Annihilation of Domain Walls in a Ferromagnetic Wire
AAnnihilation of Domain Walls in a Ferromagnetic Wire
Anirban Ghosh, Kevin S. Huang,
1, 2 and Oleg Tchernyshyov Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA Centennial High School, Ellicott City, Maryland 21042, USA
We study the annihilation of topological solitons in the simplest setting: a one-dimensional fer-romagnet with an easy axis. We develop an effective theory of the annihilation process in terms offour collective coordinates: two zero modes of the translational and rotational symmetries Z andΦ, representing the average position and azimuthal angle of the two solitons, and two conservedmomenta ζ and ϕ , representing the relative distance and twist. Comparison with micromagneticsimulations shows that our approach captures well the essential physics of the process. The dynamics of topological solitons in ferromagnets[1] poses a class of problems of fundamental interest.Time evolution of magnetization is governed by theLandau-Lifshitz-Gilbert (LLG) equation [2, 3] J ˙ m = m × h eff + α |J | ˙ m × m (1)Here m ( r , t ) = M / | M | is the unit-vector field of magne-tization, J = | M | /γ is the angular momentum density, h eff ( r ) = − δU/δ m ( r ) is the effective magnetic field ob-tained from the potential energy functional U [ m ( r )] and α (cid:28) m = m ( z, t ), with the La-grangian [1] L = (cid:90) ∞−∞ dz J (cos θ −
1) ˙ φ − U, (2)and the potential energy U = (cid:90) ∞−∞ dz (cid:0) A | m (cid:48) | + K | m × ˆ z | (cid:1) / . (3)Here θ and φ are the polar and azimuthal angles ofmagnetization m , A is the exchange constant, K isthe anisotropy, and ˆ z is the direction of the easy axis.The unit of length is the width of the domain wall (cid:96) = (cid:112) A/K and the unit of time is the inverse of theferromagnetic resonance frequency, t = 1 /ω = J /K .In what follows, we work in these natural units and set J = A = K = (cid:96) = t = 1. A topological soliton in-terpolating between the two ground states m = ± ˆ z andminimizing the potential energy (3) is a domain wallcos θ ( z ) = ± tanh ( z − Z ) , φ ( z ) = Φ . (4)The position of a domain wall Z and its azimuthal angleΦ are collective coordinates describing the zero modes ' = , ⇣ = ' = , ⇣ = ' = , ⇣ = ' = ⇡ / , ⇣ = FIG. 1. (Color online) Several configurations of a pair of do-main walls with shown values of separation ζ and twist ϕ . Thered and blue colors denote positive and negative magnetiza-tion component m z along the axis of the cylinder. The wireframes depict the local plane tangential to the magnetizationfield. Spheres on the right show the path of the magnetiza-tion field m ( z ) as z goes from −∞ to + ∞ , beginning fromand ending at the north pole (red). The south pole (blue) canonly be reached if the separation of the domain walls ζ = ∞ . associated with the translational and rotational symme-tries. Schryer and Walker showed that, in the presenceof weak perturbations, the dynamics of a domain wallreduces to a time evolution of Z and Φ. By substitutingthe domain-wall Ansatz (4) into the LLG equation (1) orinto the Lagrangian (2), one obtains an effective theoryfor this system in terms of the two collective coordinates a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Φ and Z [4]. Their equations of motion read F Φ ± Z − α ˙Φ = 0 , (5a) F Z ∓ − α ˙ Z = 0 , (5b)where the force F Z = − ∂U/∂Z and the torque F Φ = − ∂U/∂ Φ are derived from potential energy U that mayinclude perturbations beyond Eq. 3.More generally, a magnetic soliton can be describedby a set of time-dependent collective coordinates q ( t ) = { q ( t ) , q ( t ) , . . . } , whose equations of motion express thebalance of conservative, gyrotropic, and viscous forces foreach coordinate q i [5]: F i + G ij ˙ q j − Γ ij ˙ q j = 0 , (6a) F i = − ∂U∂q i , (6b) G ij = − (cid:90) dV m · (cid:18) ∂ m ∂q i × ∂ m ∂q j (cid:19) , (6c)Γ ij = α (cid:90) dV ∂ m ∂q i · ∂ m ∂q j . (6d)Here F i is the conservative force conjugate to collectivecoordinate q i , G ij is the antisymmetric gyrotropic tensor,and Γ ij is the symmetric viscosity tensor.The method of collective coordinates has been quitesuccessful in describing the dynamics of solitons in fer-romagnets [6–10] and antiferromagnets [11–14]. In mostcases, the set of coordinates q is limited to just the zeromodes associated with the global symmetries of the un-perturbed system. In such a case, weak perturbationscreate a gentle potential landscape U ( q ) that inducesslow dynamics of the formerly zero and now soft modes q , while the hard modes quickly adjust to the instanta-neous configuration of the soft modes. Including hardmodes as dynamical degrees of freedom poses significantchallenges [15].Here we apply the method of collective coordinates toa problem that requires going beyond the zero-mode ap-proximation: the annihilation of two domain walls in aone-dimensional ferromagnet with an easy axis. Whentwo domain walls are far apart, they behave like rigidobjects and can be described by two independent pairsof collective coordinates: two positions Z and Z andtwo azimuthal angles Φ and Φ . Alternatively, we mayuse the average position Z = ( Z + Z ) / + Φ ) / ζ = Z − Z and the twist ϕ = Φ − Φ .Whereas Z and Φ represent the zero modes associatedwith the symmetries of translation and rotation, the rel-ative coordinates ζ and ϕ affect the energy (3) and thusrepresent modes that harden as the domain walls getcloser and their interaction increases.Let us work with the boundary condition m ( ±∞ ) = ˆ z .We may anticipate how the annihilation proceeds in thelimit of large separation, ζ (cid:29)
1, when the domain wallsretain their individual character and are described by Eq. 5b with the top signs. The effects on the averageand relative coordinates occur at different orders in α .To zeroth order, the two domain walls exhibit rigid rota-tional and translational motion:˙Φ = − ∂U∂ζ , ˙ Z = 12 ∂U∂ϕ . (7)They acquire relative velocities at the next order:˙ ϕ = − α ∂U∂ϕ , ˙ ζ = − α ∂U∂ζ . (8)As the domain walls approach each other and beginto overlap, they lose their ideal shape (4) and Eq. 5bno longer applies. Even worse, the precise positionsand azimuthal angles of overlapping domain walls be-come ill-defined. Fortunately, we may use two conservedmomenta—angular J and linear P —as proxies for theseparation and twist. The angular momentum along the z axis is [1] J = (cid:90) ∞−∞ dz (cos θ − . (9)Here the subtraction of 1 in the brackets means that wemeasure the angular momentum relative to the uniformground state m = ˆ z . If the domain walls are far apart, ζ (cid:29)
1, cos θ ≈ − J ≈− ζ . Turning this around, we define the separation interms of the angular momentum (9), ζ ≡ − J/ m ( ±∞ ) = ˆ z ) is [1] P = (cid:90) ∞−∞ dz (1 − cos θ ) φ (cid:48) = (cid:73) (1 − cos θ ) dφ. (10)The linear momentum is the area subtended by the vector m ( z ) on the unit sphere as z goes from −∞ to + ∞ [16].For two well-separated domain walls with a twist ϕ , thisarea is 2 ϕ . Again, we turn things around and define thetwist in terms of the linear momentum (10), ϕ ≡ P/ ζ and twist ϕ are shown in Fig. 1.Unlike single domain walls, which are stable for topo-logical reasons, pairs of domain walls are unstable: min-imization of the energy (3) in the topologically trivialsector with m ( ±∞ ) = ˆ z yields a uniform ground state m ( z ) = ˆ z . To obtain a solution for a pair of domainwalls, we may rely on conservation of linear and angularmomenta and minimize the energy U at fixed P and J .This can be done through minimization of the modifiedenergy ˜ U = U − P V − J Ω , (11)where V and Ω are Lagrange multipliers. The corre-sponding Lagrangian,˜ L = (cid:90) ∞−∞ dz (cos θ − φ − V φ (cid:48) + Ω) − U, (12)describes the dynamics of magnetization in a new framemoving at the linear velocity V and rotating at the an-gular velocity Ω. Minimization of the new potential en-ergy (11) yields a static soliton in the new frame. In thestatic frame, the soliton is moving at the velocity V andis rigidly rotating at the frequency Ω. This class of dy-namic solitons was first obtained by Kosevich et al. [17]and by Long and Bishop [18]. The relation between thevelocities and momenta of these solutions is V = − P sinh 2 J ,
Ω = sin P sinh J − cos P cosh J . (13)The explicit form of the solitons is given in the Supple-mental Material [19].We are now ready to derive the equations of motionfor a pair of domain walls with four collective coordi-nates Φ, Z , ϕ , and ζ by using the general formalism (6).The gyrotropic coefficients are most easily derived fromthe Berry phase term in the effective Lagrangian for thecollective coordinates. They form two pairs of conjugatevariables, Φ and J = − ζ for rotational motion and Z and P = 2 ϕ for translational. We thus infer that theeffective Lagrangian includes the Berry-phase terms [15] L B = − ζ ˙Φ + 2 ϕ ˙ Z = A i ˙ q i . (14)From that we read off the Berry connections A Φ = − ζ , A Z = 2 ϕ , and A ζ = A ϕ = 0. The gyrotropic coefficientsare the Berry curvatures G ij = ∂ i A j − ∂ j A i . The nonzerocoefficients of the gyrotropic tensor are G Φ ζ = − G ζ Φ = G ϕZ = − G Zϕ = 2 . (15)To deduce conservative forces F i , we turn off dissipa-tion. Eq. 6 now read F i + G ij ˙ q j = 0. Conservation oflinear and angular momenta implies the absence of theexternal force and torque, F Z = F Φ = 0. The relation(13) between the velocities ˙ Z = V and ˙Φ = Ω and themomenta P = 2 ϕ and J = − ζ together with the resultsfor the gyrotropic tensor (15) yield the internal force andtorque: F ζ = 2 (cid:32) sin ϕ sinh ζ − cos ϕ cosh ζ (cid:33) , F ϕ = − ϕ sinh ζ . (16)The viscosity coefficients Γ ij are evaluated via Eq. 6dby using the explicit solutions for the solitons [19]. Wefirst focus on the simpler case of zero twist, ϕ = 0. Inthis case, only two collective coordinates, Φ and ζ , evolvein time, whereas Z and ϕ remain constant. To the lowestnon-vanishing order in α , G Φ ζ ˙ ζ − Γ ΦΦ ˙Φ = 0 , F ζ + G ζ Φ ˙Φ = 0 , (17)
500 1000 1500 2000 t π π π π φ
500 1000 1500 2000 t Z
500 1000 1500 2000 t ζ
500 1000 1500 2000 t π π - Φ FIG. 2. (Color online) Collective coordinates ζ ( t ), ϕ ( t ), Z ( t ),and Φ( t ) for initial separation ζ = 4 and initial twists ϕ = 0(red), π/ π/ α = 0 . where F ζ = − ζ for ϕ = 0. We thus need justone viscosity coefficient, Γ ΦΦ = 4 α tanh ζ (1 + ζ sinh ζ ) for ϕ = 0 [19]. The resulting equations of motion are˙Φ = F ζ / , ˙ ζ = Γ ΦΦ F ζ / . (18)From Eqs. 18 we see that the global rotation angle Φis a fast variable whose leading-order behavior is deter-mined by the dissipation-free limit (zeroth order in α ).Separation ζ is a slow variable, whose dynamics arisesat the first order in α and is dissipational in nature. Forlarge initial separation ζ (cid:29)
1, the attraction is exponen-tially suppressed, F ζ ≈ − e − ζ , and the viscosity is ap-proximately constant, Γ ΦΦ ≈ α . The separation slowlydecreases as ζ ( t ) ≈ ln( e ζ − αt ) until the walls overlap.This initial approach takes an exponentially long time t i ≈ e ζ / α . The final stage, in which the “separation”(or angular momentum) decays as ζ ( t ) ∼ Ce − αt , has acharacteristic time scale t f = 1 / α . The global rotationfrequency initially grows as ˙Φ( t ) ≈ − / ( e ζ − αt ) untilthe walls overlap, then approaches the asymptotic value˙Φ ∞ = − ζ was obtained from the angular momentum alongthe easy axis, whereas the angle Φ was measured in themiddle of the combined soliton. See Supplemental Ma-terial [19] for details. The results for the initial twist ϕ = 0, initial separation ζ = 4, and Gilbert damping α = 0 .
01 are shown as red dots (micromagnetic simula-tions) and red lines (effective theory, Eqs. 18) in Fig. 2.We find excellent agreement between the two.In the general case, with both an initial twist ϕ (cid:54) = 0and separation ζ (cid:54) = 0, the equations of motion to theleading order in α have the following form:˙Φ = F ζ / , ˙ ζ = (Γ ΦΦ F ζ − Γ Φ Z F ϕ ) / , (19a)˙ Z = − F ϕ / , ˙ ϕ = ( − Γ Z Φ F ζ + Γ ZZ F ϕ ) / . (19b)Forces F i are given in Eq. 16; components of the viscositytensor Γ ij can be found in Supplemental Material [19].During the initial approach ( ζ (cid:29) F ζ ≈ − e − ζ cos ϕ , F ϕ ≈ − e − ζ sin ϕ ,and retain their individual character, so that the dissi-pation tensor is diagonal, with Γ ΦΦ ≈ Γ ZZ ≈ α . Thetwist angle decreases slowly and linearly in time: ϕ ( t ) ≈ ϕ − αt e − ζ sin ϕ . (20)The separation evolves as ζ ( t ) ≈ ζ + ln sin ϕ ( t )sin ϕ . (21)Notably, for a large initial twist ϕ > π/
2, the force F ζ is repulsive and the domain walls initially move apartuntil ϕ decreases to π/
2. At that point, the force F ζ vanishes and the walls reach their maximum separation ζ max ≈ ζ − ln sin ϕ . This happens at t max ≈ ( ϕ − π/ e ζ α sin φ . (22)The total duration of the initial approach is t i ≈ ϕ e ζ α sin ϕ . (23)Both the linear trend in ϕ ( t ) and the backward initialrelative motion for ϕ > π/ ζ and ϕ yields U ≈ ζ + ϕ ) /ζ , Γ ΦΦ ≈ αζ ,Γ Z Φ = Γ Φ Z ≈ − αϕ , and Γ ZZ ≈ αϕ /ζ . Eqs. 19 read˙Φ ≈ − ϕ /ζ , ˙ ζ ≈ − αζ (1 + ϕ /ζ ) , (24)˙ Z ≈ ϕ/ζ, ˙ ϕ ≈ − αϕ (1 + ϕ /ζ ) . (25)During this stage, the ratio ϕ/ζ remains constant. Bothaverage velocities attain their terminal values ˙ Z ∞ = V ∞ and ˙Φ ∞ = − V ∞ /
4, where V ∞ = 2 ϕ/ζ . It is inter-esting to note that, as the domain walls annihilate andthe energy decreases toward zero, the pair does not slowdown and keeps moving and rotating at constant rates!The relative coordinates ζ ( t ) and ϕ ( t ) decay exponen-tially with the characteristic time t f ≈ α (1 + V ∞ / . (26) Again, all these trends are clear in Fig. 2. The micro-magnetic data and the effective theory (Eqs. 19) showexcellent agreement.We have considered the annihilation of two domainwalls in a ferromagnetic wire. A minimal description ofthe process requires the use of 4 physical variables. Theaverage coordinates of the combined soliton, position Z and azimuthal orientation Φ, are zero modes on accountof global translational and rotational symmetries; the rel-ative coordinates, separation ζ and twist ϕ , harden as thedomain walls merge. We obtained the equations of mo-tion for the these variables in the framework of Tretiakov et al. [5] and showed that separation ζ and twist ϕ exhibitpurely viscous dynamics, whereas the average position Z and azimuthal angle Φ are driven by the torque F ϕ ( ζ, ϕ )and force F ζ ( ζ, ϕ ), respectively. These equations of mo-tion (19) predict the dynamics of the 4 variables in ex-cellent agreement with the results of numerical micro-magnetic simulations (Fig. 2). We hope that the methodcan be successfully extended to the dynamics of othermagnetic solitons. Acknowledgments.
Research was supported by the U.S.Department of Energy, Office of Basic Energy Sciences,Division of Materials Sciences and Engineering underAward DE-FG02-08ER46544. [1] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, Phys.Rep. , 117 (1990).[2] L. D. Landau and E. M. Lifshitz, Phys. Z. Sow. , 153(1935).[3] T. Gilbert, IEEE Trans. Magn. , 3443 (2004).[4] N. L. Schryer and L. R. Walker, J. Appl. Phys , 5406(1974).[5] O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Baza-liy, and O. Tchernyshyov, Phys. Rev. Lett. , 127204(2008).[6] B. A. Ivanov and V. A. Stephanovich, Phys. Lett. A ,89 (1989).[7] K. Y. Guslienko, X. F. Han, D. J. Keavney, R. Divan,and S. D. Bader, Phys. Rev. Lett. , 067205 (2006).[8] C. H. Wong and Y. Tserkovnyak, Phys. Rev. B ,060404 (2010).[9] I. Makhfudz, B. Kr¨uger, and O. Tchernyshyov, Phys.Rev. Lett. , 217201 (2012).[10] S. K. Kim and Y. Tserkovnyak, Phys. Rev. B , 020410(2015).[11] B. A. Ivanov and A. K. Kolezhuk, Phys. Rev. Lett. ,1859 (1995).[12] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, andA. Brataas, Phys. Rev. Lett. , 127208 (2013).[13] S. K. Kim, Y. Tserkovnyak, and O. Tchernyshyov, Phys.Rev. B , 104406 (2014).[14] E. G. Tveten, T. M¨uller, J. Linder, and A. Brataas,Phys. Rev. B , 104408 (2016).[15] D. J. Clarke, O. A. Tretiakov, G.-W. Chern, Y. B. Baza-liy, and O. Tchernyshyov, Phys. Rev. B , 134412(2008). [16] F. D. M. Haldane, Phys. Rev. Lett. , 1488 (1986).[17] A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev, JETPLett. , 486 (1977).[18] K. A. Long and A. R. Bishop, J. Phys. A: Math. Gen. , 1325 (1979).[19] See Supplemental Material.[20] M. J. Donahue and D. G. Porter, National Institute ofStandards and Technology Report No. NISTIR 6376,math.nist.gov/oommf (1999). Supplemental Material: Annihilation of Domain Walls in a Ferromagnetic Wire
TWO DOMAIN WALL SOLITON
The explicit form of the class of solitons, written as static configurations as seen in the moving frame and parametrizedby the four collective coordinates Z , Φ, ζ and ϕ , looks likecos θ ( z ) − f ( z − Z ; ζ, ϕ ) (27a) φ ( z ) = Φ + g ( z − Z ; ζ, ϕ ) (27b)To express the functions f and g in a compact form we first define: a ( ϕ, ζ ) = 2 (cid:16) − cos ϕ sech ζ (cid:17) (28a) b ( ϕ, ζ ) = 2 (cid:16) ϕ csch ζ (cid:17) (28b)The expressions for f and g are then given by: f ( z ; ζ, ϕ ) = − b + b − a − ab tanh (cid:16) √ ab z (cid:17) (29a) g ( z ; ζ, ϕ ) = sin ϕ sinh ζ z + sgn ϕ tan − (cid:34)(cid:113) a ( b − b (2 − a ) tanh (cid:16) √ ab z (cid:17)(cid:35) (29b)Here ζ and ϕ are independent parameters taking values within the ranges 0 < ζ < ∞ and − π < ϕ ≤ π . BERRY PHASE IN EFFECTIVE LAGRANGIAN
Differentiating (4b) in the main text with respect to time gives˙ φ = ˙Φ + ∂g∂Z ˙ Z + ∂g∂ζ ˙ ζ + ∂g∂ϕ ˙ ϕ = ˙Φ − φ (cid:48) ˙ Z + ∂g∂ζ ˙ ζ + ∂g∂ϕ ˙ ϕ (30)Substituting this into the Berry phase part of the full Lagrangian given by (2) of the main text gives L B = ˙Φ (cid:90) dz (cos θ − − ˙ Z (cid:90) dz (cos θ − φ (cid:48) + ˙ ζ (cid:90) dz f ∂g∂ζ + ˙ ϕ (cid:90) dz f ∂g∂ϕ = − ζ ˙Φ + 2 ϕ ˙ Z + A ζ ˙ ζ + A ϕ ˙ ϕ (31)The first two terms denote the expected gyrotropic couplings between pairs of cannonically conjugate variables. Onthe other hand, the gauge connections A ζ ( ζ, ϕ ) and A ϕ ( ζ, ϕ ) indicate the gyrotropic coupling between ζ and φ . Thecorresponding curvature is F ζϕ = ∂ ζ A ϕ − ∂ ϕ A ζ = (cid:90) dz (cid:16) ∂f∂ζ ∂g∂ϕ − ∂f∂ϕ ∂g∂ζ (cid:17) (32)Since f ( g ) is an even(odd) function of z the integrand is an odd function, implying F ζϕ = 0. This basically meansthat under a transport in the ζ - ϕ plane in an infinitesimally closed loop, Berry phases gathered by the spins at equaldistances from the center on either side are equal and opposite. This is in agreement with our intuition of what shouldhappen in the large separation limit, when the two domain walls stay rigid under a small transport. Hence the Berryphase part of the effective Lagrangian is L B = − ζ ˙Φ + 2 ϕ ˙ Z (33) EQUATIONS OF MOTION
Since F i = − ∂U∂q i , equations (5a) and (5b) of the main text can be integrated to obtain U = 4(cosh ζ − cos ϕ )sinh ζ (34)The Γ ij ( ζ, ϕ ) functions form a symmetric 4 × × Z -Φ block and the ζ - ϕ block, since the remainig four off-diagonal terms are zero. To see this we firstexpress Γ ij in terms of θ ( z ) and φ ( z ). Γ ij = α (cid:90) dz (cid:16) ∂θ∂q i ∂θ∂q j + sin θ ∂φ∂q i ∂φ∂q j (cid:17) (35)As seen from (4a-b) of the main text and (29) above, θ ( z ) is even and φ ( z ) is odd. Hence if j = ζ or ϕ Γ Zj = − α (cid:90) dz (cid:16) ∂θ∂z ∂θ∂q j + sin θ ∂φ∂z ∂φ∂q j (cid:17) = 0 (36a)Γ Φ j = α (cid:90) dz sin θ ∂φ∂q j = 0 (36b)Only the Z -Φ block enters the equations of motion at O ( α ) and these functions involve only elementary integrals.Performing these integrals, one obtains Γ Z Φ = − αζ sin ϕ sinh ζ (37a)Γ ΦΦ = α (cid:104) ( b − a ) ln (cid:12)(cid:12)(cid:12)(cid:12) √ a/b − √ a/b (cid:12)(cid:12)(cid:12)(cid:12) + 2 √ ab + 4(2 − b ) tanh − (cid:112) ab (cid:105) (37b)Γ ZZ = 2 αU − Γ ΦΦ (37c)The two coupled first-order differential equations for ζ ( t ) and ϕ ( t ) can only be solved numerically. But the large- t limit, when ζ, ϕ → V = 2 ϕ/ζ Ω = ( ϕ/ζ ) − U = 2( ζ + ϕ ) /ζ (38b)Γ Z Φ = − αϕ Γ ΦΦ = 4 αζ Γ ZZ = 4 αϕ /ζ (38c)(38a) implies the constraint Ω( t ) = V ( t ) / − ϕ = − α (1 + ϕ /ζ ) ϕ (39a)˙ ζ = − α (1 + ϕ /ζ ) ζ (39b)These readily imply ˙ V = ( ζ ˙ ϕ − ˙ ζϕ ) /ζ = 0. Hence both V and Ω approach constant values V ∞ and Ω ∞ = V ∞ / − ζ and ϕ . This also implies that the two relativecoordinates ζ and ϕ are decoupled in this limit and they both decay exponentially with the same time constant τ − = 2 α (2 + Ω ∞ ). The shape of the soliton in this limit can be obtained by expanding (29) to the lowest order termsin the relative coordinates. f ( z ; ζ, ϕ ) = − (cid:16) Ω ∞ (cid:17) ζ sech (cid:104)(cid:16) Ω ∞ (cid:17) ζz (cid:105) (40a) g ( z ; ζ, ϕ ) = V ∞ z + tan − (cid:104) ϕ tanh (cid:104)(cid:16) Ω ∞ (cid:17) ζz (cid:105)(cid:105) (40b)The expression for f = m z ( z ) − w − = (1 + Ω ∞ / ζ which divergesas the uniform state is approached.Animations depicting stationary solitons (in the absence of dissipation) and annihilation processes (in thepresence of dissipation) can be found in https://sites.google.com/site/olegtjhu/research-1. OOMMF SIMULATION
Micromagnetic simulations of the annihilation process with initial conditions ζ = 4 (in natural units) and ϕ = 0 , π/ π/ A = 2 × − J / m, anisotropy constant K = 2 × J / m and magnetization M s = 10 A / m. A wire with cubic unit cell (containing a single spin)of side a = 10 nm and dimension N s a × a × a was used, N s being the number of spins in the wire. N s wastaken to be 1000 for ϕ = 0 and 2000 for ϕ = π/ π/
4. The angular momentum density of this system is J = M s /γ = 5 . × − Js / m . This gives a characteristic time scale t = J /K = 0 .
284 ns and a characteristiclength scale (exchange length) λ = (cid:112) A/K = 100 nm.Finite-size effects become important toward the end of the simulation for two reasons. Firstly, the width ofthe soliton diverges as the two domain walls merge, as seen at the end of the previous section. Once it becomescomparable to the system size, the transverse component of magnetization m ⊥ is no longer zero at the edges of thewire. Secondly, for nonzero ϕ the soliton has an overall translational motion (in the + z direction for our ϕ values),which causes it to run into one of the edges. To tackle the second issue, we started the simulation with Z = − L/ ϕ = π/ π/
4. (Here the origin is at the center of the wire of length L .) For ϕ = 0, Z = 0 was used.Moreover, to minimize the finite-size effects, we truncated the simulation data when the value of m ⊥ at the edgebecame greater than 1%.We used ∆ t = 0 .
082 ns = 0 . t for each iteration step. The four collective coordinates were extracted fromthe magnetization profile at each iteration. Since we chose the boundary condition m z ( z → ±∞ ) = 1, Z = iaλ where m zi = min j ∈{ ,...,N s } m zj . (41)Φ is the azimuthal angle of the spin at this location.cos Φ = m xi (cid:113) m xi + m yi (42)The two relative coordinates are obtained by evaluating the discretized versions of their defining expressions. ζ = − a λ N s (cid:88) j =1 ( m zj −
1) (43) ϕ = − N s − (cid:88) j =1 ( m zj − φ i +1 − φ ii