Quantum Interference at the Twist Boundary in Graphene

We explore the consequences of a rotation between graphene layers for the electronic spectrum. We derive the commensuration condition in real space and show that the interlayer electronic coupling is governed by an equivalent commensuration in reciprocal space. The larger the commensuration cell, the weaker the interlayer coupling, with exact decoupling for incommensurate rotations and in the $\theta \to 0$ limit. Furthermore, from first-principles calculations we determine that even for the smallest possible commensuration cell the decoupling is effectively perfect, and thus graphene layers will be seen to decouple for all rotation angles.

Figure 1

The inset graph displays band structure near the Dirac point for the $(AB,2a)$ $\theta =30°\pm 8.21°$ commensuration cells ($N=7$). Clearly, near the Dirac point the $\pi$ and ${\pi }^{*}$ bands are not linear. The main graph details the rapid decay, as $N$ increases, of the energy window where such deviations from linearity occur.

Figure 2

(a) Band structure of several rotated bilayer structures for $\theta =30°\pm 8.21°$ (corresponding to $N=7$); the inset shows resulting atomic structures where shading distinguishes atoms belonging to different layers. (b) Band structure of the single-layer graphene and the $ABA$ trilayer with the middle layer rotated by $\theta =21.79°$. The inset displays close-ups of structures in (a) with commensuration primitive vectors [Eqs. (7, 8)] and rotation angle [Eq. (9)] shown.