Stability of Spin-Torque Oscillators With Dual Free Layers

A system of differential equations describing the stability of a spin-torque oscillator (STO) with two free layers is introduced. The system is derived from the Landau-Lifshitz-Gilbert (LLG) equation for two freely rotating magnetic moments interacting via spin-torque. In the case of two free magnetic layers, the magnetization of each layer can precess at its own frequency, which is directly proportional to the local effective field in the layer. The spin-torque depends on the relative angle between two layers. Therefore, if the magnetization in two magnetic layers rotates with different frequencies, the spin-torque will vary in time. This can lead to instabilities of the precession angle in both the layers. When two layers oscillate at equal frequencies, instabilities due to spin-torque variations are eliminated. The requirement for the synchronization leads to a system of three differential equations for three unknowns. The stable oscillation orbits for each layer are obtained by finding fixed point solutions of the system. The results were confirmed using finite-element method (FEM) micromagnetic simulations.