A system of differential equations describing the stability of a spin–torque oscillator (STO) with two free layers is introduced. The system is derived from the Landau–Lifshitz–Gilbert (LLG) equation for two freely rotating magnetic moments interacting via spin–torque. In the case of two free magnetic layers, the magnetization of each layer can precess at its own frequency, which is directly proportional to the local effective field in the layer. The spin–torque depends on the relative angle between two layers. Therefore, if the magnetization in two magnetic layers rotates with different frequencies, the spin–torque will vary in time. This can lead to instabilities of the precession angle in both the layers. When two layers oscillate at equal frequencies, instabilities due to spin–torque variations are eliminated. The requirement for the synchronization leads to a system of three differential equations for three unknowns. The stable oscillation orbits for each layer are obtained by finding fixed point solutions of the system. The results were confirmed using finite-element method (FEM) micromagnetic simulations.

中文翻译：

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具有双自由层的自旋扭矩振荡器的稳定性

介绍了描述具有两个自由层的自旋扭矩振荡器 (STO) 稳定性的微分方程系统。该系统源自 Landau-Lifshitz-Gilbert (LLG) 方程，用于通过自旋-扭矩相互作用的两个自由旋转磁矩。在两个自由磁性层的情况下，每一层的磁化强度可以以自己的频率进动，这与层中的局部有效场成正比。自旋扭矩取决于两层之间的相对角度。因此，如果两个磁性层中的磁化以不同的频率旋转，则自旋扭矩会随时间变化。这会导致两层中进动角的不稳定性。当两层以相同的频率振荡时，消除了由于自旋扭矩变化引起的不稳定性。同步的要求导致了三个未知数的三个微分方程的系统。通过寻找系统的不动点解，得到每一层的稳定振荡轨道。使用有限元方法 (FEM) 微磁模拟证实了结果。