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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 axis). A DC external field is applied
along the axis and a circularly polarized radio-frequency
field is then applied in the 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 plane, which occurs
for small angles of
with respect to the 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 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).
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Subsections
Next: 2.5.1 Equation of motion
Up: 2. Uniformly magnetized particles
Previous: 2.4.3 Precessional switching
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Massimiliano d'Aquino
2005-11-26