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1 Introduction

Spin torque nano-oscillators (STNO) are a ferromagnet-based electronics component. In certain steady states, the magnetic moment precesses causing component resistance to oscillate [12]. Based on this oscillating resistance, an STNO can be utilized as a microwave-range voltage oscillator (see Fig. 1). However, STNOs tested till date have yet to produce adequate power. STNOs need to output at least 1 mW to be applicable [11]. The microwave power generated by an STNO was first measured in 2010 on the order of 1 nW [5]. One solution to increasing power is to electrically couple multiple oscillators. However, in experiments it has been proven that it is difficult to synchronize even two STNOs [8]. Thus, we have begun to study the dynamics of coupled STNOs to determine conditions for synchronization. In this chapter we describe the model in Cartesian coordinates and then project to complex stereographic coordinates. Next, we numerically analyze the local stability of synchronized limit cycles using transverse Lyapunov exponents (TLE). Following TLEs, we investigate global behavior by creating a two-parameter bifurcation diagram.

2 The Model

Magnetization in the free ferromagnetic layer is described by the Landau–Lifshitz equation with Gilbert damping and Slonczewski–Berger spin-torque term (LLGS) [1,2,4,6,13]

$$\frac{{\rm d}{\bf m}}{{\rm d} t} = \overbrace{-\gamma{\bf m}\times{\bf H}_{\text{eff}}}^{\displaystyle\text{precession}} + \overbrace{\lambda{\bf m}\times\frac{{\rm d} {\bf m}}{{\rm d} t}}^{\displaystyle\text{damping}} - \overbrace{\gamma a\: g{\left(P,{\bf m}\cdot{\bf M}\right)}\: {\bf m}\times\left({\bf m}\times{\bf M}\right)}^{\displaystyle\text{spin transfer torque}}\;,$$
(1)

where \({\bf m}\) represents the magnetization of the free ferromagnetic layer in Cartesian coordinates, γ is the gyromagnetic ratio, and \({\bf H}_{\text{eff}}\) is the effective external field. λ serves as the magnitude of the damping term. In the spin-torque term, a has units Oe and is proportional to the electrical current density [10]. g is a scalar function of the polarization factor P, \({\bf m}\), and the fixed-layer magnetization direction \({\bf M}\). To determine the change of field direction with respect to time, we must consider three different classes of torques acting on the field direction \({\bf m}\): effective external magnetic field \({\bf H}_{\text{eff}}\), damping λ, and spin transfer torque. \({\bf H}_{\text{eff}}\) is the sum of several factors that can be effectively represented as external fields. The factors that we consider in this fashion are exchange, anisotropy, and demagnetization. The actual external, or applied, field rounds out the sum

$$ {\bf H}_{\text{eff}} = {\bf H}_{\text{exchange}} + {\bf H}_{\text{anisotropy}} + {\bf H}_{\text{demagnetization}} + {\bf H}_{\text{applied}}.$$

We model the free layer as a single particle who’s magnetization \({\bf m}\) represents the average of the layer. Thus, there is no exchange with adjacent magnetic moments \({\bf H}_{\text{exchange}}=0\).

subsubsubparagraph Complex Stereographic Projection A spherical surface can be projected onto a plane by using the complex variable ω and the following relationships:

$$\omega=\frac{m_x+im_y}{1+m_z} \Rightarrow \quad{\bf m} =\begin{bmatrix} \displaystyle \frac{\omega+\bar{\omega}}{1+\left|\omega\right|^2} \\\displaystyle -i\frac{\left(\omega-\bar{\omega}\right)}{1+\left|\omega\right|^2} \\\displaystyle \frac{1-\left|\omega\right|^2}{1+\left|\omega\right|^2}\end{bmatrix}.$$
(2)

Here \(\bar{\omega}\) is the complex conjugate of ω. This projection maps the sphere’s north pole to the origin and the south pole to infinity. Building on [9,10], we reduce Eq. (1) to the form

$$\begin{aligned} \dot{\omega} = &\frac{\gamma}{1-i\lambda}\Bigg(-a\omega + i h_{a3}\omega + \frac{ h_{a2}}{2}(1 + \omega^2) \notag \end{aligned}$$
(3)
$$\begin{aligned} &+ im_{\Vert}\kappa\left[\cos\theta_{\Vert}\omega-\frac{1}{2}\sin\theta_{\Vert}\left({\rm e}^{i\phi_{\Vert}}-\omega^2{\rm e}^{-i\phi_{\Vert}}\right)\right] \notag \end{aligned}$$
(4)
$$\begin{aligned} &-\frac{i4\pi{S_o}}{(1+|\omega|^2)}\bigg[N_3(1-|\omega|^2)\omega-\frac{N_1}{2}(1-\omega^2-|\omega|^2)\omega \end{aligned}$$
(5)
$$\begin{aligned} & \hspace{65pt} -\frac{N_2}{2}(1+\omega^2-|\omega|^2)\omega-\frac{(N_1-N_2)}{2}\bar{\omega}\bigg]\Bigg), \notag\end{aligned}$$
(6)

where \(h_{a2}\) is the magnitude of the applied field in the y-direction and \(h_{a3}\) is the magnitudes of the applied field in the z-direction. κ is the anisotropy magnitude who’s direction is determined by the spherical-coordinate parameters \(\theta_\Vert\) and \(\phi_\Vert\). The anisotropy is scaled by \(m_\Vert = {\bf m}\cdot{\bf e}_\Vert\) where

$$\begin{aligned} {\bf e}_{\Vert}= \left[\begin{array}{c} \sin\theta_{\Vert}\cos\phi_{\Vert}\\ \sin\theta_{\Vert}\sin\phi_{\Vert}\\ \cos\theta_{\Vert}\end{array}\right].\end{aligned}$$

S 0 is the saturation magnetization. Finally, \(N_1,N_2,\) and N 3 describe the effective demagnetization field resulting from the shape of the free layer and are constrained by the relationship \(N_1+N_2+N_3=1\). The magnetic moment of an STNO is now described by two dimensions and in a polynomial-like form.

subsubsubparagraph Coupling Coupling is achieved by modeling a simple electrical circuit with STNOs arrayed in series or parallel. Figure 1 depicts the series configuration. The resistance of each STNO R i is a function of the angle θ i between M (fixed layer-green) and \({\bf m}\) (free layer-red):

$$ R_i = R_{0i} - \Updelta R_{i}\cos{\theta_i}.$$

Here, R 0 is the median resistance of an STNO and \(\Updelta R\) is the maximum variance in resistance.

Fig. 1
figure 1

Series arrayed STNOs with input current I DC and output resistance R c . The fixed ferromagnetic layer is green with magnetic moment \({\bf M}\). The free ferromagnetic layer is red and its magnetic moment is modeled by \({\bf m}\)

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3 Transverse Lyapunov Exponents

Transverse Lyapunov exponents allow us to quantify the local stability of a synchronized orbit [3,7]. Specifically, TLEs use the linearized system to measure how a small perturbation transverse to the synchronization manifold \((z_s=z_1=\cdots=z_n)\) grows or contracts. Figure 2 depicts the result of numerically calculating TLEs for two serially-coupled and identical STNOs. Here we vary the electrical current I DC with a grid step size \(\Updelta I_{DC}=10\) and the applied field angle θ h with a step size \(\Updelta \theta_h=0.05\). Initial conditions are on the synchronization manifold and close to the expected steady state. In this case, the sum of TLEs is a good representation of stability, hence the plot is color coded accordingly. A negative sum indicates stable while a positive sum indicates instability of the synchronized orbit. Additionally, the uniform red color represents areas where no oscillations were detected. These results indicate large regions of parameter space where the synchronized solution is stable. Further, there are distinct boundaries of oscillations that may trace the locus of Hopf bifurcations.

Fig. 2
figure 2

Sum of the transverse Lyapunov exponents calculated over two parameters, the electrical current I DC and the angle of the applied field θ h , with the applied field varying from 0 to \(\frac{\pi}{2}\) (left) and from \(\frac{\pi}{2}\) to π (right). Extreme red indicates no oscillations

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4 Numeric Bifurcation Diagram

We have shown that synchronized oscillations are locally stable for a large parameter space. However, simulations in the same parameter space using random initial conditions show out-of-phase steady-state behaviors [14]. Using the software package XPPAUT, we have created a two-parameter bifurcation diagram (Fig. 3) that includes the parameter spaces from Fig. 2. The diagram reveals a number of back-to-back Hopf bifurcations that are consistent with the boundary of oscillations in Fig. 2. Additionally, each pair of back-to-back Hopfs spawns one synchronized and one out-of-phase limit cycle. One-parameter bifurcation diagrams in I DC (θ h fixed) indicate that the out-of-phase solution is also locally stable for most of the oscillating region. Hence, the region of interest exhibits bistability where we expect the initial conditions to determine synchronized or out-of-phase steady-state behavior.

Fig. 3
figure 3

Two-parameter bifurcation diagram plotting input current I DC v applied field angle θ h . Dashed black lines trace saddle-node bifurcations, while solid lines are Hopfs. Green solid lines are Hopfs that spawn synchronized limit cycles and solid blue lines are Hopfs that spawn out-of-phase oscillations

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5 Remarks

The LLGS Eq. (1) is a nonlinear first-order ordinary differential equation confined to the unit sphere \(\|{\bf m}\|_2=1\). We are able to reduce the dimension of the system one third using complex-stereographic coordinates. Not only does this increase the efficiency of numerics but also simplifies integration by fixing the magnitude of \({\bf m}\) by the nature of the coordinate system.

The calculation of TLEs in Fig. 2 gives the positive result of stable synchronized oscillations in a large parameter space. However, the TLE measurement is inherently local, and therefore does not necessarily reflect global behavior. Using XPPAUT to create a two-parameter bifurcation diagram, we discover the existence of an out-of-phase limit cycle that is also locally stable. This bistability indicates that synchronization can only be achieved if the initial conditions fall within the basin of attraction of the synchronized solution.

In future work we are interested in calculating the basins of attraction, but we are also interested in the behavior of the system for many more STNOs. As we increase the number of oscillators N, we expect a nonlinear increase in the number of oscillatory steady states. We intend to leverage the symmetry-group representation of the coupling to predict the type, existence, and stability of out-of-phase oscillations. Combined with the TLE computation, this will allow us to determine if a region exists where the synchronized solution is globally stable.