Emergence of intrinsically isolated flat bands and their topology in fully relaxed twisted multi-layer graphene

Abstract

We study the electronic structure and band topology of fully relaxed twisted multi-layer graphene (TMLG). Isolated flat bands emerge in TMLG with the number of layers [M + N with M the layer number of the bottom few-layer graphene (FLG)] up to 10 and with various stacking orders, and most of them are on the hole side. The touched bands of FLGs around the Fermi level are split by the moiré coupling through the electron-hole asymmetry in low-energy bands of FLGs and by the vertical hopping between next-nearest layers. The full structural relaxation leads to global gaps that completely isolate a flat band. For TMLG with given M and N , the highest magnitude of Chern numbers (|C|) of the separable flat bands reaches M + N − 1 and can be hosted by certain isolated bands. The |C| = 9 occurs in the isolated flat valence band of several configurations with 10 layers. Such high |C| originates from the lifting of the band-state degeneracy in the weak regime of moiré coupling or from the topological phase transitions induced by the strong moiré coupling. Moreover, large orbital magnetic moments arise in isolated flat bands with high |C| and depend on the structural configurations of TMLG.