Inverse proximity effect in $s$-wave and $d$-wave superconductors coupled to topological insulators

We study the inverse proximity effect in a bilayer consisting of a thin $s$- or $d$-wave superconductor (S) and a topological insulator (TI). Integrating out the topological fermions of the TI, we find that spin-orbit coupling is induced in the S, which leads to spin-triplet $p$-wave ($f$-wave) correlations in the anomalous Green's function for an $s$-wave ($d$-wave) superconductor. Solving the self-consistency equation for the superconducting order parameter, we find that the inverse proximity effect can be strong for parameters for which the Fermi momenta of the S and TI coincide. The suppression of the gap is approximately proportional to $e^{-1/\lambda}$, where $\lambda$ is the dimensionless superconducting coupling constant. This is consistent with the fact that a higher $\lambda$ gives a more robust superconducting state. For an $s$-wave S, the interval of TI chemical potentials for which the suppression of the gap is strong is centered at $\mu_{TI} = \pm\sqrt{2mv_F^2\mu}$, and increases quadratically with the hopping parameter $t$. Since the S chemical potential $\mu$ typically is high for conventional superconductors, the inverse proximity effect is negligible except for $t$ above a critical value. For sufficiently low $t$, however, the inverse proximity effect is negligible, in agreement with what has thus far been assumed in most works studying the proximity effect in S-TI structures. In superconductors with low Fermi energies, such as high-$T_c$ cuprates with $d$-wave symmetry, we again find a suppression of the order parameter. However, since $\mu$ is much smaller in this case, a strong inverse proximity effect can occur at $\mu_{TI}=0$ for much lower values of $t$. Moreover, the onset of a strong inverse proximity effect is preceded by an increase in the order parameter, allowing the gap to be tuned by several orders of magnitude by small variations in $\mu_{TI}$.