Landau-Lifshitz-Bloch equation for domain wall motion in antiferromagnets
Z. Y. Chen,1Z. R. Yan,1M. H. Qin,1,*and J.-M. Liu1,2
1Institute for Advanced Materials, South China Academy of Advanced Optoelectronics and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
2Laboratory of Solid State Microstructures and Innovative Center for Advanced Microstructures, Nanjing University, Nanjing 210093, China
(Received 6 February 2019; revised manuscript received 6 June 2019; published 25 June 2019) In this work, we derive the Landau-Lifshitz-Bloch equation accounting for the multidomain antiferromagnetic (AFM) lattice at finite temperature, in order to investigate the domain wall motion, the core issue for AFM spintronics. The continuity equation of the staggered magnetization is obtained using the continuum approxima- tion, allowing an analytical calculation of the domain wall dynamics. The influence of temperature on the static domain wall profile is investigated, and the analytical calculations agree well with the numerical simulations on temperature-gradient-driven domain wall motion, confirming the validity of this theory. Furthermore, the decrease of the acceleration and the increase of the saturation velocity of the domain wall with the increase of temperature are uncovered for a fixed gradient. Moreover, it is worth noting that this theory could be also applied to dynamics of various wall motions in an AFM system. The present theory represents a comprehensive approach to the domain wall dynamics in AFM materials, a crucial step toward the development of AFM spintronics.
DOI:10.1103/PhysRevB.99.214436
I. INTRODUCTION
As promising materials for spintronics, antiferromagnets have attracted significant attention recently because they show fast magnetic dynamics and produce nonperturbing stray fields [1–3], especially after the effective detection and manip- ulation of antiferromagnetic (AFM) state were experimentally realized [4–6]. Theoretically, the spin dynamics in an AFM lattice can also be investigated using the Landau-Lifshitz- Gilbert (LLG) equation based on the atomistic spin models, and a number of driving mechanisms [7–16] have been pro- posed to drive effectively the domain wall (DW) in an AFM lattice. These important works not only contribute a great deal to fundamental physics but also do provide useful information for potential AFM spintronic devices.
Nevertheless, for a realistic spintronic device where the lattice size under consideration is huge, atomistic spin models are far from sufficient and an efficient computation based on such atomistic models becomes nonrealistic due to the computation capacity limit. Considering that an AFM DW may have a spatial width as large as ∼10 nm, one sees that the whole lattice used for the LLG-based micromagnetic simulation must be at least as large as∼100 nm if wall motion is considered. This makes a computation impossible due to the capacity limit, unless the lattice is cut down to∼10 nm. At a cost of physical reality, one has to set the axial anisotropies two orders of magnitude stronger than realistic values, and the DW becomes unreasonably narrow (∼1 nm). Moreover, white-noise terms are usually included into the effective field for the LLG dynamics in order to simulate temperature (T)-dependent effects, which also add huge computation cost to the simulations.
As an alternative approximation, one may utilize the coarse-grained scheme and use a macromoment mv/mk to express the two sublattice magnetizations of a finite region (called a grain) inside a AFM domain, and thus the LLG equation on the macromomentmv/mk can be used without increasing the computational cost much. However, the LLG- based simulations fail to capture the fact that the magnetiza- tion magnitude is a function ofT: usually it decreases with increasingTuntil the transition pointTN. Thus, the AFM DW dynamics at finite T especially near TN is hardly described by the LLG simulations. Furthermore, the same problem also exists in the derivation of the micromagnetic continuum equations for staggered magnetization from the LLG equation based on the coarse-grained scheme. Realistic appealing to numerical approaches is thus raised in order to treat the wall dynamics in an AFM system at finite T or T gradient, noting that theT-relevant controls, e.g.,T-gradient-driven wall motions have been often taken in the AFM spintronic devices.
In short, there is an urgent need to develop an approach in dealing with discrete and continuum models for AFM lattice at elevatedT. Compared to the LLG equation, the Landau- Lifshitz-Bloch (LLB) equation introduces the longitudinal relaxation to describe theT-dependent magnitude of the mag- netization, making it possible to reasonably investigate the DW dynamics at finiteT even nearTN. As a matter of fact, it is noted that wall motion in a FM lattice under aT-gradient field has been simulated using the LLB equation [17,18]. This computation has been proven to be efficient in large-scale mi- cromagnetic simulation of realistic spintronic devices at high T and in short time. Reasonable results on the wall motion and Walker breakdown in a multidomain FM lattice have been obtained within the framework of the LLB equation. Most recently, the results on the multidomain FM lattice suggested a linear relation between the wall velocity and T gradient.
This relation was once applied to describe the domain wall
motion in an AFM lattice. Unfortunately, this relation agrees with numerical results under small T gradient, but deviates seriously when theTgradient is large [14]. It is inconsistent with the fact that the wall velocity should be limited by the maximum spin-wave group velocity [9,10]. We would thus be highly concerned and interested to ask if the LLB scheme can be applied to track efficiently the domain wall motion in a large and multidomain AFM lattice. Indeed, the LLB equation on a ferrimagnetic monodomain lattice was recently proposed [19,20], which becomes the basis for deriving a generalized equation for a multidomain AFM lattice.
In this work, we perform a derivation of the LLB equation for a multidomain AFM lattice at finiteT; this equation would be highly efficient for large-scale micromagnetic simulation of realistic AFM spintronic devices. More importantly, a continuity equation for the staggered magnetizations can be derived from this equation using the continuum approxima- tion, which allows an analytical calculation on the domain wall motion in an AFM lattice (e.g., driven by a finite T gradient or staggered magnetic field). It is found that the theory’s predictions about several crucial issues agree well with numerical results in literature.
II. DERIVATION OF THE LLB EQUATION
We start from an AFM lattice with two intercrossing FM sublattices whose spin alignments are antiparallel. We apply the coarse-grained scheme to the whole lattice divided into a number of grains as shown in Fig.1. The grain size should be sufficiently large for high-efficiency computation but suffi- ciently small in comparison with the concerned characteristic scales in lattice, e.g., domain wall width or other anomalies in the present case. The basic strategy is to track the magnetiza- tion evolution of the two sublattices separately, which makes it possible to investigate the AFM dynamics using a method similar to that of ferromagnets [21,22]. For an arbitrary grain (i) containing two FM sublattices (v,κ), if no interaction of this grain with its neighbors is considered, the LLB equation for magnetizationmvof sublatticevis written as [19]
1 γν
dmv
dt =mν×Hν+αmν·Hν m2ν mν
−α⊥mν×(mν×Hν)
m2ν , (1)
FIG. 1. (Top) Spin configuration of atomistic regular AFM lat- tice, where the whole AFM lattice is divided into many grains (regions). (Bottom) Sublattice magnetization in a grain is described by two antiparallel macrospinsmvandmκ.
where γν is the gyromagnetic ratio, α||/α⊥ are the T- dependent longitudinal/transverse damping constants, Hν= H+HA,ν +Hνκ is the effective field including external field H, anisotropy field HA,ν and internal exchange field Hνκ, assuming the zaxis as the easy axis. The internal exchange fieldHνκaccounts for the interaction between the sublattices vandκ. They are, respectively, given by [19]
HA,ν = − 1
χ˜ν,⊥(mx,νex+my,νey),
mν =(mx,νex,my,νey,mz,νez), (2)
and
Hνκ= −J0,νκ μν
mν×(mν×mκ) m2ν −1
2 1
νν m2ν
m2e,ν −1
− 1 νκ
τκ2 τe2,κ −1
mν, (3)
whereχν,⊥is the transverse susceptibility,J0,νκis the coupling constant,μνis the saturation moment,me,νis the equilibrium magnetization, νν andνκ are the longitudinal rates,τκ = mv(mv·mκ)/m2ν,andτe,κ= |me,v·me,κ|/me,ν [19]. The first and third terms on the right side of Eq. (1) have the same forms as those in the LLG equation, and the second term describes the longitudinal relaxation depicting the magnitude variation of magnetization due to thermal fluctuations at finiteT.
It is noted that theT-dependent parameters in the two sub- lattices equal each other (e.g., γν=γκ=γ ,me,ν =me,κ= me, μν =μκ =μS,χν,⊥=χκ,⊥=χ⊥), andHνκhas a more compact form:
Hνκ = −J0
μS
mν×(mν×mκ) mν2 −1
2 1
χ˜ m2ν
m2e −1
+|J0| μS
m2ν−τκ2 m2e
mν, (4)
whereχ||is the longitudinal susceptibility,J0,νκ=J0 =NDJ, where J is the exchange coupling between the nearest- neighbor atomistic spins andNDis the coordination number.
Following the earlier works [21–23], these parametersme,χ||, and χ⊥ are reasonably estimated by numerical simulations using the stochastic LLG equation based on the atomistic model. As an example, we present the estimated parameters (empty points) and corresponding fitted results (solid lines) in Fig. 2, given the uniaxial anisotropy 0.02J. Their good consistencies confirm the estimations.
Subsequently, we discuss the effect ofT. It is noted that thermal fluctuations are less dependent on spin structures, and thus the stochastic fields for a FM system can be ap- proximately applicable to an AFM system [24–26]. This argument has been confirmed in earlier work which calculates the stochastic fields strictly using the Fokker-Planck equation [23]. When the stochastic fields are considered, the LLB
FIG. 2. Stochastic LLG simulated (a)meand (b)χ|| andχ⊥as functions of temperature and the corresponding fitting results.
equation for grain (i) now reads 1
γ dmv
dt =mν×Hν+αmν·Hν m2ν mν
−α⊥mν×[mν×(Hν+ξ⊥,ν)]
m2ν +ξ,ν (5) whereξ||,ν/ξ⊥,ν is the longitudinal/transverse stochastic field with
ξη,νa (t,r)ξη,νb (t,r)
=2Dηδabδ(t−t)δ(r−r), η=(,⊥), (6) wherea,bare the Cartesian components (=x,y,z), and the longitudinal and transverse diffusion constants D|| and D⊥ read, respectively,
D =αγkBT
MSV , and D⊥= (α⊥−α)kBT γMSVα⊥2 (7) wherekB is the Boltzmann constant,MS the saturation mag- netization, andVthe grain volume.
It is noted that in order to describe thermal fluctuations and satisfy the fluctuation-dissipation theorem, fluctuating torques and fluctuating fields are introduced into the damping term of the LLB equation. Alternatively, fluctuating fields are introduced into the precession and damping terms of the LLG equation. As a result, the LLG equation cannot be completely recovered from the LLB equation in the absence of the longitudinal dynamics. Moreover, the calculations based on the stochastic LLB equation and stochastic LLG equation are expected to be consistent with each other at low T far below the Néel temperature (TN). However, the thermal effects at rather high T cannot be well investigated based on the stochastic LLG equation because it fails to capture the T- dependent magnetization magnitude.
Actually, any grain must have coupling with its neighbors and an inclusion of the coupling is a prerequisite to consider a multidomain AFM system. We discuss the intergrain ex- change field between grain (i) and grain (j), using the same approach as given in Ref. [22] to extend the LLB equation.
For two neighboring grains (i) and (j), the intergrain exchange
interactionHexi jreads Hexi j= −J
k,l
Sk·Sl
= −J F 2a2l
mν,i
mν,i
· mκ,j
mκ,j
+mκ,i
mκ,i
·mν,j
mν,j
, (8) where k,l sums all the nearest-neighbor pairs connecting the two grains, S is the normalized atomistic spin,F is the interface area, and al is the lattice constant, mv,i/mk,j is the magnetization of sublattice v/κ in grain i/j. Then, we obtain the intergrain exchange field to sublattice v of grain (i) imposed by sublatticeκ in grain (j):
Hex,ν,i= − 1 MSV/2
∂Hexi j
∂mν,i = 2A(0)
aldMSm2e(mκ,j+mν,i), (9) where A(0)=J/2al is the exchange stiffness at zeroT, and d is the grain dimension. It is noted that Eq. (9) is ob- tained on the assumption that the two sublattices’ magneti- zations in grains (i) and (j) can be described as macrospins mv,i and mk,j. This would overestimate the intergrain ex- change coupling. Following the earlier work, a correction factor al/d should be taken into account to diminish the overestimation [22].
Moreover, considering the thermal fluctuations, the ex- change stiffness is alsoTdependent, given byA(T)=A(0)m2e if the thermal average spin moment is equal to the equilibrium magnetizationme. Thus, the total intergrain exchange field of sublatticeνin grainireads
Hex,ν,i= 2A(T) d2MSme2
j
(mκ,j+mν,i), (10) where the sum is over all the nearest-neighboring grains.
To this stage, we have successfully obtained the LLB equation applicable to a multidomain AFM lattice, in par- ticular, to describe the domain wall dynamics. Certainly, a more explicit form of the LLB equation using the continuum approximation would be appreciated [16]. In proceeding, we define the total magnetizationmi=mν,i+mκ,iand staggered magnetization ni=mν,i−mκ,i for grain (i) to replace mν,i
andmκ,i. The effective fields for grains (i) and (j) are then written asHν,i=Hm,i+Hn,i, andHκ,i=Hm,i−Hn,i, where Hm,i andHn,iare, respectively, the effective fields related to miandni. Noting that the longitudinal relaxation of sublattice magnetization is much faster than the transverse relaxation, and the magnetization is nearly identical to the equilibrium one, i.e.,|mν,i| =me,i[18,19], one has the alternative expres- sions of the LLB equations after necessary substitutions and continuum approximation:
dm
dt =γ(m×Hm+n×Hn)− α⊥
2m2e
m×dm
dt +n×dn dt
+γ α
2m2e[(m·Hn)n+(n·Hm)n], (11) and
dn
dt =γn×Hm, (12)
with the effective fieldsHmandHn(see Ref. [27] for detailed derivation). Here, Eq. (11) has been transformed into the Gilbert form, and particular damping terms are safely omitted as done in the LLG scheme [16,28,29], which hardly affects our main results.
For an AFM system below TN, one has m·n∼0, and n2 ∼4m2e which is also T dependent due to the fact that the longitudinal relaxation is generally much faster than the transverse one. Under zero applied field,mas a function ofn can be derived from Eq. (12) [16,29]:
m=
dn dt ×n
4γm2e J0/μS+2NDA/d2MSm2e =Am
dn
dt ×n, (13) where parameter Am is introduced for brevity. Substituting Eq. (13) into Eq. (11) and taking the cross product withn, we obtain
Amn×d2n dt2 ×n
=n×
− γ
2 ˜χ⊥nzez+γA(T)
MSme2∇2n+ α⊥
2m2e dn
dt
×n,(14) where nz is the zcomponent of n. Specifically, all parame- ters including exchange, magnetic anisotropy, and damping parameters are T dependent in Eq. (14). More importantly, the magnitude of n also depends on T by introducing the longitudinal relaxation, which is basically different from the equation derived from the LLG equation. We have obtained an analytical expression of the staggered magnetization for an AFM lattice, whose magnitude and orientation are spatially inhomogeneous andTdependent. It thus allows one to track various stimuli-driven domain structure evolution and wall motion in a multidomain AFM system.
By using Eqs. (13) and (14), we can perform the analytical calculations within the framework of the LLB equation for an AFM system. Here, the second-order derivative ofnwith respect to time is essential in distinguishing the magnetic dynamics in an AFM system from that in a FM one [30].
In particular, the parameters and magnitude for the staggered magnetization (n) areTdependent, allowing one to investigate the magnetic dynamics at finiteT, including the domain wall motion in ultralarge scale. Furthermore, the domain wall motion in an AFM lattice, as driven by various stimuli such as temperature gradient [13–15], external field [28,31,32], and Néel spin-orbit torque [10,33], can be similarly calculated using Eq. (14).
III. APPLICATION OF THE LLB EQUATION For the validity of this continuum LLB theory on AFM lattice, one looks to several well-known facts for checking.
As an initial check, we discuss the static solutions. One of the special solutions to Eq. (14) is the static Néel wall config- uration with the polar angle of the staggered magnetization θ=2arctan[exp(z−z0)/λ], where z0 is the position of the wall center, andλis theT-dependent wall width:
λ(T)=
2 ˜χ⊥|A(T)|
MSme2 . (15)
FIG. 3. (a) Numerical and analytical calculatedλas a function of T, and the three components of the magnetization versusycoordinate at (b)T =0 and (c)T =1.4J/kB, and (d) the estimatedhzandht as functions ofT. The sketches of circular and elliptical DWs are also presented, respectively, in the insets of (b) and (c).
One observes thatλ(0) is exactly the same as that derived from the LLG equation [16]. Moreover, λ increases with increasingTand ultimately becomes divergent atTN, as shown in Fig.3(a), which gives the numerical and analytical calcu- latedλas a function of T. The analytical data well coincide with the numerical results, both based on the LLB equation, supporting the validity of this continuum theory.
It is noted that the AFM DW profiles have important influence on the wall dynamics and relevant magnetoresis- tance, while theirT dependences are still unclear so far. We numerically study the effect of temperature on the Bloch DW profiles using Eq. (5) on an 8al×8al×200al sys- tem. Similar to ferromagnets [34–36], three types of walls including circular, elliptical, and linear walls are observed.
The circular wall emerges at zero T, as shown in Fig. 3(b) which gives the three components of the magnetization versus y coordinate. Figure 3(c) presents the components at T =1.4J/kB, which clearly demonstrates an elliptical wall. Similarly, the wall profiles can be described by the hyperbolic functions nz(T)=hz(T)tanh[(y−y0)/λ(T)] and nt(T)=ht(T)sech[(y−y0)/λ(T)], where nt is the trans- verse component of n, and hz/ht is the amplitude of easy axis/transverse magnetization. The estimatedhz(T) andht(T) are summarized in Fig. 3(d)where ht is smaller than hz at finite T, demonstrating the existence of elliptical walls. In addition, for Th <T <TN, the domain wall is linear with a finitehz and zeroht. This effect can be understood from the influence of thermal fluctuations on the DW. The spins in the wall usually deviate from the easy axis and have large exchange and anisotropy energies, and thus they are more sensitive to thermal fluctuations than the spins inside the domain, resulting in the fact thatht decreases more quickly thanhz asTincreases, as confirmed in our simulations. Fur- thermore, the difference between hz(T) andht(T) increases with the increasing anisotropy (the corresponding results are not shown here), the same as in FM systems [34,35]. As a
matter of fact, earlier work claimed that the FM and AFM domain walls share common static properties at zeroT[37].
Here, it is clearly demonstrated that this behavior also exists at finiteTeven nearTN.
Given the validity of the developed LLB theory, we intend to solve Eq. (14) using the approach with polar coordinates proposed in earlier work to investigate the thermally driven DW motion for an AFM lattice in a finiteTgradient [10,38].
As has been clarified in the earlier works [13–15], the compe- tition between the entropy torque and Brownian force under aT gradient determines the motion of AFM DW. Here, we pay particular attention on the entropy torque-driven DW motion where the stochastic field can be safely neglected.
Furthermore, we assume thatTis rather belowTNand the DW structure is robust during its motion [9,38–40]. In this case, the staggered magnetization is a function of the composite variableZ =z−vt:
dnx
dt = −vnx, dnz
dt = −vnz, dθ
dt = −vθ= −vsinθ λ ,
(16) where v is the wall velocity, θ is the angle between the staggered magnetization andzaxis, and the prime represents the derivative with respect toZ. We obtain the velocity of wall motion under a temperature gradient:
v=−α11+
α112+4α2
2α2
, (17)
whereα1andα2 areT-dependent variables (see Ref. [27] for details). One may note that an anisotropy gradient could be induced by theTgradient [see theχ⊥(T) curve in Fig.2(b)], which also contributes to the DW motion [41,42]. However, comparing with the effect of the strong exchange interaction, the effect of the anisotropy term on the DW dynamics can be safely ignored. More interestingly, it is also demonstrated that the DW velocity is limited by
vmax= γalJ√ 2ND
μS
me=c(T), (18) wherec(0) is the group velocity of spin wave at zeroT(see Ref. [27] for details), further confirming the fact that the limi- tation of the DW velocity originates from the emission of spin wave [9,16]. With the increasingT, the enhanced thermal fluc- tuations effectively weaken the exchange interaction and in turn suppressc(T) andvmax. In Fig.4(a), the LLG simulated, the LLB simulated, and analytically calculated velocities are presented, and the good coincidence of these results confirms the validity of the effective theory. Interestingly,∼1000 CPU hours are needed for the LLG simulations performed on our computer cluster, while only∼2 CPU hours are needed for the LLB simulations, demonstrating the high efficiency of the theory in dealing with the antiferromagnetic dynamics.
More importantly, one may perform the LLB simulations to investigate the AFM dynamics at high temperatures even near TN. In Fig. 4(b), the LLB-simulated DW positions as functions of t under ∇T =0.003J/kBal for various T0 (the lowest temperature of the system) are presented, which demonstrates the difference of the DW dynamics for highT0
from lowT0. On the one hand, the DW quickly accelerates
FIG. 4. LLG-simulated (empty triangles), the LLB-simulated (empty circles), and analytically (solid line) calculated DW velocities as functions of ࢟T (a), and the LLB-simulated DW positions as function oft for variousT0 (b) under ∇T =0.003J/kBal, and (c) under the effective staggered fieldHN=0.0005J/μS, and (d) the LLB simulated (empty circles) and analytically calculated (solid line) velocities as functions ofT0underHN =0.0005J/μS. The LLG simulated results in (a) are reproduced from Ref. [14].
to the saturation velocity for low T0, and the accelerating time significantly increases with the increase ofT0, as clearly shown in the movies in the Supplemental Material [27]. It is noted that the exchange interaction between neighboring spins is effectively reduced as T0 increases, contributing to the decrease of the acceleration. On the other hand, higher T0 generally results in stronger changes of the magnetization and driving torque, resulting in a larger saturation velocity, as shown in the simulations.
In order to better understand the temperature effect on the DW dynamics, we also investigated the DW motion driven by an effective staggered fieldHNalong thezaxis (HN and−HN
are applied onvandκsublattices, respectively), which could be induced by electric current in CuMnAs and Mn2Au. The LLB-simulated DW positions as functions oft for variousT0 are shown in Fig.4(c), which clearly shows that both the ac- celeration and saturation velocity are significantly suppressed with the increase of T0, attributed to the reduction of the effective exchange interaction. Similarly, the velocity of wall motion under the staggered field could be also analytically calculated:
vN =λHN
α⊥ me. (19)
Figure4(d)gives the LLB-simulated and analytically calcu- lated DW velocities. The results for T0<2TN/3 well coin- cide with each other, while deviating from each other for T0 >2TN/3. It is noted that the divergence of the longitudinal relaxation time atT ∼TN is hardly captured by the approx- imate condition n2∼4me2, resulting in the deviation of the
analytical results from the numerical results. However, the physics has been clearly uncovered by the LLB simulations, and the investigation is far beyond the capacity of the present LLG method, which fails to describe the temperature depen- dence of the magnetization magnitude.
IV. DISCUSSION AND CONCLUSION
So far, the validity of the dynamic equation for staggered magnetization in an AFM lattice has been well confirmed by checking the static domain wall profiles andT-gradient-driven wall motion which are well consistent with the numerical results. Thus, the two major issues (AFM wall motion at finite Tin large-scale system) which are hardly reached in the LLG- based simulations have been removed if the LLB equation and derived continuum equation are utilized. More importantly, we would like to point out that this essential equation can be also used to investigate the AFM dynamics driven by other stimuli [43–45]. For example, a large-scale system is needed to generate Gauss T field, which is hardly reached by the conventional LLG simulations [46]. As a matter of fact, the analytical calculation has been performed, and the corresponding results will be reported elsewhere.
In conclusion, we have derived the LLB equation with intergrain and stochastic fields for AFM systems, which
allows one to investigate the magnetic dynamics at finite temperatures using multiscale approaches. Moreover, the con- tinuity equation of the staggered magnetization has been also derived using the continuum approximation. The derivations have been used to investigate the influence of temperature on the static AFM domain wall, which reveals a similar behavior to FM systems. The analytical calculation of the temperature-gradient-driven AFM domain wall motion well agrees with the numerical results and reproduces successfully the saturation velocity, well confirming the validity of our derivations. More interestingly, physics-related DW dynamics under temperature gradient has been predicted by the LLB simulations. Importantly, this theory could be applied to other wall driving mechanisms such as Néel spin-orbit torques and spin-transfer torques as well.
ACKNOWLEDGMENTS
The work is supported by the National Key Projects for Basic Research of China (Grant No. 2015CB921202), the Natural Science Foundation of China (Grant No. 11204091), the Science and Technology Planning Project of Guangzhou in China (Grant No. 201904010019), and the Natural Science Foundation of Guangdong Province (Grant No.
2016A030308019).
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