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Dynamics of vortices in the two-dimensional anisotropic Heisenberg model with magnetic fields.
(2003)
- The subject of this work is the dynamics of a vortex in a classical 2-dimensional spin system with anisotropic exchange interaction under the combined action of magnetic fields and damping. Static as well as dynamic magnetic fields were employed (as dynamical field we used a homogeneous field which is rotating in the XY-plane). The most important goal of this work was to demonstrate that there is a coupling between the inner and translational freedom degrees of the vortex, coupling which is responsible for at least 2 phenomena that we study in detail in this Thesis: 1. the switching or flipping of the vortex polarization (for negative field frequency), and 2. the formation of stable orbits of the vortex center around the center of the system driven by the rotating field (for positive frequency). It was known to us that the polarization can change abruptly its sign under the action of a field rotating in the XY-plane, for p omega < 0 and appropriate field amplitudes. In the Chapter 4 we have investigated the possible underlying mechanisms for this phenomenon. Our main results can be summarized as follows: a) The flipping times do not depend essentially on the size of the system, provided that the lattice is large enough (radius L >~ 36 lattice constants). In other words, the switching of the vortex polarization is not much affected by the presence of boundaries. b) In our numerical simulations we observed a clear correlation between the core magnetization dynamics (the oscillations of the core spins in the out-of-plane direction) and the velocity of the vortex center in the plane of the lattice. c) A diagram of flipping events as a function of the field parameters, from extensive numerical simulations with an OP vortex in a rotating magnetic field, was presented. We found out that in the (omega, h) parameters space there is no well-defined curve which separates the regime where the flips do not occur from the regime where they do. We found intervals ("windows'') of intermittent flip and non-flip events. d) The switching of the vortex polarization can be achieved also by applying a static magnetic field with both in-plane (IP) and out-of-plane (OP) components. The IP component of the field sets the vortex into translational movement in the XY-plane, while the OP component breaks the vertical symmetry favoring one of the two possible orientations. e) The switching dynamics may be described in terms of a core model which takes into account a coupling between the vortex polarization dynamics and the motion of the vortex center. We showed that a reduced core model, which is valid near the threshold of the IP-OP vortex instability (lambda ~ lambda c), can be mapped to a generalized Thiele equation with an inertial term. f) It is plausible that the phenomenon of switching we described will not be essentially affected by the inclusion of a dipole-dipole interaction. The experimental works on nanodisks mentioned in the Introduction of this Thesis reported the observation of vortices in either of two polarization states, and the switching between them was forced by means of static fields perpendicular to the plane of the disks. Rotating magnetic fields might be used as well static fields with both IP and OP components to make this switching more favorable. In the Chapter 5 we turned to the study of the movement of the vortex in the XY plane, in the presence of the IP rotating field. Attention was directed to the existence of stable orbits, where the vortex stays inside the system in a stationary movement, forming circular limit cycles. We discussed then the failure of the conventional Thiele approach to describe this phenomenon, and this motivated us to formulate an extended collective coordinate Theory, which leads to a qualitative agreement with the results of the simulations. A diagram of the different types of trajectories, as a function of the field parameters, showed the presence of non-monotonous effects and "windows'', like in the case of the switching diagram. We are led to conclude that for some regions of the field parameters space, the system exhibits chaos -which is typical for many-body systems-, though no particular tool of the chaos theory was used to study our discrete and collective coordinate models, from this viewpoint. Our theoretical work qualitatively suggests that it would be interesting to apply in the experiments weak rotating fields like those used here, to control both the mean position of a vortex in larger magnetic dots (where the vortex center could show dynamics) and at the same time the sign of the out-of-plane core magnetization. Future directions of this work may include the use of inhomogeneous fields, particularly with a gaussian localization in a small region of the lattice or "spot'', as a model of the field of a laser beam.
