Electronic structure of lattice relaxed alternating twist tNG-multilayer graphene: from few layers to bulk AT-graphite

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. We 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 degrees for twisted bilayer graphene to up to 0.2 degrees 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 and how the decoupling of tNG multilayers bands allows mapping onto those of periodic AT-graphite's at different k(z) values.