Level attraction and exceptional points in a resonant spin-orbit torque system

Level attraction can appear in driven systems where instead of repulsion, two modes coalesce in a region separated by two exceptional points. This behavior was proposed for optomechanical and optomagnonic systems, and recently it was observed for dissipative cavity magnon-polaritons. We demonstrate that such a regime exists in a spin-orbit torque system where a magnetic oscillator is resonantly coupled to an electron reservoir. An instability mechanism necessary for mode attraction can be provided by applying an electric field. The field excites interband transitions between spin-orbit split bands leading to an instability of the magnetic oscillator. Two exceptional points then appear in the oscillator energy spectrum and the region of instability. We discuss conditions under which this can occur, and we estimate the electric field strength necessary for reaching the attraction region for a spin-orbit torque oscillator with Rashba coupling. A proposal for experimental detection is made using magnetic susceptibility measurements.

Figure 1

Level repulsion and attraction. (a) Typical hybridization picture between energy levels $\mathrm{\Delta }=1$ and $\mathrm{\Omega }$ in the case of $g^2>0$. (b) Level attraction picture with $Re\left(g^2\right)<0$ and $Im\left(g^2\right)=0$ when $\mathrm{\Gamma }=\gamma$. The energy modes ${\omega }^{(\pm )}$ coalesce in the region $(\mathrm{\Omega }-\mathrm{\Delta })<4g^2$ bounded by two exceptional points. (c) Mode attraction is broken when $Im\left(g^2\right)\ne 0$. One exceptional point, which satisfies Eqs. (3), is realized. Dashed lines show the imaginary parts of ${\omega }^{(\pm )}$.

Figure 2

Schematic picture of a magnetic oscillator $\mathit{S}\left(t\right)$ coupled to the material (Pt) with Rashba spin-orbit interaction $\frac{\alpha }{\hslash }(\mathit{p}\times \widehat{\mathit{z}})·\mathit{\sigma }$ via $s\text{−}d$ exchange coupling. (a) In equilibrium, $\mathit{S}\left(t\right)$ precesses around the quantization $z$ axis, which coincides with the direction of the magnetization of conduction electrons $\langle \mathit{s}\rangle$. (b) Applying electric field $\mathit{E}$ in noncentrosymmetric material creates a transverse spin accumulation $\langle \delta \mathit{s}\rangle$ that modifies the dynamics of the magnetic oscillator.

Figure 3

Real parts of ${\omega }^{(+)}$ and ${\omega }^{(-)}$ for the spin-orbit torque oscillator with the Rashba spin-orbit interaction on the $(E_x,E_y)$ plane. The branch cut, which is determined by Eq. (3b), terminates at the exceptional point given by Eq. (3a). A closed loop around the exceptional point is shown by the red line.

Figure 4

Magnetic susceptibility for the Rashba spin-orbit torque oscillator (arbitrary units). The solid (dashed) line shows the imaginary (real) part of $\chi \left(\omega \right)$. (a) Two absorption peaks for $E<E_c$. (b) Resonance and antiresonance in the case of $E>E_c$ and weak interaction $|\mathrm{\Omega }-\mathrm{\Delta }|\gtrsim |ReK|$. (c) Strongly interacting case for $E>E_c$ and $|\mathrm{\Omega }-\mathrm{\Delta }|\lesssim |ReK|$.