Topological Josephson ${\varphi }_0$ junctions

We study the effect of a magnetic field on the current-phase relation of a topological Josephson junction formed by connecting two superconductors through the helical edge states of a quantum spin-Hall insulator. We predict that the Zeeman effect along the spin quantization axis of the helical edges results in an anomalous Josephson relation that allows for a supercurrent to flow in the absence of superconducting phase bias. We relate the associated field-tunable phase shift ${\varphi }_0$ in the Josephson relation of such a ${\varphi }_0$ junction to the existence of a so-called helical superconductivity, which may result from the interplay of the Zeeman effect and spin-orbit coupling. We analyze the dependence of the magneto-supercurrent on the junction length and discuss its observability in suitably designed hybrid structures subject to an in-plane magnetic field.

Figure 1

Anomalous Josephson effect in short $[L=0.1\xi$, panels (a) and (b)] and long $[L=10\xi$, panels (c) and (d)] S-QSHI-S junctions. Panels (a) and (c) show the current-phase relation at temperature $T/{\mathrm{\Delta }}_0={10}^{-3}$ for different values of the applied magnetic field $h$. Panels (b) and (d) show the anomalous Josephson current at $\varphi =0$ as a function of $h$.

Figure 2

(a) Proposed setup to detect the ${\varphi }_0$ junction behavior in a S-QSHI-S hybrid system: The magnetic field $\mathbf{B}$ is applied in the plane of the quantum well. The edge states on both sides of the sample contribute to the Josephson current. A net anomalous Josephson effect remains, if the junctions have unequal lengths, $L_1\ne L_2$. (b) The anomalous Josephson current at ${\varphi }_0$ as a function of the Zeeman splitting $h={\mu }_B{g}_{\mathrm{eff}}\left|\mathbf{B}\right|/2$ in the case $L_2=10\xi$ and $L_1=L_2/3$ for various temperatures.

Figure 3

The slope of the anomalous Josephson current in short junctions as a function of the magnetic field $h_{\perp }$ perpendicular to the spin quantization axis and the chemical potential. The slope remains close to its quantized value as long as $h_{\perp }\ll \sqrt{{\mathrm{\Delta }}_0^2+{\mu }^2}$.