Computational Design of the Rare-Earth Reduced Permanent
A. Kovacsa, J. Fischbachera, M. Gusenbauera, H. Oezelt a,
H. C. Herperb, O. Yu. Vekilova b, P. Nievesc, S. Arapanc,d, T. Schrefla,*
a Department for Integrated Sensor Systems, Danube University Krems, Austria
b Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden
c ICCRAM, University of Burgos, Burgos, Spain
dIT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic
* Corresponding author. E-mail: firstname.lastname@example.org, Tel: +43 2622 23420 20, Fax: 43 2622 23420 99, Address: Donau-Universität
Krems, Department für Integrierte Sensorsysteme, Viktor Kaplan Str. 2 E, 2700 Wiener Neustadt, Austria
Column of article (Research) Type of article (Article) Subject: Rare earth permanent magnets
ABSTRACT Multiscale simulation is a key research tool for the quest for new permanent magnets. Starting with first
principles methods, a sequence of simulation methods can be applied to calculate the maximum possible coercive field
and expected energy density product of a magnet made from a novel magnetic material composition. Fe-rich magnetic
phases suitable for permanent magnets can be found by adaptive genetic algorithms. The intrinsic properties computed
by ab initio simulations are used as input for micromagnetic simulations of the hysteresis properties of permanent
magnets with realistic structure. Using machine learning techniques, the magnet’s structure can be optimized so that the
upper limits for coercivity and energy density product for a given phase can be estimated. Structure property relations
of synthetic permanent magnets were computed for several candidate hard magnetic phases. The following pairs
(coercive field (T), energy density product (kJ/m³)) were obtained for Fe3Sn0.75Sb0.25: (0.49, 290), L10 FeNi: (1, 400),
CoFe6Ta: (0.87, 425), and MnAl: (0.53, 80).
KEYWORDS rare-earth, permanent magnets, micromagnetics
Permanent magnets are widely used in modern society. Important markets for permanent magnets  are wind power,
hybrid and electric vehicles, electric bikes, air conditioning, acoustic transducer, and hard disk drives. With the growing
demand for permanent magnets in environmental friendly transport and power generation  there is an ongoing quest
to reduce the rare-earth content or use alternative rare-earth efficient or rare-earth free hard magnetic phases. Some of
the considered hard magnetic phases may bridge the gap between ferrites and high performance NdFeB magnets .
In this work we present an overview on how the magnetic properties of a virtual magnet can be predicted starting from
first principles. The materials modelling workflow in this paper is an example for traditional multiscale simulations
with parameter passing. Several physical models are linked together in order to compute the hysteresis properties of
permanent magnets: Genetic algorithms in combination with density functional theory guide the search for stable
uniaxial ferromagnetic phases. This process may be assisted by mining materials databases. Then density functional
theory is applied, in order to compute intrinsic magnetic properties such as spontaneous magnetization, magneto-
crystalline anisotropy energy, and exchange integrals. The results feed into atomistic spin dynamics models for the
computation of the magnetization, the anisotropy constant, and the exchange constant as function of temperature. These
temperature dependent properties are then used as input for micromagnetic simulations. Numerical optimization tools
help to tune the microstructure such that the coercive field or the energy density product is maximized for a given set
of intrinsic magnetic properties.
In addition, we computed the reduction of coercivity owing to thermal fluctuations . Analyzing the results of the
micromagnetic simulations, we can identify how much different effects such as misorientation, demagnetizing fields,
and thermal fluctuations reduce the coercive field with respect to the anisotropy field of the material.
The main focus of this paper is to predict the potential for various rare-earth free or rare-earth reduced permanent
magnetic phases with respect to the expected extrinsic magnetic properties such as coercivity and energy density
product. A sufficiently high coercive field and a sufficiently high energy density product are the key for the application
of a new phase. These properties result from the interplay between the intrinsic magnetic properties of the magnet, the
magnet’s microstructure, and thermal fluctuations. Therefore, the main part of the paper will cover micromagnetic
results for the hysteresis properties, which will be presented for well-known magnetic phases (L10 FeNi,
Nd0.2,Zr0.8Fe10Si2, Sm0.7Zr0.3Fe10Si2) as well as for phases predicted by genetic algorithms and density functional theory
(Fe5Ge, CoFe6Ta). For some of the phases the intrinsic magnetic properties were computed by first principle simulations
(Fe5Ge, CoFe6Ta, Fe3Sn0.75Sb0.25) and atomistic spin dynamics (MnAl).
The results presented in this paper are centered on the micromagnetic computation of the expected performance of
various hard magnetic phases. For details on the adaptive genetic algorithms for the search for new magnetic phases we
refer the reader to recent articles applying the method to Fe3Sn , CoFe2P , and magnetic phases in the L10 structure
. First principle simulations of magnetic properties are review in . An overview of essential micromagnetic
techniques to compute the influence of microstructure on the coercivity and on the energy density product is given by
Fischbacher et al. .
An adaptive genetic algorithm  in combination with the ab initio package VASP  was used to scan the phase
space for Fe rich compounds that are non-cubic and stable. The magnetic properties were calculated with help of the
full-potential linear mufﬁn-tin orbital (FP-LMTO) method implemented in the RSPt code . Synthetic micro-structures
were constructed with the open-source 3D polycrystal generator tool Neper .
A Python script controlling the open-source CAD software Salome  introduces the grain boundary phase with a
specific thickness and produces the finite element mesh. For these synthetic microstructures the demagnetization curve
is computed through minimization of the micromagnetic energy with a preconditioned nonlinear conjugate gradient
method . The search for higher coercive fields, µ0Hc, and energy density products, (BH)max, is managed via the
open-source optimization framework Dakota . Thus, the optimal structure for a given hard-magnetic phase can be
found. For characterizing the magnet, we use the M(Hext)-loop, which gives the magnetization as function of the external
field. The demagnetization curve is then corrected by the demagnetizing field of the sample. A similar procedure is
done in experiments when the hysteresis curves are not measured in a closed circuit. Then we transform the
magnetization to the magnetic induction, B, in order to obtain the B(Hint)-loop and the energy density product. Here Hint
is the internal field.
Finally, we consider the reduction of coercivity by thermal activation. We compute the critical value of the external
field that reduces the energy barrier for nucleation to 25 kBT. The system is assumed to overcome this energy barrier
within a waiting time of one second owing to thermal fluctuations . We use a modified string method  to
compute the energy barriers for different values of the external field. The computation of reduction of the coercive field
through thermal activation gives the limits of the coercive field  of a certain hard magnetic phase.
3.1 Rare-earth free phases
Using an adaptive genetic algorithm  the crystal phase space of Fe-Co-Ta was searched for non-cubic systems with
high stability. For CoFe6Ta we performed the two simulations starting from the scratch with 8 and 16 atoms/cell.
Various non-cubic stable phases could be identified. Some of the most stable non-cubic phases were tetragonal (space
group 115), rhombohedral (space group 160), orthorhombic (space group 38), and orthorhombic (space group 63, where
a and b lattice parameters are very similar), and with an enthalpy of formation of -0.07033 eV/atom, -0.06353 eV/atom,
-0.06025 eV/atom and -0.05929 eV/atom, respectively. Data and calculations details of these theoretical phases can be
found in the Novamag database  , see the following links . The lowest ground state energy was found for a
monoclinic system (space group 8) with enthalpy of formation of -0.07488 eV/atom . These results correspond to a
high-throughput DFT calculations (at zero-temperature) using AGA, where similar default settings were used for all of
them with the Generalized Gradient Approximation (GGA). To analyze in more detail the stability of these phases is
recommended to compute the free energy at finite temperature including electronic, phononic and magnetic terms .
In space groups 63 and 160, CoFe6Ta shows a uniaxial magnetocrystalline anisotropy. The complete theoretical study
of these phases is in progress and it is planned to be reported in the near future, so here we just selected and mentioned
some preliminary results. Using the RSPt code  we calculated the anisotropy constant and the spontaneous
magnetization for CoFe6Ta in space group 63 to be K = 1 MJ/m³ and µ0Ms = 1.82 T.
Figure 1 shows the micromagnetically computed B(Hint) loop for different nanostructures made of CoFe6Ta. The grains
have approximately the same volume of 34 × 34 × 146 nm3, 56 × 56 × 56 nm3, and 72 × 72 × 34 nm3 for the columns,
equiaxed polyhedra, and platelets, respectively. The macroscopic shape of the magnet is cubical with an edge length of
300 nm. The volume fraction of a non-magnetic grain boundary phase is 18 percent. The energy density product is
Figure 1 Magnetic induction as function of the internal field for nanostructured CoFe6Ta. Nanostructuring is essential to obtain a high
coercive field. The coercive field increases with increasing aspect ratio of the grains. The aspect ratios of the columnar, equiaxed, and
platelet shaped grains 4.3, 1, and 0.47, respectively
Fe-rich materials with non-cubic uniaxial crystal structures are promising candidates for rare-earth free permanent
magnets. Because of the hexagonal crystal structure and its high spontaneous magnetization, Fe3Sn compounds were
considered. However, Fe3Sn shows an easy-plane anisotropy  both in simulations and experiment. Substituting Sn
by Sb changes the easy-plane anisotropy to uniaxial anisotropy. The results show a uniaxial anisotropy constant
K = 0.33 MJ/m³ and spontaneous magnetization µ0Ms = 1.52 T for Fe3Sn0.75Sb0.25 . These properties were assigned
to the grains of a synthetically generated structure whereby the average grain size was 50 nm. An exchange stiffness
constant A = 10 pJ/m was used. The grains were separated by a weakly ferromagnetic grain boundary (gb) phase with
a magnetization of µ0Ms,gb = 0.81 T and an exchange stiffness constant Agb = 3.7 pJ/m. The micromagnetic simulation
of the reversal process (see Figure 2) shows that multidomain states remain stable after irreversible switching owing to
domain wall pinning at the grain boundaries. The computed energy density product is coercivity limited. Its maximum
value of 290 kJ/m³ may only be achieved for nanostructured systems with a grain size smaller than 50 nm. Unfortunately,
Fe3Sn0.75Sb0.25 is not stable. Attempts to stabilize the phase by small additions of Mn were successful. However, owing
to the change of the electronic structure and the number of valence electrons the anisotropy flipped back to in-plane
Figure 2 Domain wall – microstructure interaction in a Fe3Sn0.75Sb0.25 magnet. Left hand side: Grain structure. Right hand side: At an
internal field of µ0Hint = 0.49 T the flower like magnetic state (1) breaks into a two-domain state (2) with a domain wall pinned at the grain
boundaries. Images reproduced from .
3.3 Microstructure optimization
For computing the influence of the microstructure on the hysteresis properties we varied the grain size, the grain shape,
the thickness of the grain boundary phase, and the magnetization in the grain boundary phase. The design space was
sampled with the help of the software tool Dakota .
In order to obtain a general trend on how microstructural features influence the coercive field we use dimensionless
units. The coercive field is given in units of the anisotropy field, 2K/(µ0Ms). The grain boundary magnetization is
measured in units of the magnetization of the main hard magnetic phase, Ms,bulk. Grain size and grain boundary thickness
are measured in units of the Bloch parameter 0 = (A/K)1/2, which is the characteristic length in hard magnetic materials.
The results presented in the Figure 3 and Figure 4 were obtained by varying the microstructure for magnets made of the
L10 FeNi (bulk), MnAl, and Nd0.2Zr0.8Fe10Si2 (see Table 1). The granular structure used for the simulations is shown in
the top row of Figure 3. Because we used dimensional units, the influence of grain boundary phase, grain size, and grain
aspect ratio on coercivity for other hard magnetic phases can be derived from the presented data.
Figure 3 Coercive field (left hand side) and the energy density product (right hand side) as function of grain boundary properties.
The design space for analysis of the influence of grain boundary properties on coercivity and energy density product
was spanned by the grain boundary thickness and the magnetization of the grain boundary. We varied the thickness of
the grain boundary from 1.1 δ0 to 4.4 δ0, while keeping the size of the magnet constant. The magnetization of the grain
boundary phase was varied from 0.05 Ms,bulk to 0.55 Ms,bulk. The exchange stiffness constant of the grain boundary phase
is assumed to be proportional to its magnetization squared  according to Agb = Abulk(Ms,gb/Ms,bulk)². Thus, the grain
boundary phase changes from almost non-magnetic to ferromagnetic. The polycrystalline structure used for the
simulations is shown in Figure 3. The average grain size is 37 δ0.
Clearly the maximum coercive field is reached for a thin, almost non-magnetic grain boundary phase. Both, increasing
the grain boundary thickness or increasing the grain boundary magnetization reduces the coercive field. The
magnetization of the grain boundary phase contributes to the total magnetization. Therefore, the maximum energy
density product occurs for thin grain boundaries and a moderately high magnetization in the grain boundary. We can
conclude that excellent hysteresis properties can be achieved even for ferromagnetic grain boundaries, given that its
thickness is sufficiently small. For example, a coercive field of 0.4 × 2K/(µ0Ms) is reached for a grain boundary thickness
of 2 δ0, when the magnetization in the grain boundary phase is about ½ of its bulk value.
The weakly soft magnetic grain boundary phase acts as soft magnetic defect. Detailed micromagnetic studies show that
at such grain boundaries magnetization reversal is initiated . We see that the coercive field decreases with increasing
spontaneous magnetization of the grain boundary phase. Furthermore, the coercive field decreases with increasing
thickness of the grain boundary phase. Though the structure is more complicated for polycrystalline magnets with a
weakly soft magnetic grain boundary phase, the effect is similar to that reported by Richter et al.  who showed a
similar dependence of the nucleation field on the size of a soft defect in a one-dimensional micromagnetic model. The
energy to form the domain wall of the reversed nucleus increases with decreasing thickness of the soft magnetic defect.
In magnets with a thin grain boundary phase the domain wall of the nucleus extends into the main hard magnetic phase
and the domain wall energy increases. Therefore, magnets with a thinner grain boundary phase show a higher coercive
Figure 4 Influence of grain size and grain shape. The contours give the coercive field as function of grain size and aspect ratio. The different
panels refer to different saturation magnetization of the grain boundary phase with a thickness of 0.
We now modified the design space. We kept the grain boundary thickness at δ0 and varied the magnetization in the
grain boundary phase, the size of the grains, and the aspect ratio of the grains. An aspect ratio greater 1 refers to
elongated, needle like grains; an aspect ratio smaller 1 refers to platelet like grains.
The panels of Figure 4 show the coercive field as function of the grain size and the aspect ratio for different
magnetization in the grain boundary phase. For an almost non-magnetic grain boundary phase the coercive field
increases with increasing aspect ratio. This means that magnets with needle like grains show a higher coercive field
than magnets with platelet like grains. This effect diminishes when the magnetization of the grain boundary phase is
increased. For Ms,gb = 0.4 Ms,bulk, there is hardly any change of the coercive field with aspect ratio. For large
magnetization in the grain boundary phase the trend is reversed and platelet-shaped grains show a slightly higher
coercive field than needle like grains. The grain size effect on coercivity is more pronounced in platelet shaped grains.
The results of Figure 3 also show that the highest coercive field can be achieved for an almost non-magnetic grain
boundary (0.05 Ms,bulk). The coercive field is a factor of 4.5 higher than for a grain boundary phase with a spontaneous
magnetization of 0.55 Ms,bulk. Figure 4 shows that the coercivity increases with decreasing grain size. We can conclude
that magnets with small, exchange-decoupled grains show the highest coercive field. Indeed, the highest coercive field
is found for the top left point on the top left subplot for Figure 4 with Ms,gb = 0.02 Ms,bulk: Here we have a nanostructured
systems with exchange isolated grains with a grain size which is smaller than 20 0.
3.4 Coercivity limits
Using numerical micromagnetics we can separate the effects that lead to a reduction of coercivity with respect to the
anisotropy field of the magnet. We compute the demagnetization curve but switch off the magnetostatic field. When the
computed coercive field is less than the anisotropy field the reduction has to be attributed to misalignment of the grains
or secondary soft magnetic phases. In a second step, we switch on magnetostatic interactions and simulate the
demagnetization curve again. The resulting decrease of the coercive field has to be attributed to demagnetizing effects.
Finally, we can simulate how the system escapes from a metastable state over the lowest energy barrier. This gives the
temperature dependent coercive field .
Figure 5 Limits of coercivity. Effects that reduce the coercive field in permanent magnets for different candidate phases. The symbols give
the coercive field. stars: anisotropy field, x: micromagnetics without magnetostatics, +: full micromagnetics, o: micromagnetics with
thermal activation. Please note the different scale for the µ0H axis for (Sm,Zr)Fe10Si2.
In the following analysis we did not assume any soft magnetic secondary phases. The external field was oriented one
degree off the easy axes of a small cube with an edge length of 40 nm. The computed effects that reduce the anisotropy
fields are (1) misorientation, (2) demagnetizing effects, and (3) thermal fluctuations. Here the coercive field was
computed for an ideal structure: The grain size is very small (40 nm) and there are no defects. Thus, the computed
coercive field is an upper limit for coercivity for a given hard magnetic phase.
We applied this procedure to several candidate phases for rare-earth free or rare earth reduced magnets. For each phase
we show the anisotropy field, the reduction owing to misorientation, the reduction by demagnetizing effects, and the
reduction by thermal fluctuations (see Figure 5). The intrinsic magnetic properties used for the simulations are listed in
Table 1. The anisotropy constant, the spontaneous magnetization, and the exchange constant for MnAl were obtained
from atomistic spin dynamics at T = 300 K. Fe5Ge is an Fe-rich binary phase predicted by an adaptive genetic algorithm.
The anisotropy constant and the spontaneous magnetization for Fe5Ge, Fe3Sn0.75Sb0.25, and CoFe6Ta were obtained from
first principle simulations at T = 0. The exchange constant for Fe5Ge and CoFe6Ta was taken to be proportional to the
spontaneous magnetization squared (A = cMs²) whereby c was taken from Ms and A of -Fe. The intrinsic material
parameters for L10 FeNi, Nd0.2Zr0.8Fe10Si2, and Sm0.7Zr0.3Fe10Si2 are experimental data for T = 300 K were taken from
literature. If no other source for the value of the exchange constant was available we used A = 10 pJ/m .
The results clearly show that we cannot expect a coercive field greater than 1 T in most rare earth free magnets. For
FeNi (bulk) a high degree of uniform chemical order was assumed. Experimentally synthesized L10 FeNi particles may
contain patches where the chemical order is reduced locally. The corresponding local reduction of magnetocrystalline
anistropy will deteriorate coercivity. Similarly, crystal defects such as twins or antiphase boundaries reduce the coercive
field in MnAl magnets . Rare-earth magnets in the ThMn12 structure with Zr substitution have a low rare-earth
content. Moreover, the magnetocrystalline anisotropy – especially that of the (Sm,Zr)Fe10Si2 magnet – is sufficiently
high to support a reasonable coercive field. For Nd0.2Zr0.8Fe10Si2, and Sm0.7Zr0.3Fe10Si2 the coercive field computed
with thermal activation (dots in Figure 5) is 70 percent of the anisotropy field.
Table 1 Anisotropy constant K, spontaneous magnetization Ms, and exchange constant A used for the simulations presented in Figure 5.
L10 FeNi (Si substrate)
L10 FeNi (bulk)
We showed how to exploit materials simulations for the computational design of the next generation rare-earth reduced
permanent magnets. Based on the results presented above we can draw the following conclusions.
• Nanostructuring is essential to achieve a high coercive field in rare-earth free compounds with moderate magneto-
• Coercivity decreases with increasing magnetization in the grain boundary phase and with increasing thickness of the
grain boundary phase.
• However, excellent permanent magnetic properties can be achieved even for moderately ferromagnetic grain
boundary phases provided that the grain boundary is thin enough.
• The shape of the grains is only important for nearly non-magnetic grain boundaries. For systems in which
ferromagnetic Fe containing grain boundaries are expected, the grain shape plays a minor role.
• Thermal fluctuations may considerably reduce the coercive field. Thus, even in perfect structures the coercive field
is well below the anisotropy field.
This work was supported by the EU H2020 project NOVAMAG (Grant no 686056), the Austrian Science Fund FWF
(I3288-N36), and by the European Regional Development Fund in the IT4Innovations national supercomputing center
- path to exascale project, project number CZ.02.1.01/0.0/0.0/16_013/0001791 within the Operational Programme
Research, Development and Education.
Compliance with ethics guidelines
All authors declare that they have no conflict of interest or financial conflicts to disclose.
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