Effect of magnetic field and Rashba spin-orbit interaction on the Josephson tunneling between superconducting nanowires

We calculate the Josephson current between two one-dimensional (1D) nanowires oriented along $x$ with proximity induced $s$-wave superconducting pairing and separated by a narrow dielectric barrier in the presence of both Rashba spin-orbit interaction (SOI) characterized by strength $\alpha$ and Zeeman fields ($h$ along $\widehat{z}$ and $\mathbf{B}$ in the $x\text{−}y$ plane). We formulate a general method for computing the Andreev bound states energy, which allows us to obtain analytical expressions for the energy of these states in several asymptotic cases. We find that in the absence of the magnetic fields the energy gap between the Andreev bound states decreases with increasing Rashba SOI constant leading eventually to touching of the levels. In the absence of Rashba SOI, the Andreev bound states depend on the magnetic fields and display oscillatory behavior with the orientational angle of B, leading to the magneto-Josephson effect. We also present analytic expressions for the dc Josephson current charting out their dependence on $\mathbf{B}$, $h$, and $\alpha$. We demonstrate the existence of finite spin-Josephson current in these junctions in the presence of external magnetic fields and provide analytic expressions for its dependence on $\alpha$, $\mathbf{B}$, and $h$. Finally, we study the ac Josephson effect in the presence of the SOI (for $\left|\mathbf{B}\right|=h=0$) and an external radiation and show that the width of the resulting Shapiro steps in such a system can be tuned by varying $\alpha$. We discuss experiments that can test our theoretical results.

Figure 1

Two $s$-wave superconductors separated with a $\delta$-like dielectric potential under magnetic fields $\mathbf{h}$ and $\mathbf{B}$ co-planar and perpendicular to spin-orbit interactions, respectively. The bulk $s$-wave superconductor that induces superconductivity in the wires is not shown for clarity.

Figure 2

Dependence of the branch energies, given by Eq. (15), on the order parameter phase difference at $\varphi =0.5$ and (a) $B=h=0$, $D=0.9$; (b) $h=0.1$, $B=0.9$, $D=0.3$.

Figure 3

A plot of the energy spectrum [Eq. (18)] as a function of the order parameter phase difference at (a) $\alpha =0.1$ and $D=0.6$, and (b) $\alpha =0.205$ and $D=1$.

Figure 4

Plot of energy dispersion of different energy branches as a function of order parameter phases difference at (a) $D=0.2$ and different values of the SOI strength $\alpha$, and (b) $\alpha =0.1$ and different values of the transmission coefficient $D$.

Figure 5

(a) A plot of $E_{++}$ as a function of $\phi$ at different $\tilde{h}$. Amplitude of the energy oscillation increases with $\tilde{h}$; (b) mutual optimal values of $\tilde{h}$ and $D$ at which electron- and hole-energy branches are crossed, creating a zero-energy mode; and (c) the Andreev bound state energies $E_{h\pm }$ touch at particular values of $D=0.5$ and $\tilde{h}=0.653281$ (thin curves), and of $D=0.5$ and $\tilde{h}=0.270590$ (thick curves), which may make the oscillation period $4\pi$ in the Landau-Zenner sense.

Figure 6

Current-phase relation at magnetic field $h=0.1$ according to Eqs. (24, 25, 26).

Figure 7

Spin current as a function of magnetic field orientation at $D=0.3$, $h=0.1$, $\alpha =0$, $\phi =0.5$ and two values of magnetic filed $B=0.9$ (curve 1) and $B=2$ (curve 2). Calculations are done using Eqs. (27), (28), and (15).

Figure 8

Transformation of current-phase relation with parameter of spin-orbital coupling $\alpha$ at $D=0.5$ [Eq. (29)]. Numbers indicate the values of parameter spin-orbital coupling $\alpha$ used for different curves.

Figure 9

(a) Effect of transparency and spin-orbit coupling on the $\phi$ dependence of the Shapiro step width [Eq. (39)]. (b) Demonstration of $\alpha$ dependence of Shapiro step width in different approximations as given by Eqs. (39, 40, 41).

Figure 10

$I-V$ curve at $D=0.5$, $\alpha =0.1$ without radiation (curve 1) and under external radiation (curve 2).

Figure 11

Effect of spin-orbital coupling on the amplitude dependence of (a) Shapiro step width and (b) critical current.

Figure 12

The $\alpha$ dependence of $I_c$ for $D=0.5$ and $D$–dependence of $I_c$ for $\alpha =0.2$ at $\omega =0.5$ and $A=0.5$.