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 disorderrobust s-wave pairing symmetry, we find both odd-frequency (ω) 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-ω 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-ω superconductivity but also the interplay between different odd-ω states.

We demonstrate theoretically that proximity-induced superconductivity in silicene offers the possibility to exert strong quantum ground state control. We show that electrically controlled 0-π transitions occur in Josephson junctions in the presence of an exchange field due to the buckling of the silicene lattice. We also discover that zigzag-oriented interfaces, featuring intervalley scattering, cause a ϕ0 state with an applied electric field. Finally, we demonstrate that Majorana bound states along the silicene edge are tunable via the edge orientation, electric, and in-plane spin exchange fields.PACS numbers: 74.50.+r, 74.45.+c, 73.61.Wp, 71.10.Pm The discovery of new low-dimensional materials, where the electron bands have topological properties, has attracted a large amount of interest in recent years. A particularly intriguing material is silicene [1], which consists of an atomically thin, buckled layer of Si atoms arranged in a honeycomb lattice. This material has theoretically been shown to host both different topological phases and has individually tunable mass gaps for each spin σ at each valley η [2-5]. These properties make silicene ideal for envisaging various types of device functionality related to both spintronics and valleytronics, a quest also significantly fueled by its compatibility with existing Si semiconductor technology. On the experimental side, silicene has already been studied on metallic substrates, including ZrB 2 [6] and notably Ag(111) [7][8][9], as well as for nonmetallic hosts, where a Si nanosheet grown on MoS 2 bulk crystals has recently been reported [10].A particularly exciting prospect is to consider the manifestation of superconducting correlations in silicene, with the unique properties of silicene likely leading to an advanced interplay between spintronics and superconductivity [11]. Recent experimental progress has enabled the study of superconductivity in atomically thin materials, such as in graphene [12][13][14] and transition metal dichalcogenides [15,16], through the proximity effect from external superconductors. Motivated by this, we set out to determine how superconductivity is manifested in silicene, especially focusing on the external control of unusual phenomena via an electric field.We demonstrate that proximity-induced superconductivity in silicene allows for a strong quantum ground state control. By creating superconductor-ferromagnetsuperconductor (SFS) Josephson junctions in bulk silicene, we find that the exchange field in the F region gives rise to electrically controlled 0-π transitions, due to the buckling of the silicene lattice. We also discover that zigzag-oriented SF interfaces, which host notable intrinsic intervalley scattering, result in an exotic ϕ 0 state, directly tunable by electric field. Finally, we demonstrate that the existence of Majorana bound states (MBS) at SF junctions on silicene edges is controlled by edge orientation and the strength of electric and in-plane exchange fields.First, we compute the supercurrent in bulk silicene S...

Superconductor-ferromagnetic heterostructures have been suggested as one of the most promising alternatives of realizing odd-frequency superconductivity. In this work we consider the limit of shrinking the ferromagnetic region to the limit of a single impurity embedded in a conventional superconductor, which gives raise to localized Yu-Shiba-Rusinov (YSR) bound states with energies inside the superconducting gap. We demonstrate that all the sufficient ingredients for generating odd-frequency pairing are present at the vicinity of these impurities. We investigate the appearance of all possible pair amplitudes in accordance with the Berezinskii SP * OT * = −1 rule, being the symmetry under the exchange of spin, spatial, orbital (in our case O = +1) and time index, respectively. We study the spatial and frequency dependence of of the possible pairing amplitudes, analyzing their evolution with impurity strength and identifying a reciprocity between different symmetries related through impurity scattering. We show that the odd-frequency spin-triplet pairing amplitude dominates at the critical impurity strength, where the YSR states merge at the middle of the gap, while the even components cancel out close to the impurity. We also show that the spin-polarized local density of states exhibits the same spatial and frequency behavior as the odd-ω spin-triplet component at the critical impurity strength.

Magnetic chains on superconducting systems have emerged as a platform for realization of Majorana bound states (MBSs) in condensed matter systems with possible applications to topological quantum computation. In this work, we study the MBSs formed in magnetic chains on twodimensional honeycomb materials with induced superconductivity. We establish chemical potential vs Zeeman splitting phase diagrams showing that the topological regions (where MBSs appear) are strongly dependent on the spiral angle along the magnetic chain. In some of those regions, the topological phase is robust even for large values of the local Zeeman field, thus producing topological regions which are, in a sense, "unbounded" in the large-field limit. Moreover, we show that the energy oscillations with magnetic field strength due to MBS splitting can show very different behaviors depending on the parameters. In some regimes, we find oscillations with increasing amplitudes and decreasing periods, while in the other regimes the complete opposite behavior is found. We also find that the topological phase can become dependent on the chain length, particularly in topological regions with a very high or no upper bound. In these systems, we see a very smooth evolution from MBSs localized at chain end points to in-gap Andreev bound states spread over the full chain. arXiv:1808.07402v2 [cond-mat.mes-hall]