Electronic structure of graphene twist stacks

We investigate the electronic structure of graphene stacks having an ordered sequence of pairs of twisted layers—the graphene twist stack. We find that this remarkable system entails a fundamental mixing of dimensionalities: While the twist stack spectrum is generated by an ensemble of independent effective twist bilayer Hamiltonians, the wave functions are products of bilayer wave functions and standing electron waves in the stacking direction, and thus extend over many layers of the stack. These have the property that of the ensemble of Dirac cones that constitute the twist stack band structure, it is those topologically closest to single layer graphene that dominate the surface region. We further examine the impact of stacking disorder, finding that these results are robust for moderate stacking fault density.

Figure 1

Schematic illustration of the grouping of graphene stack atoms into one-dimensional chains along with hopping processes that lead to the inter- ($t_{\perp i}^{\alpha \beta }$) and intralayer ($t_{\parallel i}^{\alpha \beta }$) matrix elements of $H^{\alpha \beta }$.

Figure 2

Band structure of an 11-layer twist stack ($\theta =21.{78}^{°}$) ideal system shown in (a) while in (b) a single rotation fault has been introduced. In both panels colored dashed lines indicate the band structure given by Eq. (4), with full black lines the directly calculated twist stack band structure. Insets in panel (a) show the layer-integrated density $\rho (j)$ for the stack states arising from bands indicated by arrows. Lower inset panel displays $\rho (j)$ of the defect states associated with the defect band indicated by arrow, with upper inset panel similar but for a 101-layer stack with seven randomly distributed twist faults. See text for details.

Figure 3

Numerical values for the coefficients $c_{n\mathbf{k}}^{(m)}(\sigma )$ in Eq. (5). Unless indicated all systems have 19 layers, with rotation angle displayed in caption.

Figure 4

Weight of stack states at the first layer for a 101-layer ideal twist stack (filled symbols) and the same system with seven randomly distributed twist faults (open symbols). See text for details.