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We study the superconducting proximity effect induced in the surface statesof the 3-d topological insulator Bi$_2$Se$_3$ by a singlet, s-wavesuperconductor deposited on its surface. To this effect, the$\mathbf{k}\cdot\mathbf{p}$-Hamiltonian of Bi$_2$Se$_3$ and the BCS-Hamiltonianare mapped onto tight-binding chains which we couple through atransfer-Hamiltonian at the interface. We then employ the Recursive Green'sFunction technique to obtain the local spectral function and infer thedispersion of the interface-states from it. In agreement with earliermicroscopic studies of this problem, we find that the Fu-Kane model is areasonable approximation at energies $\epsilon\ll \Delta_{\rm SC}$. However,for energies close to the SC bulk gap, the Fu-Kane model is expected to breakdown. Indeed, our numerical calculations show strong modifications of theinterface-state dispersion for $\epsilon \gtrsim \Delta_{\rm SC} $. We findthat the proximity effect can be strong enough to induce a gap in the surfacestate that is comparable to the superconducting gap. An analysis of the spatialprofile of the states shows that their weight shifts towards the SC as thecoupling strength increases. We conclude that an intermediate coupling is idealfor realising the Fu-Kane scenario.

A Numerical Study of the Superconducting Proximity Effect in Topological Surface States