Pauli Matrix Based Formulation of Classical Spin Wave Dispersion Theory

Ferromagnetic resonance (FMR) analyses are typically based on magnetostatic spin wave theory. In this approximation, only the moderate wave number (k) dipole-exchange spin wave band is taken into account, while the low k and other bands are ignored. Such an incomplete treatment leads to problems when, for example, one analyzes the FMR response in metals or attempts to explain off resonance losses. The full theory can be done classically through the use of the magnetic torque equation and the Maxwell equations, but a full treatment that includes conductivity, damping, and a general anisotropy is generally tedious and complicated. In this work, a new approach to the classical theory of electromagnetic spin wave excitations has been developed, based on the use of Pauli matrices sigma_x, sigma_y, and sigma_z. One can analyze the low k propagation region and the high k exchange region, without approximation, and obtain relatively simple working equations in closed form. Based on this Pauli matrix formulation, one can write down simple expressions for the spin wave dispersion and mode polarizations, and solve the requisite boundary value problems for finite geometry problems in a simple way.