Electronic structure of turbostratic graphene

TL;DR

This paper investigates how the electronic properties of bilayer graphene change with the twist angle, revealing a decrease in Fermi velocity and complex band warping as the layers become aligned, with implications for understanding graphene's electronic behavior.

Contribution

It provides a theoretical analysis of the electronic structure of twisted bilayer graphene, deriving a formula for Fermi velocity dependence on twist angle and exploring the singular limit as the angle approaches zero.

Findings

Fermi velocity decreases as twist angle approaches zero

Agreement with continuum models for angles greater than approximately 5°

Strong band warping occurs as the twist angle approaches zero

Abstract

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity $v_F$ of the Dirac cones decreases as $\theta\to 0^\circ$; the form we derive for $v_F(\theta)$ agrees with that found via a continuum approximation in Phys. Rev. Lett., 99:256802, 2007. From tight binding calculations for structures with $1.47^\circ \le \theta < 30^\circ$ we find agreement with this formula for $\theta \gtrsim 5^\circ$. In contrast, for $\theta \lesssim 5^\circ$ this formula breaks down and the Dirac bands become strongly warped as the limit $\theta \to 0$ is approached. For an ideal system of twisted layers the limit as $\theta\to0^\circ$ is singular as for $\theta > 0$ the Dirac point is fourfold degenerate, while at $\theta=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined \emph{globally}, in contrast to the 'nearsightedness' [W. Kohn. Phys. Rev. Lett., 76:3168, 1996.] of electronic structure generally found in condensed matter.