Phase-tunable Majorana bound states in a topological N-SNS junction

We theoretically study the differential conductance of a one-dimensional normal-superconductor-normal-superconductor (N-SNS) junction with a phase bias applied between the two superconductors. We consider specifically a junction formed by a spin-orbit coupled semiconducting nanowire with regions of the nanowire having superconducting pairing induced by a bulk $s$-wave superconductor. When the nanowire is tuned into a topologically nontrivial phase by a Zeeman field, it hosts zero-energy Majorana modes at its ends as well as at the interface between the two superconductors. The phase-dependent splitting of the Majorana modes gives rise to features in the differential conductance that offer a clear distinction between the topologically trivial and nontrivial phases. We calculate the transport properties of the junction numerically and also present a simple analytical model that captures the main properties of the predicted tunneling spectroscopy.

Figure 1

(a) Cartoon of the N-SNS setup we consider. A semiconducting nanowire with epitaxially grown $s$-wave superconductor covering two separate parts of the wire. The two superconductors $S1$ and $S2$ are connected by a grounded superconducting loop through which a flux $\mathrm{\Phi }$ can be threaded, resulting in a phase difference between $S1$ and $S2$. An external magnetic field ${\mathbf{B}}_{\mathrm{Z}}$ is applied that couples to the electronic spins inside the wire, but does not add to $\mathrm{\Phi }$. The wire is tunnel coupled to a probe lead at the left end which can be used for tunneling spectroscopy by applying a bias voltage $V$ to it while measuring the current $A$. (b) More schematic view of the setup. The wire is oriented along the $z$ axis and ${\mathbf{B}}_{\mathrm{Z}}$ lies in the $yz$ plane, where the effective spin-orbit field is assumed to point along the $y$ axis.

Figure 2

Differential conductance of the nanowire in units of $e^2/h$ for $\theta =\pi /2$. With this orientation of the Zeeman field the system is always in the trivial regime. (a) Conductance as a function of bias voltage $V$ and $V_{\mathrm{Z}}$ for $\varphi =0$. (b) Conductance as a function of $V$ and $\varphi$ for $V_{\text{Z}}=8.3E_{\text{SO}}$, indicated by the red line in (a).

Figure 3

Differential conductance of the wire in units of $e^2/h$ for $\theta =0$. Now, the wire can enter a topological phase when increasing $V_{\mathrm{Z}}$. (a, left column) Conductance as a function of $V$ and $V_{\mathrm{Z}}$ for $\varphi =0,\pi /2$, and $\pi$. (b, right column) Conductance as a function of $V$ and $\varphi$ for $V_{\mathrm{Z}}=E_{\mathrm{SO}},5.5E_{\mathrm{SO}}$, and $8.3E_{\mathrm{SO}}$, indicated by the red lines in the top panel of (a).

Figure 4

The differential conductivity of the wire in units of $e^2/h$ as a function of $V$ and position of the normal probe lead. We have set the phase difference to $\varphi =0,\pi /2$, and $\pi$ (from top to bottom), the Zeeman energy to $V_{\text{Z}}=5.5E_{\text{SO}}$, and the coupling strength to the lead to ${\gamma }_W=E_{\text{SO}}$.

Figure 5

Differential conductance in units of $e^2/h$ for the case that $L_1\ne L_2$. We took $\theta =0$ and $\varphi =\pi$, and show $dI/dV$ as a function of $V$ and $V_{\mathrm{Z}}$ (a), (b) and of $V$ and the potential of the central normal region $V_m$ (c), (d). In (a) and (c) we have set $L_1=825$ nm and $L_2=525$ nm and in (b) and (d) we took the opposite $L_1=525$ nm and $L_2=825$ nm. In (a) and (b) we did not include an extra potential offset in the central region $V_m=0$, and in (c) and (d) we fixed $V_{\mathrm{Z}}=5.5E_{\mathrm{SO}}$, as indicated by the red lines in the top plots.

Figure 6

(a) Sketch of a one-dimensional $p$-wave superconductor centered around $z=0$. We expect Majorana bound states to form at the ends of the wire and close to the phase jump at $z=0$ (b). The phase of the pairing potential ${\mathrm{\Delta }}_p\left(z\right)$ as a function of $z$ (c).

Figure 7

The spectrum of $H_{\mathrm{M}}$ for $\mu =0,{\mathrm{\Delta }}_{\mathrm{ind}}=4E_{\mathrm{SO}}$, and $L=20l_{\mathrm{SO}}$. (a) Level structure as a function of $V_{\mathrm{Z}}$ for $\varphi =0$ (top) and $\varphi =\pi$ (bottom). (b) Spectrum as a function of $\varphi$ for $V_{\mathrm{Z}}=8.8E_{\mathrm{SO}}$ (top) and $V_{\mathrm{Z}}=9.25E_{\mathrm{SO}}$ (bottom), as indicated by the red lines in (a).