Numerical studies of confined states in rotated bilayers of graphene

Rotated graphene multilayers form a new class of graphene-related systems with electronic properties that drastically depend on the rotation angles. It has been shown that bilayers behave like two isolated graphene planes for large rotation angles. For smaller angles, states in the Dirac cones belonging to the two layers interact resulting in the appearance of two Van Hove singularities. States become localized as the rotation angle decreases and the two Van Hove singularities merge into one peak at the Dirac energy. Here we go further and consider bilayers with very small rotation angles. In this case, well-defined regions of AA stacking exist in the bilayer supercell and we show that states are confined in these regions for energies in the [$-{\gamma }_t$, $+{\gamma }_t$] range with ${\gamma }_t$ the interplane mean interaction. As a consequence, the local densities of states show discrete peaks for energies different from the Dirac energy.

Figure 1

Van Hove singularities of a $(4,5)$ bilayer, $\theta =7.{34}^{\circ }$. (a) Schematic drawing. (b) $(4,5)$ band structure and total DOS.

Figure 2

Energies of the Van Hove singularities (the two peaks closest to $E_D$) in ($n,m$) bilayers versus $\theta$. Lines are guides for the eyes.

Figure 3

Gap at P (point M of the supercell) as a function of $\theta$. Line is a guide for the eyes.

Figure 4

Band structures and DOS for bilayers with rotation angles that lead to Van Hove singularities close to the limit of the linear variation. (a) $(19,20)$, $\theta =1.{67}^{\circ }$; (b) $(31,32)$, $\theta =1.{05}^{\circ }$; (c) $(45,46)$, $\theta =0.{73}^{\circ }$. Left: Dispersion of monolayer graphene is shown in black dashed line for comparison. Right: (Red line) Local DOS at the center of AA zone, (black dashed line) total DOS.

Figure 5

TB DOS in bilayers. (a) Total DOS. Local DOS: (b), (c) On atom at the center of an AA zone, (d) on atom at the center of an AB zone (at. A: with an atom directly on top or below; at. B: without an atom on top or below). ($n$,$m$) Bilayers: Values $(100,101)$ $\theta =0.{33}^{\circ }$, $(200,201)$ $\theta =0.{16}^{\circ }$, $(300,301)$ $\theta =0.{11}^{\circ }$, $(400,401)$ $\theta =0.{08}^{\circ }$.

Figure 6

Variation of the energy of the first peak ($E_1$) away from the Dirac point of the local DOS at the center of AA region as a function of $\theta$.

Figure 7

Band structure of an AA and an AB stacked bilayer in the vicinity of the Dirac energy.

Figure 8

Velocity of a bilayer divided by the velocity of a monolayer, at the Dirac point, as a function of $\theta$. The insert shows a zoom on very small angles and on angles close to 60${}^{\circ }$. Black/green point: TB calculation, red cross: ab initio calculation, blue line: law proposed by Lopes dos Santos et al. (Ref. 22), orange line: model proposed by Bistritzer and MacDonald (Ref. 29) for very small angles. The latter line is rescaled to obtain a first minimum of the velocity at $1.{13}^{\circ }$ as in TB calculations (see text). Ab initio calculations and most of the TB calculations for $\theta >1.{05}^{\circ }$ are from Ref. 31.

Figure 9

Band structure $E\left(\vec{k}\right)$ calculated from ab initio vasp and TB: (a) graphene, (b) AB bilayer, (c) AA bilayer. ${\gamma }_t=0.34$ eV.

Figure 10

Density of states. (a) Ab initio [linear muffin tin orbitals (Ref. 42)] and TB total DOS in graphene. (b) TB total DOS in AB (Bernal) and AA bilayers. (Inset: Total DOS around the $E_{\mathrm{D}}=0$. ${\gamma }_t=0.34$ eV.)

Figure 11

Coupling term $t$ between two $p_z$ orbitals versus the distance $r$: (full line) in two layers (interlayer coupling); (dashed line) in the same layer (intralayer coupling). Here, the smooth cutoff function $F_c$ is not taken into account [i.e., large $r_c$ value in (A7)].