Numerical Computation of Spin-Transfer Torques for Antiferromagnetic Domain walls
Hyeon-Jong Park, Yunboo Jeong, Se-Hyeok Oh, Gyungchoon Go, Jung Hyun Oh, Kyoung-Whan Kim, Hyun-Woo Lee, Kyung-Jin Lee
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Numerical Computation of Spin-Transfer Torques for AntiferromagneticDomain Walls
Hyeon-Jong Park, Yunboo Jeong, Se-Hyeok Oh, Gyungchoon Go, JungHyun Oh, Kyoung-Whan Kim, Hyun-Woo Lee, and Kyung-Jin Lee
1, 4, ∗ KU-KIST Graduate School of Converging Science and Technology,Korea University, Seoul 02841, Korea Department of Semiconductor Systems Engineering,Korea University, Seoul 02841, Korea Department of Nano-Semiconductor and Engineering,Korea University, Seoul 02841, Korea Department of Materials Science and Engineering,Korea University, Seoul 02841, Korea Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Korea Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea (Dated: January 23, 2020)
Abstract
We numerically compute current-induced spin-transfer torques for antiferromagnetic domain walls, basedon a linear response theory in a tight-binding model. We find that, unlike for ferromagnetic domain wallmotion, the contribution of adiabatic spin torque to antiferromagnetic domain wall motion is negligible,consistent with previous theories. As a result, the non-adiabatic spin-transfer torque is a main driving torquefor antiferromagnetic domain wall motion. Moreover, the non-adiabatic spin-transfer torque for narrowerantiferromagnetic domain walls increases more rapidly than that for ferromagnetic domain walls, which isattributed to the enhanced spin mistracking process for antiferromagnetic domain walls. ∗ Electronic address: [email protected] . INTRODUCTION Antiferromagnetic spintronics has recently attracted considerable interest because of the im-munity against external magnetic fields and the potential for high frequency dynamics [1–3].Antiferromagnets produce no stray field and do not couple to external magnetic fields because ofzero net magnetic moment, which is advantageous for high-density device integration. Moreover,in contrast to ferromagnets, the resonance frequency of antiferromagnets for the zero wavevectormode is related to the exchange interaction, which results in terahertz magnetic excitations [4, 5] andmay find use in terahertz spintronic devices [6–9]. For domain wall dynamics, it was predicted thatspin-orbit torques enable much faster antiferromagnetic domain wall motion than ferromagneticcounterpart [10, 11]. This fast domain wall dynamics is caused by the complete decoupling betweenthe domain wall position and domain wall angle because the gyrotropic coupling is proportionalto the net spin density [12–16], which is zero in antiferromagnets.Antiferromagnets can also be electrically manipulated by conventional spin-transfer torques inthe absence of the spin-orbit interaction [17–29]. Previous studies on conventional spin-transfertorques can be classified into two groups. The first group [17–23] is for spin-valve-like structures inwhich an antiferromagnet is interfaced with a normal metal and the spin-transfer torque consists ofdamping-like and field-like components through real and imaginary spin-mixing conductances atthe antiferromagnet/normal metal interface [22, 30]. The second group [23–29] is for continuouslyvarying antiferromagnetic spin textures such as domain walls for which the spin-transfer torqueconsists of adiabatic and non-adiabatic torques. An ab initio study [23] computed the adiabaticspin torque for a domain wall in a system where two ferromagnetic layers are antiferromagneticallycoupled in the thickness direction. A microscopic calculation based on the Green’s functionformulation of Landauer-Büttiker transport theory [24] reported that non-equilibrium spin densitycorresponding to the adiabatic torque for antiferromagnetic spin textures is finite but does notlead to domain wall motion. Phenomenological theories based on spin pumping and Onsagerreciprocity [25, 26] predicted that the main driving torque for antiferromagnetic domain wallmotion is the non-adiabatic torque. For antiferromagnetic domain walls, therefore, no microscopiccomputation of both torque components on equal footing has been reported. In order to understanddistinct features of adiabatic and non-adiabatic spin-transfer torques acting on antiferromagneticspin textures, it is important to microscopically compute these two mutually orthogonal torquecomponents. 2or ferromagnetic domain walls, a number of theoretical [31–35, 37–43] and experimentalstudies [44–52] have investigated the non-adiabaticity β of spin currents, a key parameter ofnon-adiabatic spin-transfer torque. Different mechanisms of β arise depending on the relativelength scale of domain wall. One mechanism that is independent of the domain wall length iscaused by the spin relaxation [32–34, 38–40]. This mechanism predicts β/α ≈ where α is thedamping parameter, which is related to the spin relaxation in equilibrium. Another mechanismthat is independent of the domain wall length is the intrinsic spin torque due to the perturbationof the electronic states when an electric field is applied [43]. As domain walls get narrower thanthe length scale of spin precession around the exchange field, other mechanisms become moredominant. For narrow domain walls, the conduction electron spins are unable to follow a rapidchange in the magnetization, i.e. the ballistic spin mistracking, which contributes to the non-adiabaticity [35–37]. When domain walls are atomically thin, the reflection of conduction electronspins from the domain wall becomes non-negliglible, resulting in the momentum transfer [31].A recent experiment [53] reported a large β/α for a domain wall in an antiferromagneticallycoupled ferrimagnet, suggesting that mechanisms beyond the spin relaxation may take effect inantiferromagnetic domain walls.In this paper, we compute non-equilibrium spin density based on a linear response theory ina tight binding model. From the computed non-equilibrium spin density that is defined at eachsublattice, we calculate local and effective spin-transfer torques, which can be decomposed toadiabatic and non-adiabatic torques. Here local spin-transfer torque is a torque exerting on aspin moment at each sublattice whereas effective spin-transfer torque is obtained by integratinglocal spin-transfer torque over the antiferromagnetic domain wall profile. Therefore, the effectivespin-transfer torque is the experimentally measurable quantity. As the leading-order contribution,we find that the effective adiabatic torque is zero for antiferromagnetic domain walls. On the otherhand, the effective non-adiabatic torque is large and increases significantly as the domain wallwidth decreases.The paper is organized as follows. In Sec. II, we present a linear response theory to computethe local spin-transfer torque at sublattices of an antiferromagnetic domain wall. In Sec. III, wepresent a continuum approximation of the equations of motion of an antiferromagnetic domain walland describe how to obtain the effective spin-transfer torques integrated over the antiferromagneticdomain wall profile from the local spin-transfer torques. In Sec. IV, we show numerical results ofthe local and effective spin-transfer torques for antiferromagnetic domain walls and compare the3esults with ones for ferromagnetic domain walls. Finally, Sec. V concludes this work. II. MICROSCOPIC APPROACH TO COMPUTE SPIN-TRANSFER TORQUES
For an antiferromagnet, we consider two sublattices, A and B , alternating in the x -directionalong which the magnetization profile has a texture (i.e., domain wall) and an electric field isapplied. We model a magnetic system with random impurities under an external electric field as H = P k [ H ( k ) + V imp + H ext ( k , t )] , where V imp is the impurity potential and k is a wave vectorin the transverse direction, and H ( k ) = X i (cid:16) C † iA ( k ) C † iB ( k ) (cid:17) ǫ ( k ) σ − ∆ A σσσ · m iA ǫ ( k ) σ − ∆ B σσσ · m iB C iA ( k ) C iB ( k ) − t H X i h C † iA ( k ) σ C iB ( k ) + C † iB ( k ) σ C i +1 ,A ( k ) i − t H X i h C † iB ( k ) σ C iA ( k ) + C † i +1 ,A ( k ) σ C i,B ( k ) i , (1)where ∆ η is an exchange strength at an atomic sublattice ( η = A, B ), m iη is the unit vector alongthe magnetic moment at a sublattice η in the i -th cell, t H is a hopping energy between sublattices, σσσ is the Pauli spin matrix, and σ is a × identity matrix. Because our model assumes alternatingsublattices in the x -direction, we consider the nearest-neighbor hopping between two sublattices.We specify creation operators with C † iη ( k ) = ( C † iη ↑ ( k ) , C † iη ↓ ( k )) at a sublattice η in the i -th cellwith spins ( ↑ or ↓ ) and annihilation operators with C iη ( k ) = ( C iη ↑ ( k ) , C iη ↓ ( k )) T , respectively.We assume that, along the transverse directions (i.e., the y - and z -directions), the system keepsa periodic structure so that quantum states in these transverse directions are described by a wavevector k = ( k y , k z ) and eigenenergy ǫ ( k ) = − t H (cos k y d + cos k z d ) , where d is the atomicspacing.On the other hand, H ext ( k , t ) describes the external electric field; H ext ( k , t ) = − i | e | t H d ¯ h X i ˆx · A i ( t ) h C † iA ( k ) σ C iB ( k ) + C † iB ( k ) σ C i +1 ,A ( k ) i + i | e | t H d ¯ h X i ˆx · A i ( t ) h C † iB ( k ) σ C iA ( k ) + C † i +1 ,A ( k ) σ C i,B ( k ) i (2)where A i ( t ) = − ˆx E sin ω p t/ω p is a vector potential with the electric field E in the x direction. Fora DC case, we set a frequency ω p to be zero at the final stage of calculation.4y integrating out electronic degrees of freedom { C † , C } on the Keldysh contour, a local spintorque at a site iη is given by [54] τττ iη = ∆ η m iη ( t ) × δ s ( t ) iη = ∆ η m iη ( t ) × X k Tr spin {− i ¯ h G < ( k ; t ) σσσ } iη,iη . (3)Here, G < ( k ; t ) is a lesser Green function of the full Hamiltonian, G ( k , t ) = [ i ¯ h∂ t − H ( k ) − V imp − H ext ( k , t )] − , and the non-equilibrium spin density is determined by a linear part of a fullGreen function on a vector potential A ( t ) as, δ s iη ( t ) = P k Tr spin {− i ¯ h ( G H ext G ) < ( k ; t ) σσσ } where G = [ E − H − V imp ] − is the unperturbed Green function averaged over impurities. A. Linear response approximation
We treat the external electric field perturbatively because the energy change between adjacentatoms, | e |E d , is much smaller than the hopping energy, t H . Along this scheme, the lesser Greenfunction of the system is written as [55] G
We average the Green functions over impurity configurations [57, 58]. The correspondingself-energy in the coordinate representation is given by Σ imp ( r , r ′ ; E ) = G R ( r , r ′ ; E ) h V imp ( r ) , V imp ( r ′ ) i (9)6here h V imp ( r ) , V imp ( r ′ ) i is the correlation function of impurities and G R is a retarded Greenfunction averaged over impurities. For random and short-ranged impurities with a screening length /k s , the correlation function is found to be proportional to [57] h V imp ( r ) V imp ( r ′ ) i ∝ k s e − k s | r − r ′ | , (10)indicating that Σ imp is also a short-ranged function. Assuming such short-ranged impurities we set Σ imp ii ′ ( E ) = δ ii ′ V π X k G Rii ( k , E ) (11)where a summation over k means that the impurity self-energy is also local over the transversedirections. Thus, by including the self-energy from impurity, the Green function forms the Dysonequation; X i h [ g R ( k , E )] − ii − Σ imp ii ( E ) i G Ri i ′ ( k , E ) = × δ ii ′ (12)with a × identity matrix × . A solution of the equation is not trivial due to a self-consistency of G R and requires a large computation burden. For simplicity, because the modification of electronicstructure by external fields occurs mainly near the chemical potential, we take into account theself-energy at the chemical potential µ in a whole energy as Σ imp ( E ) ≈ Σ imp ( E ) (cid:12)(cid:12)(cid:12) E = µ , namely the self-energy independent of energy. We solve the Dyson equation self-consistently toobtain Σ imp of Eq. (11), and obtain the non-equilibrium spin density by using Eqs. (5) and (6).This locally defined non-equilibrium spin density gives a local spin torque through Eq. (3). III. EFFECTIVE SPIN-TRANSFER TORQUES ACTING ON AN ANTIFERROMAGNETIC DO-MAIN WALL
To investigate the role of spin-transfer torques in domain wall motion, one has to find out theeffective spin-transfer torques, which are obtained by integrating local spin-transfer torques overthe domain wall profile. This section presents the equations of motion and associated effectivespin-transfer torques acting on an antiferromagnetic domain wall as follows.Local magnetic moment at each sublattice in the i th unit cell is assumed to be m iη =sign η (cos φ iη sin θ iη , sin φ iη sin θ iη , cos θ iη ) with η = A, B . Here, θ iη and φ iη are polar and7zimuthal angles at a sublattice η in the i th unit cell. We introduce a prefactor sign η to de-scribe antiferromagnet or ferromagnet (i.e., sign A = 1 and sign B = − for antiferromagnet and sign A = sign B = 1 for ferromagnet). We adopt the Walker’s ansatz for a domain wall profile [59]as, θ iη = 2 tan − { exp[( X − x iη ) /λ DW ] } , φ iη = π , (13)where X and λ DW are the position and the width of the domain wall, respectively. For antiferro-magnetic domain walls (see Fig. 1 for a domain wall profile), we introduce the total and staggeredmagnetic moments as M i ≡
12 ( m AFM iA + m AFM iB ) , n i ≡
12 ( m AFM iA − m AFM iB ) . (14)The equations of motion for antiferromagnetic domain walls are obtained with the second-orderexpansion of small parameters ( ∂/∂x , ∂/∂t , M , and spin-torque terms). The free energy U of thesystem is written as [11, 12, 62] U = Z dx " a | M | + A (cid:18) ∂ n ∂x (cid:19) + L M · ∂ n ∂x , (15)where a ( A ) is homogeneous (inhomogeneous) exchange parameter and L is a parity-breakingexchange strength [62, 64]. The Euler-Lagrange equation with respect to M and n is given by ∂ L ∂ M ( n ) − ∂∂t (cid:18) ∂ L ∂ ˙ M ( ˙ n ) (cid:19) = ∂ R ∂ ˙ M ( ˙ n ) , (16)where the Lagrangian density L and the Rayleigh function R are respectively given by [12, 25, 65–68] L = s ˙ n · ( n × M ) − U , R = sα ˙ n , (17)with α the Gilbert damping parameter, and s ( ≡ M s /γ ) the averaged angular momentum for twosublattices, where M s is the saturation magnetization, and γ is the gyromagnetic ratio.From Eqs. (15)-(17), we obtain the equations of motion for n and M as ˙ n = 1 s f M × n + τττ n , ˙ M = 1 s f n × n − α ˙ n × n + τττ M , (18)8here f M ( n ) is an effective field on M ( n ) , f M ≡ − a M − L ∂ n ∂x , f n ≡ A ∂ n ∂x + L ∂ M ∂x , and τττ M and τττ n are respectively spin-transfer torques acting on M and n , which we define below.In ferromagnets, the torques can be decomposed to adiabatic and non-adiabatic componentsas [31–33] τττ FM = τ aFM ∂ m ∂x − τ naFM m × ∂ m ∂x , (19)with τ a and τ na , the magnitudes of adiabatic and non-adiabatic torques. Likewise, the spin-transfertorques in each sublattice of an antiferromagnet can be written as τττ A = τ a A ∂ m AFM A ∂x − τ na A (cid:18) m AFMA × ∂ m AFM A ∂x (cid:19) ,τττ B = τ a B ∂ m AFM B ∂x − τ na B (cid:18) m AFM B × ∂ m AFM B ∂x (cid:19) . (20)Using the total and staggered magnetizations, Eq. (20) with the second-order expansion of smallparameters becomes [12, 25] τττ n = τ aAFM ∂ n ∂x ,τττ M = − τ naAFM n × ∂ n ∂x , (21)where τ aAFM = τ a A + τ a B , τ naAFM = τ na A + τ na B . (22)We note that in Eq. (21), τττ n and τττ M are obtained from τττ A and τττ B , which are computed using thelinear response theory described in the above section. One can then obtain τ aAFM and τ naAFM fromEq. (21), which are local quantities defined in a unit cell.To obtain effective spin-transfer torques, we integrate the local spin torques over the domainwall profile as follows. Using the collective coordinate approach with respect to the domain wallposition X and the domain wall angle φ , the equations of motion of an antiferromagnetic domainwall are readily obtained as [11, 53, 63] ρ ¨ X + 2 αs ˙ X = − s ˜ c AFM J ,ρ ¨ φ + 2 αs ˙ φ = 0 , (23)where ρ ≡ s /a , and ˜ c AFM J is the effective non-adiabatic spin-transfer torque integrated over theantiferromagnetic domain wall profile, given as ˜ c AFM J = − λ DM Z ∞−∞ dx (cid:20) τ naAFM ∂ n ∂x · ∂ n ∂X (cid:21) . (24)9
150 -75 0 75 150-1.00.01.0 m x m y m z m F M i (a) -50 -25 0 25 50-1.00.01.0 m x m y m z (b) m A F M i -75 -50 -25 0 25 50 75-1.00.01.0 x i / 2dx i / 2d x i / dx i / d n x n y n z (c) n A F M i -75 -50 -25 0 25 50 75-0.10.00.1 M x M y M z M x M y M z (d) M A F M i FIG. 1: (color online) Domain wall profiles of (a) ferromagnetic and (b) antiferromagnetic domain walls,where d is the atomic spacing and the domain wall width λ DW is 20 d . For the antiferromagnetic domain wall(b), the components of the staggered n and total M magnetic moments are shown in (c) and (d), respectively. By the same way, one can calculate the effective adiabatic spin-transfer torque, ˜ b AFM J , integratedover the antiferromagnetic domain wall profile, as ˜ b AFM J = L a Z ∞−∞ dx (cid:20) τ aAFM (cid:18) n × ∂ n ∂x (cid:19) · ∂ n ∂φ (cid:21) . (25)However, this adiabatic torque contribution, which is the third order of small parameters, is absentin Eq. (23) because Eq. (23) is obtained by expanding up to the second order. The adiabatictorque contribution appears only when the higher order terms are considered as in Ref. [24]. Thismeans that the adiabatic torque contribution to the antiferromagnetic domain wall motion is muchweaker than the non-adiabatic torque contribution. Therefore, as a leading-order contribution,the antiferromagnetic domain wall velocity in the steady state (i.e., ¨ X = 0 ) is determined by theeffective non-adiabatic torque, given as v DW = − ˜ c AFM J α . (26)10n the other hand, the equations of motion of ferromagnetic domain wall are given as [31, 33] ˙ φ + αλ DW ˙ X = − ˜ c FM J λ DW , − λ DW ˙ X + α ˙ φ = ˜ b FM J λ DW , (27)where [75] ˜ c FM J = − λ DW Z ∞−∞ dx (cid:20) τ naFM ∂ m ∂x · ∂ m ∂X (cid:21) . ˜ b FM J = 12 Z ∞−∞ dx (cid:20) τ aFM (cid:18) m × ∂ m ∂x (cid:19) · ∂ m ∂φ (cid:21) . (28)As well known, for ferromagnetic domain walls, both adiabatic and non-adiabatic contributionsappear in the equations of motion at the same order, in contrast to the case of antiferromagneticdomain wall motion.In the next section, we present numerical results of local and effective spin-transfer torques forferromagnetic and antiferromagnetic domain walls. IV. RESULTS AND DISCUSSIONA. Local spin-transfer torques
For numerical computation, we choose the number of unit cell N cell = 600 , the atomic spacing d = 0 . nm, the hopping parameter t H = 1 eV , and the number of k point in the transversedirection is × , which guarantees converged results. We use the Fermi energy E F = 0 . ,the exchange strength ∆ = 1 . , and the impurity scattering energy parameter V = 3 . ,unless specified.We calculate the non-equilibrium spin density at each atomic site iη using Eq. (6) and decomposeit into local adiabatic and non-adiabatic torques using Eqs. (19) and (21). Calculated localspin-transfer torques are shown in Fig. 2 for (a) ferromagnetic and (b) antiferromagnetic domainwalls. An interesting observation is that the local non-adiabatic torque for a relatively narrowantiferromagnetic domain wall changes its sign near the domain wall center ( x i = 0 ), indicated byan orange arrow. This negative local torque originates from spatial oscillation of non-equilibriumspin density near the domain wall, which results from the spin mistracking process [35, 75]. Wenote that for the same domain wall width ( λ DW = 6 d ), the local torques for ferromagnetic domainwalls do not show such sign change, suggesting that the spin mistracking is more pronouncedfor antiferromagnetic domain walls than for ferromagnetic domain walls. This enhanced spin11
100 -50 0 50 100-2.0-1.5-1.0-0.50.0 na F M | m x m | [ s - ] a F M | x m | [ s - ] DW -100 -50 0 50 100-0.2-0.10.0 a A F M | x n | [ s - ] (b)(a) DW -50 -25 0 25 500.00.51.01.52.0 x i / dx i / d x i / 2dx i / 2d na A F M | n x n | [ s - ] DW -50 -25 0 25 50-0.10.00.10.20.30.4 DW FIG. 2: (color online) Computed results of local spin-transfer torques. (a) The adiabatic τ a (left panel) andnon-adiabatic τ na (right panel) torques for ferromagnetic domain walls. (b) The adiabatic τ a (left panel)and non-adiabatic τ na (right panel) torques for antiferromagnetic domain walls. In (a) and (b), we compareresults for several domain wall widths, λ DW = 6 d, d and d . The orange arrow in the right panel of (b)shows negative local non-adiabatic torques for an antiferromagnetic domain wall. mistracking for antiferromagnetic domain walls may be understood as follow. According toRef. [35], for ferromagnetic domain walls, the non-adiabaticity due to the spin mistracking processis proportional to exp( − κλ DW /ζ ) where κ is a constant, ζ = E F / (∆ k F ) , and k F is the Fermiwave vector. Therefore, the non-adiabaticity increases exponentially with decreasing the exchangeinteraction ∆ . In antiferromagnets, the effective exchange interaction averaged over two sublatticesis zero. As a result, it is expected that the characteristic length scale ζ is very long. We note that thelarge non-adiabaticity or long characteristic length scale of transverse spin currents were recentlyreported in experiments using antiferromagnetically coupled ferrimagnets [53, 74].Another interesting observation is that the local adiabatic torque is sizable for both ferromagnetic[left panel of Fig. 2(a)] and antiferromagnetic [left panel of Fig. 2(b)] domain walls. We will12iscuss the relation between this non-zero local adiabatic torque and effective adiabatic torque forantiferromagnetic domain walls in the next section. Finally, in Fig. 2, it is observed that the signsof the torque are different for ferromagnetic and antiferromagnetic domain walls. However, thissign difference is found to depend on the parameters (not shown), which may depend on banddetails [38]. B. Effective spin-transfer torques for antiferromagnetic domain walls (a) b F M J [ m / s ] E F - 0.5 eV 0.0 eV 0.5 eV DW / d c F M J [ m / s ] e ff (b) b A F M J [ m / s ] E F - 0.5 eV 0.0 eV 0.5 eV DW / d c A F M J [ m / s ] FIG. 3: (color online) In (a), the effective adiabatic torque ˜b J (left panel) and the effective non-adiabatictorque ˜c J (right panel) for ferromagnetic domain walls are plotted as a reduced domain wall widths λ DW /d .We compare results for several Fermi energies, E F = − . eV, . eV, and . eV. In (b), under thesame condition we examine the effective adiabatic torque (left panel) and the effective non-adiabatic torque(right panel) for antiferromagnetic domain walls. The inset in the right panel of (a) shows the effectivenon-adiabaticity β eff in the ferromagnet case. In this section, we discuss effective adiabatic spin-transfer torques ( ˜ b FM J and ˜ b AFM J ) and effectivenon-adiabatic spin-transfer torques ( ˜ c FM J and ˜ c AFM J ), which are calculated by integrating the local13orques over the domain wall profile [see Eqs. (24), (25), and (28)].For ferromagnetic domain walls [Fig. 3(a)], the effective adiabatic ( ˜ b FM J ; left panel) and non-adiabatic ( ˜ c FM J ; right panel) torques are almost constant regardless of the domain wall widthranging from d to d . Even with a variation of E F , this insensitivity to the domain wall widthis maintained. Since both ˜ b FM J and ˜ c FM J are finite, one can define the effective non-adiabaticity β eff ( ≡ ˜ c FM J / ˜ b FM J ), which is almost a constant of the order of 0.05 in our model, consistent withprevious works [48–51]. In contrast, the effective torques for antiferromagnetic domain walls showtwo distinct features in comparison to those for ferromagentic domain walls. First, the effectiveadiabatic torque [ ˜ b AFM J ; left panel of Fig. 3(b)] is almost zero regardless of the Fermi energy anddomain wall width. Given that the local adiabatic torque for antiferromagnetic domain walls isfinite [left panel of Fig. 2(b)], this nearly zero effective adiabatic torque results from the symmetryof (cid:16) n × ∂ n ∂x (cid:17) · ∂ n ∂φ , which is zero when integrating over a whole domain wall profile [see the integralof Eq. (25)]. It also supports that the adiabatic torque contribution to the antiferromagnetic domainwall motion is almost absent.Second, the effective non-adiabatic torque [ ˜ c AFM J ; right panel of Fig. 3(b)] increases rapidly withdecreasing the domain wall width, which is consistent with that expected for the spin mistrackingprocess. To further validate the spin mistracking process as a main origin of the enhanced ˜ c AFM J for a narrower wall, we compute ˜ c FM J and ˜ c AFM J with varying the exchange parameter ∆ (Fig. 4).We find that ˜ c AFM J increases more rapidly than ˜ c FM J . These results support that the spin mistrackingprocess is responsible for the enhanced ˜ c AFM J for a narrower wall, especially in antiferromagnets. V. CONCLUSION
In this paper, we numerically compute the adiabatic and non-adiabatic spin-transfer torques forantiferromangetic domain walls. We find that the effective adiabatic torque in antiferromagneticdomain walls is almost zero, which means that the adiabatic torque does not affect dynamicsof antiferromagnetic domain walls. This negligible contribution of the adiabatic spin torque toantiferromagnetic domain wall motion is consistent with previous theories [25, 26] based on spinpumping and Onsager reciprocity. It is also consistent with a recent experiment [53] showing thatthe adiabatic torque contribution on the velocity of ferrimagnetic domain wall is proportional to theequilibrium net spin density δ s and is thus almost zero near the angular momentum compensationtemperature T A . 14 .0 1.1 1.2 1.3 1.4-0.14-0.13-0.12-0.11-0.10 1.0 1.1 1.2 1.3 1.40.100.110.120.130.14 c A F M J [ m / s ] (a)(b) [eV] c F M J [ m / s ] FIG. 4: (color online) The effective non-adiabatic torque ˜ c J with various exchange parameter ∆ in (a)ferromagnets and (b) antiferromagnets at domain wall width λ DW = 8 d . We also find that the effective non-adiabatic torque for antiferromagnetic domain walls canbe sizable and increases more rapidly with decreasing the domain wall width in comparison tothat for ferromagnetic domain walls. Our result supports that the rapid increase of non-adiabatictorque for antiferromagnetic domain walls is caused by the spin mistracking process, which is morepronounced in antiferromagnets than in ferromagnets.As a final remark, given that the effective adiabatic torque ˜ b AFM J is almost zero while the effectivenon-adiabatic torque ˜ c AFM J is finite, it is unphysical to define the non-adiabaticity ( β = ˜ c AFM J / ˜ b AFM J )for antiferromagnetic domain walls. For the same reason, the question about whether or not β isclose to the damping constant α , which has been a long-standing debate for ferromagnetic domainwalls [38, 76–79], is not justified for antiferromagnetic domain walls.15 cknowledgments This work was supported by the National Research Foundation of Korea (NRF) (NRF-2015M3D1A1070465, NRF-2017R1A2B2006119) and by the Korea Institute of Science andTechnology (KIST) Institutional Program (project no. 2V05750, 2E29410). G.G. was supportedby NRF-2019R1I1A1A01063594. H.-W.L. was supported by NRF-2018R1A5A6075964. [1] A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A , 3098 (2011).[2] R. Duine, Nat. Mater. , 344 (2011).[3] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotechnol. , 231 (2016).[4] T. Satoh, S.-J. Cho, R. Iida, T. Shimura, K. Kuroda, H. Ueda, Y. Ueda, B. A. Ivanov, F. Nori, and M.Fiebig, Phys. Rev. Lett. , 077402 (2010).[5] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. Mahrlein, T. Dekorsy, M. Wolf, M. Fiebig, A.Leitenstorfer, and R. Huber, Nat. Photonics , 31 (2011).[6] R. Cheng, D. Xiao, and A. Brataas, Phys. Rev. Lett. , 207603 (2016).[7] Ø. Johansen and J. Linder, Sci. Rep. , 33845 (2016).[8] R. Khymyn, I. Lisenkov, V. Tiberkevich, B. A. Ivanov, and A. Slavin, Sci. Rep. , 43705 (2017).[9] D.-K. Lee, B.-G. Park, and K.-J. Lee, Phys. Rev. Applied , 054048 (2019).[10] O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. Lett. , 017202 (2016).[11] T. Shiino, S.-H. Oh, P. M. Haney, S.-W. Lee, G. Go, B.-G. Park, and K.-J. Lee, Phys. Rev. Lett. ,087203 (2016).[12] S. K. Kim, K.-J. Lee, and Y. Tserkovnyak, Phys. Rev. B , 140404(R) (2017).[13] K.-J. Kim, S. K. Kim, T. Tono, S.-H. Oh, T. Okuno, W. S. Ham, Y. Hirata, S. Kim, G. Go, Y.Tserkovnyak, A. Tsukamoto, T. Moriyama, K.-J. Lee, and T. Ono, Nat. Mater. , 1187-1192 (2017).[14] L. Caretta, M. Mann, F., Büttner, K. Ueda, B. Pfau, C. M. Günther, P. Hessing, A. Churikoa, C. Klose,M. Schneider, D. Engel, C. Marcus, D. Bono, K. Bagschik, S. Eisebitt, and G. S. D. Beach, Nat.Nanotechnol. , 1154-1160 (2018).[15] S. A. Siddiqui, J. Han, J. T. Finley, C. A. Ross, and L. Liu, Phys. Rev. Lett. , 057701 (2018).[16] S.-H. Oh and K.-J. Lee, J. Magn. , 196 (2018).[17] A. S. Núñez, R. A. Duine, P. Haney, and A. H. MacDonald, Phys. Rev. B , 214426 (2006).
18] Z. Wei, A. Sharma, A. S. Núñez, P. M. Haney, R. A. Duine, J. Bass, A. H. MacDonald, and M. Tsoi,Phys. Rev. Lett. , 116603 (2007).[19] S. Urazhdin and N. Anthony, Phys. Rev. Lett. , 046602 (2007).[20] P. M. Haney and A. H. MacDonald, Phys. Rev. Lett. , 196801 (2008).[21] H. B. M. Saidaoui, A. Manchon, and X. Waintal, Phys. Rev. B , 174430 (2014).[22] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. , 057601 (2014).[23] Y. Xu, S. Wang, and K. Xia, Phys. Rev. Lett. , 226602 (2008).[24] A. C. Swaving and R. A. Duine, Phys. Rev. B , 054428 (2011).[25] K. M. D. Hals, Y. Tserkovnyak, A. Brataas, Phys. Rev. Lett. , 107206 (2011).[26] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and A. Brataas, Phys. Rev. Lett. , 127208 (2013).[27] X. Zhang, Y. Zhou, and M. Ezawa, Sci. Rep. , 24795 (2016).[28] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. , 147203 (2016).[29] Y. Yamane, J. Ieda, and J. Sinova, Phys. Rev. B , 054409 (2016).[30] Y. Tserkovnyak, and H. Ochoa, Phys. Rev. B , 100402(R) (2017).[31] G. Tatara and H. Kohno, Phys. Rev. Lett. , 086601 (2004).[32] S. Zhang and Z. Li, Phys. Rev. Lett. , 127204 (2004).[33] A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys. Lett. , 990 (2005).[34] Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B , 144405 (2006).[35] J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B , 054428 (2006).[36] J.-i. Ohe and B. Kramer, Phys. Rev. Lett. , 027204 (2006).[37] G. Tatara, H. Kohno, J. Shibata, Y. Lemaho, and K.-J. Lee, J. Phys. Soc. Japan , 054707 (2007).[38] I. Garate, K. Gilmore, M. D. Stiles, and A. H. MacDonald, Phys. Rev. B , 104416 (2009).[39] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev. Lett. , 027204 (2007).[40] K. Gilmore, I. Garate, A. H. MacDonald, and M. D. Stiles, Phys. Rev. B , 224412 (2011).[41] A. Manchon and K.-J. Lee, Appl. Phys. Lett. , 022504 (2011).[42] C. A. Akosa, W.-S. Kim, A. Bisig, M. Kläui, K.-J. Lee, and A. Manchon, Phys. Rev. B , 094411(2015).[43] K.-W. Kim, K.-J. Lee, H.-W. Lee, and M. D. Stiles, Phys. Rev. B , 224426 (2015).[44] M. Hayashi, L. Thomas, Ya.B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, and S.S.P. Parkin, Phys. Rev.Lett. , 197207 (2006).[45] R. Moriya, L. Thomas, M. Hayashi, Y.B. Bazaliy, C. Rettner, and S.S.P. Parkin, Nat. Phys. , 368 , 066603 (2008).[47] O. Boulle, J. Kimling, P. Warnicke, M. Kläui, U. Rüdiger, G. Malinowski, H.J.M. Swagten, B.Koopmans, C. Ulysse, G. Faini, Phys. Rev. Lett. , 216601 (2008).[48] M. Eltschka, M. Wötzel, J. Rhensius, S. Krzyk, U. Nowak, M. Kläui, T. Kasama, R.E. Dunin-Borkowski, L.J. Heyderman, H.J. van Driel, R.A. Duine, Phys. Rev. Lett. , 056601 (2010).[49] C. Burrowes, A.P. Mihai, D. Ravelosona, J.-V. Kim, C. Chappert, L. Vila, A. Marty, Y. Samson, F.Garcia-Sanchez, L.D. Buda-Prejbeanu, I. Tudosa, E.E. Fullerton, J.-P. Attané, Nat. Phys. , 17 (2010).[50] K. Sekiguchi, K. Yamada, S.-M. Seo, K.-J. Lee, D. Chiba, K. Kobayashi, and T. Ono, Phys. Rev. Lett. , 017203 (2012).[51] J.-Y. Chauleau, H. G. Bauer, H. S. Körner, J. Stigloher, M. Härtinger, G. Woltersdorf, and C. H. Back,Phys. Rev. B , 020403(R) (2014).[52] A. Bisig, C. A. Akosa, J.-H. Moon, J. Rhensius, C. Moutafis, A. von Bieren, J. Heidler, G. Kiliani,M. Kammerer, M. Curcic, M. Weigand, T. Tyliszczak, B. Van Waeyenberge, H. Stoll, G. Schütz, K.-J.Lee, A. Manchon, and M. Kläui, Phys. Rev. Lett. , 277203 (2016).[53] T. Okuno, D.-H. Kim, S.-H. Oh, S. K. Kim, Y. Hirata, T. Nishimura, W. S. Ham, Y. Futakawa, H.Yoshikawa, A. Tsukamoto et al. , Nat. Electron. , 239 (2019).[54] P. Baláž, V. K. Dugaev, and J. Barnaś, Phys. Rev. B. , 024416 (2012).[55] R. Lake, G. Klimeck, R. C. Bowen, and D. Jovanovic, J. Appl. Phys. , 7845 (1996).[56] S. Datta, ”Electronic Transport in Mesoscopic Systems” (Cambridge University Press, Cambridge)1997.[57] J. H. Oh, M. Shin, and S.-H. Lee, J. Appl. Phys. , 233706 (2013).[58] A. Kamenev and A. Andreev, Phys. Rev. B. , 2218 (1999).[59] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, Course of Theoretical PhysicsVol. 8 (Pergamon, Oxford, 1960).[60] S.-H. Oh, S. K. Kim, J. Xiao, and K.-J. Lee, Phys. Rev. B , 174403 (2019).[61] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T. Shinjo, Phys. Rev. Lett. , 077205(2004).[62] E. G. Tveten, T. Müller, J. Linder, and A. Brataas, Phys. Rev. B , 104408 (2016).
63] O. A. Tretiakov, D. Clarke, G.-W. Chern, Ya. B. Bazaliy, and O. Tchernyshyov, Phys. Rev. Lett. ,127204 (2008).[64] N. Papanicolaou, Phys. Rev. B , 15062 (1995).[65] S.-H. Oh, S. K. Kim, J. Xiao, and K.-J. Lee, Phys. Rev. B. , 174403 (2019).[66] A. F. Andreev and V. I. Marchenko, Sov. Phys. Usp. , 21 (1980).[67] A. Chiolero and D. Loss, Phys. Rev. B , 738 (1997).[68] B. A. Ivanov and A. L. Sukstanskii, Solid State Commun. , 523 (1984).[69] Y. Aharonov and A. Stern, Phys. Rev. Lett. , 3593 (1992).[70] M. D. Stiles and A. Zangwill, J. Appl. Phys. , 6812 (2002).[71] X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph, Phys. Rev. B , 12317 (2000).[72] M. D. Stiles and A. Zangwill, Phys. Rev. B , 014407 (2002).[73] P. Merodio, A. Kalitsov, H. Bea, V. Baltz, and M. Chshiev, Appl. Phys. Lett. , 122403 (2014).[74] J. Yu, D. Bang, R. Mishra, R. Ramaswamy, J. H. Oh, H.-J. Park, Y. Jeong, P. V. Thach, D.-K. Lee, G.Go et al. , Nat. Mater. , 29 (2019)[75] K.-J. Lee, M. D. Stiles, H.-W. Lee, J.-H. Moon, K.-W. Kim, and S.-W. Lee, Phys. Rep. , 89 (2013).[76] O. Boulle, G. Malinowski, M. Kl´’aui, Mater. Sci. Eng. R-Rep. , 159 (2011).[77] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn. , 133706 (2006).[78] R. A. Duine, A. S. Núñez, J. Sinova, and A. H. MacDonald, Phys. Rev. B , 214420 (2007).[79] Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W. Bauer, Phys. Rev. B , 144405 (2006)., 144405 (2006).