Electronic spectrum of twisted bilayer graphene

TL;DR

This paper investigates the electronic spectrum of twisted bilayer graphene using a tight-binding model, revealing angle-dependent gaps, their sensitivity to twist angle variations, and the transition from metallic to gapped phases.

Contribution

The study introduces a detailed tight-binding model considering neighboring atom positions and numerically evaluates the spectrum for various twist angles, highlighting the non-monotonous gap behavior.

Findings

Gaps appear at certain angles above ~1.89° with values up to 80 meV.

The gap exhibits exponential sensitivity to small angle variations.

For small angles, the system transitions from metallic to gapped phases.

Abstract

We study the electronic properties of twisted bilayers graphene in the tight-binding approximation. The interlayer hopping amplitude is modeled by a function, which depends not only on the distance between two carbon atoms, but also on the positions of neighboring atoms as well. Using the Lanczos algorithm for the numerical evaluation of eigenvalues of large sparse matrices, we calculate the bilayer single-electron spectrum for commensurate twist angles in the range $1^{\circ}\lesssim\theta\lesssim30^{\circ}$. We show that at certain angles $\theta$ greater than $\theta_{c}\approx1.89^{\circ}$ the electronic spectrum acquires a finite gap, whose value could be as large as $80$ meV. However, in an infinitely large and perfectly clean sample the gap as a function of $\theta$ behaves non-monotonously, demonstrating exponentially-large jumps for very small variations of $\theta$. This sensitivity to the angle makes it impossible to predict the gap value for a given sample, since in experiment $\theta$ is always known with certain error. To establish the connection with experiments, we demonstrate that for a system of finite size $\tilde L$ the gap becomes a smooth function of the twist angle. If the sample is infinite, but disorder is present, we expect that the electron mean-free path plays the same role as $\tilde L$. In the regime of small angles $\theta<\theta_c$, the system is a metal with a well-defined Fermi surface which is reduced to Fermi points for some values of $\theta$. The density of states in the metallic phase varies smoothly with $\theta$.