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

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.