Superconductivity in monolayer and few-layer graphene. II. Topological edge states and Chern numbers

Abstract

We study the emergence of electronic edge states in superconducting monolayer, bilayer, and trilayer graphene for both spin-singlet and spin-triplet superconducting order parameters. We focus mostly on the gapped chiral p + ip -and d + id -wave superconducting states that show a nonzero Chern number and a corresponding number of edge states. For the p + ip -wave state, we observe a rich phase diagram for the Chern number when tuning the chemical potential and the superconducting order parameter amplitude, which depends strongly on the number of layers and their stacking and is also modified by trigonal warping. At small parameter values, compared to hopping energy, we observe a region whose Chern number is unique to rhombohedrally stacked graphene and is independent of the number of layers. Our results can be understood in relation not only to the superconducting order parameter winding as expected but also to the normal state band structure. This observation establishes the importance of the normal state characteristics for understanding the topology in superconducting graphene systems.

Long-range pairing in monolayer NbSe2 facilitates the emergence of topological superconducting states

“…CN = ±6, ±5, ±4, ±3, ±2, ±1) appearing and the TSC phase regions with CNs = 3 and −2 start to shrink when p A1 is greater than 0.14. The CN displays a sensitive dependence on V z and µ and distributes irregularly between CN = 3 and CN = −2 phases due to long-range pairing, which has also been observed in previous studies [19,20,22,24,[26][27][28][29]. It is notably that the sensitive non-zero CNs appear even under V z = 0 when p A1 is greater than or equal to 0.16 (see figures 2(c)-(f), 3 and 4), suggesting that there may be more intrinsic superconductors (e.g.…”

“…Two-dimensional TSCs with nontrivial properties exhibit topological edge states (TESs) along their edges, as dictated by the well-established bulk-boundary correspondence (BBC) principle, which associates the topological invariant with the number of TESs [2,18]. However, recent research progress on topological materials has revealed the sensitivity and richness of the CN phase diagram to variations in the chemical potential (µ), magnetic field (V z ), and superconducting order parameter amplitude [19][20][21][22][23]. Furthermore, there is evidence of a mismatch between the CN and the number of TESs, indicating a deviation from the BBC principle [19,[21][22][23][24][25][26][27][28][29][30].…”

Superconductivity in monolayer and few-layer graphene. I. Review of possible pairing symmetries and basic electronic properties

“…Overall, our project consists of three parts: in the present paper we focus on calculations of the band structure and density of states, with a particular focus on the gap closing points in the energy spectrum, as well as on evaluating the effects of trigonal warping present in bi-and trilayer graphene. In the upcoming two works we will focus first on the topological properties of the SC state in monolayer and multilayer graphene and on the corresponding edges states [79] and, secondly, on the impurity-induced [21,[80][81][82][83] subgap states appearing in the presence of impurities [84]. We show that the results of these calculations depend strongly on the underlying SC order parameter, and thus a comparison with experimental measurements would allow us to determine the SC symmetry as well as type of SC pairing.…”