arXiv:1112.0911v1 [cond-mat.mes-hall] 5 Dec 2011
Vortex core magnetization dynamics induced by thermal excitation
Tiago S. Machado1, Tatiana G. Rappoport2, and Luiz C. Sampaio1
1Centro Brasileiro de Pesquisas F´ ısicas, Xavier Sigaud,
150, Rio de Janeiro, RJ, 22.290-180, Brazil
2Instituto de F´ ısica, Universidade Federal Fluminense, Rio de Janeiro, RJ, 24.210-346, Brazil
(Dated: December 6, 2011)
We investigate the effect of temperature on the dynamic properties of magnetic vortices in small
disks. Our calculations use a stochastic version of the Landau-Lifshitz-Gilbert (LLG) equation, valid
for finite temperatures well below the Curie critical temperature. We show that a finite temperature
induces a vortex precession around the center of the disk, even in the absence of other excitation
sources. We discuss the origin and implications of the appearance of this new dynamics. We also
show that a temperature gradient plays a role similar to that of a small constant magnetic field.
The control and manipulation of the magnetization in
magnetic materials is one of the most interesting chal-
lenges in the field. In the last 10 years it has been shown
that spin polarized current [1, 2], light , and electric
field  can be used to modify a magnetic state, pro-
viding new opportunities for technological applications.
More recently, it has also been observed that heat can act
as an excitation source, producing a change in the mag-
netization. When different temperatures are applied at
opposite ends of a magnetic material, leading to a tem-
perature gradient, a pure spin current is generated, an
effect known as the spin Seebeck effect (SSE) . Numer-
ical simulations also indicate that a temperature gradient
can move domain walls in nanowires .
Depending on their length and thickness, microsized
disks made by magnetic materials like Permalloy (Py)
can exhibit a vortex in its center [7, 8]. Under excitation
of an in-plane magnetic field or a spin polarized current
in the form of short pulses or an a.c. resonant excitation,
the vortex core moves around the center of the disk, and
depending on the excitation intensity, the core magneti-
zation can be reversed.
In this paper we investigate the effect of the temper-
ature on the vortex core magnetization dynamics in Py
disks. We show that even in the absence of a temperature
gradient, heat can induce dynamics in the system. The
vortex core rotates around the center of the disk and the
amplitude of the trajectory depends on the temperature.
We discuss the consequences of this effect on the hys-
teresis curve and also address the effect of a temperature
gradient in the dynamical process.
In order to simulate the vortex core magnetization dy-
namics we used the Landau-Lifshitz-Gilbert (LLG) equa-
tion. The effect of the temperature is introduced in the
calculations by including a stochastic term in the total
field . The new stochastic term mimics random fluc-
tuations induced by the interaction of the nanomagnet
with a thermal bath.
The LLG equation is then given by
dτm(t) = −γ0m(t)×h−αm(t)×[m(t)×h], (1)
where α is the Gilbert damping constant, γ0 is the gy-
temperatures T1 and T2 where T2 = T1+∆T. (b) mx as a
function of an in-plane magnetic field hx for disks at different
temperatures T = T1 = T2. (c) Nucleation hn and annihila-
tion han fields as a function of temperature.
(a) Permalloy disk with electrical contact pads at
romagnetic ratio, m is the normalized magnetization of
a cell, and h=heff+νhr, heff being the effective field
and νhrthe noise field. heff contains the exchange and
dipole-dipole interactions, and the applied magnetic field.
The noise term has the form of a random mag-
netic torque −νm×hr , where hr is the random
vector whose components are independent, and ν2=
sV is a parameter which measures the in-
tensity of thermal noise; kB is the Boltzmann constant,
T is the temperature, µ0 is the magnetic permeability,
Msis the magnetization saturation and V is the volume
of the cell. It is important to note that this approach is
valid for temperatures well below the critical temperature
of the magnet. In this regime the exchange interaction
preserves the magnetization uniformity inside each cell
despite of the thermal perturbation. As a consequence,
the random thermal torque conserves the magnetization
magnitude in the stochastic dynamics.
Figure 1(b) illustrates our setup. We consider a Py
disk exhibiting a vortex and we connect two thermal con-
tacts at opposite sides of the disk. The thermal contacts
at different temperatures: (a) ∆T = 0 and (c) ∆T ?= 0.
Snapshot of the magnetic configuration of the disk where the
arrows represent the in-plane magnetization and the colors
represent out-of-plane magnetization. The black line repre-
sents the core trajectory: (b) ∆T = 0 and (d) ∆T ?= 0.
Time evolution of the mx component of the disk
can in principle be at two different temperatures T1and
T2. In order to calculate the temperature distribution
inside the disk, we used a relaxation method to solve the
Laplace equation, as was done previously to obtain the
voltage drop for the same geometry .
The disk has a diameter of 300 nm and thickness of
20 nm, and was discretized in cells of 5 × 5 × 5 nm3.
The parameters associated with the LLG equation are
the saturation magnetization Ms= 8.6 × 105A/m, the
exchange coupling A = 1.3 × 10−11J/m and the Gilbert
damping constant α = 0.01. Given the temperature of
the thermal contacts, we solve the LLG equation numeri-
cally using the fourth-order Runge-Kutta approximation
with a code we have written for this purpose.
Let us initially consider the disk with T = 0 in both
contacts, which corresponds to the usual LLG equation.
As it is expected, we find a magnetic vortex structure
with the vortex core at the center of the disk, as shown
in Figure 1(b). We now consider the two contacts at same
temperature T ?= 0 and calculate the hysteresis curve for
an in-plane external magnetic field (see Fig. 1(a)). When
applying a magnetic field, the vortex core moves towards
the disk edge. Its expulsion occurs at a critical anni-
hilation field han, leading the disk to a mono-domain
state. For decreasing fields, the vortex nucleates again
at a lower nucleation field hn where a sharp transition
is observed from the uniform state to the vortex state.
One can observe that both hanand hnare temperature
dependent. However, while handecreases monotonically
with temperature, hnincreases with T ( Fig. 1 c). This
result is consistent with experiments  for intermedi-
ate T, which corroborate the way our model deals with
An interesting picture emerges when we calculate the
time evolution of the vortex core magnetization at a finite
temperature (T ?= 0). Surprisingly, the thermal fluctua-
tions play a role similar to an external excitation such as
magnetic field or spin polarized current. In Fig. 2(b), we
can observe that the thermal noise displaces the vortex
core from the center of the disk. The vortex precesses
around it, producing an orbital trajectory. The preces-
sion frequency is the natural oscillatory frequency, here
approximately 200 MHz. Figure 2(a) shows the time vari-
ation of the mxcomponent of the magnetization at 10,
100 and 300 K. One sees that mxhas a sinusoidal depen-
dence with time. Both amplitude and variance increase
with temperature while the oscillating frequency remains
constant. The system has a transient in which the am-
plitude increases with time, decreasing again later.
We can use a simplified model to understand the role
of the thermal fluctuations in the dynamics of the vor-
tex core. Let us consider Thiele’s formulation for the
dynamics of a vortex core : G × ˙ r + Fd+ Fu= 0
where r is the vortex’s center and G = −Gˆ z is the
gyroforce determined by the vortex non-uniform mag-
netization distribution . Fd= −Dv is the dissipa-
tion force and Fu= −?∇U(r) where U(r) is the mag-
netic potential that includes the exchange, anisotropy
and magneto elastic energy. In the linear approximation,
U = 1/2kr2. In the presence of a random field νhr,
which mimics thermal fluctuations, the equation has an
extra force Fr = −µ(ˆ z × νh), where µ = 2/3πRhMsc,
R is the disk radius, h is the disk thickness and c is
the vortex quirality . It is possible to decouple the
ˆ x and ˆ y components of the coupled first order differen-
tial equations  and the resulting equations for the
x and y components are equivalent to an underdamped
harmonic oscillator in the presence of a random force:
¨ x+2γ ˙ x+ω2
0x = F(t). This stochastic equation was first
solved by Chandrasekhar  for different initial condi-
tions. In our case, the initial conditions are such that
r(t = 0) = v(t = 0) = 0. These initial conditions are
also a solution of Thiele’s differential equation in the ab-
sence of thermal fluctuations where both kinetic and po-
tential energies are equal to zero. However, when these
fluctuations are included, during a transient the system
gains an extra energy kBT which reflects in an increase
of both potential and kinetic energies. As a result, the
vortex moves away from the disk center and acquires a
velocity. The vortex then begins to precess around the
disk center with its natural oscillatory frequency, with the
initial precession radius proportional to the temperature
(and hence, to the amplitude of the fluctuations). After
the transient, the vortex has the behavior observed in
Chandrasekhar’s solution. ?r? follows an underdamped
circular motion and the variance of the velocity increases
with time, approaching a constant value proportional to
For the LLG dynamics, even though the random term
acts differently on each individual cell, we observe a be-
havior very similar to the one predicted by Thiele’s equa-
tion. However, for high temperatures we see a small shift
of the frequency of the circular motion to values that are
lower than the natural frequency of the system. A similar
shift was observed in recent experiments . The dis-
crepancy between the LLG dynamics and Thiele’s equa-
tion for high temperatures is expected, since the mag-
netic structure of the disk is modified in this temperature
In our simulations the oscillating radius is about 2-
3 nm at room temperature. However, larger values of
orbital radius could be measured by heating the sam-
ple above room temperature. For instance, it is possi-
ble to use techniques based on x-ray magnetic circular
dichroism in time-resolved x-ray microscopy experiments
to observe the trajectory. Recently, St¨ ohr et al. have
presented space- and time-resolved images of the mag-
netic vortex resonant movement in a spin-valve nanopil-
lar . Driven by a dc spin current, the spin-torque
effect gives the vortex a resonant movement with a ra-
dius of ∼ 10 nm. They also measured the vortex position
without spin current. This was done via independent
measurements, which should show the static position of
the vortex. However, the position is not static, following
a trajectory with a radius of ∼ 2-3 nm. We suggest that
this could be related to the the thermal effect we observe
in our simulations.
Returning to the hysteresis, the decrease of han with
temperature can now be better understood. It occurs
because in addition to the applied magnetic field that
moves the vortex core from the center of the disk along
the y axis, the thermal excitation produces an extra shift
of the vortex core. Since the radius increases with T, a
smaller field is then needed to allow the vortex core to
reach the disk edge leading to the vortex annihilation.
It is known that a thermal gradient can be mapped
onto an additional torque in the LLG equation . So,
the effect of the thermal gradient should be similar to
that of an external field. On the other hand, here we
have seen that the temperature itself also modifies the
dynamics of the vortex core. To investigate the influence
of the thermal gradient on the dynamics, we keep one side
of the magnet at T = T1 with the opposite side at T2.
For ∆T varying from 25 to 75 K we found that the vortex
core also oscillates at the natural frequency (200 MHz)
and with increasing amplitudes (see Fig 2d). However
mxis shifted to positive values, meaning the orbit center
is displaced from the center of the disk to the positive y-
axis, which is perpendicular to the heat flow (see Fig 2c).
These data show that ∆T plays the same role of a small
magnetic field applied in the x direction, hx.
To further investigate the similarities between the tem-
perature gradient and a constant magnetic field, we com-
pared it with a system in the presence of an external
magnetic field at a constant temperature T ?= 0. We cal-
culated the vortex core speed considering the presence of
hxor ∆T independently, as can be seen in Fig 3(a) and
(b) respectively. Both dynamics are very similar. For a
fixed temperature T2, the external field displaces the vor-
tex core, increasing the potential energy and decreasing
the kinetic energy of the core. This results in a decrease
of the vortex speed (Fig 3(a)). Similarly, for a fixed T2
FIG. 3: Vortex core speed in function of temperature with
an applied magnetic field hxand ∆T = 0 (a), and without
magnetic field and ∆T ?= 0 (b).
a decrease in T1by ∆T also results in a decrease of the
kinetic energy and speed (Fig 3(b))
In conclusion, we show that for small disks a constant
temperature can drive the system away from its equilib-
rium and produce a precession of the vortex core around
the center of the disk. As a result, temperature can be
used, in conjunction with other external excitations, to
manipulate the magnetic state in specific geometries.
We would like to thank M. A. de Menezes for use-
ful discussions. This work was supported by CNPq and
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