Inertia-Free Thermally Driven Domain-Wall Motion in Antiferromagnets
Severin Selzer, Unai Atxitia, Ulrike Ritzmann, Denise Hinzke, and Ulrich Nowak
Department of Physics, University of Konstanz, D 78457 Konstanz, Germany
Domain wall motion in antiferromagnets triggered by thermally induced magnonic spin currents is
studied theoretically. It is shown by numeric al calculations based on a classical spin model that the wall
moves towards the hotter regions, as in ferromagnets. However, for larger driving forces the so called
Walker breakdown which usually speeds down the wall is missing. This is due to the fact that the wall
is not tilted during its motion. For the same reason antiferromagnetic walls have no inertia and, hence,
no acceleration phase leading to higher effective mobility.
The interest in antiferromagnetic and ferrimagnetic
materials has increased recently for several reasons. One
is the more complex spin structures which lead to addi-
tional spin wave modes with higher frequencies and,
consequently, faster spin dynamics than in ferromagnets
(FMs). Possible applications are in the field of ultrafast spin
dynamics [1,2]. Also, ferrimagnets and antiferromagnets
(AFMs) have attracted a lot of attention as low-damping,
insulating magnets in the emerging field of spincalori-
tronics [3–5] which is on the combined transport of spin
and heat. Finally, antiferromagnets are also discussed as
future material for antiferromagnetic spintronics, since it
has been shown that despite their lack of a macroscopic
magnetization their magnetic state can be controlled via
spin torque switching and can be read out via their
magnetoresistive properties [6]. Spintronic phenomena call
for exploitation in devices with magnetic storage function-
alities, where a magnetic nanostructure has to be controlled
efficiently and fast. The information can be stored in
magnetic domains, in isolated magnetic nanoparticles, or
even in domain walls (DWs) [7]. For the latter case
synthetic AFMs have been shown to pave a new road
towards higher DW mobility [8].
For a ferromagnetic system, in Ref. [9] the existence of
thermally driven domain-wall motion in temperature gra-
dients was demonstrated by computer simulations based on
different approaches, an atomistic spin model as well as a
micromagnetic model based on the Landau-Lifshitz-Bloch
(LLB) equation of motion. A thermodynamic explanation
for this kind of DW motion rests on the minimization of the
free energy of the DW (or the maximization of entropy).
For a DW at finite temperature, the free energy is
ΔFðTÞ¼ΔU − TΔS, where ΔU is the internal energy
and ΔS the entropy of the DW. It is a monotonically
decreasing function of temperature [9–11]. This rather
general argument explains a DW motion towards the hotter
parts of the sample where the free energy is lower [11–13]
and it can be expected to hold for other magnetic textures as
well. Furthermore, it has been shown by Schlickeiser et al.
that the DW motion is caused by a so-called entropic
torque. The exchange stiffness is weaker for higher temper-
ature and therefore, an effective torque on the DW is created
driving it towards the hotter region [11].
A more microscopic explanation for DW motion in
temperature gradients rests on the continuous stream of
thermally excited magnons from the hotter towards the
colder region with a transfer of angular momentum pushing
the wall in the direction opposite to the magnonic spin
current [14]. Theoretical investigations based on these
arguments show that the magnonic torque should be
analogous to the macroscopic entropy torque [15].
Like ferromagnets, antiferromagnets are materials with
magnetic long-range order, but while neighboring atomic
magnetic moments are aligned parallel in a FM, in an AFM
they are antiparallel and compensate each other. Thus,
there is no net magnetization. The antiferromagnetic order
parameter, which is referred to as staggered magnetization,
is hence defined as 2n ¼ m
1
− m
2
, where m
1;2
are the
magnetizations of the sublattices.
Although an AFM differs from a FM on a microscopic
level, their thermodynamic equilibrium properties, as, e.g.,
the free energy, are the same. This is due to the fact that
these properties result solely from the Hamiltonian of the
system. For the simple case of a two-sublattice AFM with
only nearest-neighbor interaction the identical transforma-
tion to a model with reversed sign of the exchange constant
J and of all spins in one sublattice leads directly to a
ferromagnetic model. Accordingly, the temperature
dependence of the exchange stiffness should be identical,
which suggests that antiferromagnetic DWs might behave
as ferromagnetic ones. Note, however, that the argument
above is solely classical and quantum mechanical correc-
tions may lead to additional effects where equilibrium
properties of FMs and AFMs deviate.
On the other hand, the dispersion relations of the spin
waves in FM and AFM are clearly different. In particular, in
an AFM there exist circularly and linearly polarized spin
waves [16,17]. Based on a microscopic spin wave model, it
107201-1
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Erschienen in: Physical Review Letters ; 117 (2016), 10. - 107201
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was argued recently that the interaction of spin waves with
DWs in AFMs—and consequently their dynamics—should
be distinct from the DW dynamics in FMs. The mechanism
of DW motion driven by antiferromagnetic spin waves
should at the microscopic level vary with the excited spin
wave mode: while circularly polarized spin waves carrying
angular momentum cause the DW to precess and get
reflected pushing the DW away from the source, linearly
polarized spin waves pass through the wall without
reflection dragging the DW towards the source [17,18].
Following these arguments a rather complex behavior of
AFM DWs could be expected.
In this Letter, we study the thermally driven DW motion
in AFMs using an atomistic approach by solving the
stochastic Landau-Lifshitz-Gilbert equation of motion for
an atomistic spin model. Comparing our results with
the DW dynamics in FMs we find similarities as well as
clear differences. For small temperature gradients the wall
velocity in AFMs is identical to the one in FMs while for
larger driving forces the so-called Walker breakdown—
which usually speeds down the DW—is missing.
Furthermore, the DW is not tilted during its motion so
that it has no inertia and, hence, no acceleration phase
leading to higher effective mobility.
For our study we first simulate the dynamics of a DW in
a thermal gradient based on a model of classical spins in
the form of normalized magnetic moments S
i
¼ μ
i
=μ
s
on a
simple cubic lattice. We consider isotropic Heisenberg
exchange interactions for nearest neighbors and a biaxial
anisotropy. The Hamiltonian reads
H ¼ −J
X
hi;ji
S
i
S
j
− d
x
X
i
ðS
x
i
Þ
2
− d
z
X
i
ðS
z
i
Þ
2
; ð1Þ
with J<0 for antiferromagnetic order. Here d
z
>d
x
> 0
are anisotropy constants defining an easy axis in the
z direction and an intermediate axis in the x direction.
The dynamics of the normalized magnetic moments
at finite temperature are given by the stochastic Landau-
Lifshitz-Gilbert equation,
∂S
i
∂t
¼ −
γ
μ
s
ð1 þ α
2
Þ
S
i
× ½H
i
þ αðS
i
× H
i
Þ; ð2Þ
where γ is the gyromagnetic ratio and H
i
¼ −∂H=
∂S
i
þ ζ
i
ðtÞ. The first term represents a precession of the
magnetic moments around an effective field H
i
, while the
second includes phenomenological relaxation with damp-
ing constant α [19,20]. The temperature in this model is
included within the framework of Langevin dynamics
by an additional white noise term ζ
i
ðtÞ in the effective
field [21,22].
The simulations are based on the numerical integration of
the stochastic Landau-Lifshitz-Gilbert equation using Heun’s
method [22] with time step Δt ¼ 1.76 × 10
4
μ
s
=ðγJÞ.
We simulate an elongated three-dimensional model with
16 × 16 × 256 spins featuring open boundary conditions and
an easy axis with d
z
¼ 0.1 J while d
x
is varied. After
initialization and relaxation of the DW, a linear thermal
gradient is applied to the system, where the temperature
increases along the z direction from T ¼ 0 to
T ¼ 0.57 T
c
=a,whereT
c
is the critical temperature and
a the lattice parameter. The damping constant is set to
α ¼ 0.01 and absorbing boundary conditions are introduced
at the cold end to avoid spin wave reflection.
We start with a comparison of thermally driven DW
motion in ferromagnetic and antiferromagnetic systems
(Fig. 1). The first result we obtain is that in both systems the
DW moves towards the hotter region, as expected from the
thermodynamic considerations. But there is a significant
difference in the behavior of the DW during its motion.
The dynamics of the ferromagnetic DW profile reveals an
acceleration phase during which the DW is forced out of
its easy x-z plane. This tilting depends strongly on the
anisotropy d
x
, the damping parameter α, and the strength of
the thermal gradient. After that acceleration the DW moves
with constant velocity and constant tilting angle. For larger
gradients eventually the wall starts to rotate while it is
moving and the velocity decreases (see Fig. 4). This is the
so-called Walker breakdown, which was observed for
field-, current- and also thermally driven DW motion in
FMs [9,23–25].
In case of the AFM there is no acceleration phase
and no tilting of the plane containing the DW. Instead,
the wall remains in its initial plane and moves with
(a)
(b)
(c)
FIG. 1. Motion of the DW in a thermal gradient. Linear
temperature gradient ΔT=Δz ¼ 0. 56 × 10
−3
T
c
=a (a), and pro
file of the DW at different times in a ferromagnetic (b), and
antiferromagnetic (c) system with d
x
¼ 0.02 J.
107201-2
120
~
..!-
80
~
~
<I
40
0 .
0
...
§
,
,.
•
!
.
I
I
I
!
2 4
d.,
= 0.00 J -
----
0.
01
J-
----
0.04 J -
---
-
6 8
timet (10
3
~)
10
FIG.
2.
Displacement of the DW versus time. Comparison
between ferromagnetic (solid) and antiferromagnetic (dash
ed
)
systems with different intermediate anisotropies
dx
in a thermal
gradient
of
b..T
I
b,_
z = 1.67 X w-
3
Tel a.
constant
ve
locity.
Note
that
the
movement
of
the
DW
in
Fig. 1 shows sli
ght
d
eviat
ions from a
pur
e
balli
stic motion
with
consta
nt
ve
lo
city
due
to an additional diffusive
com
pon
ent.
Th
e tilting
(or
the
la
ck
of
it
) is crucial for the dynamics
of
the
DW
as
can
be
seen in Fig. 2. F
or
th
e ferromagnetic
system
th
ree cases
ca
n
be
distinguished.
In
the
case
of
high
anisotropy dx = 0.
04
J, the motion
of
the
DW
after the
acceleration
pha
se
is
linear and the vel
oc
ity is maximal.
In
the
case
of
a
sm
all
er
anisotropy
of
dx = 0.01 J, the
DW
motion is accompanied
by
oscillations that are typi
ca
l for
the Walk
er
breakdown, where the
DW
starts to
pr
ecess and
the mean velocity decreases. For
dx = 0 there is no favored
orientation
of
th
e wall and
th
e
DW
rotates
co
ntinuously and
moves
constan
tl
y w
ith
reduced velocity.
Th
e
DW
motion in
the
AFM,
ho
weve
r, is independ
ent
of
the anisotropy.
Th
ere
is
no
tilting
of
the
DW
and therefore
no
precession
of
the
DW
and
no
Walk
er
breakdown.
Th
e
DW
mobility is much
hi
gh
er
in
AFM
s for
lar
ge
driving forces above
th
e Walk
er
breakdown.
Th
e absence
of
a Walk
er
breakdown resu
lt
s from
the
symmetry
of
th
e
to
rques acting on the DW.
In
a temperatme
gradi
en
t
the
exc
h
ange
st
iffn
ess
(t
he
th
ermodynamic
aver
-
age
of
th
e microscopic
exc
han
ge
inte
ra
ct
ion) is s
pa
ce
dependent and decreases w
ith
increasing
tem
peram
re,
vanishing at the Curie point. Considering the
ce
ntral plane
of
the
DW
the s
pa
ce
dependent
exchange
fie
ld
results
in t
wo
torques foUowing from the equation
of
motion
rEq. (2) l, one
cont
ributi
on
(the
d
oub
le
cross
product) which
dri
ves
th
e
wa
ll and
one
contribution (the cross product)
w
hi
c
h-in
a
FM-
wou
ld
tilt the
wa
lJ
(see Fig. 3).
In
an
AFM
the first term is antisymmetric, pointing in an
opposite dir
ect
i
on
on
the
two
different
sub
lattices,
as
it
is necessary
to
drive the DW.
Th
e second
to
rque, however,
is
symme
tric pointing into the
same
dir
ect
ion on the two
sub
lani
ces. In a
FM
it
wou
ld
tilt the wall but in an
AFM
these two
co
ntributions cancel, which explains the fact th
at
FIG. 3. Sketch of
the
torques acting on
the
central plane
of
a
DW in a thermal gradient The
two sublattices
of
th
e AFM are
occupied with blue and yellow s
pin
s,
respectively.
the
DW
slid
es
easily with
out
leaving the easy plane. The
DW
in
an
AFM
is displaced
but
not tilted
in
a t
em
perature
gradient. Consequently, it
ha
s
no
in
ertia and
no
Walk
er
breakdown occurs.
As
Fi
g. 4 shows, the
DW
motion in an
AFM
is not
affected
by
an
intermediate anisotropy d
x,
si
nce
the
wal
l
always stays in .its easy plane.
Th
e
ve
l
oc
ity
of
th
e
wa
ll
is
then
always
proportional to
th
e temperature
gradient
Th
ese
findings
are
in clear contrast to the
movement
of
the
DW
in
the
FM
, w
he
re
th
e vel
oc
it
y
co
llapses at a certain point,
th
e
Walk
er
breakdown.
Th
e position
of
the breakdown could
X
0.
12
FM
X
0.09
J 0.06
(b) 0.
15
.----.------r---,.-
.,..--
--,
,/
0.
12
0.09
J 0.06
,
,,(
~
-
/
,/
, ,.)( X
~
/
/
,,,'
,,
/ ' X
,'
X X
X
X
0,03 ,
x'
~
0
0
0
/'
·
--
-- -- -- .
.9
,
___
_
"_
_"
'
-
~-
-
--~
- -
~
-
-
--~-
-
0 0.6 1.2 1.8 2,4
temperature gradient
~~
( w-
3
~)
FIG. 4. DW velocity
as
a function of temperature gradient in
an
antiferromagnetic (red) and a ferromagnetic (blue) system with
(a)
dx
= 0.02 J and (b)
dx
=
O.Ql
J.
Comparison between
numerical results (poin
ts
) and analytical estimates Oines).
107201-3
not be determined exactly, since close to the breakdown the
precession of the DW is very slow and the wall can reach
the end of the system before rotating once. Hence, the
velocity seems to level off before collapsing, which is a
numerical effect.
Since below the Walker threshold the wall velocities in
FMs and AFMs agree a comparison with earlier results for
DW motion in FMs appears useful. Schlickeiser et al. [11]
derived an analytical estimation of the DW velocity and
Walker threshold for DW motion in a temperature gradient
within the framework of the micromagnetic Landau-
Lifshitz-Bloch (LLB) equation in the one-dimensional
limit. For the magnitude of the DW velocity they obtained
v
DW
¼
2γ
M
s
1
α
⊥
∂A
∂z
1 þ
α
2
⊥
m
2
e
−
γδ
w
m
e
ð1 − kÞ
2
~
χ
⊥
∂A=∂z
~
A
w
2
− 1
s
: ð3Þ
Here, the first line describes the linear increase of v
DW
while the second line, related to a rotation of the wall,
contributes only above the Walker threshold
~
A
w
, which is
given by
~
A
w
¼
m
e
M
s
α
⊥
δ
w
ð1 − kÞ
4
~
χ
⊥
: ð4Þ
M
s
is the saturation magnetization (at zero temperature),
while the micromagnetic exchange stiffness AðTÞ, the
transverse susceptibility
~
χ
⊥
ðTÞ, the transverse damping
parameter α
⊥
ðTÞ, and the reduced equilibrium magnetiza-
tion m
e
ðTÞ are temperature dependent, thermodynamic
equilibrium quantities. The factor k ¼ 1 − d
x
=d
z
breaks
the symmetry in the x-y plane.
Regarding an AFM we can neglect the second term;
hence, we obtain
v
LLB
DW
¼
2γa
3
μ
s
α
∂T
∂z
∂A
∂T
; ð5Þ
with α
⊥
¼ α, M
s
¼ μ
s
=a
3
, and atomic distance a. The
temperature dependence of the transverse damping
constant within the LLB approach is weak and neglected
here as well as the term quadratic in α
⊥
. To include the
temperature dependence of the micromagnetic exchange
stiffness a simple, linear estimate is AðTÞ¼Að0Þ=T
c
,
where it is Að0Þ¼J=2a in our spin model. This approxi-
mation was used in [11]. In the following, we use a better
low temperature approximation which was calculated with
the classical spectral density method [26,27].
There, it was found that the micromagnetic exchange
AðTÞ scales with the equilibrium magnetization m
e
ðTÞ,
Aðm
e
Þ¼Að0Þm
2
e
Dðm
e
Þ, where m
2
e
dependence comes out
from the free magnon gas approximation (equivalent to the
mean field approximation), and Dðm
e
Þ stands for a small
correction coming from the magnon-magnon interactions.
At low to intermediate temperatures, one can use the well-
known low temperature relation for the equilibrium temper-
ature, m
e
¼ 1 − Ω, where Ω ¼ Wk
B
T=J
0
is the thermally
averaged spin wave occupation defined by the Watson
geometric sum, NW ¼
P
k
1=ð1 − γ
k
Þ, and the exchange
interaction, J
0
¼ zJ, z being the number of nearest neigh-
bors and N the number of spins in the lattice, whereas, the
structure factor reads zγ
k
¼
P
r
ij
e
ikr
ij
, where r
ij
is the
connecting to the nearest neighbor vector. As for the small
correction, Dðm
e
Þ¼1 − Gk
B
T=J
0
, where G is the geomet-
rical sum, NG¼
P
k
γ
k
=ð1 − γ
k
Þ. Hence, the explicit
temperature dependence of m
e
and Dðm
e
Þ can directly
be used to calculate ∂A=∂T, which ultimately determines
the DW velocity [Eq. (5)]. We note here that both
geometrical integrals, W and G, depend on both, the
structure of the exchange interactions and the system size
through γ
k
[28]. In the present work we use W ¼ 1.65 and
G ¼ 0.5, related to the finite size of the atomistic system
utilized in our computer simulations. Excellent agreement
between our analytical estimations of the v
AFM
DW
and
computer simulations is found for a range of thermal
gradients, which allow us to conclude that in AFMs,
as for FM, the so-called entropic torques drive the DW
motion. Note, that for very high driving forces the DW
velocity cannot further increase, but more likely saturates,
as it should be limited by the maximum group velocity of
the excited magnons [18,29]. This limit cannot be further
explored due the finite system size.
Our discussion above rests on Eq. (3), which was derived
for a FM. Though our numerical results suggest agreement
with this equation a strict derivation of the DW velocity
of AFMs in temperature gradients should rest on a
two-sublattice version of the LLB equation [30], an effort
which is beyond the scope of this work. Furthermore, our
discussion above is purely classical. Taking into account
quantum effects the entropic torques in ferro- and anti-
ferromagnets are not necessarily identical. Nevertheless,
their symmetry will be the same and thus our main result,
the inertia-free behavior of the antiferromagnetic DW, will
still hold true. The effects of monochromatic spin waves
[17,18] which can indeed move the AFM DW wall in either
direction vanish in a thermally excited spin wave spectrum.
While circularly polarized spin waves cause the DW to
precess and are reflected, for linearly polarized spin waves
which can be seen as a combination of right- and left-
circularly polarized spin waves there is no net rotation of
the wall. In a temperature gradient right- and left-circularly
polarized spin waves are excited equally. Hence, on average
there is no (net) rotation of the DW and therefore no
reflection.
In conclusion, DWs in antiferromagnets are driven by
thermal gradients and move towards the hotter region. On a
macroscopic level this is explained by the minimization of
107201-4
the free energy and agrees with older results for thermally
driven DW motion in ferromagnets. However, in contrast
to FMs the dynamics of the wall is independent of the
intermediate anisotropy which defines the easy plane
containing the wall magnetization. The reason for this is
the fact that those torques which—in a FM—tilt the wall
cancel each other in the two sublattices of the AFM. This
has two consequences. One is the absence of the Walker
breakdown in AFMs leading to a higher wall mobility. This
is increasingly important in the limit of low intermediate
anisotropy where DWs in FMs always must precess with
low DW velocity. The other consequence of the lack of DW
tilting is that DW in AFMs are massless. Because of the
lack of inertia they do not accelerate but reach their
stationary DW velocity instantaneously. This is very
important for experiments and technical applications using
pulsed driving forces, since one could operate on much
shorter time scales. Furthermore, the argument of counter-
acting torques leading to massless DWs is not restricted to
temperature gradients but also applies to other driving
mechanisms in antiferromagnets featuring staggered tor-
ques, like Néel spin-orbit torques [29] or current driven DW
motion in synthetic antiferromagnets [8,31].
We thank the DFG for financial support through the
SFB 767 and SPP 1538. U. A. gratefully acknowledges
support from EU FP7 Marie Curie Zukunftskolleg Incoming
Fellowship Programme, University of Konstanz (Grant
No. 291784).
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