Phase and amplitude noise spectra in spin-transfer nano-oscillators

Abstract

Analytical expressions of the phase and the amplitude noise spectra under the thermal fluctuation fields (i=x, y, z) are derived for a simplified model of spin-transfer oscillators within a small amplitude approximation. The line broadening caused by the transverse fluctuations hxT and hyT is calculated by using the linear phase-noise theories developed for electronic oscillators and that caused by the longitudinal fluctuation hzT is treated as a random walk problem well known in the theory of magnetic resonance. It is found that the linewidth caused by hxT and hyT is proportional to (αT∕MV)(1∕A2), where T is the temperature, α, M, and V are the Gilbert damping parameter, the saturation magnetization, and the volume of the free layer, respectively, and A is the amplitude of oscillation. On the other hand, the width caused by hzT is independent of the amplitude and is simply proportional to αT∕MV.