Self-consistent superconducting proximity effect at the quantum spin Hall edge

We study self-consistently a microscopic interface between a quantum spin Hall insulator (QSHI) and a superconductor (SC), focusing on properties related to Majorana fermion creation. For an $s$-wave SC we show that odd-in-momentum, or $p$-wave, order parameters exist for all doping levels of the QSHI and that they can be related to different spinless Cooper pair amplitudes. Despite this, the induced superconducting gap in the QSHI always retains its $s$-wave character, validating the commonly used effective model for superconductivity in a topological insulator. For a $d_{\mathit{xy}}$-wave SC, we show that a Majorana mode is only created at finite doping and that there is no excitation gap protecting this mode.

Figure 1

Microscopic details of the QSHI-SC interface for the (1,0) surface of the SC and with sites $a_{i\alpha }$ (dark) and $b_{i\alpha }$ (light).

Figure 2

(a) $F_U$ across the QSHI-SC interface for the combinations $({\mu }_{\mathrm{I}},\mathrm{t̃})=(0,0.2)$ in (black $\times$), $(0,0.6)$ in (blue $$), $(0.75,0.2)$ in (red $ˆ$), and $(0.75,0.6)$ in (green $◊$). The vertical dashed line marks the interface, and the distance is measured in number of unit cells. Inset: $F_U$ as function of $E_g$ at site I1 for $0⩽{\mu }_I<1$, $\mathrm{t̃}⩽0.6$, and the bulk order parameter ${\Delta }_U=0.1$ (black) and 1 (red). Crosses mark data, and lines are linear fits. (b) $-{\Delta }_U(\mathrm{S}1)$ and $E_g(\mathrm{I}1)$ as function of ${\mu }_{\mathrm{I}}$ for $\mathrm{t̃}=0.2$ (black) and 0.6 (red). Crosses mark $E_g$(I1) reached in a non-self-consistent calculation. The dashed line indicates $\left|{\Delta }_U\right|=E_g$ in the SC bulk.

Figure 3

(a) $F_U$ (thick black), $F_U^o$ (thin black), $F_{J1}^o$ (thin red), and $F_{J2}^o=F_{J3}^o$ (thin dashed red) across the QSHI-SC interface for ${\mu }_{\mathrm{I}}=\mathrm{t̃}=0.5$. The vertical dashed line marks the interface and the distance is measured in number of unit cells. Inset: $k$ dependence at site I1 for $F_{\mathit{Uk}}^o$ (black) and $F_{J1k}^o$ (red). (b) $F_U$ (thick) and $F_U^o$ (thin) on site I1 and $F_J^o$ (thin dashed) on the $\mathrm{t̃}$ bond for $\mathrm{t̃}=0.2$ (black) and 0.6 (red) as a function of ${\mu }_{\mathrm{I}}$.

Figure 4

Bogoliubov band structure for the QSHI-$d_{\mathit{xy}}$ SC structure with ${\mu }_{\mathrm{I}}=\mathrm{t̃}=0.5$ and $J=1$. Unaffected by the interface are the Dirac cone on the left-hand side of the QSHI (red, close to $k=-\pi$) and Andreev bound states on the right-hand side of the SC (green). Andreev bound states at the QSHI-SC interface (blue, crosses) (resolution in $k$ is much larger than the crosses indicate) with a single Majorana fermion mode (arrow). Inset: Zoom-out at higher energies. The conical structure at $k=\pm \pi$ is owing to the nodes in the $d_{\mathit{xy}}$-wave order parameter.