Spin-wave modes in ferromagnetic nanodisks, their excitation via alternating currents and fields, and auto-oscillations

Abstract

The excitation of the linear spin wave modes of a soft ferromagnetic free layer of a nanopillar structure through dc-ac currents that traverse the structure is studied, as well as with ac magnetic fields. There is interest in understanding the magnetization dynamics in these structures since they may be used as microwave sources when these nano-oscillators enter into auto-oscillatory regimes. The free layer is a soft ferromagnet, like Permalloy, in the shape of a circular disk, with a very small thickness in the range of the exchange length. Using a description of the magnetization dynamics in terms of a Hamiltonian for weakly interacting waves, we determine the spin wave modes of the structure under two approximations: a very thin film limit, and under a model that includes the effect of the full magnetostatic interaction. We consider direct and parametric excitations of different spin wave modes with ac currents, i.e., with exciting frequency approximately equal to the frequency of the mode or to twice its value, respectively. The Oersted field mainly plays a role in the direct resonant excitation of the modes. Our main conclusion is that for a dc current below the critical value necessary for the development of auto-oscillations, using parametric excitation, a very high value of the ac current is necessary to reach the auto-oscillatory behavior in this geometry. However, if the out-of-plane component of the spin transfer torque is high enough, the ac critical current for auto-oscillations is significantly reduced, leading to a signature for its detection. We comment on parallel pumping and transverse excitation using ac magnetic fields.