Eur. Phys. J. B 33, 133–145 (2003) DOI: 10.1140/epjb/e2003-00150-3 T HE EUROPEAN P HYSICAL JOURNAL B Cycloidal vortex motion in easy-plane ferromagnets due to interaction with spin waves A.S. Kovalev 1 , F.G. Mertens 2, a , and H.J. Schnitzer 2 1 Institute for Low Temperature Physics and Engineering, 47 Lenin Ave., 61103, Kharkov, Ukraine 2 Physikalisches Institut, Universit¨at Bayreuth, 95440 Bayreuth, Germany Received 9 December 2002 Published online 4 June 2003 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2003 Abstract. The dynamics of a non-planar vortex in a two-dimensional easy-plane ferromagnet of finite size is studied. Spin dynamics simulations show small cycloidal oscillations of the vortex around its mean path. In contrast to an earlier phenomenological theory we give a physical explanation: The oscillations occur due to the interaction of the vortex with coherent spin waves which are excited by this vortex at the moment when it starts to move, in order to conserve the total energy and angular momentum. The calculation of these quantities yields the frequencies and amplitudes of the cycloidal oscillations in good agreement with the simulation data. PACS. 75.10.Hk Classical spin models – 05.45.Yv Solitons – 75.40.Mg Numerical simulation studies 1 Introduction During the past years much attention has been given to the investigation of the structure and dynamics of vor- tices in magnetic materials of different types [1–17]. These nonlinear topological excitations play an active role in resonance properties of magnets and in the thermody- namics of the Kosterlitz-Thouless vortex-unbinding tran- sition in 2D and quasi-2D magnetic systems (such as mag- netic lipid layers, organic intercalated compounds (e.g. (CH) n (NH 3 ) 2 CuCl 4 ) and layered magnets), they are also important for the problem of the Bloch lines in domain walls. Recently the direct experimental visualization of magnetic vortices in magnetic nanodots by magnetic force and Lorentz microscopies measurements [18–20] gave a new impulse to the investigations in this field of the physics of magnetism. Intensive experimental study of ar- tificial 2D lattices of magnetic nanodots in vortex config- urations [21] leads to potential applications of these ob- jects in magnetic memory devices. Now even a delicate phenomenon like a shift of the vortex position from the center of a nanodot in an external magnetic field is ob- served experimentally [20,22]. However, many particular features of vortex dynamics may be studied now only by analytical methods or numerical simulations. The dynamics of vortices is quite different in ferro- and antiferromagnets and depends essentially on the type and value of the magnetic anisotropy. From the theoret- ical point of view, the situation in easy-plane ferromag- a e-mail: franz.mertens@uni-bayreuth.de nets with small exchange or single-ion anisotropy seems to be most interesting, because in this case Galilei’s law is not valid for a vortex and therefore it exhibits a non- Newtonian dynamics. The structure of this magnetic vor- tex is in a certain sense similar to that of vortices in a nonideal Bose gas described by Pitaevskii [23] and its dy- namics is similar in first approximation to the dynamics of vortices in superfluids. Such a nontrivial dynamics takes place only if the anisotropy is smaller than some critical value and the vortex has a so-called out-of-plane (OP) structure with nonzero components of the magnetization in the hard-axis direction. For this vortex the so-called gyrovector which was defined by Thiele in [1,2] is nonzero and a first-order equation of motion for the vortex center can be derived [3], similar to the Thiele equation for mag- netic bubbles and Bloch lines in easy-axis ferromagnets. In an infinite medium without gradients of the magne- tization field such an isolated vortex cannot move [5]. A motion is possible only in the presence of a spatial rotation of spins in the easy plane (spin flux) [5]. This is similar to the situation in hydrodynamics, where a vortex in a nondissipative medium can move only with the velocity of the medium, i.e. it is “frozen” in the liquid. In the real situation of a magnet with a finite den- sity of vortices and other magnetic excitations (such as spin waves, for example) and in a confined geome- try magnetic vortices can move. In the hydrodynamic approach the interaction with other vortices and the boundaries is simple and leads to vortex motion with average velocity V ∼ 1/ √ n, where n is the density of vor- tices. In a finite ferromagnet with a characteristic size L,