Oscillatory edge modes in two dimensional spin-torque oscillator arrays

Spin-torque oscillators (STOs) are dissipative magnetic systems that provide a natural platform for exploring non-Hermitian phenomena. We theoretically study a two-dimensional (2D) array of STOs and show that its dynamics can be mapped to a 2D, non-Hermitian Su-Schrieffer-Heeger (SSH) model. We calculate the energy spectrum and identify the one-dimensional (1D) edge states of our model, corresponding to auto-oscillation of STOs on the boundary of the system while the bulk oscillators do not activate. We show that tuning the Gilbert damping, injected spin current, and coupling between STOs allows for exploring the edge-state properties under different parameter regimes. Furthermore, this system admits 1D edge states with nonuniform probability density, and we explore their properties in systems of different sizes. Additional symmetry analysis indicates that these states are not topologically protected but are nevertheless confined to the edge of the system, as the bulk is protected by $\mathcal{P}\mathcal{T}\text{symmetry}$. These results indicate that 2D arrays of STOs may be useful to explore novel edge-state behavior in dissipative systems.

Figure 1

Schematic of the effective tight-binding model of a 2D STO array, showing a system of size $4\times 8$, where there are 4 unit cells per row. Red (dark color) sites are the A sublattice and green (light color) sites are the B lattice.

Figure 2

The dependence of the real (a)–(b) and imaginary (e)–(f) parts of the energy spectrum of $H$ on $J/|\tilde{J}|$ for a system of $10\times 20$ STOs for $\omega =\tilde{J}, \alpha =0.2, J=0.2\tilde{J}$, and $J_2=0.01\tilde{J}$ (a),(e) and $J_2=0.1\tilde{J}$ (b),(f). Modes with $\mathrm{Im}\left(E\right)>0$ correspond to auto-oscillation of STOs on the edge of the system, appearing for $\left|J\right|<\tilde{J}$. These modes correspond to the flat in-gap bands at $\mathrm{Re}\left(E\right)-\omega \approx 0$ in the real spectrum. Panels (c)–(d), (g)–(h) show the real (c)–(d) and imaginary (g)–(h) parts of the energy spectrum for a system of $50\times 100$ STOs, with $J_2=0.01\tilde{J}$ (c),(g) and $J_2=0.1\tilde{J}$ (d),(h). Apart from the different system size, all the other parameters are same as in (a)–(b) and (e)–(f). In the larger system there is some activation of bulk STOs in the $\mathcal{P}\mathcal{T}$-broken regime, indicated in (g)–(h) by the additional states with $\mathrm{Im}\left(E\right)>0$ in the region $\alpha \omega >|\tilde{J}-J|$.

Figure 3

Spatial distribution of 1D edge modes in a 2D STO array. (a)–(d) Color density plots showing ${\left|\psi \right|}^2$ for four of the the flat-band modes with $\mathrm{Im}\left(E\right)>0$. The states are localized on the left edge, with ${\left|\psi \right|}^2=0$ everywhere in the bulk. The insets show the 1D distribution along the edge. (e) 1D plot showing the spatial distribution of all 10 lasing edge states and the corresponding real part of the energy $\mathrm{Re}\left(E\right)-\omega$. The edge states are not uniform across the system due to the coupling $J_2$ between rows. States appear in pairs where states with equal magnitude and opposite sign of $\mathrm{Re}\left(E\right)-\omega$ have the same spatial distribution (dashed and solid lines overlap).

Figure 4

Example of the spatial distribution of 10 unique lasing edge states (out of 50) and the corresponding real part of the energy $\mathrm{Re}\left(E\right)-\omega \times {10}^{-1}$ for a system of $50\times 100$ sites. The larger system shows greater heterogeneity in the spatial distribution of edge modes. The top row of the legend corresponds to the left column and the bottom row corresponds to the right column; values increase downward in each column. For example $\mathrm{Re}\left(E\right)-\omega =1.0$ corresponds to the bottom-left panel.

Figure 5

Energy spectrum as a function of $k_y$ using periodic boundary conditions in the vertical direction. Subplots (a) and (b) have parameters $J_2/\tilde{J}=0.01$ and $J_2/\tilde{J}=0.1$, respectively while $J/\tilde{J}=0.2$, as in Fig. 2. The edge states are well separated from the bulk states and have nonzero group velocity.