Spin imbalance in hybrid superconducting structures with spin-active interfaces

We consider a heterostructure consisting of a normal metal and a superconductor separated by a spin-active interface. At finite-bias voltages, spin-filtering and spin-mixing effects at the interface allow for an induced magnetization (spin imbalance) on the superconducting side of the junction, which relaxes to zero in the bulk. Such interfaces are also known to host a pair of in-gap Andreev bound states which were recently observed experimentally. We show that these states are responsible for the dominant contribution to the induced spin imbalance close to the interface. Motivated by recent experiments on spin-charge density separation in superconducting aluminum wires, we propose an alternative way to observe spin imbalance without applying an external magnetic field. We also suggest that the peculiar dependence of the spin imbalance on the applied bias voltage permits an indirect bound-state spectroscopy.

Figure 1

(a) A normal metal–superconductor (N–S) junction with a spin-active interface is characterized by an intrinsic magnetic moment along $\widehat{\mu }$. The closed trajectory indicates formation of Andreev surface bound states. (b)–(d) Local density of states in S as function of distance from the interface for spin-mixing angles $\vartheta =0$, $0.49\pi$, and $0.83\pi$. The interface transparencies are $D_{\uparrow }=D_{\downarrow }=0.06$. Here, $\theta$ is the incidence angle, ${\xi }_0$ is the superconducting coherence length, $T_C$ the critical temperature, and $N_F$ the density of states at the Fermi energy in the normal state.

Figure 2

(a) Interface value $M_{eq}\left(0\right)$ as function of spin-mixing angle. (b) Semilog plot of $M_{eq}\left(z\right)/M_{eq}\left(0\right)$ as function of distance from the interface. (c), (d) Spin-down and spin-up local density of states $N_{\downarrow ,\uparrow }(\epsilon ,z)$ as function of distance for $\vartheta =0.66\pi$. $D_{\uparrow }=D_{\downarrow }=0.06$, and $T=0.01T_C$.

Figure 3

Nonequilibrium part of spin imbalance. (a) Interface value as a function of bias voltage for $\vartheta =0$, $D_{\uparrow }=0.06$, and $D_{\downarrow }=0.02$. (b) Interface value for $\vartheta =0.32\pi$ (black circles), $\vartheta =0.49\pi$ (blue rectangles), $\vartheta =0.66\pi$ (red diamonds), $\vartheta =0.83\pi$ (green triangles), and $D_{\uparrow }=D_{\downarrow }=0.06$. (c) Semilog plot of $M_{ne}\left(z\right)/M_{ne}\left(0\right)$ for $eV=3.2k_B{T}_C$ (solid lines) and $eV=1.7k_B{T}_C$ (dashed lines). The gray rectangle in (a) and (b) depicts the subgap region. Temperature $T=0.01T_C$.

Figure 4

Derivative of the anomalous part of spin imbalance with respect to bias voltage for (a) $\vartheta =0$, $D_{\uparrow }=0.06$, and $D_{\downarrow }=0.02$; (b) $\vartheta =0.32\pi$ and $D_{\uparrow }=D_{\downarrow }=0.06$; (c) $\vartheta =0.66\pi$ and $D_{\uparrow }=D_{\downarrow }=0.06$. Solid, dashed, and dotted lines correspond to $z=0,{\xi }_0$, and $2{\xi }_0$, respectively. The gray rectangle depicts the subgap region. Temperature $T=0.05T_C$.