Inverse proximity effect in a topological-insulator–superconductor hybrid system

The recent experiments on ultrathin superconductors (SCs), in proximity with some substrates, have unveiled the possible enhancement and suppression of superconducting order parameters; yet their mechanisms are still ambiguous. Here we examine the inverse proximity effect between an ultrathin SC and the surface state of a topological insulator. We show the effect of the renormalized electron-phonon ($e$-ph) coupling constant $\ensuremath{\lambda}$ on superconducting order parameters is determined by two important mechanisms: the modified density of states (DOS) $N({\ensuremath{\epsilon}}_{\mathrm{F}})$ from the occupied bands near the Fermi surfaces, and the renormalized interaction ${U}_{\mathrm{eff}}$ from the unoccupied bands. We demonstrate the enhanced order parameter near the bottom of the double-well bands with appropriate interface coupling due to the enhanced DOS; while in the strong coupling regime we observe the suppressed order parameter even with finite occupation in the Fermi surface, which is the result of the conjunct influence of the suppressed DOS and the renormalized ${U}_{\mathrm{eff}}$. This picture is applied to multiple SC layers, showing that the oscillation of order parameter can be well ascribed to the oscillation of an $e$-ph coupling constant, instead of solely the DOS. Our results not only provide a theoretical basis for tuning of the order parameters in the SC by chemical potential and interface coupling, but also provide some new insight into the order parameters in a large number of SC-substrate hybrid systems via the inverse proximity effect.