Proximity effect at the superconductor–topological insulator interface

We study the excitation spectrum of a topological insulator in contact with an $s$-wave superconductor, starting from a microscopic model, and develop an effective low-energy model for the proximity effect. In the vicinity of the Dirac cone vertex, the effective model describing the states localized at the interface is well approximated by a model of Dirac electrons experiencing superconducting $s$-wave pairing. Away from the cone vertex, the induced pairing potential develops a $p$-wave component with a magnitude sensitive to the structure of the interface. Observing the induced $s$-wave superconductivity may require tuning the chemical potential close to the Dirac point. Furthermore, we find that the proximity of the superconductor leads to a significant renormalization of the original parameters of the effective model describing the surface states of a topological insulator.

Figure 1

(Color online) Band structure for a slab with a (1, 1, 1) surface of a model TI described by Eq. (2). The blue (dark gray) areas represent bulk states and the in-gap lines are surface states. The right panel shows a zoom-in of the anisotropic Dirac cone at the M point of the surface Brillouin zone.

Figure 2

(Color online) Spectrum of the full Hamiltonian (1) for a TI slab with (1, 1, 1) surfaces having an interface with an $s$-wave superconductor. Bulk TI states are green (light gray) and SC states are blue (dark gray areas). The proximity effect induces a gap in the surface states spectrum at the interface (red/dark gray in-gap lines) while the states on the opposite surface have an unperturbed Dirac cone dispersion (orange/light gray lines). The light blue (light gray) in-gap points are calculated using an effective theory for the interface described by Eqs. (11, 14, 15, 16).

Figure 3

(Color online) Surface state localized at the interface between a TI and an $s$-wave SC. For zero tunneling (not shown), the state resides entirely inside the TI and decays exponentially away from the boundary (up to an even-odd oscillatory factor). At finite transparency, the surface state penetrates inside the SC. The spin-down (lower amplitude) component vanishes on the top TI layer, thus it does not propagate inside the SC if only nearest-neighbor tunneling is considered.