▎ 摘　　要

Flat bands with extremely narrow bandwidths on the order of a few millielectron 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 flat bands 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 2 in the same parameter space of twist angle theta and interlayer coupling., making it in principle simpler to tailor narrow bandwidth flat bands. Application of vertical pressure can enhance the first magic angle in minimal models at theta similar to 1.05 degrees to larger values of up to theta similar to 1.5 degrees when P similar to 2.5 GPa, where theta proportional to omega/nu(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 flat bands 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 degrees or 180 degrees alignments.