Synchronization of spin torque nano-oscillators

Synchronization of spin torque nano-oscillators (STNOs) has been a subject of extensive research as various groups try to harness the collective power of STNOs to produce a strong enough microwave signal at the nanoscale. Achieving synchronization has proven to be, however, rather difficult for even small arrays while in larger ones the task of synchronization has eluded theorists and experimentalists altogether. In this work we solve the synchronization problem, analytically and computationally, for networks of STNOs connected in series. The procedure is valid for networks of arbitrary size and it is readily extendable to other network topologies. These results should help guide future experiments and, eventually, lead to the design and fabrication of a nanoscale microwave signal generator.

Figure 1

Left: Schematic representation of a nanopillar STNO. A spin-polarized current can exert a torque on the magnetization of the free layer and lead to steady precession. Right: A circuit array of STNOs connected in series.

Figure 2

Top: Loci of Hopf bifurcations of synchronized oscillations. Bottom: Stability of synchronization manifold (red, supercritical Hopf and stable synchronization manifold; black, subcritical Hopf and unstable synchronization manifold; and blue, supercritical Hopf and unstable synchronization manifold). The combined results of these two plots reveal the optimal region to synchronize a series array of nanopillar STNOs: the first quadrant of parameter space $(I_{\text{dc}},{\theta }_h)$. Parameters [21] are $N_1=1, N_2=0, \gamma =2.2\times {10}^5\mathrm{m}{\mathrm{A}}^{-1}{\mathrm{s}}^{-1}, \alpha =0.008, \kappa =45$ Oe, $\mu =0.992, h_a=300$ Oe, ${\beta }_{\mathrm{\Delta }R}=5.95\times {10}^{-4}$.

Figure 3

Locking into synchronization with $N=1000$ STNOs. Start at high $I_{\text{dc}}$ and let the system lock into the common equilibrium. Then sweep down $I_{\text{dc}}$ until the common equilibrium vanishes and synchronized oscillations appear. Top inset: Zoom-in on the top part of the oscillation showing a high level of synchronization between all the STNOs. Bottom inset: Zoom-in on the set of random initial conditions for the $N=1000$ STNOs and evolution for small time values showing rapid convergence to a synchronized equilibrium.

Figure 4

Frequency response of an array of $N$ STNOs connected in series. The observed dips in frequency correspond to switching between out-of-plane and in-plane oscillations. Parameters are the same as in Fig. 2, with ${\theta }_h=3\pi /4$.

Figure 5

Linewidth. The observed dips in frequency correspond to switching between out-of-plane and in-plane oscillations. Parameters are the same as in Fig. 2, with ${\theta }_h=3\pi /4$.