Electronic structure of lattice relaxed alternating twist tG-multilayer graphene and primary and secondary gaps in G/hBN using a reparametrized two-center approximation TB model

  Alternating twist (AT) multilayer graphene systems are at the heart of recent research efforts on flat band superconductivity and therefore precise descriptions of their atomic and electronic structures are desirable. After introducing the methodology behind our reparametrized two-center (TC) tight-binding (TB) model, we firstly present the electronic structure of AA’AA’… stacked AT N-layer (tNG) graphene for $N = $3-10, 20 layers and bulk AT graphite systems where the atomic structure is relaxed using a molecular dynamics simulation code. The low energy bands depend sensitively on the relative sliding between the layers but we show explicitly up to N = 6 that the highly symmetric AA’AA’. . . stacking is energetically preferred among all interlayer sliding geometries of each added layer, justifying why experimental devices consistently show results compatible with this geometry. It is found that lattice relaxation enhances electron-hole asymmetry, and leads to small reductions of the magic angle values with respect to analytical or continuum model calculations with fixed tunneling strengths that we quantify from few layers to bulk AT-graphite. The twist angle error tolerance near the magic angles obtained by maximizing the density of states of the nearly flat bands expand progressively from 0.05° for twisted bilayer graphene to up to 0.2° for AT-graphite, hence allowing a greater twist angle flexibility in multilayers. We further comment on the role of perpendicular electric and magnetic fields in modifying the electronic structure of the system. We secondly illustrate the reparametrized TC-model approach to assess the impact of the substrate relaxation on the primary gap at charge neutrality and secondary valence band gap of graphene on hexagonal boron nitride (G/BN) as a function of twist angle where we observe a substrate-induced reduction of the primary gap and the closing of the primary and secondary gaps for finite angles as well as a slight initial increase for the primary gap at ~ 0.5 degree that coincides with a shallow energetic stabilization of the atomic structure away from perfect alignment.