Tunneling spin Hall effect induced by unconventional $p$-wave magnetism

TL;DR

This work establishes a tunneling spin Hall effect in a normal metal/$p$-wave magnet/superconductor junction, where spin-dependent Andreev reflection in the normal lead is strongly asymmetric in transverse momentum due to $p$-wave magnetism. Using nonequilibrium Green's function methods, the authors derive a transverse spin conductance that depends on the orientation of the spin-split Fermi surfaces, yielding a finite spin current with zero net charge. The spin Hall angle can be as large as $\mathcal{O}(0.2)$ under optimal alignment ($\phi=\pm\pi/2$), indicating highly efficient charge-to-spin conversion and promising spintronic applications in superconducting heterostructures. This mechanism provides a Berry-curvature-independent route to pure spin currents arising from momentum filtering induced by unconventional magnetic order. All mathematical expressions are presented with $...$-style delimiters for clarity in reproducing and leveraging the results.

Abstract

We propose a tunneling spin Hall effect in a normal metal/$p$-wave magnet/superconductor junction. It is found that the Andreev reflection in the normal lead is spin-dependent and exhibits strong asymmetry with respect to the transverse momentum, giving rise to a pure transverse spin Hall current with zero net charge. The transverse spin conductance is analytically derived using the nonequilibrium Green's function approach, revealing that the predicted spin Hall effect is governed by the direction of the Fermi surface splitting in the $p$-wave magnet. A finite transverse spin current with a large spin Hall angle arises when the line connecting the centers of the spin-split Fermi surfaces is perpendicular to the normal direction of the junction, which indicates a highly efficient charge-to-spin conversion, suggesting potential applications in spintronic devices.

Figures (4)

• Figure 1: (Top panel) Schematic image of the two-dimensional normal metal/$p$-wave magnet/superconductor junction. The longitudinal direction of the junction is along the $x$ axis. The central $p$-wave magnet region has a length $L$ and is located at $0<x<L$. The semi-infinite normal lead (N) and superconducting lead (S) are located at $x<0$ and $x>L$, respectively. In the central region, the Fermi surfaces for the spin-up and -down electron states are denoted by the red and blue circles, respectively. The angle between the line connecting the centers of the spin-polarized Fermi surfaces and the $x$-axis is represented by $\phi$. (Bottom panel) The two-dimensional square lattice on which the junction in (a) are discretized.
• Figure 2: Zero-bias Andreev reflection probabilities versus the transverse momentum $k_y$ ($-\pi<k_y<\pi$) for $\phi=0$ (a), $\phi=\pi$ (b) $\phi=\pi/10$ (c) and $\phi=\pi/2$ (d). The differences of the chemical potential are set as $\delta\mu=0$ [(a), (c) and (d)] and $\delta\mu=0.5t$ (b). The other parameters are $M=10$, $\mu=1.2t$, $\mathcal{J}=0.6t$ and $\Delta=0.015t$. The Andreev reflection coefficients for spin-up and -down channels are denoted by red and blue lines, respectively.
• Figure 3: The transverse conductance (a) and longitudinal conductance (b) versus the $p$-wave exchange coupling strength $\mathcal{J}$ for spin-up and -down channels, which are denoted by red and blue lines, respectively. The numerical parameters are $M=10$, $E=0$, $\mu=1.2t$, $\delta\mu=0.3t$, $\mathcal{J}=0.6t$ and $\Delta=0.018t$.
• Figure 4: The spin Hall angle $\gamma$ as a function of $\phi$. The numerical parameters are $M=10$, $E=0$, $\mu=3.2t$, $\delta\mu=0$ and $\Delta=0.018t$.