Probing Phase Coupling Between Two Spin-Torque Nano-Oscillators with an External Source

Phase coupling between auto-oscillators is central for achieving coherent responses such as synchronization. Here we present an experimental approach to probe it in the case of two dipolarly coupled spin-torque vortex nano-oscillators using an external microwave field. By phase locking one oscillator to the external source, we observe frequency pulling on the second oscillator. From coupled phase equations we show analytically that this frequency pulling results from concerted actions of oscillator-oscillator and source-oscillator couplings. The analysis allows us to determine the strength and phase shift of coupling between two oscillators, yielding important information for the implementation of large interacting oscillator networks.

Figure 1

(a) Schematics of the sample and electrical circuit. (b)–(g) Power spectral density maps of auto-oscillation modes in log scale. Four different polarity states of the two thick Py vortex layers are shown: (b) $⟨1\uparrow 2\uparrow ⟩$, (c) $⟨1\downarrow 2\downarrow ⟩$, (d) $⟨1\uparrow 2\downarrow ⟩$, and (e) $⟨1\downarrow 2\uparrow ⟩$. (f),(g) Enlarged power spectral density data of $⟨1\uparrow 2\uparrow ⟩$ and $⟨1\downarrow 2\uparrow ⟩$ for the red box regions of (b) and (e), respectively. Mutual synchronization is observed between ${\mu }_{0}H=17.5$ and 26.0 mT in the $⟨1\downarrow 2\uparrow ⟩$ state.

Figure 2

(a) Location of spectra at the $⟨1\downarrow 2\uparrow ⟩$ state for the microwave study. (b) Auto-oscillation spectra as a function of microwave field frequency for ${\mu }_{0}H=31.9\mathrm{mT}$, indicated in (a). Signal from the source appears as the oblique narrow line. The microwave power is $-23\mathrm{dBm}$, corresponding to ${\mu }_0{h}_{\mathrm{rf}}=0.08\mathrm{mT}$. White arrows show the phase-locking bandwidths. Black dashed curves are the fits to Eq. (4). (c) Vector diagram of ${\mathrm{\Omega }}_{12}$ and ${\mathrm{\Delta }}_{2e}$ when oscillator 1 is phase locked to the microwave field.

Figure 3

Probing dipolar coupling in various conditions. (a),(b) The $⟨1\downarrow 2\uparrow ⟩$ state for ${\mu }_{0}H=31.9\mathrm{mT}$, with microwave powers of (a) $-28\mathrm{dBm}$ and (b) $-18\mathrm{dBm}$. (c),(d) The $⟨1\downarrow 2\downarrow ⟩$ state for ${\mu }_{0}H=-61.6\mathrm{mT}$ with microwave power of (c) $-28\mathrm{dBm}$ and (d) $-23\mathrm{dBm}$. White arrows show the phase-locking bandwidths. Black dashed curves are the fits to Eq. (4). The fit parameters are listed in Table 1.