2.5 LLG dynamics under circularly polarized field

The Landau-Lifshitz-Gilbert (LLG) equation has also played a central role in the interpretation of ferromagnetic resonance (FMR) phenomena [38]. Typical experiments involve small particles and thin-disks with rotational symmetry with respect to an axis (say $z-$axis). A DC external field is applied along the $z-$axis and a circularly polarized radio-frequency field is then applied in the $x-y$ plane. In this condition it has been shown that the absorbed power exhibits a maximum for a suitable resonance frequency. Analytical derivation of the resonance frequency for uniformly magnetized ellipsoidal particles was found by Kittel in 1948 [39] under the hypothesis of harmonic magnetization motion in $m_x,m_y$ plane, which occurs for small angles of $\textbf{{m}}$ with respect to the $z-$axis. In generic conditions the LLG equation has to be solved numerically. In fact, exact analytical solution can be derived in few cases and are generally obtained by linearizing the equation around some given state. In a new approach recently proposed [40], exact analytical solutions were derived for the full nonlinear LLG equation with damping in the case when the magnetic body is an ellipsoidal particle with rotational symmetry around a certain axis and the external field is circularly polarized. In this situation, one can prove that exact solutions of LLG equation always exist. These solutions are characterized by uniform magnetization rotating at the angular velocity $\omega$ with certain lag angle with the respect to the rotating applied field. The rotational invariance of this system and the fact that LLG equation conserves magnetization amplitude, permit one to reduce the problem to the study of an autonomous dynamical system on the unit sphere. This reduction is achieved by introducing an appropriate rotating frame of reference. The resulting autonomous dynamical system may exhibit various phase portraits characterized by equilibrium points and limit cycles [40]. The limit cycles in the rotating frame correspond to uniform quasiperiodic magnetization motions in the laboratory frame, deriving from the combination of the rotation of the frame and the periodicity of the limit cycle. The study of these quasiperiodic solutions is then reduced to the study of limit cycles of a vector field on the sphere. Despite the simplicity of the system, this study is extremely difficult and no general technique is available. In this respect, our purpose is to present a technique to predict the existence, the number, the form and the stability of these limit cycles (and therefore of the quasiperiodic magnetization modes) in the special case, often encountered in the applications, of small value of the damping constant in the LLG equation. The analysis is carried out by using an appropriate perturbation technique which is generally referred to as Poincaré-Melnikov function technique (see Appendix B and Ref. [43]).

Figure 2.15: Trajectories of magnetization on the unit sphere in the laboratory (left) and the rotating frames (right).