Multiple odd-frequency superconducting states in buckled quantum spin Hall insulators with time-reversal symmetry

We consider a buckled quantum spin Hall insulator (QSHI), such as silicene, proximity coupled to a conventional spin-singlet $s$-wave superconductor. Even limiting the discussion to the disorder-robust $s$-wave pairing symmetry, we find both odd-frequency ($\omega$) spin-singlet and spin-triplet pair amplitudes, both of which preserve time-reversal symmetry. Our results show that there are two unrelated mechanisms generating these different odd-$\omega$ pair amplitudes. The spin-singlet state is due to the strong interorbital processes present in the QSHI. It exists generically at the edges of the QSHI, but also in the bulk in the heavily doped regime if an electric field is applied. The spin-triplet state requires a finite gradient in the proximity-induced superconducting order along the edge, which we find is automatically generated at the atomic scale for armchair edges but not at zigzag edges. In combination these results make superconducting QSHIs a very exciting venue for investigating not only the existence of odd-$\omega$ superconductivity but also the interplay between different odd-$\omega$ states.

Figure 1

Schematic picture of the system with a conventional spin-singlet $s$-wave superconductor substrate (blue) and a honeycomb QSHI sheet on top with $A$ site atoms (black) and $B$ sites (gray), as well as ZZ or AC edges indicated. The top semitransparent electrode is a gate controlling ${\lambda }_{\mathrm{v}}$.

Figure 2

Pair amplitudes for a metallic QSHI as a function of sublattice staggering ${\lambda }_{\mathrm{v}}$. Here the parameters are ${\lambda }_{\mathrm{SO}}/t=0.50, U/t=2.0$, and $\mu /t=0.70$. (a) ES on-site $s$-wave and NN $s^{+}$-wave. (b) OS NN $s^{+}$-wave.

Figure 3

ES on-site $s$-wave pair amplitude for a ZZ ribbon. We use the same parameters as in the metallic bulk case, except the chemical potential is $\mu /t=0.05$. (a) Spatial intensity profile of the pair amplitude for the whole ribbon at ${\lambda }_{\mathrm{v}}/{\lambda }_{\mathrm{SO}}=0.0$. The radius of each circle is proportional to the magnitude, while positive (negative) sign is coded in blue (red). (b) Evolution of pair amplitudes with ${\lambda }_{\mathrm{v}}$ on edge sites ($A$, left; $B$, right) and one layer deeper ($A$, right; $B$, left).

Figure 4

ES NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 3, apart from ${\lambda }_{\mathrm{v}}/{\lambda }_{\mathrm{SO}}=0.65$ in (a).

Figure 5

OS NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 3.

Figure 6

ES on-site $s$-wave pair amplitude for an AC ribbon for the same parameters as Fig. 3. (a) Spatial intensity profile of the pair amplitude for the whole ribbon at ${\lambda }_{\mathrm{v}}/{\lambda }_{\mathrm{SO}}=0.0$. (b, c) Evolution of the pair amplitudes with ${\lambda }_{\mathrm{v}}$ on the top (b) and bottom (c) edge on edge sites (index 1) and one layer beyond (index 2).

Figure 7

ES NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 6, apart from ${\lambda }_{\mathrm{v}}/{\lambda }_{\mathrm{SO}}=0.2$ in (a).

Figure 8

OS NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 6.

Figure 9

OT on-site $s$-wave pair amplitude for the same parameters and geometry as in Fig. 6.

Figure 10

OT NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 6, apart from ${\lambda }_{\mathrm{v}}/{\lambda }_{\mathrm{SO}}=0.35$ in (a).

Figure 11

ET NN bond $s^{+}$-wave pair amplitude for the same parameters and geometry as in Fig. 6.