Magic Angles In Equal-Twist Trilayer Graphene

TL;DR

This paper discovers that equal-twist trilayer graphene exhibits perfectly flat electronic bands at specific 'magic angles' related to bilayer graphene, with analytical proof and numerical analysis exploring deviations from ideal conditions.

Contribution

It analytically demonstrates the relation between flat bands in trilayer and bilayer graphene at magic angles and explores their spectral properties.

Findings

Flat bands occur at magic angles equal to bilayer angles divided by √2.

Analytical proof of the relation between trilayer and bilayer flat bands.

Numerical analysis of spectrum away from the chiral limit.

Abstract

We consider a configuration of three stacked graphene monolayers with equal consecutive twist angles $\theta$. Remarkably, in the chiral limit when interlayer coupling terms between $\textrm{AA}$ sites of the moir\'{e} pattern are neglected we find four perfectly flat bands (for each valley) at a sequence of magic angles which are exactly equal to the twisted bilayer graphene (TBG) magic angles divided by $\sqrt{2}$. Therefore, the first magic angle for equal-twist trilayer graphene (eTTG) in the chiral limit is $\theta_{*} \approx 1.05^{\circ}/\sqrt{2} \approx 0.74^{\circ}$. We prove this relation analytically and show that the Bloch states of the eTTG's flat bands are non-linearly related to those of TBG's. Additionally, we show that at the magic angles, the upper and lower bands must touch the four exactly flat bands at the Dirac point of the middle graphene layer. Finally, we explore the eTTG's spectrum away from the chiral limit through numerical analysis.