Strongly Coupled Modes of M and H for perpendicular resonance

We apply the equations for M and H to study the two doubly-degenerate coupled modes of M and H for a semi-infinite ferromagnet, conductor or insulator, magnetized normal to the plane (perpendicular resonance), for wavevector normal to the plane. With dimensionless damping constant α and dimensionless transverse susceptibility χ_⊥=M₀/H_e (H_e≡H_app−M₀), an analytic expression for the eigenmodes shows that for perturbation theory to apply the condition α >> χ_⊥ must hold. This is violated in the ferromagnetic regime, so perturbation theory does not apply there. This generalizes numerical results of Ament and Rado for parallel resonance. Emphasizing the conductor permalloy, we study the eigenvalues and eigenmodes, as well as the dissipation rate due to absorption both from the total effective field (including, for the first time, the contribution of the non-uniform exchange energy term) and from Joule heating. Using these modes we then apply, for a semi-infinite ferromagnet, a range of boundary conditions (i.e. surface anisotropies) on M_⊥ to find the reflection coefficient R and reflectivity |R|². We show that the absorption is given, not by the imaginary part of the susceptibility, but of an effective susceptibility that includes the effect of the non-uniform exchange coefficient A.