Several physical properties of major current interest for magnetic nano-devices are determined by the small amplitude magnetization dynamics within a finite bandwidth, in the vicinity of an extremum of the micro-magnetic energy functional. Among the most salient examples in this class of problem we shall mention: spin wave band structures in magnonic crystals, (linear) resonance spectra (FMR, MRFM, spin diodes…), thermal magnetization fluctuations (magnoise) around a stationary state, and the onset of the dynamic instability under spin transfer or spin orbit torque. In such cases one can safely linearize the magnetization equation of motion around the considered equilibria. The resulting linear dynamic equations can be further cast into an eigenvalue (spectral) problem, for a strictly differential linear operator, after Fourier transform in the time variable and introduction of a magneto static potential. We will discuss how this is then amenable to extremely efficient finite element numerical implementations, taking advantage of state of the art linear algebra libraries. Numerical results for various physical examples will be presented and discussed, illustrating the versatility and power of this approach. In some cases orders of magnitude improvements in computational time are achieved compared to more traditional approaches (time domain solution of the nonlinear LLG equations + FFT in time). If time allows, the relevance of spectral calculations in the context of some fully non-linear problems will be also outlined (amplitude equations, string methods…).
MAINZ Graduate School and SPICE Center, Johannes Gutenberg University, Mainz, Germany