Magnetic-Field-Driven Antiferromagnetic Domain Wall Motion
JJournal of the Physical Society of Japan
FULL PAPERS
Magnetic-Field-Driven Antiferromagnetic Domain Wall Motion
Jotaro J. Nakane and Hiroshi Kohno
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
We theoretically study the antiferromagnetic domain wall motion actuated by an inhomogeneous external magneticfield. The Lagrangian and the equations of motion of antiferromagnetic spins under an inhomogeneous magnetic field arederived, first in terms of the N´eel vector, and then using collective coordinates of the domain wall. A solution is found thatdescribes the actuation of a domain wall by an inhomogeneous field, in which the motion is initiated by a paramagneticresponse of wall magnetization, which is then driven by a Stern-Gerlach like force. The e ff ects of pinning potential arealso investigated. These results are in good agreement with atomistic simulations. While the present formulation containsthe so-called intrinsic magnetization associated with N´eel texture, a supplementary discussion is given to reformulatethe theory in terms of physical magnetization without the intrinsic magnetization.
1. Introduction
Antiferromagnets have recently gained high expectationsas a candidate for next generation spintronic devices, ow-ing to its robustness to external magnetic field, THz rangespin dynamics, and the variety of hosting materials.
Thedemonstration of antiferromagnetic domain switching usingthe N´eel spin-orbit torque encouraged researchers to pursueantiferromagnetic spintronics. Still, despite its appealing fea-tures and high publicity, the manipulation and detection ofspin textures in antiferromagnets remain a challenge due toits unsusceptible nature.A domain wall is one of the topological objects in antifer-romagnets that may prove useful in memory devices, and itscreation, manipulation and detection has been the scope in nu-merous studies. Some pioneering theoretical works show thatantiferromagnetic domain walls can be driven by spin waves and spin-orbit torques, though experimental reports on an-tiferromagnetic domain wall motion are currently limited toindirect observations. Domain walls in materials similar toantiferromagnets, such as synthetic antiferromagnets andferrimagnets around the angular momentum compensationtemperature, have also been studied, which allow for aneasier observation and manipulation of domain walls.Recently, it was proposed that antiferromagnetic spin tex-tures give rise to intrinsic magnetization.
It was demon-strated that this intrinsic magnetization couples to externalmagnetic fields, and may be used to actuate antiferromag-netic domain wall motion. In this paper, we reinvestigate thisproblem starting from the same model. We found an addi-tional coupling of the N´eel vector to the inhomogeneous mag-netic field, similar to the one in Ref., which nullifies thee ff ect of the above intrinsic magnetization. With the new La-grangian obtained, we find an alternative mechanism for do-main wall motion actuated by an inhomogeneous externalmagnetic field.This paper is organized as follows. After presenting inSec. 2 the Lagrangian and equations of motion for the anti-ferromagnetic order parameter (N´eel vector) under an inho-mogeneous magnetic field, we derive in Sec. 3 the equationsof motion in terms of collective coordinates of a domain wall.By solving the equations, we find a solution in which the do- main wall position grows exponentially with time. Interest-ingly, there is no domain wall motion in the absence of damp-ing. We also study the e ff ects of pinning introduced by a lo-cal modulation of easy-axis magnetic anisotropy. Finally, weperform an atomistic simulation to test the analytical results,and see that they are in good agreement. As a supplementarydiscussion, we identify the physical magnetization and refor-mulate the theory therewith.
2. Model
In this section, we derive an e ff ective Lagrangian thatdescribes low-frequency, long-wavelength spin dynamics ofan antiferromagnet, starting from a lattice spin model. Weclosely follow the procedure described in Ref., except thatwe consider the inhomogeneity of the external magnetic fieldfrom the beginning of the formulation. We start with a Heisenberg Hamiltonian for classical spinson a one-dimensional lattice with antiferromagnetic exchangecoupling J >
0, easy-axis anisotropy K >
0, and Zeemancoupling, H = J (cid:88) i S i · S i + − K (cid:88) i ( S i · e z ) + γ (cid:126) (cid:88) i H i · S i , (1)where γ is the gyromagnetic ratio. The localized spins andmagnetic field at lattice site i are written as S i and H i , respec-tively. In a typical easy-axis antiferromagnet, the exchangeenergy dominates J (cid:29) K , giving rise to relatively thick do-main walls (e.g. 150nm for NiO ). Therefore, we work inthe exchange approximation J (cid:29) K (and J (cid:29) γ (cid:126) | H i | ), andfocus on spin textures with slow spatial / temporal variations.Let us write the antiferromagnetic spins in terms of theN´eel and uniform moments, n n = S n − S n + S , l n = S n + S n + S , (2)respectively, where | S i | = S is a constant. ( i is the site index,and n is the unit-cell index.) The original spins are retrievedby S n = S ( l n + n n ) , S n + = S ( l n − n n ) . (3) a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b . Phys. Soc. Jpn. FULL PAPERS
With n n and l n , the Hamiltonian is written as H = (cid:80) n h n ,where h n = JS (cid:40) l n − n n ) − ( l n − l n − ) + ( n n − n n − ) + ( n n − n n − ) · l n − n n · ( l n − l n − ) (cid:41) − KS (cid:8) ( l zn ) + ( n zn ) (cid:9) + γ (cid:126) S (cid:110) ( H n + H n + ) · l n − ( H n + − H n ) · n n (cid:111) . (4)We adopt the continuum approximation and write n n − n n − (cid:39) a ∂ x n and l n − l n − (cid:39) a ∂ x l , where a is the lattice constant.The magnetic field is also assumed slowly-varying (having nostaggered component) and thus H n + H n + (cid:39) H , H n + − H n (cid:39) a ∂ x H , and the summation is replaced by an integration (cid:88) n = (cid:90) dx a . Thus the Hamiltonian is written as H = JS (cid:90) dx a (cid:26) l − n ) + (2 a ) (cid:2) ( ∂ x n ) − ( ∂ x l ) (cid:3) + (2 a ) (cid:2) l · ( ∂ x n ) − n · ( ∂ x l ) (cid:3)(cid:27) − KS (cid:90) dx a (cid:8) ( l z ) + ( n z ) (cid:9) + γ (cid:126) S (cid:90) dx a (cid:8) H · l − a ( ∂ x H ) · n (cid:9) . (5)As we shall see later, | l | = O ( a /λ ) in the exchange approxima-tion ( J (cid:29) K ), where λ = a √ J / K is the typical length scaleof spatial variation. We discard the terms which are of higherorder in l , such as Kl = O ( Jl ) and ( a ∂ x l ) = O ( l ). Usingthe constraints, n + l = , n · l = , (6)or n · ∂ x l = − ( ∂ x n ) · l , which follow from | S i | = const . , weobtain H (cid:39) JS (cid:90) dx a (cid:26) l + a ∂ x n ) + a l · ( ∂ x n ) − K J ( n z ) (cid:27) + γ (cid:126) S (cid:90) dx a (2 H · l − a ( ∂ x H ) · n ) . (7)As seen, the magnetic field couples not only to l but also to n . To derive the Lagrangian, L = L − H , we next look at itskinetic part, L = (cid:126) S (cid:88) i ˙ φ i cos θ i = (cid:126) S (cid:88) n (cid:16) ˙ φ n cos θ n + ˙ φ n + cos θ n + (cid:17) , (8)where the spins are expressed as S i = S (sin θ i cos φ i , sin θ i sin φ i , cos θ i ) . (9)Let θ n + = π − ( θ n + δθ n + ) and φ n + = π + ( φ n + δφ n + ), sothat the neighboring spins are totally antiparallel when δθ = δφ =
0. To leading order in δθ and δφ , one finds L = (cid:126) S (cid:90) dx a l · ( n × ˙ n ) , (10)up to a total time derivative. This shows that 2 (cid:126) S ( l × n ) isthe canonical momentum conjugate to n . As a side note, theemergent gauge field, A AF , i = l · ( ∂ i n × n ), demonstrated in for canted antiferromagnets complies with this kinetic term.Damping is taken into account by Rayleigh’s dissipationfunction, W = α (cid:126) S (cid:88) i ˙ S i = α (cid:126) S (cid:88) n (˙ l n + ˙ n n ) . (11)In the continuum approximation, we write W = (cid:126) S (cid:90) dx a (cid:32) α l ˙ l + α n ˙ n (cid:33) , (12)where we introduced two damping constants, α l and α n , formore generality. By noting the constraints, Eq. (6), the equations of motionare obtained as ˙ n = (cid:32) s δ H δ n + α n ˙ n (cid:33) × l + (cid:32) s δ H δ l + α l ˙ l (cid:33) × n ˙ l = (cid:32) s δ H δ n + α n ˙ n (cid:33) × n + (cid:32) s δ H δ l + α l ˙ l (cid:33) × l , (13)in agreement with Ref. Here, we defined the angular mo-mentum density s = (cid:126) S / (2 a ). Note that these equations ofmotion respect the constraints, Eq. (6). The first equation ofEq. (13) can be written as l = (cid:126) JS ( n × ˙ n − γ H ⊥ ) − a ∂ x n ) , (14)to leading order in l , where H ⊥ = n × ( H × n ) is the compo-nent perpendicular to n . The first term ( ∼ n × ˙ n ) embodies themomentum nature of l conjugate to n , namely, it is propor-tional to the “velocity” ˙ n . The second term is due to cantinginduced by H . The linear dependence on the field H ⊥ indi-cates that the response to it is paramagnetic. The last term( ∼ ∂ x n ) is referred to in Ref. as the intrinsic magnetic mo-ment induced by the N´eel texture. Substituting this result intothe second equation of Eq. (13), one obtains the equation ofmotion written solely by the N´eel vector, − (cid:126) J ¨ n × n = (cid:20) − JS a ( ∂ x n ) − KS ( n · ˆ z ) ˆ z + (cid:126) S α n ˙ n + γ (cid:126) J ( H · n ) ˙ n × n + ( γ (cid:126) ) J ( H · n ) H − γ (cid:126) J ( ˙ H × n ) (cid:21) × n . (15)As an important observation, the terms with ∂ x H have beencanceled out. To see what happened, let us go back to theLagrangian L , or its density, L = s l · ( n × ˙ n ) − s γ (cid:26) H · l − a ∂ x H ) · n (cid:27) − JS a (cid:26) l + a ∂ x n ) + a l · ( ∂ x n ) − K J ( n z ) (cid:27) . (16)Since this is quadratic in l , one can “integrate out” l and obtain
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FULL PAPERSFig. 1. (Color online) This figure illustrates that the uniform moment,Eq. (2), in the presence of antiferromagnetic spin texture depends on thechoice of unit cell. The long arrows represent the atomic spins, S i . The topred arrows and the bottom green arrows represent the “uniform moments”locally defined by the average of the two neighboring spins, ( S i + S i + ) / the one written solely by the N´eel vector, L = (cid:126) Ja ˙ n − JS a ∂ x n ) + KS a ( n z ) + ( γ (cid:126) ) Ja ( H × n ) + γ (cid:126) S ∂ x ( H · n ) − s n × ˙ n ) · (cid:34) a ∂ x n + γ (cid:126) JS H (cid:35) . (17)Here, the Zeeman coupling of the “intrinsic magnetization”( ∼ H · ∂ x n ) has been combined with the additional term ( ∼ n · ∂ x H in Eq. (5)), forming a total derivative, ∂ x ( H · n ). Thisis why the intrinsic moment does not appear in the equationof motion, Eq. (15). Intuitively, this can be understood fromFig. 1, which shows that the texture-induced uniform momentdepends on the unit-cell choice; if another choice is made,it changes sign. This means that the texture-induced uniformmoment is an artifact of the parametrization of Eq. (2), anddoes not appear in physical phenomena.Two small notes. First, the quadratic term in H , namely, thefifth term ∼ ( H · n ) H in Eq. (15) or the fourth term ∼ ( H × n ) in Eq. (17), has the same form as the magnetic anisotropyterm, hence the magnetic field acts as a hard-axis anisotropy.(But the e ff ect is small, see below.) Second, the sixth term ∼ ( n × ˙ n ) · ∂ x n in Eq. (17) is “topological”, and does notcontribute to the equation of motion, hence can be omitted.
3. Domain Wall Motion
In this section, we study the domain wall motion in an in-homogeneous magnetic field using collective coordinates. Wetake the magnetic field to be in the easy-axis (ˆ z -) directionwith magnitude linearly varying in space, H = H z ˆ z = ( H + H x ) ˆ z . (18) To work with collective coordinates of the antiferromag-netic domain wall, one must first obtain a static domain wallsolution. Dropping the time derivative terms, Eq. (15) be-comes0 = (cid:34) − Ja ( ∂ x n ) − K ( n · ˆ z ) ˆ z + ( γ (cid:126) ) JS ( H · n ) H (cid:35) × n . (19) With Eq. (18) for the magnetic field,0 = (cid:104) − Ja ( ∂ x n ) − K (cid:48) ( n · ˆ z ) ˆ z (cid:105) × n , (20)where K (cid:48) = K − ( γ (cid:126) H z ) / (8 JS ). Note that the Zeeman cou-pling γ (cid:126) H z is typically of the order of few kelvins (for H z of few tesla), similar to the anisotropy energy. Thus, the dif-ference between K (cid:48) and K is of order O ( K / J ) which is dis-missed under our current approximation, K (cid:48) (cid:39) K . We writethe N´eel vector using the polar and azimuthal angles, n = (sin θ cos φ, sin θ sin φ, cos θ ) , (21)under the assumption that the uniform moment is small(namely, 1 = n + l (cid:39) n ). Assuming φ is spatially uniform,and noting that ˆ z = n cos θ − e θ sin θ , the static texture satisfies0 = − λ ∂ x θ + cos θ sin θ, (22)where λ = a √ J / K . A domain wall solution is given by cos θ = tanh (cid:18) ± x − X λ (cid:19) , (23)and thus sin θ = (cid:104) cosh x − X λ (cid:105) − , where X is the domain wallposition. The ± sign is the topological charge of the domainwall. The position of the wall, X , and the angle of the wallplane, φ ( = const.), will be promoted to dynamical variablesin the next subsection. Using the domain wall solution for n , Eq. (23), the kineticpart of the Lagrangian, L in Eq. (10), is obtained as L DW , = s (cid:0) ± l φ ˙ X − λ l θ ˙ φ (cid:1) , (24)where we defined l θ = (cid:90) dx λ e θ · l ( x )cosh x − X λ , l φ = (cid:90) dx λ e φ · l ( x )cosh x − X λ . (25)Equation (24) indicates that l θ and l φ are canonical momentaconjugate to φ and X , respectively, which should be consid-ered as new collective variables. Note that a uniform momentinduced by the (longitudinal) field, Eq. (18), is localized atthe domain wall (see Eq. (14)). In a more systematic treat-ment, this corresponds to expanding l with some complete setof functions { ϕ n ( x ) } , l ( x ) = ( l θ e θ + l φ e φ ) ϕ ( x ) + (cid:88) k l k ϕ k ( x ) , (26)where ϕ ( x ) = x − X λ , (27)and retain the first two terms. See Ref. for other ϕ k ( x )’s,which, together with ϕ ( x ), form a complete orthogonal basis.The Hamiltonian, Eq. (7), then becomes H DW = JS λ a (cid:40)(cid:18) l θ ∓ a λ (cid:19) + l φ + a λ (cid:41) − γ (cid:126) S (cid:18) λ a l θ ∓ (cid:19) ( H + H X ) . (28)The dissipation function Eq. (12), dismissing the α l ˙ l
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FULL PAPERS term, is given by W DW = s α n λ (cid:18) ˙ X λ + ˙ φ (cid:19) . (29)These results lead to the following four equations of mo-tion, ± ˙ l φ = − α n ˙ X /λ + γ H λ (cid:18) l θ ∓ a λ (cid:19) , (30)˙ l θ = α n ˙ φ, (31)˙ φ = − (4 JS / (cid:126) ) (cid:18) l θ ∓ a λ (cid:19) + γ ( H + H X ) , (32)˙ X = ± (4 JS / (cid:126) ) λ l φ . (33)Note that ( X , l φ ) and ( φ, l θ ) are coupled via H . However, l θ and l φ can be eliminated, resulting in two coupled equationsfor X and φ , or χ and ϕ defined by χ = X λ + H H λ , ϕ = φ + φ , (34)as (cid:40) ¨ χ = − ˜ α ˙ χ + ˜ H ˜ αϕ ˙ ϕ = ˜ H χ − ˜ αϕ . (35)Here, we defined˜ α = S J (cid:126) α n , ˜ H = γ H λ, (36)and φ = α − n ( l θ ∓ a / λ ) − φ , where l θ and φ are initial valuesintroduced when Eq. (31) is integrated in time. It is readilyseen that the acceleration of the domain wall is absent if thereis no damping, ˜ α =
0. This feature is not seen in Refs.
Writing the equations of motion in matrix form, ddt ˙ χχϕ = − ˜ α H ˜ α H − ˜ α ˙ χχϕ , (37)and assuming the solution of the form ∼ e ε t , the problem re-duces to an eigenvalue problem with determinant, ε ( ε + ˜ α ) − ˜ α ˜ H = . (38)Since ˜ α is positive, this equation has one real positive root ε ,and two complex roots ε and ε ( = ε ∗ ) with negative realparts. Using Cardano’s method, the roots are written as ε n = − α + ω n (cid:114) q + (cid:113) q − p + ω n (cid:114) q − (cid:113) q − p , n = , , p = ˜ α , q = ( ˜ α + ˜ H ) ˜ α , ω = − + √ i , and thereal branch of the cube roots are chosen. (Note that q − p ≥ χ = C e ε t + Re (cid:104) C e ε t (cid:105) , (40) ϕ = C ˜ H ˜ α + ε e ε t + Re (cid:34) C ˜ H ˜ α + ε e ε t (cid:35) , (41)where a real constant C and a complex constant C are deter-mined by initial conditions. Thus, we find that an inhomoge-neous magnetic field drives domain wall motion, X ∼ λ C e ε t ,that grows exponentially in time. ˜ α (s − )10 ˜ H ( s − ) − ε ( s − ) Fig. 2. (Color online) The real positive eigenvalue ε [Eq. (39)] plotted asa function of ˜ α and ˜ H . It increases with ˜ H as expected. When ˜ α (cid:29) ˜ H , the roots can be given as ε (cid:39) H ˜ α , ε (cid:39) − ˜ α + i ˜ H , (42)to leading order of ˜ α/ ˜ H . We expect most antiferromagnetsunder magnetic field satisfy this condition. For example, for2 S α n = − , J = K, γ (cid:126) H = / cm, λ = S =
1, then, ˜ α = s − , ˜ H = s − , and ε = − . A plotof ε in Eq. (39) is shown in Fig. 2 as a function of ˜ α and ˜ H .The above solutions (and equations) are limited to the case, | l θ | , | l φ | (cid:28)
1, in order for the exchange approximation to bevalid. Since l θ /χ ∼ α n ˜ H / ( ˜ α + ε ), this requires | χ | (cid:28) ( ˜ α + ε ) / ( α n | ˜ H | ), or | X − X | /λ (cid:28) max { S J / (cid:126) , ε /α n } / | ˜ H | ≡ b ,namely, the domain wall position should be within the dis-tance ∼ b λ from the position X ≡ − H / H of vanishingexternal field, H z =
0. Beyond this point, the nonlinearity of l θ may not be neglected. With parameters given above, wehave b ∼ , hence b λ is much larger than the domain wallwidth, and the above condition is always satisfied. We notethat the above condition is equivalent to γ (cid:126) | H z | (cid:28) J at thedomain wall position, which is practically always satisfied. We introduce a pinning potential by a local modulation δ K of the anisotropy K at the origin, K → K − a δ ( x ) δ K . Thepinning potential of the domain wall is then given by V pin = − δ KS ( X /λ ) (cid:39) δ KS (( X /λ ) − Θ ( λ − | X | ) , (43)which we approximated by a truncated parabola. ( Θ is theHeaviside step function.) When | X | < λ , the equation of mo-tion is altered as ddt ˙ χχϕ = − ˜ α − δ ˜ K ˜ H ˜ α H − ˜ α ˙ χχϕ , (44)
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FULL PAPERS . . . . D o m a i n w a ll p o s i t i o n ( X / a ) Time (10 − s)SimulationLine of best fit Fig. 3. (Color online) Domain wall position as a function of time undera linearly-varying magnetic field, Eq. (18). The simulation result (orangeline) is fitted by an exponential curve (blue dashed line), with exponent ε = . × s − . The used parameters are as follows; J = K, K = γ (cid:126) = / T, a = − m, H = / a , H + H X (0) = H λ × − (field at the initial position X (0)), α = − , and S =
1. The system is one-dimensional and has N = spins (so the system size is L = a = µ m),and the spins at both ends are fixed upwards ( S = S N = + ˆ z ). The startingconfiguration is Eq. (23) with initial position X (0) = λ (cid:39) a . A timediscretization of dt = × − s is used. where δ ˜ K = δ KS a λ J (cid:126) , (45)and we redefined χ = X λ − γ H ˜ H δ ˜ K − ˜ H , (46) ϕ = φ + φ − α γ H δ ˜ K δ ˜ K − ˜ H . (47)The determinant is now given by( ε + ε ˜ α + δ ˜ K )( ε + ˜ α ) − ˜ α ˜ H = . (48)A positive real root exists when δ ˜ K < ˜ H , (49)giving us the depinning condition. The analytical expressionof the real root of the cubic equation is given by Eq. (39) with p = ˜ α − δ ˜ K and q = ( ˜ α + ˜ H − δ ˜ K ) ˜ α .
4. Numerical Simulation
To test the validity of the approximations made above, suchas the continuum description, the discarding of higher-orderterms in l , and the use of collective coordinates, we performan atomistic simulation based on the equation of motion foreach S i ,˙ S i + α S S i × ˙ S i = (cid:126) − (cid:104) J ( S i − + S i + ) − KS zi ˆ z + γ (cid:126) H i (cid:105) × S i . (50)Using the approximate domain wall solution (23) as an initialconfiguration, we solved Eq. (50) under an inhomogeneousmagnetic field, Eq. (18). The position of the domain wall isdetermined by linear interpolation as the point at which theprofile of the staggered component ( − i S zi vanishes, and it . . . . D o m a i n w a ll p o s i t i o n ( X / a ) Time (10 − s) δK = 2 × − K δK = 1 × − K Fig. 4. (Color online) Domain wall position as a function of time in thepresence of pinning potential. Using the same parameters as in Fig. 3, weadded a pinning potential δ K on the neighboring two sites at the initial posi-tion of the domain wall (i.e. 10 λ from the left end). The domain wall remainspinned for δ K = × − K [purple (or light) line], whereas it is depinned for δ K = × − K [blue (or dark) line]. The inset shows a closeup. is plotted in Fig. 3 as a function of time. The values of theparameters used are described in the caption of Fig. 3. Thewidth of the domain wall is λ = a √ J / K (cid:39) . a . In accordwith our analysis, the domain wall position changes exponen-tially with time. The exponent obtained from the simulation, ε = . × s − , is very close to the analytical result, ε = . × s − [Eq. (39)]. The domain wall moves in thedirection of stronger magnetic field.We next simulate the motion of the domain wall with pin-ning potential located at the initial position of the domainwall. With the parameter values described in the caption ofFig. 3, Eq. (49) is satisfied when δ K ≤ . × K. Totest this value, we simulate the domain wall motion with“strong pinning” δ K = × − K, and “weak pinning” δ K = × − K. As shown in Fig. 4, the former pins thedomain wall, while the latter cannot stop the exponential in-crease of the domain wall position.
5. Physical Picture
Here, we discuss the physical mechanism of the domainwall actuation by an inhomogeneous magnetic field.First, consider a static solution. Then, Eqs. (30), (32), and(33) lead to l θ = ± a / λ , l φ =
0, and X = − H / H . The firsttwo relations show that only the artifactual texture-induceduniform moment is present, while the third relation tells usthat the domain wall must be positioned where the magneticfield vanishes. When the domain wall is placed in a finite mag-netic field, the N´eel vector starts to precess ( ˙ φ (cid:44)
0) accord-ing to Eq. (32), and the uniform moment l θ develops throughdamping [see Eq. (31)]. As seen from Eqs. (30) and (33), thedevelopment of l θ in conjunction with the field gradient ap-plies a force on the domain wall; like in the Stern-Gerlach ex-periment, the domain wall feels a force towards the directionwith stronger magnetic field to gain Zeeman energy. Thus, theantiferromagnetic domain wall can be thought of as a param-agnetic particle under an (inhomogeneous) applied magneticfield. Without damping, the uniform moment is not inducedby the field, hence there is no actuation of domain wall mo-tion. The present mechanism involves a dissipative process,
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FULL PAPERS hence is di ff erent from the purely reactive mechanisms dis-cussed in Refs.
6. Reformulation
We have seen that the Zeeman coupling of the intrinsicmagnetization ( ∼ H · ∂ x n ) is nullified by the coupling of theN´eel vector to the field gradient ( ∼ n · ∂ x H in Eq. (5)). Thereasoning behind this was given intuitively through Fig. 1,which indicates that the intrinsic magnetization is not a phys-ical quantity.In this section, we reformulate the theory in terms of “phys-ical magnetization”, eliminating the intrinsic magnetization.We first identify the physical magnetization by reexaminingthe interaction with external magnetic field, and therewith ex-press the Lagrangian (Sec. 6.1). The result is applied to thecollective coordinates of a domain wall (Sec. 6.2). Finally,the procedure is extended to general lattices (Sec. 6.3). To identify the physical magnetization, we look at the in-teraction with the external magnetic field [the second line ofEq. (7)], and rewrite it as2 γ (cid:126) S (cid:90) dx a (cid:26)(cid:16) l + a ∂ x n (cid:17) · H − a ∂ x ( H · n ) (cid:27) . (51)The first term is the Zeeman coupling in the bulk, and the sec-ond (total-derivative) term describes that at the edges. There-fore, the physical magnetization is identified to be − γ s ˜ l , with˜ l ≡ l + a ∂ x n ) . (52)This is the uniform moment with the intrinsic magnetizationsubtracted, and agrees with Haldane’s definition (accordingto the analysis made in Ref. ). In terms of ˜ l , the Lagrangiandensity, Eq. (16), is simplified as L = s ˜ l · ( n × ˙ n − γ H ) − s JS (cid:126) (cid:40) ˜ l + a ∂ x n ) − K J ( n z ) (cid:41) . (53)Here, we dropped the “topological” term ∼ ∂ x n · ( n × ˙ n ) since itdoes not a ff ect the equation of motion. Note that the exchangesti ff ness constant of the N´eel vector (the coe ffi cient of ( ∂ x n ) )has been reproduced correctly (without eliminating the uni-form moment), and the “sublattice symmetry” (˜ l , n ) → (˜ l , − n )has been recovered. Note also that the constraints are pre-served, ˜ l · n = n + ˜ l =
1, within the exchange ap-proximation. These suggest that the theory is simplified if re-formulated in terms of ˜ l .To complete this program, we need to examine the dampingterm. If the dissipation function has the form, W = s (cid:90) dx (cid:32) α l ˙˜ l + α n ˙ n (cid:33) , (54)in terms of ˜ l [instead of l as in Eq. (12)], the damping termis also simplified. To show that this is indeed the case, it issu ffi cient to observe that the spins S i couple to other degreesof freedom (“environment”) through (˜ l , n ) rather than ( l , n ).As an example, let us consider the s-d exchange coupling to conduction electrons, H sd = − J sd (cid:88) i S i · σ i , (55)where σ i is the electron spin at site i , and J sd is the couplingconstant. This has the same form as the Zeeman coupling( H i → σ i ), and we can proceed in exactly the same way asEq. (51). By noting that σ i may have staggered component aswell, we obtain the s-d coupling in the continuum approxima-tion as H sd = − J sd S (cid:90) d r (cid:26)(cid:16) l + a ∂ x n (cid:17) · σ l + n · σ n (cid:27) , (56)where σ l and σ n are the uniform and staggered components ofthe electron spin density. (For simplicity, we dropped the totalderivative terms.) As seen, there is a “correction” a ∂ x n / l , n ), instead of ( l , n ). (Precisely speaking, the “corrections”arise symmetrically between l and n , but in the second term,we adopted the exchange approximation, n + a ∂ x l / (cid:39) n .)Therefore, the resulting Gilbert damping, or the dissipationfunction, should have the form of Eq. (54) through (˜ l , n ).Recently, we have conducted explicit calculations ofGilbert damping and spin-transfer torques, and found thatthe expectation value of σ n is odd in n while that of σ l iseven. Thus, the spin torques resulting from the s-d ex-change interaction also possess the sublattice symmetry under(˜ l , n ) → (˜ l , − n ). For a domain wall in the collective coordinate description,the physical magnetization, Eq. (52), is given by˜ l θ ≡ l θ ∓ a λ , (57)and l φ (not altered). With these, the equations of motion arewritten as ± ˙ l φ = − α n ˙ X /λ + γ H λ ˜ l θ , (58)˙˜ l θ = α n ˙ φ, (59)˙ φ = − (4 JS / (cid:126) ) ˜ l θ + γ ( H + H X ) , (60)˙ X = ± (4 JS / (cid:126) ) λ l φ . (61)The sign ± represents the topological charge. We see that astatic domain wall has ˜ l θ = γ (cid:126) H / JS for H =
0, and ˜ l θ = H (cid:44) The procedure described in Sec. 6.1 can be generalized toarbitrary bipartite lattices. We assume a nearest-neighbor ex-change interaction J and a uniaxial magnetic anisotropy K .Taking the unit cell along the x direction, the Hamiltoniandensity (without Zeeman coupling terms) is calculated as H D ,ν = s ν JS (cid:126) (cid:40) ˜ l + a D D (cid:88) i = ( ∂ i n ) − K ν J ( n z ) (cid:41) , (62)where ˜ l is given by Eq. (52), D is the space dimensional-ity, and ν is the number of nearest-neighbor sites. For exam-ple, ( D , ν ) = (2 , , (2 ,
6. Phys. Soc. Jpn.
FULL PAPERS is the same as Eq. (51), hence ˜ l is identified as the physi-cal magnetization. The Lagrangian density is given by L = s ˜ l · ( n × ˙ n − γ H ) − H D ,ν .
7. Summary
We have investigated the motion of an antiferromagneticdomain wall under inhomogeneous magnetic field. Startingfrom the lattice Heisenberg model with antiferromagneticexchange coupling, easy-axis anisotropy, and Zeeman cou-pling, we constructed a continuum model by closely fol-lowing Ref.
We first retrieved a term that was missing inRef., in which the N´eel vector couples to the field gradi-ent. We have shown that this retrieved term nullifies the pre-viously demonstrated coupling of the intrinsic magnetization,attributed to the N´eel spin texture, to the magnetic field, andfound an alternative mechanism for domain wall motion actu-ated by an inhomogeneous field.As a supplemetary discussion, we pointed out that the uni-form moment l defined by Eq. (2) contains unphysical com-ponent (intrinsic magnetization). We have reformulated thetheory by properly defining the physical magnetization.We thank T. Funato, Y. Imai, K. Nakazawa, T. Yamaguchi,A. Yamakage, K. Yamamoto, and Y. Yamazaki for helpful dis-cussion. This work is supported by JSPS KAKENHI GrantNumbers JP15H05702, JP17H02929 and JP19K03744. JJNis supported by a Program for Leading Graduate Schools “In-tegrative Graduate Education and Research in Green NaturalSciences” and Grant-in-Aid for JSPS Research Fellow GrantNumber 19J23587.
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