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We consider a buckled quantum spin Hall insulator (QSHI), such as silicene,proximity-coupled to a conventional spin-singlet, s-wave superconductor. Evenlimiting the discussion to the disorder-robust s-wave pairing symmetry, we findboth odd-frequency ($\omega$), spin-singlet and spin-triplet pair amplitudesand where both preserve time-reversal symmetry. Our results show that there aretwo unrelated mechanisms generating these different odd-$\omega$ pairamplitudes. The spin-singlet state is due to the strong inter-orbital processespresent in the QSHI. It is exists generically at the edges of the QSHI, butalso in the bulk in heavily doped regime if an electric field is applied. Thespin-triplet state requires a finite gradient in the proximity-inducedsuperconducting order along the edge, which we find is automatically generatedat the atomic scale for armchair edges but not at zigzag edges. In combinationthese results make superconducting QSHIs a very exciting venue forinvestigating not only the existence of odd-$\omega$ superconductivity, butalso the interplay between different odd-$\omega$ states.

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