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

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 h(x)(T) and h(y)(T) is calculated by using the linear phase-noise theories developed for electronic oscillators and that caused by the longitudinal fluctuation h(z)(T) is treated as a random walk problem well known in the theory of magnetic resonance. It is found that the linewidth caused by h(x)(T) and h(y)(T) is proportional to (alpha T/MV)(1/A(2)), where T is the temperature, alpha, 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 h(z)(T) is independent of the amplitude and is simply proportional to alpha T/MV. (c) 2007 American Institute of Physics.