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

TL;DR

This study reveals the formation of isolated flat bands with high topological Chern numbers in fully relaxed twisted multi-layer graphene, highlighting their dependence on stacking order and structural relaxation.

Contribution

It demonstrates the emergence of intrinsically isolated flat bands with high Chern numbers in fully relaxed twisted multi-layer graphene, a novel finding in the field.

Findings

Isolated flat bands appear in TMLG with up to 10 layers.

Maximum Chern number magnitude reaches M+N-1 in certain configurations.

High Chern numbers (up to 9) are associated with topological phase transitions.

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 moir\'{e} 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 moir\'{e} coupling or from the topological phase transitions induced by the strong moir\'{e} coupling. Moreover, large orbital magnetic moments arise in isolated flat bands with high $|C|$ and depend on the structural configurations of TMLG.