Theory of spin motive force in one-dimentional antiferromagnetic domain wall
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Journal of the Physical Society of Japan
Theory of Spin Motive Forcein One-Dimensional Antiferromagnetic Domain Wall
Akira Okabayashi ∗ and Takao Morinari † Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
We present the theory of the spin motive force in antiferromagnets. We consider a one-dimensional antiferromagneticdomain wall strongly coupled with conduction electrons via an exchange interaction. We carry out a unitary transfor-mation that rotates the spin coordinate system of the conduction electron locally, so that the quantization axis is in thedirection of the localized spin. By numerically solving the time dependent Schr¨odinger equation, we clearly demon-strate that the spin motive force acts on the conduction electron. The result suggests that there is no distinction betweenantiferromagnets and ferromagnets from the viewpoint of the basic phenomenon relevant to spintronics.
In the field of spintronics, one of the technical issues iscreating spin-polarized current. A metal with localized spinsforming a spin texture, such as a domain wall, can create aspin current under a magnetic field in the presence of strongcoupling between conduction electrons and localized spins. The precession of localized spins induced by the magneticfield leads to a time dependent Berry phase e ff ect. ThisBerry phase e ff ect gives rise to a motive force, which iscalled the spin motive force, and creates spin-polarized cur-rent. Electrical voltage generated by such a Berry phase ef-fect has been confirmed experimentally. On the otherhand, owing to the conservation of spin angular momentum,a torque on localized spins, the so-called spin-transfer torquee ff ect,
14, 15 is created by the spin-polarized current.Most spintronics research focuses on ferromagnets, but an-tiferromagnets can also be used to manipulate spin informa-tion. The important di ff erence between ferromagnets andantiferromagnets is that the smoothly varying field is not lo-cal magnetization but local staggered magnetization. By us-ing the smoothly varying local staggered magnetization, current-induced torque e ff ects in antiferromagnets have beenpredicted theoretically
20, 21 and confirmed experimentally
The reverse of this e ff ect, that is, the spin motive force forantiferromagnets, has been formulated by a semiclassical ap-proximation.
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Starting from the band picture, the spatialvariation of spins is included within the semiclassical approx-imation. An important issue is whether the semiclassical ap-proximation is justified in the case of the antiferromagnets.In this work, we develop a theory for the spin motive forcecreated by the staggered magnetization dynamics without re-lying on the semiclassical approximation. We consider a sys-tem consisting of a one-dimensional antiferromagnetic do-main wall and a conduction electron with the exchange in-teraction between localized spins and the conduction electron.By solving the time dependent Schr¨odinger equation for a sin- ∗ E-mail: [email protected] † E-mail: [email protected] gle conduction electron, we show that, under a magnetic fieldalong the system, the rigidly precessing domain wall givesrise to the spin motive force in the strong coupling limit of theexchange interaction. This spin motive force is not describedby the semiclassical approximation. There is a combined ef-fect of the sign change of the potential created by the timedependent Berry phase and the spin flip at each hopping ofthe conduction electron. This combined e ff ect leads to the netspatial gradient of the e ff ective potential associated with thestaggered potential, and so the spin motive force acts on theconduction electron.We consider a one-dimensional antiferromagnet. The co-ordinate axes are defined as shown in Fig. 1. The localizedmoment with spin S at site j is represented by S j = S (sin θ j cos φ j , sin θ j sin φ j , cos θ j ) . We assume that there is a static antiferromagnetic domainwall,
19, 25, 26 with θ j = π (cid:18) − j ℓ (cid:19) , as shown in Fig. 1. Here, ℓ is the length scale of the domainwall and we take the lattice constant to be unity.A conduction electron at site j interacts with the localizedspin at the same site via the exchange interaction J . We applya magnetic field along the z -axis, B = (0 , , B ). Under thismagnetic field, localized moments precess about the z -axis.Here, we assume that the domain wall is rigidly precessing sothat φ j = π t / T . Here, 1 / T ≡ g µ B B / (2 π ) with g as the g-factor and µ B as theBohr magneton. To focus on the spin motive force, we do notconsider the spin transfer torque created by the current.The Hamiltonian of the system is given by H = − η X j ( c j † c j + + h . c . ) − J X j S j · ( c j † σ c j ) . (1)
1. Phys. Soc. Jpn.
Fig. 1. (Color online) Antiferromagnetic domain wall. Arrows representlocalized moments constituting the domain wall. We take the coordinate sys-tem as shown in the figure. The thick arrow represents the applied magneticfield. Under the magnetic field, the domain wall precesses about the z -axis. θ is the angle between the direction of the local moment and the z -axis. Wedefine θ at each site and denote it by θ j at site j . Here, η is the hopping parameter for the conduction electron.The operator c † j is a creation operator of the conduction elec-tron at site j in the spinor form, c † j = (cid:16) c † j ↑ c † j ↓ (cid:17) . The vector σ has components of the Pauli matrices: σ = (cid:16) σ x , σ y , σ z (cid:17) . Note that the Hamiltonian does not contain the dynamics ofthe domain wall because it is completely determined as men-tioned above. The time dependence of the localized momentsenter the Hamiltonian through the time dependence of S j .Now, we take the strong coupling limit with respect to J . Inthe strong coupling limit, the spin of the conduction electronat site j is in the direction of S j . We rotate the spin coordi-nate system of the conduction electron locally, so that the z -direction is in the direction of S j . Such a rotation is carriedout by the following unitary transformation, c j → U j c j , with U j = d j · σ , Here, d j = p + S jz ) (cid:16) S jx , S jy , + S jz (cid:17) . After carrying out this transformation, the staggered poten-tial ( − j JS σ z acts on the conduction electron. This potentialterm is JS σ z for j even sites, i.e., sublattice A, and − JS σ z for j odd sites, i.e., sublattice B. To remove the sign di ff erence inthis potential term, we perform an additional unitary transfor-mation, c j → i σ y c j , at sublattice B. Thus, the HamiltonianEq. (1) is rewritten as H = − η X j ∈ A (cid:16) c j † U † j U j + i σ y c j + + c j † U † j U j − i σ y c j − + h . c . (cid:17) − JS X j c j † σ z c j + i ~ X j ∈ A c † j (cid:16) U † j ∂ t U j (cid:17) c j − i ~ X j ∈ B c † j (cid:16) U † j ∂ t U j (cid:17) c j . (2) V / s t ! z Fig. 2. (Color online) Staggered potential V st as a function of z . Becauseof the antiferromagnetic nature, the time dependent Berry phase creates thestaggered potential V st Eq. (6), as described in the text.
The third and fourth terms have been added so that the Hamil-tonian is consistent with the equation of motion of the creationand annihilation operators.The e ff ect of the domain wall dynamics on the conductionelectron is conveniently represented by the gauge field a j = − i ~ U j † ∂ t U j , (3) a j = i ~ U j † ∇ j U j . (4)Here, we define ∇ j as ∇ j f j = ( f j + − f j − ) / f j . The e ff ective electric field is defined by e j = − ∂ t a j − ∇ a j = − ~ σ · (cid:16) ∂ t d j × ∇ d j (cid:17) . (5)If we define a staggered potential by V st = ( − j h a j i ↑↑ = − ( − j h a j i ↓↓ , (6)then V st is a smooth function with respect to z , as shown inFig. 2. In the case of ferromagnets, the e ff ective electric fieldgiven by Eq. (5) leads to the spin motive force. However, inthe case of antiferromagnets, it is not clear whether the fieldgiven by Eq. (5) leads to the spin motive force. Nevertheless,as will be shown later, the field (5) leads to the spin motiveforce in the case of antiferromagnets as well, and that is con-firmed by numerical simulation.To verify the creation of the spin motive force in the sys-tem, we solve the time dependent Schr¨odinger equation forthe conduction electron with the Hamiltonian Eq. (2). We rep-resent the eigenstate of the system by | ψ ( t ) i . The wave func-tion ψ j σ ( t ) of the conduction electron at site j with spin σ isdefined by: | ψ ( t ) i = X j , σ ψ j σ ( t ) c † j σ | i . (7)Here, | i denotes the vacuum state. We solve the time evo-lution of ψ j σ ( t ) using the fourth-order Runge-Kutta methodunder an open boundary condition. The time evolution of thewave function is shown in Fig. 3 in the case of J /η =
2. Phys. Soc. Jpn. !! ! " t () j ! " t () j ! " t () j ! " t () j = t (a) 0 = t (c) 200 = t (d) 400 !! ! ! ! " = t (b) 50 z zzz Fig. 3. (Color online) Time evolution of the wave packet. The number of lattice sites is 600. We take ℓ =
200 and J /η =
5. For T , we take η T / ~ =
5. As theinitial state at t =
0, we put a spin-up state at j = z -direction [(c) and (d)]. The unit of time is taken as ~ /η . As an initial condition at t =
0, we take the spin-up stateat j = ψ j σ (0) = C δ σ, ↑ exp (cid:16) − j /λ (cid:17) with C as the normalization con-stant [Fig. 3(a)]. We consider a localized state at the originand take λ = .
5. The initial localized spin-up state quicklyexpands, as shown in Fig. 3(b). The wave packet consists ofspin-up states at A sublattice and spin-down states at B sublat-tice. This feature is understood from the potential term men-tioned above. As time elapses, the wave packet clearly movesto the direction of the positive z -axis [Figs. 3(c) and 3(d)].This motion of the wave packet demonstrates the presence ofthe spin motive force in the antiferromagnet. We carried out anumerical simulation for J /η = J , which a ff ects the direction, the direc-tion of the wave function propagation is reversed as well. Wealso note that the initial spin state does not a ff ect the propaga-tion direction. This is because the spin state of the conductionelectron takes the lower energy state upon propagation due tothe potential energy associated with the exchange interaction.The time dependence of the wave packet motion becomes more clearly seen from the expectation value of the positionof the conduction electron, X j j (cid:12)(cid:12)(cid:12) ψ j σ ( t ) (cid:12)(cid:12)(cid:12) . (8)This quantity is shown in Fig. 4. When the wave packet is farfrom the edges of the domain wall, the motion of the wavepacket is in agreement with the motion of the correspondingclassical particle under a constant acceleration. We find X j j (cid:12)(cid:12)(cid:12) ψ j σ ( t ) (cid:12)(cid:12)(cid:12) ≃ π m ℓ T t , (9)for t ≪ ℓ . This is understood from the force acting on theconduction electron created by the e ff ective electric field Eq.(5). From the fitting of the data, we find that m ≃ .
5, whichis consistent with the picture above.We carried out a similar calculation in the ferromagneticdomain wall case and found the same result. Within the strongcoupling limit, there is no di ff erence.The appearance of the spin motive force in the precess-ing antiferromagnetic domain wall is clearly understood asfollows. The key is the staggered potential V st . A naive ex-pectation is that the spin motive force cancels out because ofthe sign di ff erence in the spin motive force Eq. (6) at A and
3. Phys. Soc. Jpn. !!!! ==== ! ! t / ! ! " j t () j j Fig. 4. (Color online) Time evolution of the expectation value of the posi-tion of the conduction electron for ℓ = , , and t ∼ ℓ when the wave packet reaches the edge ofthe domain wall. B sublattices. However, the spin of the conduction electronflips at each hopping. The spin state of the conduction elec-tron changes at each hopping through the term U † j U j + i σ y .We note that U † j U j + is close to the unit matrix because ofthe smooth variation of the staggered magnetization with re-spect to j and t . Because of the presence of the term i σ y , thespin of the conduction electron flips. The spin flip leads tothe sign change in V st , while there is a sign di ff erence in V st between A and B sublattices. Therefore, there are two signchanges, and thus no cancellation occurs for the spin motiveforce created by the staggered potential V st . We note that thegauge field appearing in the components of the matrix U † j U j + does not play an important role. In a one-dimensional antifer-romagnetic domain wall case, the e ff ect of the gauge field in U † j U j + vanishes under the precession created by the mag-netic field. The vector potential given by Eq. (4) does not playan important role, which is clearly understood as follows. Inthe one-dimensional antiferromagnetic domain wall, which isshown in Fig. 1, the z -component of the vector potential iswritten as[ a j ] z = − ~ σ z ( ∇ j φ ) sin θ j j ∈ A ) − ~ ( − i σ y ) σ z ( ∇ j φ ) sin θ j ! ( i σ y ) ( j ∈ B ) . We note that the azimuthal angle φ j is independent of site j .Therefore, Eq. (4) is negligible.Now, we comment on the semiclassical approximation.
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One can derive an e ff ective Hamiltonian by starting from theband picture and then including the spatial variation of thespins. In such an approximation, there appears a term thatdepends on the dynamics and spatial variation of the spins.However, the term vanishes in the strong coupling limit. Thisis simply because the semiclassical approximation is not jus- tified in the strong coupling limit because the spatial variationof the spins is not properly taken into account. In contrast, thespatial variation of the spins is exactly taken into account inour formulation based on the unitary transformation.To summarize, we have numerically demonstrated the cre-ation of the spin motive force in a rigidly precessing anti-ferromagnetic domain wall. We have solved the time depen-dent Sch¨odinger equation and clearly shown the motion ofthe wave packet of the conduction electron that reflects thepresence of the spin motive force. The motion of the con-duction electron is understood by the combined e ff ect of theforce created by the staggered potential V st and the spin flip ateach hopping. Our numerical simulation suggests that there isno qualitative distinction between antiferromagnets and ferro-magnets in the strong coupling limit with respect to the spinmotive force. Acknowledgements
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