Majorana state on the surface of a disordered three-dimensional topological insulator

We study low-lying electron levels in an “antidot” capturing a coreless vortex on the surface of a three-dimensional topological insulator in the presence of disorder. The surface is covered with a superconductor film with a hole of size $R$ larger than coherence length, which induces superconductivity via proximity effect. Spectrum of electron states inside the hole is sensitive to disorder, however, topological properties of the system give rise to a robust Majorana bound state at zero energy. We calculate the subgap density of states with both energy and spatial resolution using the supersymmetric $\sigma$ model method. We identify the presence of the Majorana fermion with symmetry class $B$. Tunneling into the hole region is sensitive to the Majorana level and exhibits resonant Andreev reflection at zero energy.

Figure 1

A sketch of the considered setup. Three-dimensional topological insulator is covered by a superconducting film with a hole of radius $R$. The gap in the density of states is induced on the covered surface forming an antidot and confining surface excitations inside the hole. External magnetic field induces an Abrikosov vortex inside the hole. The spectrum of energy eigenstates inside the hole acquires a Majorana zero level which can be accessed in the tunneling measurement.

Figure 2

Global density of states at low energies as a function of energy $E$. Solid curves show DOS for different values of the coupling to the tunneling probe, characterized by the normal tunnel conductance $G_t$ in units $e^2/h$. The Majorana level at zero energy repels low-lying finite energy levels as seen from the dip in DOS next to the zero-bias Majorana peak. The dashed gray curve shows the DOS for the system of class $D$ that does not host a Majorana state. The dashed red curve is the quasiclassical result that disregards effects of level correlation. It shows no oscillations and does not distinguish classes $B$ and $D$.

Figure 3

Global density of states as a function of energy $E$. The solid blue oscillating curve represents the low-energy result (19) for $G_t=0.1e^2/h$, the dashed curve shows the high-energy asymptotics (20), and the red curve presents the numerical solution interpolating between the two limits.

Figure 4

Schematic plot of the conductance as a function of voltage $eV$ for an arbitrary disorder realization. Each discrete level produces a resonance peak. At zero temperature (solid blue curve), the heights of the peaks are $\le e^2/\pi \hslash$ with the equality achieved for the Majorana peak at $V=0$. This indicates perfect Andreev reflection by the Majorana level. Widths of the peaks are proportional to $G_t$. At moderate temperatures $G_t\lesssim T/{\omega }_0\lesssim 1$ (solid black curve) peaks are broadened but retain their spectral weight. At yet higher temperatures $T\gg {\omega }_0$ the DOS is completely smeared (dashed curve) and the average value $\simeq \nu$ is observed.

Figure 5

Spatial dependence of the density of states for different values of $E$. At lowest energies, the result (16) is reproduced, while at energies well above $E_{\mathrm{Th}}$ the density of states is depleted only close to the hole edge within the strip $\sim R\sqrt{E_{\mathrm{Th}}/E}$ in agreement with Eq. (B11). Remarkably, the density of states is first increased above its normal value before dropping to zero at $r=R$.