Determination of allowed standing exchange modes in nonmetallic ferromagnetic films

A secular equation is derived which determines the allowed standing spin waves in thin nonmetallic ferromagnetic films in terms of surface anisotropy (which may differ at the two surfaces) and of bulk exchange, anisotropy, and applied-field parameters. The method is restricted to cases in which the exchange dominates the surface anisotropy, so that the equilibrium magnetization can be approximated as uniform across the film. Within the same approximation, exchange modes only and not magnetostatic ones are examined. It is shown how modes may exist with different surface and bulk ellipticities; these involve superposition of degenerate Larmor and counter-Larmor precessions. By variously passing to the continuum, setting ellipticities equal to unity, letting the two surfaces become identical, and considering one surface only, many existing methods of attacking the problem are reproduced and interrelated.