Nonlocal Spin Transport Mediated by a Vortex Liquid in Superconductors

Departing from the conventional view on superconducting vortices as a parasitic source of dissipation for charge transport, we propose to use mobile vortices as topologically stable information carriers. To this end, we start by constructing a phenomenological theory for the interconversion between spin and vorticity, a topological charge carried by vortices, at the interface between a magnetic insulator and a superconductor, by invoking the interfacial spin Hall effect therein. We then show that a vortex liquid in superconductors can serve as a spin-transport channel between two magnetic insulators by encoding spin information in the vorticity. The vortex-mediated nonlocal signal between the two magnetic insulators is shown to decay algebraically as a function of their separation, contrasting with the exponential decay of the quasiparticle-mediated spin transport. We envision that hydrodynamics of topological excitations, such as vortices in superconductors and domain walls in magnets, may serve as a universal framework to discuss long-range transport properties of ordered materials.

Figure 1

(a) A schematic of spin-vorticity transmutation at the interface between a magnetic insulator and a superconductor. The blue arrow represents spin in a magnetic insulator and the red arrows depict the phase of a superconducting vortex. (b) A schematic of the mean-field phase diagram of bulk type-II superconductors [2], which comprises the Meissner phase at low magnetic fields $H<H_{\mathrm{c}1}$ with complete expulsion of the magnetic flux, the mixed phase at intermediate fields $H_{\mathrm{c}1}<H<H_{\mathrm{c}2}$, where the magnetic flux penetrates into a superconductor in the form of vortices, and the normal-metal phase at high fields $H>H_{\mathrm{c}2}$, where superconductivity is destroyed. $T_c$ is the critical temperature for the onset of superconductivity. Vortices in the mixed phase can form several states of matter, including vortex liquid.

Figure 2

Schematics of experimental setups for probing spin-vorticity transmutation at the interface between a magnetic insulator (MI) and a superconductor (SC) subjected to an external magnetic field $\mathbf{H}$. (a) The torque $\mathit{\tau }$ [Eq. (2)] on MI exerted by the vorticity flux ${\mathbf{j}}_v$ in SC can be probed by performing ferromagnetic resonance (FMR) measurements on MI, while applying a transverse current to SC, which generates a longitudinal vorticity flux ${\mathbf{j}}_v$ via the Lorentz force [13]. (b) The vorticity-dependent work $W$ [Eq. (2b)] performed by magnetic dynamics $\dot{\mathbf{n}}$ on vortices creates the nonequilibrium vorticity accumulation at the interface and thereby induces a diffusive vorticity flux ${\mathbf{j}}_v$. The vorticity flux ${\mathbf{j}}_v$ can be probed by measuring a transverse voltage drop across the SC, which is induced by ${\mathbf{j}}_v$ via the Josephson effect [13], while driving FMR dynamics in the MI. In addition, the energy dissipation through vortex generation suggests a new channel for a Gilbert damping of MI, which can be manifested through the enhancement FMR linewidth [14].

Figure 3

(a) The vorticity-current density ${\mathbf{j}}_v$ induces the charge-current density ${\mathbf{j}}_c$ [Eq. (3)] in SH via the Josephson effect, which in turn exerts the torque $\mathit{\tau }$ [Eq. (4)] on the MI via the spin Hall effect. (b) The dynamics of the MI $\dot{\mathbf{n}}$ induces the normal charge-current density ${\mathbf{j}}_c$ [Eq. (5)] in SH via the inverse spin Hall effect. The counterpropagating supercurrent then performs the vorticity-dependent work $W$ [Eq. (6)] on a vortex via the Lorentz force.

Figure 4

A schematic of the nonlocal spin transport between two MIs mediated by a vortex liquid in the interconnecting SC. The dynamics of the magnetization $\mathbf{n}$ of the left MI pumps the vorticity current ${\mathbf{j}}_v$ into the SC. Some of the pumped vortices travel across the SC by thermal diffusion and reach the interface to the right MI, exerting the torque $\mathit{\tau }$ to it. The width of the SC needs to be much larger than the length to prevent vortices from escaping through sides.