Analytical investigation of modulated spin torque oscillators in the framework of coupled differential equations with variable coefficients

Abstract

Modulation of Spin Torque Oscillators (STOs) is investigated by analytically solving the time-dependent coupled equations of an auto-oscillator. A Fourier series solution is proposed, leading to the coefficients being determined with a linear set of equations, from which a Nonlinear Amplitude and Frequency Modulation (NFAM) scheme is obtained. In this framework, the NFAM features are related to the intrinsic STO parameters, revealing a frequency-dependence of the harmonic-dependent modulation index that allows a modulation bandwidth to be defined for these devices. The presented results expose a rich parameter space, where the modulation and the STO's operation conditions define the observed modulation features. The Fourier-series representation of the time signal is suitable for studying periodic perturbations on the auto-oscillator equation.