LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
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The small oscillation modes in complex micromagnetic systems around an equilibrium are numerically evaluated in the frequency domain by using a novel formulation, which naturally preserves the main physical properties of the problem. The Landau-Lifshitz-Gilbert (LLG) equation, which describes magnetization dynamics, is linearized around a stable equilibrium configuration and the stability of micromagnetic equilibria is discussed. Special attention is paid to take into account the property of conservation of magnetization magnitude in the continuum as well as discrete model. The linear equation is recast in the frequency domain as a generalized eigenvalue problem for suitable self-adjoint operators connected to the micromagnetic effective field. This allows one to determine the normal oscillation modes and natural frequencies circumventing the difficulties arising in time-domain analysis. The generalized eigenvalue problem may be conveniently discretized by finite difference or finite element methods depending on the geometry of the magnetic system. The spectral properties of the eigenvalue problem are derived in the lossless limit. Perturbation analysis is developed in order to compute the changes in the natural frequencies and oscillation modes arising from the dissipative effects. It is shown that the discrete approximation of the eigenvalue problem obtained either by finite difference or finite element methods has a structure which preserves relevant properties of the continuum formulation. Finally, the generalized eigenvalue problem is solved for a rectangular magnetic thin-film by using the finite differences and for a linear chain of magnetic nanospheres by using the finite elements. The natural frequencies and the spatial distribution of the natural modes are numerically computed.