Bifurcation analysis of strongly nonlinear injection locked spin torque oscillators

We investigate analytically and numerically the dynamics of an injection locked in-plane uniform spin torque oscillator for several forcing configurations at large driving amplitudes. For the analysis, the spin wave amplitude equation is used to reduce the dynamics to a general auto oscillator equation in which the forcing is a complex valued function F(p,ψ)∝ϵ1(p)cos(ψ)+iϵ2(p)sin(ψ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}(p,\psi ) \propto \epsilon _1(p)cos(\psi )+i \epsilon _2(p)sin(\psi )$$\end{document}. Assuming that the oscillator is strongly non-isochronous and/or forced by a power forcing ( |νϵ1/ϵ2|≫1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\nu \epsilon _1 / \epsilon _2|\gg 1$$\end{document}), we show that the parameters ϵ1,2(p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{1,2}(p)$$\end{document} govern the main bifurcation features of the Arnold tongue diagram: (i) the locking range asymmetry is mainly controlled by the derivative dϵ1/dp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\epsilon _1/dp$$\end{document}, (ii) the loss of stability when the frequency mismatch between the generator and the oscillator increases occurs for a power threshold depending on ϵ1,2(p)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{1,2}(p)$$\end{document} and (iii) the frequency hysteretic range is related to the transient regime at zero mismatch frequency. Then, the model is compared with the macrospin simulation for driving amplitudes as large as 100-103A/m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^0-10^3 A/m$$\end{document} for the magnetic field and 1010-1012A/m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{10}-10^{12} A/m^2$$\end{document} for the current density. As predicted by the model, the forcing configuration (nature of the driving signal, applied stimuli direction, harmonic orders) affects substantially the oscillator dynamic. However, some discrepancies are observed. In particular, the prediction of the frequency and power locking range boundaries may be misestimated if the hysteretic boundaries are of same magnitude order. Moreover, the misestimation can be of two different types according to the bifurcation type. These effects are a further manifestation of the complexity of the dynamics in non-isochronous auto-oscillators.