Spin phase regulated spin Josephson supercurrent in topological superconductor

Without applied bias voltage, a superconducting phase difference can drive a charge Josephson supercurrent in a superconductor junction. In analogy, we here theoretically propose a spin phase that intrinsically generates spin Josephson supercurrent, and this spin Josephson effect is studied in a junction of superconducting nanowire (SNW). We show that spin-orbit coupling and magnetic field give rise to spin-triplet superconductivity in the SNW, thus allowing the superfluid of both charge and spin. Next we introduce the concept of a spin phase that can be generated by controls of spin current, magnetic field, or electric field. By the analysis of pairing correlations and Ginzburg-Landau-type theory, it is shown that the spin phase makes spin-up and spin-down $S=1$ Cooper pairs get opposite phases and move oppositely in a Josephson junction, so a dissipationless pure spin current is induced. We also derive the formula that the spin current is equal to the derivative of an Andreev bound state to the spin phase, which is analogous to that of charge Josephson effect. At last, our calculation verifies the existence of spin supercurrent inside the superconducting gap in the topologically nontrivial phase. Our paper provides a view on spin Josephson effect and indicates the potential combination of topological superconductivity and spintronics.

Figure 1

Schematic plots of energy bands for (a) topologically trivial phase and (b) nontrivial phase. $ex$ ($e\overline{x}$) and $hx$ ($h\overline{x}$) approximately indicate electron and hole bands with $x$ ($-x$) spin. In (b), a high magnetic field gives rise to the overlap between $e\overline{x}$ and $h\overline{x}$, and their gap is opened by the SOC. Therefore, a Cooper pair is formed by two electrons with the same spin $\overline{x}$, and the spin-triplet component is greatly enhanced. (c) shows the equal spin Andreev reflection coefficient $S_{h\overline{x}e\overline{x}}$ of the NNW-SNW junction versus the incident energy $\epsilon$. Here we set $\mathrm{\Delta },\hslash ,m=1$ as units, $\mu =0$, and $\alpha =0.6$ adapted from the experiment [28]. The chemical potential of NNW is ${\mu }_{\mathrm{NNW}}=50$. Without further explanation, the parameters are the same in other figures.

Figure 2

The devices to induce spin phase differences. (a) Applying a spin current gives rise to a spin phase difference parallel to spin direction of the spin current. (b), (c) The spin phase differences of direction $z$ and $x$ can also be caused by rotating the magnetic field (b) and regulating the electric field (c) on the right side, respectively.

Figure 3

The ABSs (a), (c) and supercurrent (b), (d) as functions of the $x$-direction (a), (b) and $z$-direction (c), (d) spin phase difference with $B=1.25$ and NNW length $d\to 0$. Inset in (b) shows the Cooper pairs almost polarized at $\overline{x}$ direction, so in (b) the charge and spin supercurrents are nearly opposite with the ratio $-2e/\hslash$. Inset in (d) shows that Cooper pairs can be equally decomposed into $z$ and $\overline{z}$ polarized components moving oppositely and a pure spin supercurrent is realized in (d). When SNWs are in trivial phase, the ABSs versus $x$ and $z$ direction spin phases are shown in (e), (f) respectively. The ABSs are independent of spin phase, so the subgap spin supercurrent is nonexistent according to Eq. (16). For $B=0.75$, there is no ABS, so the corresponding curve is absent.