Magnon Spin Current Induced by Triplet Cooper Pair Supercurrents

At the interface between a ferromagnetic insulator and a superconductor there is a coupling between the spins of the two materials. We show that when a supercurrent carried by triplet Cooper pairs flows through the superconductor, the coupling induces a magnon spin current in the adjacent ferromagnetic insulator. The effect is dominated by Cooper pairs polarized in the same direction as the ferromagnetic insulator, so that charge and spin supercurrents produce similar results. Our findings demonstrate a way of converting Cooper pair supercurrents to magnon spin currents.

Figure 1

A superconductor ($S$) can carry (a) a charge supercurrent consisting of an equal number of spin-up and spin-down triplet Cooper pairs traveling in the same direction or (b) a spin supercurrent where the spin-up and spin-down triplets travel in opposite directions. Because of a coupling between the spins in the superconductor and in an adjacent ferromagnetic insulator (FI), these supercurrents induce a magnon spin current in the FI (green). The induced spin current can be measured through a normal-metal contact via the inverse spin-Hall effect (ISHE). In order to couple the currents, we only require that the Cooper pairs must carry a net spin along the easy axis of the FI which is fixed along the $z$ axis. The orientation of the interface is not restricted by our model, so that the Cooper pair-induced magnon spin current also occurs if the FI is instead polarized in plane, in that case with an interface in the $xz$ plane.

Figure 2

Both a charge ($Q^{\uparrow }=Q^{\downarrow }=Q$) and a spin ($Q^{\uparrow }=-Q^{\downarrow }=Q$) supercurrent induce the same finite magnon spin current density $j^z$ in an adjacent ferromagnetic insulator. This is because spin-up Cooper pairs dominate in inducing the magnon spin current. The supercurrents induce the magnon spin current through an asymmetry in the eigenenergy spectrum ${\mathrm{\Omega }}_q$ of the new boson operators with respect to inversion of the momentum $q$ (inset). The results are robust to changes in the parameters, and for this particular figure we have chosen $\mu =-0.5$, $\lambda ={10}^{-4}$, $f_{\uparrow }=f_{\downarrow }={10}^{-4}$, $S=1/2$, $J={10}^{-2}$, $K={10}^{-3}$ for a 1D chain of $N=365$ lattice sites at temperature $T=0.001$. For a square lattice with a $p_x$- or $p_y$-wave symmetry, the results are qualitatively the same as in 1D. A supercurrent of spin-up Cooper pairs induces a current of spin-down magnons in the opposite direction, while transverse currents are not allowed. All energies, as well as the magnon spin current density, are scaled by the hopping parameter $t$. The temperature is scaled by $t/k_B$, where $k_B$ is the Boltzmann constant. The momenta $q$ and $Q$ are both scaled by $\hslash /a$, where $\hslash$ is the reduced Planck constant and $a$ is the lattice constant.