Proposal for the detection and braiding of Majorana fermions in a quantum spin Hall insulator

We show how a quantum dot with a ballistic single-channel point contact to a superconductor can be created by means of a gate electrode at the edge of a quantum spin Hall insulator (such as an InAs/GaSb quantum well). A weak perpendicular magnetic field traps a Majorana zero mode, so that it can be observed in the gate-voltage-averaged differential conductance $\langle dI/dV\rangle$ as a $4e^2/h$ zero-bias peak above a $(\frac{2}{3}{\pi }^2-4)e^2/h$ background. The one-dimensional edge does not permit the braiding of pairs of Majorana fermions, but this obstacle can be overcome by coupling opposite edges at a constriction, allowing for a demonstration of non-Abelian statistics.

Figure 1

Left panel: Andreev quantum dot, created by a gate electrode at the edge of a quantum spin Hall (QSH) insulator in a perpendicular magnetic field $B$. A current $I$ is passed between metallic and superconducting contacts, and the differential conductance $G=dI/dV$ is measured as a function of the bias voltage $V$ for different gate voltages. Right panel: Band structure of an InAs/GaSb quantum well, for the parameters used in the computer simulations. The helical edge states appear inside the gap, connecting conduction and valence bands. Below the gate, the conduction band is pushed through the Fermi level at $E_{\mathrm{F}}$, to create a metallic puddle. Inside the superconducting contact, a gap $2\Delta$ opens around the Fermi level.

Figure 2

(a) Zero-temperature differential conductance $G=dI/dV$ as a function of bias voltage $V$, calculated numerically for the system of Fig. 1. The bottom of the conduction band in the gated region ($200\mathrm{nm}\times 200\mathrm{nm}$) is $E_{\mathrm{c}}=-1.5\mathrm{meV}$ below the Fermi level. The black curve is for a single disorder realization, the red curve is the disorder average. The calculated background conductance $G_{\mathrm{base}}$ from Eq. (6) is indicated (arrow). For comparison, the conductance without the gate electrode is also shown (green dashed curve). The Majorana resonance is then fully absorbed in the background and invisible. (b) Enlargement of the Majorana resonance from the left panel, to show the difference in line shape.

Figure 3

Differential conductance, averaged over gate voltages ($-4.5\mathrm{meV}⩽E_{\mathrm{c}}⩽-1.5\mathrm{meV}$) for a single disorder realization. All system parameters are the same as in Fig. 2, except for the size of the gated region, which is $100\mathrm{nm}\times 100\mathrm{nm}$. The solid curve is at zero temperature and the dashed curve at a temperature of $0.1\mathrm{K}$.

Figure 4

(a) Constriction in a quantum spin-Hall insulator, contacted along three edges by a superconducting island. If the fourth edge is blocked by a gate electrode, the constriction traps a Majorana fermion. Each superconducting island is connected to a superconducting ground by a split Josephson junction enclosing a magnetic flux, indicated schematically by a $\times$ symbol and shown expanded for one of the islands. (b) Two constrictions in series form a $\pi$-shaped circuit that can be used to braid the Majorana fermions (green circles). The flux through each split Josephson junction controls the coupling of adjacent Majoranas. (c) Schematic of the braiding operation in the $\pi$ circuit.[38] Coupled Majoranas are connected by a solid line, uncoupled Majoranas by a dashed line.