Persistent Spin Oscillations in a Spin-Orbit-Coupled Superconductor

Quasi-two-dimensional superconductors with tunable spin-orbit coupling are very interesting systems with properties that are also potentially useful for applications. In this Letter we demonstrate that these systems exhibit undamped collective spin oscillations that can be excited by the application of a supercurrent. We propose to use these collective excitations to realize persistent spin oscillators operating in the frequency range of 10 GHz–1 THz.

Figure 1

(a) Response of a Cooper pair in the $\lambda =+\mathrm{\text{chirality}}$ subband of a Gor’kov-Rashba superconductor subjected to an oscillating magnetic field in the $\widehat{\mathit{y}}$ direction. The solid circle is the Fermi surface and the black dot is the origin of momentum space. The arrows labeled by “1”, “2”, and “0” describe the orientation of the spins under the action of a magnetic field that points up, down, or vanishes. Spontaneous oscillations are sustained, in the absence of a magnetic field, by the internal exchange field. (b) A supercurrent boosts the Fermi surface in the $\widehat{\mathit{x}}$ direction (solid line) and creates a magnetic field in the $\widehat{\mathit{y}}$ direction. As a result, spins begin to oscillate around the new equilibrium orientation, indicated by the thick red arrows on the boosted Fermi surface.

Figure 2

Panels (a)–(c) The in-plane dynamical spin susceptibility ${\chi }_{\parallel }(0,\omega )$ [in units of the 2D density-of-states $m/(2\pi )$] as a function of $\omega$ (in units of $\mu$) for increasing values of $\Delta$ (in units of $\mu$) and $V=0$. The solid line represents $\Re e{\chi }_{\parallel }(0,\omega )$, while the dashed line represents $\Im m{\chi }_{\parallel }(0,\omega )$. Note that for finite $\Delta$, $\Re e{\chi }_{\parallel }(0,\omega )$ diverges at $\omega ={\omega }_1$ and that $\Im m{\chi }_{\parallel }(0,\omega )=0$ for $0<\omega <{\omega }_1$. Panel (d) The quantity $E_{+}(k)+E_{-}(k)$ as a function of $k$ (in units of $k_0$). In this figure we have fixed $\alpha =0.2\mu /k_0$ with $k_0=\sqrt{2m\mu }$.

Figure 3

Panel (a) The $\omega \to 0$ limit of ${\chi }_{\parallel }(0,\omega )$ as a function of $\alpha k_0/\Delta$. The solid line is the result obtained from Eq. (11) while the filled circles are the result of Ref. 6. Panel (b) A 2D color plot of the frequency ${\omega }_{\parallel }$ of the in-plane collective spin mode (in units of $\Delta$) as a function of the inverse of the strength of electron-electron repulsions ($V_0/V$, with $V_0=4\pi /m$) and SOC ($\alpha k_0/\Delta$). In this plot $\Delta =0.1\mu$. The top contour line is for ${\omega }_{\parallel }=2\Delta$ while the bottom contour line defines the boundary of the region in which ${\omega }_{\parallel }=0$. The collective spin mode is undamped when it lies within the superconducting gap ($0<{\omega }_{\parallel }<2\Delta$), i.e., when ${\omega }_{\parallel }$ falls in the region enclosed by the two contour lines. Panels (c) and (d) represent 1D cuts of the plot in panel (b).