

Heat flow solver, LLB and 2-sublattice LLB

• S. Lepadatu, “Interaction of Magnetization and Heat Dynamics for Pulsed Domain

Wall Movement with Joule Heating” Journal of Applied Physics 120, 163908 (2016)

• S. Lepadatu “Emergence of transient domain wall skyrmions after ultrafast

demagnetization” Physical Review B 102, 094402 (2020)

Spin transport solver

• S. Lepadatu, “Unified treatment of spin torques using a coupled magnetisation

dynamics and three-dimensional spin current solver” Scientific Reports 7, 12937

(2017)

• S. Lepadatu, “Effect of inter-layer spin diffusion on skyrmion motion in magnetic

multilayers” Scientific Reports 9, 9592 (2019)

• C.R. MacKinnon, S. Lepadatu, T. Mercer, and P.R. Bissell “Role of an additional

interfacial spin-transfer torque for current-driven skyrmion dynamics in chiral

magnetic layers” Physical Review B 102, 214408 (2020)

• C.R. MacKinnon, K. Zeissler, S. Finizio, J. Raabe, C.H. Marrows, T. Mercer, P.R.

Bissell, and S. Lepadatu, “Collective skyrmion motion under the influence of an

additional interfacial spin-transfer torque” Scientific Reports 12, 10786 (2022)

• S. Lepadatu and A. Dobrynin, “Self-consistent computation of spin torques and

magneto-resistance in tunnel junctions and magnetic read-heads with metallic pinhole

defects” Journal of Physics: Condensed Matter 35, 115801 (2023)

Elastodynamics solver with thermoelastic effect and magnetostriction

• S. Lepadatu, “All-optical magneto-thermo-elastic skyrmion motion”

arXiv:2301.07034v2 (2023)

Boris Computational Spintronics User Manual

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