Topological flat bands without magic angles in massive twisted bilayer graphenes

Twisted bilayer graphene (TBG) hosts nearly flat bands with narrow bandwidths of a few meV at certain magic twist angles. Here we show that in twisted gapped Dirac material bilayers, or massive twisted bilayer graphenes (MTBG), isolated nearly flat bands below a threshold bandwidth Wc are expected for continuous small twist angles up to a critical θc depending on the flatness of the original bands and the interlayer coupling strength. Narrow bandwidths of Wâ30meV are expected for θâ3â for twisted Dirac materials with intrinsic gaps of ∼2eV that finds realization in monolayers of gapped transition metal dichalcogenides (TMDC), silicon carbide (SiC) among others, and even narrower bandwidths in hexagonal boron nitride (BN) whose gaps are ∼5eV, while twisted graphene systems with smaller gaps of a few tens of meV, e.g., due to the alignment with hexagonal boron nitride, show the vestiges of the magic angle behavior in the bandwidth evolution. The phase diagram of finite valley Chern numbers of the isolated moire bands expands with increasing difference between the sublattice selective interlayer tunneling parameters. The valley contrasting circular dichroism for interband optical transitions is constructive near 0â and destructive near 60â alignments and can be tuned through electric field and gate-driven polarization of the mini valleys. Combining massive Dirac materials with various intrinsic gaps, Fermi velocities, and interlayer tunneling strengths suggests optimistic prospects of increasing θc and achieving correlated states with large U/W effective interaction versus bandwidth ratios.