Nearly flat bands in twisted triple bilayer graphene

We investigate the electronic structure of alternating-twist triple Bernal-stacked bilayer graphene (t3BG) as a function of interlayer coupling ω, twist angle θ, interlayer potential difference Δ, and top-bottom bilayers sliding vector τ for three possible configurations AB/AB/AB, AB/BA/AB, and AB/AB/BA. The parabolic low-energy band dispersions in a Bernal-stacked bilayer and gap opening through a finite interlayer potential difference Δ allows the flattening of bands in t3BG down to ∼20 meV for twist angles θ≲2∘ regardless of the stacking types. The easier isolation of the flat bands and associated reduction of Coulomb screening thanks to the intrinsic gaps of bilayer graphene for finite Δ facilitate the formation of correlation-driven gaps when it is compared to the metallic phases of twisted trilayer graphene under electric fields. We obtain the stacking dependent Coulomb energy versus bandwidth U/W≳1 ratios in the θ and Δ parameter space. We also present the expected K-valley Chern numbers for the lowest-energy nearly flat bands.