Flatbands in twisted double bilayer graphene

Flatbands with extremely narrow bandwidths on the order of a few mili-electron volts can appear in twisted multilayer graphene systems for appropriate system parameters. Here we investigate the electronic structure of a twisted bi-bilayer graphene, or twisted double bilayer graphene, to find the parameter space where isolated flatbands can emerge as a function of twist angle, vertical pressure, and interlayer potential differences. We find that in twisted bi-bilayer graphene the bandwidth is generally flatter than in twisted bilayer graphene by roughly up to a factor of two in the same parameter space of twist angle $\theta$ and interlayer coupling $\omega$, making it in principle simpler to tailor narrow bandwidth flatbands. Application of vertical pressure can enhance the first magic angle in minimal models at $\theta \sim 1.05^{\circ}$ to larger values of up to $\theta \sim 1.5^{\circ}$ when $ P \sim 2.5$~GPa, where $\theta \propto \omega/ \upsilon_{F}$. Narrow bandwidths are expected in bi-bilayers for a continuous range of small twist angles, i.e. without magic angles, when intrinsic bilayer gaps open by electric fields, or due to remote hopping terms. We find that moderate vertical electric fields can contribute in lifting the degeneracy of the low energy flatbands by enhancing the primary gap near the Dirac point and the secondary gap with the higher energy bands. Distinct valley Chern bands are expected near $0^{\circ}$ or $180^{\circ}$ alignments.