Nonlinear magnetization dynamics in ferromagnetic materials
PhD. student: Dr. Roberto Bonin (*), Politecnico di Torino & IEN Galileo Ferraris
Tutors: Prof. Piero Mazzetti, Politecnico di Torino, Dipartimento di Fisica
Dr. Giorgio Bertotti, IEN Galileo Ferraris, Settore Materiali
The study of magnetization dynamics is of interest to many scientific areas:
• Physics: the nature of magnetization processes can be known in more detail and on this basis new
experimental techniques can be developed to measure magnetization processes;
• Mathematics: solving the equations that describe the behavior of the magnetization involves many
mathematical tools from the theory of nonlinear dynamical systems;
• Materials science: new ferromagnetic materials are created and their structure is studied;
• Computer and information technology: ferromagnetic devices for data storage (MRAM).
The starting point for the study of high-frequency magnetization processes is Landau-Lifshitz-Gilbert
(LLG) equation:
dm
dm
= −m × hef f + αm ×
(1)
dt
dt
This equation describes the behavior of the magnetization m when it is subject to the effective field hef f ;
the dynamics is dissipative and the dissipation constant is α. Exact analytical solutions can be obtained
for a spheroidal ferromagnetic particle, in which the magnetization is considered uniform in space, placed
in an external field, circularly polarized in the plane perpendicular to the symmetry axis of the particle
and constant along this axis [1]. Deviations from spatial uniformity can be studied by considering two
kinds of perturbations:
• Spin-wave perturbations: the perturbation wavelength is much smaller than the sample dimensions [1];
• Magnetostatic modes: the perturbation varies over a scale comparable to the sample dimensions
and magnetostatic boundary conditions must be taken in account; for this case we have obtained
the frequency spectrum of the perturbations [2].
Other general classes of phenomena can be modeled by finding the exact analytical solutions of LLG
equation:
• magnetization relaxation to equilibrium in switching processes [3];
• spin transfer given by a spin polarized current flowing through the ferromagnetic body [4]. In this
case the LLG equation must be corrected to take in account the effect of the current:
µ
¶
dm
m × ep
dm
− αm ×
= −m × hef f − β
(2)
dt
dt
1 + c p m · ep
[1] G. Bertotti, I.D. Mayergoyz, and C. Serpico, Phys. Rev. Lett., 86, 724 (2001); Phys. Rev. Lett., 87, 217203
(2001)
[2] G. Bertotti, R. Bonin, I.D. Mayergoyz, and C. Serpico, Journ. Appl. Phys, 95, 11, 7046 (2004)
[3] G. Bertotti, R. Bonin, A. Magni, I.D. Mayergoyz, and C. Serpico, Energy equation for the analysis of magnetization relaxation to equilibrium, to be published in JMMM ; Analytical description of magnetization relaxation
to equilibrium, to be presented at the MMM04 conference
[4] G. Bertotti, A. Magni, R. Bonin, I.D. Mayergoyz, and C. Serpico, Bifurcation analysis of magnetization dynamics driven by spin injection, to be published in JMMM
(*) bonin@ien.it
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