Robust synchronization of an arbitrary number of spin-torque-driven vortex nano-oscillators

Nonlinear magnetization dynamics in ferromagnetic nanoelements excited by a spin-polarized dc current is one of the most intensively studied phenomena in solid-state magnetism. Despite immense efforts, synchronization of oscillations induced in several such nanoelements [spin-torque-driven nano-oscillators (STNO)] still represents a major challenge from both the fundamental and technological points of view. In this paper we propose a system where synchronization of any number of STNOs, represented by magnetization vortices inside squared nanoelements, can be easily achieved. Using full-scale micromagnetic simulations we show that synchronization of these STNOs is extremely dynamically stable due to their very large coupling energy provided by the magnetodipolar interaction. Finally, we demonstrate that our concept allows robust synchronization of an arbitrary number of STNOs (arranged either as a one-dimensional chain or as a two-dimensional array), even when current supplying nanocontacts have a broad size distribution.

Figure 1

Magnetization dynamics of a single STNO. [(a) and (b)] Snapshots of magnetization configurations in low- and high-current oscillation regimes [solid circles define the nanocontact area, the dotted line in (a) is the vortex trajectory]. (c) Spectral amplitude of magnetization oscillations under the contact.

Figure 2

Vortex profiles ($m_{\perp }$ along the line connecting the vortex core and the point contact centers) for various current strengths demonstrate strong vortex deformations for large currents.

Figure 3

Magnetization dynamics of two dipolar-coupled STNOs. (a) Snapshot of a magnetization configuration in a synchronized regime (note opposite oscillation phases under the contacts); (b) time dependencies of the average magnetization under the first [$m_1^x\left(t\right)$, red (medium gray) line] and second [$m_2^x\left(t\right)$, blue (dark gray) line] contacts and their sum [$m_{\mathrm{tot}}^x\left(t\right)=m_1^x\left(t\right)+m_2^x\left(t\right)$, green (light gray) line]; (c) spectral amplitude of the total magnetization $m_{\mathrm{tot}}^x$; (d) oscillation power of the difference $m_{\mathrm{diff}}^x\left(t\right)=m_1^x\left(t\right)-m_2^x\left(t\right)$, showing the large power increase in the synchronized regime if the phase shift of $\pi$ is introduced between the ac currents generated by the process. It can be clearly seen that when thermal fluctuations are included, the dependence $P\left(I\right)$ [panel (d)] nearly does not change.

Figure 4

Magnetization dynamics of the two STNOs connected by a “bridge” (partial exchange coupling), shown in same way as in Fig. 3. STNOs are synchronized in the almost whole current region, where their steady-state gyration is observed.

Figure 5

Magnetization dynamics of the two STNOs with a full exchange coupling, presented the same way as in Fig. 3. Vortex gyration is out of phase for almost all current values, leading to in-phase magnetization oscillations under the contacts. Numbers within double-sided arrows correspond to numbers of panels in Fig. 6, where the snapshots of magnetization dynamics for corresponding currents are displayed.

Figure 6

Snapshots of the $m_{\perp }$ projection for the four different synchronized states of two STNOs with a full exchange coupling (see Fig. 5): (1) in-phase oscillations of vortices for $I$ = 3.5 mA, (2) out-of-phase oscillations of vortices for $I$ = 5.0 mA, (3) nearly in-phase oscillations of vortices for $I$ = 8.5 mA accompanied by the polarity reversal of the half-antivortex at the upper system border [shown with red (dark gray) arrows], and (4) oscillations of vortices for $I$ = 9.2 mA accompanied by the polarity reversal of both half-antivortices at the upper and lower [green (light gray) arrows] system borders, whereby the phase difference between the vortex precessions is neither 0 nor $\pi$.

Figure 7

Coupling energy vs current strength in the first nanocontact $I_1$ (for the same $I_2=5$ mA) for systems of two vortex STNOs: (gray triangles) nanodisks with dipolar coupling only, (red triangles) nanosquares with dipolar coupling only [Fig. 3], (green squares) “bridge” configuration [Fig. 4], and (blue circles) full exchange and dipolar coupling [Fig. 5].

Figure 8

To the explanation of the coupling energy dependence on the current $I_1$. [(a) and (a${}^{\prime }$)] Snapshots of the magnetization configurations as arrows; [(b) and (b${}^{\prime }$)] volume charges; and [(c) and (c${}^{\prime }$)] surfaces charges along the adjacent nanosquare surfaces [marked in (b) and (b${}^{\prime }$) with blue and red lines] for two different currents $I_1=1.75$ mA (left column of images) and $I_1=3.0$ mA (right column of images). For the configuration with the “bridge” (see Fig. 4), surface charges between vertical dashed lines shown in parts (c) and (c${}^{\prime }$) would not exist.

Figure 9

Snapshots of the in-plane magnetization distributions for 10 nano-oscillators in the transient ($t=15$ ns, upper panel) and synchronized ($t=116$ ns, lower panel) regimes.

Figure 10

Total oscillation power (normalized to the oscillation power of one STNO) of the alternating sum of the in-plane components inside the point contacts $m_{\mathrm{alt}}^x\left(t\right)={\sum }_i{(-1)}^{i-1}{m}_i^x\left(t\right)$, which takes into account the opposite oscillation phases of the magnetization in adjacent contacts.

Figure 11

Magnetization dynamics of six STNOs connected by the “bridges” that have the same widths $l=200$ nm as in Fig. 4 (partial exchange coupling) and shown in the same way as in previous figures. Configuration of vortices shown in (a) demonstrates the state of the system few nanoseconds after the initial in-plane field pulse expelled the vortices out of contacts area (so the stationary synchronized state is not achieved yet). The perfect synchronization is demonstrated by comparing the relation of the total power to the power of one vortex oscillator $P_{\mathrm{tot}}/P_1$ [red circles in the bottom graph (c)] with the square of the STNOs number $N^2=36$ (blue horizontal line at the same plot).

Figure 12

Snapshots of the vortex annihilation stages in a system of fully exchange-coupled nanosquares on the top and bottom system borders due to the absence of the confinement from side borders of nanosquares.

Figure 13

Snapshots of the out-of-plane magnetization projection for a 2D system of STNOs ($6\times 6$ lattice) shortly after the current has been switched on (upper image) and after a perfect synchronization is achieved (lower image). Currents through all contacts are equal to $I=5$ mA.

Figure 14

Magnetization dynamics of a $6\times 6$ lattice of STNOs shown in Fig. 13, presented in same way as in Fig. 11. In this case the perfect synchronization is also demonstrated by comparing the relation $P_{\mathrm{tot}}/P_1$ [red circles in the bottom graph (b)] with the square of the STNOs number ${36}^2=1296$ (blue horizontal line in the same graph).