Dipole-exchange spin waves in perpendicularly magnetized discs: Role of the Oersted field

Abstract

We develop the theory of the exchange dipole spin waves in thin circular discs for the case where the magnetization is nominally perpendicular to the plane. Our interest is in the circumstance where a transport current is injected into the disc, with current also perpendicular to the plane of the disc. Such a current creates an azimuthal magnetic field, referred to often as the Oersted field. We develop the theory of the influence of the Oersted field on the spin-wave spectrum of the disc. This field produces a vortex state. We suggest that this vortex state is stable down to zero applied field. If the external applied field ${H}_{0}$ is in the $+z$ direction, perpendicular to the plane of the disc, the vortex state has magnetization at the center of the disc also parallel to $+z$ always. This is the case even when ${H}_{0}l4\ensuremath{\pi}{M}_{S}$, where the magnetization at the center of the disc is antiparallel to the local field ${H}_{0}\ensuremath{-}4\ensuremath{\pi}{M}_{S}$ there. We present calculations of the current dependence of spin-wave frequencies of several modes as a function of applied magnetic field. We also address an issue overlooked in previous studies of spin waves in thin discs. This is that for quantitative purposes, it is not sufficient to describe internal dipole fields generated by the spin motions simply by adding an effective internal field $\ensuremath{-}4\ensuremath{\pi}{m}_{z}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{z}$ to the equations of motion, with ${m}_{z}$ the component of dynamic magnetization normal to the surface. For samples of present interest, we derive terms we call gradient corrections, and these play a role quantitatively comparable to exchange itself in the analysis of the spin-wave frequencies. Quantitative studies of spin dynamics in such samples thus must include the gradient corrections.