Abstract
We introduce a complete physical model for the single-particle electronic structure of twisted bilayer graphene (tBLG), which incorporates the crucial role of lattice relaxation. Our model, based on $k \cdot p$ perturbation theory, combines the accuracy of DFT calculations through effective tight-binding Hamiltonians with the computational efficiency and complete control of the twist angle offered by continuum models. The inclusion of relaxation significantly changes the bandstructure at the first magic-angle twist corresponding to flat bands near the Fermi level (the "low-energy" states), and eliminates the appearance of a second magic-angle twist. We show that minimal models for the low-energy states of tBLG can be easily modified to capture the changes in electronic states as a function of twist angle.
Highlights
The discovery of correlated phases in twisted bilayer graphene (TBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring the nature of unconventional superconductivity [1,2]
Since the discovery of correlated phases in TBLG, many simplified n-band models have been proposed for the flat bands, usually based on localized functions of the BMD model
One such minimal model consists of ten bands [20], and we argue that it can accurately capture both the band structure (Fig. 1) and the electronic effects of the different stacking regions that emerge after relaxation
Summary
The discovery of correlated phases in twisted bilayer graphene (TBLG) has generated much interest in this structurally and compositionally rather simple system; it has emerged as a new platform for tunable electronic correlations, and for exploring the nature of unconventional superconductivity [1,2]. Our model reproduces the results of DFT-quality tight-binding Hamiltonians but at a smaller computational cost and, more importantly, it applies to all twist angles near the magic-angle value Such a single-particle model is a prerequisite for a physically meaningful prediction of correlation effects, as the presence of unphysical features in the single-particle band structure causes uncontrolled errors in many-body calculations. Our model has three key ingredients: (1) relaxation of the bilayer system [25], including the out-of-plane relaxation of different regions as well as the in-plane strain corrections to the Hamiltonian of the individual monolayers; (2) terms beyond the first shell of couplings in the k·p continuum model, which are necessary to capture the changes in stacking order at small angles; and (3) inclusion of k-dependent terms, which allow the k·p model to reproduce more accurately the particlehole asymmetry of realistic ab initio band structures. These terms have a smooth dependence on θ , allowing for interpolation between the specific twist angles that correspond to finite supercells, to generate a model valid for any desired angle in that range
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