NORMAL MODES AND VORTEX DYNAMICS IN 2D EASY-AXIS FERROMAGNETS 1

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1 IC/96/234 United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NORMAL MODES AND VORTEX DYNAMICS IN 2D EASY-AXIS FERROMAGNETS 1 F.Kh. Abdullaev Physical-Technical Institute of the Uzbek Academy of Sciences, G. Mavlyanova str. 2b, Tashkent, Uzbekistan and A.S. Kirakosyan 2 International Centre for Theoretical Physics, Trieste, Italy. ABSTRACT Magnon modes, localized on vortices in easy-axis 2-D ferromagnets are studied. There are local modes for all values of quantum numbers m in a definite frequency range, determined by the strength of anisotropy. Connected with the translational mode, the vortex dynamics problem is considered. The momentum of individual moving vortex is calculated based on the exact solution of equation for vortex motion in the isotropic case. The momentum balance equation with the Magnus-type force for the system of vortices is derived. PACS Hk Ch MIRAMARE - TRIESTE November Submitted to JETP Letters. 2 Permanent address: Physical-Technical Institute of the Uzbek Academy of Sciences, G. Mavlyanova str. 2b, Tashkent, Uzbekistan. 1

2 Quasilocal magnon modes in isotropic [?], easy-plane Heisenberg [?],[?], XY-type [?] 2D ferromagnets (FM) have been investigated. Literally local mode with m = 0 in easyplane 2D Heisenberg antiferromagnets (AFM) was studied in [?]. In the recent work of [?] two modes were found in 2D easy-plane AFM within the continuous spectrum. The problem of spin wave scattering on the soliton in 2-D isotropic Heisenberg FM has been solved and quasilocal modes have been investigated in [?]. In the local rotated coordinate frame, connected with the magnetization vector of vortex, linearized Landau- Lifshitz equations(lle) have been reduced to the 2-D Schrodinger equation. In the static limit, this equation possess an exact solution due to scale invariance. The vortex- magnon scattering (S) matrix calculation showed, that the most strong scattering takes place in the region of local modes(lm). Considerable physical interest is taken in the study of such problem in easy-axis FMs. Another important problem, closely connected with the translational mode of the vortex, is the calculation of a vortex distortion, caused by translational motion. Vortex, considered in this paper, has an instanton-type core structure. The scale invariance is the unique property, providing the possibility to integrate equations of motion for the vortices. Due to the exact solution of equation, describing the dynamics of moving vortex in isotropic ferromagnet, we may construct a solution in the anisotropic case for solitons with small radius. There is some connection between dynamics of vortices in easyplane and easy-axis 2D ferromagnets. In both cases, vortices are "frozen" in environment according to the Helmholtz theorem as in hydrodynamics. It is bound up with the fact, that the ratio of force, acting on the vortex, to the vortex velocity is proportional to the difference of the magnetization projection at the vortex axis and at an infinity (hard axis for easy-plane ferromagnets and easy axis for easy axis ones)[?]- [?]. That is why, if the vortex is not acted by any force it can't move. Interaction between solitons in isotropic ferromagnets is absent. But interaction arises in the anisotropic case. Local Modes. - Let us go on to the investigation of LM in anisotropic FM. It is known, that in the anisotropic case, a soliton with small radius (soliton with localization length much less than magnetic length l 0 = α/ β) precesses with frequency ω = ω 0 / ν, where α, β - constant of inhomogeneous exchange and anisotropy respectively, Ω0 = βgm 0, ν - topological charge of the soliton, g - giromagnetic ratio [?]. It is suitable to study small oscillations of the magnetization vector of vortex in the coordinate frame connected with easy axis anisotropy (z axis). We introduce variables m z = M o $ sin θ 0,

3 m + = m x + ımy = M 0 ('dcos9o where M 0 - nominal magnetization, i) = θ Θ0, <pi = <p <po and ^o = νχ + ωt + χ0, and for θ 0 we have the following equation J\ d ( d9 0 \ u 2 sm9 0 cos9 0 \ - sinθ0 cosθ 0 + sinθ 0 ω = 0. (1) 0r dr dr j r At r <ti l0 Eq. (1) has the solution accurate to small parameter (lr) tg 20 =r A The lacing method gives the approximate solution, which had been used in [?] 2 In the present work we will use a linearized coupled set of dynamic L-L equations. Far from vortex we will reduce this set to the 2D Schrodinger equation. L-L equations (LLE) have the following form 1 2 O (A9 - (V^)2 sinθcosθ) -sm9cos9 + ^ sin θ = 0, 1 39 l02div(sin 2 6V<p) sin θ = 0. UJQ ut Let us introduce a new variable \i = sin 9o<pi. The variables \i and i) are the projections of the M on the local axes e\ and e* 2 connected with the vortex: /j, = Me\, $ = Me 2 - The axis e*3 coincides with M direction in the unperturbed vortex. Then we get the coupled set of two partial differential equations for i) and \i Here 2z/cos6 l o ctyx 1 dfx r 2 &x W O /Q dt, [- +U 2 {r)\n - 0 =. (2) U 2 (r) = ctg9 0 A9 0 - (V9 0 ) 2. Multiplying (1) to r 2 dθdr0 and integrating within the limits 0 to r and using the boundary conditions θ 0 = π at r = 0 and θ 0 = 0 at r = oo we get

4 -4 r(sin 2 θ 0 - (1 - cosθ 0 ))ρdρ. (3) It is easy to show by direct integration, that far from the vortex the r.h.s. of Eq. (3) may be neglected for all r. Thereby we have the following approximate expression for U2(r) U 2 (r) =(^ + l) cos2θ 0 -^2(3cosθ 0-2). Far from the vortex "potentials" approach asymptotically to each other. The equation for ψ = i) + i/j, is derived from (2) and has the form For r ^$> R we have U(r) = (1 Ω)/L02. Eq. (4) differs from having been derived in [?] by an additional term "anisotropy potential" U(r). For r <^i l0 Eq. (4) with the accuracy lr) 2 has the solution, describing magnon modes, localized on the soliton [?] ' H\ ma ) Here σ = ν, ν < m < ν,ω1- frequency of small oscillations. For the sake of convenience let ν > 0. It is evident from (4), that there are LMs for all m at frequences uji < Ω0 ω. The use of Macdonald's function K n (x) and the modified Bessel's function I n (x) gives the approximate solutions. Due to the lacing method used in [?],[?] we have for F m (r) at the distances R < i r < i 1, and m > v For m < v kr). (6) y L J \K0 / {'lit -t iy 1)'. l\ u {Kor) " hvj V 2 ; V^o7 \m + u\\ I\ m+v \(kr) Here k 2 = (1 ω ~)/^o> ^o = (1 ~ ω)/l02. Thereby we have the exponential decay of intensity of spin wave in the region of LMs. Similarly, the approximate solutions for quasilocal modes at the frequences ω1 > Ω 0 Ω may be constructed and the S matrix computed. Vortex Dynamics. - Now let us investigate the distortion of the system of spins, caused by motion of vortex in isotropic FM (Belavin-Polyakov vortex) [?]. This problem is an

5 auxiliary stage in the study of the dynamics in the anisotropic case. When the vortex moves with stationary velocity V the small additives to ipo and Θ0 will occur [?]. To find these additives we represent ip and θ in the form where e x and e* r are the unit vectors of appropriate polar coordinates frame. In the limit of small velocities of the vortex equation of motion has the form of linearized LLE for LMs with the additional term, which is proportional to the vortex velocity. We get for ψ the following equation from (4) and L-L equations = - A + 2 cos 2θ 0 ] ψ 1ı2 0 02, r 2 r 2 dx oj o lor where Vexp(ıζ) = V y + ıv x. We look for the solution in the form ( 8 ) ψ(χ, r ) = exp(ıζ + ıχ)g(r). The equation for G(r) can easily be integrated. We may exclude the singularity of the momentum density and momentum flow one of the vortex at its axis by an appropriate choice of the integration constant. Then we have Vr / 1 / r \ 2l A 4cj o to V v + l\nj ) The expression for the full momentum of magnetization field accounted for one atomic layer has the form +00 2TT P = - J J (1-cos 6)Vprdrd X. (10) 00 Now if we will compute the momentum of moving vortex, using Eqs. (9) and (10) we will see that it diverges in the infinite medium. This is connected with the abovementioned "frozenness" of vortex in environment. Belavin-Polyakov vortices describe the static state of isotropic magnet and already for this reason they neither can move nor interact with each other. Nevertheless, when the anisotropy is taken into account, interaction occurs. There is a characteristic length, when a space power law growth of distortion turns into exponential decay. This length is a cutoff parameter for integration. Calculation of the momentum of moving vortex in anisotropic case at account of vortex distortion ψ gives p = M ΠA 0 ΝV g(ujuj)(l + ν).

6 The Eq. (11) valid in the limit of slow motion of vortex. Vortex momentum is proportional to the square of the cut-off parameter, which may be called a vortex soft core. The soft core size is equal to 1/k0 = l0/(1 ~~ ω)1/ 2. At ν = 1 precession frequency of soliton Ω = Ω0 is accurate up to (lr0) 4. In this case the vortex momentum becomes anomalously big and dependent on vortex radius R. Vortex size R depends on the number of magnons in 2D magnetic soliton. The last one is integral of motion. That is why it does not change when vortex moves. In the soft core region we have the reconstruction of scale invariance. So we can use the solution (11) for anisotropic ferromagnet. Direct differentiation of (10) with regard to time gives the following expression for the momentum balance Here F is a force, acting on vortex from the remaining vortices and r.h.s. of Eq. (12) is the Magnus force, acting on the vortex from the homogeneous magnet. Calculation of the force F is the subject for a separate discussion. In conclusion we would like to note that in our work we showed that in easy-axis ferromagnets there are local modes on the two-dimensional topological solitons for all values of quantum number m. This is different from the situation in easy-plane FMs, where only quasilocal modes can exist. In the limits of small velocities one can construct the vortex dynamics due to the scale invariance. Acknowledgments. - We wish to thank Ilia Krive and Alexander Nersesian for clarifying discussions. This work was supported by the Fund of Fundamental Researches of the Uzbek Academy of Sciences. Finally, we gratefully acknowledge the sponsoring support of S.K. Abduvaliev.

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