Topological flat bands without magic angles in massive twisted bilayer graphenes*

We show that in massive twisted bilayer graphenes, isolated nearly flat bands below a threshold bandwidth W_c are expected for continuous small twist angles up to a critical θ_c depending on the flatness of the original bands and the interlayer coupling strength. 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 0° and destructive near 60° 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, interlayer tunneling strengths suggest optimistic prospects of increasing θ_c and achieving correlated states with large U/W effective interaction versus bandwidth ratios.