Proximity-induced unconventional superconductivity in topological insulators

We study and classify the proximity-induced superconducting pairing in a topological insulator (TI)-superconductor (SC) hybrid structure for SCs with different symmetries. The Dirac surface state gives a coupling between spin-singlet and spin-triplet pairing amplitudes as well as pairing that is odd in frequency for $p$-wave SCs. We also find that all SCs induce pairing that is odd in both frequency and orbital (band) index, with oddness in frequency and orbital index being completely interchangeable. The different induced pairing amplitudes significantly modify the density of states in the TI surface layer.

Figure 1

Proximity-induced pairing amplitudes in the TI surface state from an $s$-wave (a) and (b), $d$-wave (c) and (d), and $p_x+i{p}_y$-wave (e) and (f) SC as a function of orbital tunneling ratio ${\tilde{t}}_1/{\tilde{t}}_2$ for $\tilde{t}=0.9$, ${\Delta }_0=0.5$, and ${\mu }_{\mathrm{SC}}=-0.5$ eV. Symmetry of the SC amplitudes (a), (c), and (e) and triplet $p$-wave amplitudes (b) and (d) are divided into even-intraorbital (thick black), odd-intraorbital (thin black), even-interorbital (dashed black), and odd-frequency odd-orbital (thin red) pairing amplitudes. For the $p_x+i{p}_y$-wave SC, the induced spin-triplet $s$-wave (line) and $d$-wave (crosses) amplitudes (f) are divided into odd-orbital (black), odd-frequency even-intraorbital (thick red), odd-intraorbital (thin red), and even-interorbital (dashed red) amplitudes.

Figure 2

LDOS (states/eV/unit cell) in the TI surface state as a function of energy for a spin-singlet SC (a) with $s$-wave (black) and $d$-wave (red) symmetries and for a spin-triplet SC (b) with $p$-wave A${}_{1\mathrm{u}}$ (green line), A${}_{2\mathrm{u}}$ (black line), B${}_{1\mathrm{u}/2\mathrm{u}}$ (red line with circles), and E${}_{2\mathrm{u},+/-}$ (blue crosses) symmetries. Here, ${\Delta }_0=5$, ${\mu }_{\mathrm{SC}}=-0.3$, and $\tilde{t}=(0,0.6)$. The Dirac point is found at higher energies due to doping from the SC. Ripples are due to finite $\mathbf{k}$-point sampling.