Anomalous Josephson effect in semiconducting nanowires as a signature of the topologically nontrivial phase

We study Josephson junctions made of semiconducting nanowires with Rashba spin-orbit coupling, where superconducting correlations are induced by the proximity effect. In the presence of a suitably directed magnetic field, the system displays the anomalous Josephson effect: a nonzero supercurrent in the absence of a phase bias between two superconductors. We show that this anomalous current can be increased significantly by tuning the nanowire into the helical regime. In particular, in a short junction, a large anomalous current is a signature for topologically nontrivial superconductivity in the nanowire.

Figure 1

(a) The schematic setup involving a nanowire (light gray) placed in proximity with two superconductors (dark gray). The effective magnetic field of Rashba SOC is along the $z$ axis, i.e., in the plane of the substrate and perpendicular to the nanowire. (b), (c) The normal-state single-particle spectrum of the infinite nanowire described by the Hamiltonian (1) with $h_z=0$ and with $|h_x|>m{\alpha }^2$ (b) and $|h_x|<m{\alpha }^2$ (c).

Figure 2

The anomalous Josephson current $I_{\mathrm{an}}$ in units of the maximal anomalous current $I_h=e{h}_z/\pi$ as a function of $\mu /\mathrm{\Delta }$ in a short junction in the presence of a small $h_z$. The results are shown for $h_x/\mathrm{\Delta }=3$ and several values of $m{\alpha }^2/\mathrm{\Delta }$ (a) and for $m{\alpha }^2/\mathrm{\Delta }=25$ and several values of $h_x/\mathrm{\Delta }$ (b).

Figure 3

The anomalous current $I_{\mathrm{an}}$ in a short junction at $\mu =0$ in units of $I_{\mathrm{\Delta }}=e\mathrm{\Delta }/\pi$ as a function of the angle $\theta$ that parametrizes the Zeeman field, $\mathbf{h}=h(cos\theta ,0,sin\theta )$. The results are shown for $h/\mathrm{\Delta }=3$ and several values of $m{\alpha }^2/\mathrm{\Delta }$ (a) and for $m{\alpha }^2/\mathrm{\Delta }=25$ and several values of $h/\mathrm{\Delta }$ (b). Note the cusp at $h_z=hsin{\theta }_c=\mathrm{\Delta }$.

Figure 4

Left panels: The anomalous current $I_{\mathrm{an}}/I_h$ at $h_x/\mathrm{\Delta }=5$ and $h_z/\mathrm{\Delta }=0.1$ as a function of ${\mu }_S/\mathrm{\Delta }$ for $m{\alpha }^2/\mathrm{\Delta }=1$ (a) and $m{\alpha }^2/\mathrm{\Delta }=25$ (b), shown for several values of ${\mu }_N/\mathrm{\Delta }$. Right panels: The anomalous current $I_{\mathrm{an}}$ as a function of the barrier height $|{\mu }_N-{\mu }_S|/\mathrm{\Delta }$ for ${\mu }_S/\mathrm{\Delta }=0$ (solid lines) and ${\mu }_S/\mathrm{\Delta }=10$ (dashed lines), using the same other parameters as in panel (b). The current is shown in units of $I_h$ (c) and in units of $I_c{h}_z/\mathrm{\Delta }$ (d), where the critical current $I_c$ is calculated at $h_z=0$.

Figure 5

Normal-state transmission $T$ through a Dirac potential as a function of the energy $E$ of the incident particle. The results are shown in the helical gap ($\left|E\right|<|h_x|$) for several values of the barrier height $U$ for $|h_x|>m{\alpha }^2$ (left panel) and for $|h_x|<m{\alpha }^2$ (right panel).

Figure 6

Ratio between the anomalous and critical current, $I_{\mathrm{an}}/(I_c{h}_z/\mathrm{\Delta })$, as a function of the height of the electrostatic barrier, in units of $({\mu }_S-{\mu }_N)/\mathrm{\Delta }$, at ${\mu }_S=0$. The black dots are the numerical results from a tight-binding Hamiltonian with bandwidth $t/\mathrm{\Delta }=25$. The red curves are the result of the helical edge model, Eq. (13), with the transmission $T=T(E=0)$ obtained from the continuous model. Left and right panels correspond to two different sets of parameters $h_x/\mathrm{\Delta }$ and $m{\alpha }^2/\mathrm{\Delta }$. The black dots in the right panel correspond to the solid line in Fig. 4.

Figure 7

The supercurrent $I\left(\phi \right)$ in units of $I_{\mathrm{\Delta }}$ flowing in a short junction ($L=0$) as a function of the phase difference $\phi$. The top panels correspond to the topologically nontrivial regime ($\mu /\mathrm{\Delta }=0$), whereas the bottom panels correspond to the topologically trivial regime ($\mu /\mathrm{\Delta }=10$). The results are shown for several values of $h_z\le \mathrm{\Delta }$ for $m{\alpha }^2<\left|h_x\right|$ (left panels) and $m{\alpha }^2>\left|h_x\right|$ (right panels).

Figure 8

The anomalous current $I_{\mathrm{an}}/I_h$ at $h_z/\mathrm{\Delta }=0.01$ for junctions with finite length $L/l_{\mathrm{\Delta }}$. (a) $I_{\mathrm{an}}/I_h$ vs $\mu /\mathrm{\Delta }$ for a junction with $L/l_{\mathrm{\Delta }}=60$ for the same parameters as in Fig. 2. In the topological regime, this length corresponds to $L/{\xi }_{\alpha }\approx 8.5$. (b) $I_{\mathrm{an}}/I_h$ vs $L/l_{\mathrm{\Delta }}$ calculated at $\mu =0$ for $m{\alpha }^2/\mathrm{\Delta }=25$ and several values of $h_x/\mathrm{\Delta }$. The value of $L/l_{\mathrm{\Delta }}$ for which $L/{\xi }_{\alpha }\approx 1$ is shown by a vertical arrow.

Figure 9

The anomalous current $I_{\mathrm{an}}/I_{\mathrm{\Delta }}$ vs $\theta$ in a long junction at $\mu =0$ and $h/\mathrm{\Delta }=1.5$. The results are shown for $m{\alpha }^2/\mathrm{\Delta }=25$ and several values of $L/l_{\mathrm{\Delta }}$ (a) and for $L/l_{\mathrm{\Delta }}=20$ and several values of $m{\alpha }^2/\mathrm{\Delta }$ (b). In panel (a), the result for a short junction ($L/l_{\mathrm{\Delta }}=0$) is shown for comparison.