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

TL;DR

This paper analytically investigates the modulation behavior of Spin Torque Oscillators using coupled differential equations, deriving a nonlinear modulation scheme that relates intrinsic device parameters to modulation bandwidth and features.

Contribution

It introduces a Fourier series-based analytical approach to model and understand the nonlinear amplitude and frequency modulation of STOs, linking modulation characteristics to device parameters.

Findings

Derived a linear set of equations for Fourier coefficients of STO signals

Established a frequency-dependent modulation index for STOs

Mapped the modulation features to intrinsic device parameters

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.