Abstract

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 \(\mathcal {F}(p,\psi ) \propto \epsilon _1(p)cos(\psi )+i \epsilon _2(p)sin(\psi )\). Assuming that the oscillator is strongly non-isochronous and/or forced by a power forcing ( \(|\nu \epsilon _1 / \epsilon _2|\gg 1\)), we show that the parameters \(\epsilon _{1,2}(p)\) govern the main bifurcation features of the Arnold tongue diagram: (i) the locking range asymmetry is mainly controlled by the derivative \(d\epsilon _1/dp\), (ii) the loss of stability when the frequency mismatch between the generator and the oscillator increases occurs for a power threshold depending on \(\epsilon _{1,2}(p)\) 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 \(10^0-10^3 A/m\) for the magnetic field and \(10^{10}-10^{12} A/m^2\) 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.

Graphical Abstract

中文翻译：

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强非线性注入锁定自旋矩振荡器的分岔分析

摘要

我们通过分析和数值研究了大驱动振幅下几种强制配置的注入锁定面内均匀自旋扭矩振荡器的动力学。为了进行分析，使用自旋波振幅方程将动力学简化为一般的自动振荡器方程，其中力是复值函数\(\mathcal {F}(p,\psi ) \propto \epsilon _1(p )cos(\psi )+i \epsilon _2(p)sin(\psi )\)。假设振荡器是强非等时的和/或受到动力强迫（\(|\nu \epsilon _1 / \epsilon _2|\gg 1\) ），我们表明参数\(\epsilon _{1 ,2}(p)\)控制阿诺德舌图的主要分叉特征：(i) 锁定范围不对称性主要由导数\(d\epsilon _1/dp\)控制，(ii) 稳定性损失当发生器和振荡器之间的频率失配增加时，功率阈值取决于\(\epsilon _{1,2}(p)\)和 (iii) 频率迟滞范围与零失配时的瞬态状态相关频率。然后，将该模型与宏自旋模拟进行比较，以驱动磁场幅度高达\(10^0-10^3 A/m\)和\(10^{10}-10^{12} A/ m^2\)为电流密度。正如模型预测的那样，强制配置（驱动信号的性质、施加的刺激方向、谐波阶数）对振荡器动态有显着影响。然而，也观察到一些差异。特别地，如果滞后边界具有相同的量级，则频率和功率锁定范围边界的预测可能被错误估计。此外，根据分叉类型，误估计可以有两种不同类型。这些效应进一步体现了非等时自振荡器动力学的复杂性。

图形概要