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 $\ensuremath{\omega}$, twist angle $\ensuremath{\theta}$, interlayer potential difference $\mathrm{\ensuremath{\Delta}}$, and top-bottom bilayers sliding vector $\mathbit{\ensuremath{\tau}}$ 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 $\mathrm{\ensuremath{\Delta}}$ allows the flattening of bands in t3BG down to $\ensuremath{\sim}20$ meV for twist angles $\ensuremath{\theta}\ensuremath{\lesssim}{2}^{\ensuremath{\circ}}$ 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 $\mathrm{\ensuremath{\Delta}}$ 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 $U/W\ensuremath{\gtrsim}1$ ratios in the $\ensuremath{\theta}$ and $\mathrm{\ensuremath{\Delta}}$ parameter space. We also present the expected $K$-valley Chern numbers for the lowest-energy nearly flat bands.