Phenomenological models of magnetization damping

Abstract

In the last 70 years many models have been published for the relaxation of magnetization precession in ferromagnetic materials. Such models are important to predict energy loss for three phenomena: the free precession decay, the precession driven by an applied microwave field, and the noise of a system in equilibrium. Five models of magnetization damping are typical and were selected for this thesis: Landau-Lifshitz (LL) damping, Bloch-Bloembergen (BB) damping, Codrington-Olds-Torrey (COT) damping, Gilbert (G) damping and Modified Bloch-Bloembergen (MBB) damping. Traditionally, these models have been characterized in the small-signal limit by their susceptibility tensors, which relate the complex amplitudes of driving field and the magnetization response in a linear manner. This work categorizes the five damping models according to which field or magnetization components drive the damping. The models are compared with regard to their relaxation rate of the free decay, the geometrical shape of their trajectory, their susceptibility tensor, their energy loss for the precession driven by an external microwave field, and their thermal noise in equilibrium. The energy loss of the driven precession is determined both as time-averaged loss and as instantaneous loss. The analyses in this work take advantage of the Smith matrix form of the equations of motion for the magnetization precession in the small-signal limit. The Smith matrix form is the general form of a linearized system of coupled harmonic oscillators, solved for the external driving force. Traditionally, energy loss and noise power spectra have been calculated with the components of the susceptibility tensor involving bulky ratios of complex expressions. In this work, all calculations are carried out with the real-valued matrices that multiply the magnetization and its derivative in the Smith form of the equations of motion. The three main advantages are the following: a) The calculations are simpler, b) Many results about energy loss and thermal noise are expressed in simple form in terms of a single matrix, the symmetric part of the damping matrix, c) The Smith matrix form allows calculation of instantaneous, time-dependent loss, where traditionally only time-averaged loss was considered. The power spectra of field noise as well as magnetization noise are analyzed with the fluctuation-dissipation theorem and the Wiener-Khinchine theorem. In this work a classical proof of the fluctuation-dissipation theorem is given, which is more elementary than most found in the literature and uses Boltzmann statistics and ordinary differential equations.