Atomic corrugation and electron localization due to Moiré patterns in twisted bilayer graphenes

We report on unprecedentedly large-scale density-functional calculations that clarify atomic and electronic structures of twisted bilayer graphene (BLG). We find the existence of the critical twist angle from either the AB or the AA stacking BLG, above which the two graphene layers are essentially decoupled and below which the atomic planes are corrugated and the Dirac electrons are localized. We also find a magic angle at which the Fermi velocity of the Dirac electron vanishes. We clarify that the Moiré pattern in tBLG with a tiny twist angle generates inhomogeneity for the electron systems and thus causes the drastic modification of the electronic properties, leading to flat bands at the Fermi level. Sensitivity to the Moiré of the valence-electron density and the electron state near the Fermi level is discussed.

Figure 1

The side and top views of the optimized tBLG with the twist angle $\theta$ of (a) ${29.4}^{\circ }$, (b) ${8.26}^{\circ }$, and (c) ${3.89}^{\circ }$. In (b) and (c), the corrugation $\Delta$ is visible with the longest distance $d_{\mathrm{far}}$ and the shortest distance $d_{\mathrm{near}}$. In (d), $d_{\mathrm{far}}$ and $d_{\mathrm{near}}$ as functions of the twist angle $\theta$ are shown. The crosses are the calculated values. The lines are guides for eyes. The orange (light) and blue (dark) circles in (b) and (c) depict the AA and AB stacking regions, respectively.

Figure 2

Binding energy $E_{\mathrm{b}}$ per atom as a function of $\theta$. The crosses show calculated values. The line is a guide for eyes.

Figure 3

Energy bands of tBLGs with $\theta =29.4{}^{\circ }$ (a), ${2.88}^{\circ }$ (b), and ${0.99}^{\circ }$ (c). The Fermi level $E_{\mathrm{F}}$ is set to be 0. The enlargement near $E_{\mathrm{F}}$ for $\theta =0.{99}^{\circ }$ is shown in (d). The energy scale is different in each figure. Also the size of the Brillouin zone is different since the numbers of the atoms in each (hexagonal) unit cell are 388 (a), 1588 (b), and 13 468 (c),(d), respectively. The Dirac bands of a SLG are also shown by blue (dashed) lines.

Figure 4

Fermi velocity normalized to that of SLG $v_{\mathrm{F}}^0$, as a function of $\theta$: the velocities in the optimized structures ($v_{\mathrm{F}}$); those for the flat structures with the interlayer distances of 3.61 Å ($v_{\mathrm{F}}^{\mathrm{far}}$) and 3.34 Å ($v_{\mathrm{F}}^{\mathrm{near}}$). Black circles are experimental data [8].

Figure 5

The squared Kohn-Sham orbitals at the Fermi level $E_{\mathrm{F}}$, summed over fourfold degenerate states. The $\theta$ is (a) ${29.4}^{\circ }$ and (b) ${0.99}^{\circ }$. The AA and AB in (b) depict the local AA and AB stacking regions, respectively.

Figure 6

The unit cell of a commensurate $(M,N)$ tBLG. Suppose a graphene sheet with a superperiodicity defined by the two $(N,M)$ and $(-M,N+M)$ vectors (a) and rotate it by $+\theta /2$ (b). Similarly, take a graphene sheet with periodicity defined by the $(M,N)$ and $(-N,M+N)$ vectors (c) and rotate it by $-\theta /2$ (d). The commensurate tBLG (e) with the twist angle $\theta$ and the periodicity $L_{\mathrm{cell}}$ is obtained by stacking the thus obtained two layers, (b) and (d).

Figure 7

Moiré patterns in tBLGs schematically drawn for the twist angle $\theta$ of (a) $5^{\circ }$, (b) ${10}^{\circ }$, (c) ${15}^{\circ }$, (d) ${20}^{\circ }$, (e) ${25}^{\circ }$, (f) ${30}^{\circ }$, (g) ${35}^{\circ }$, (h) ${40}^{\circ }$, (i) ${45}^{\circ }$, (j) ${50}^{\circ }$, and (k) ${55}^{\circ }$. The pattern for the tBLG with ${60}^{\circ }-\theta$ is essentially the same as that with $\theta$.

Figure 8

Atomic structures of $(N,N-1)$ tBLGs with the twist angle $\theta$ of (a) $6.{01}^{\circ }$ ($N=6$), (b) $3.{89}^{\circ }$ ($N=9$), and (c) $0.{99}^{\circ }$ ($N=34$). The structures are obtained by full structural optimizations for (a) and (b). The structure in (c) is obtained by the extrapolation (see Appendix pp4). (d) A Fourier coefficient ${\alpha }_1$ [Eq. (D1)] is calculated as a function of $\theta$, being fitted by the solid curve for extrapolation to the smaller twist angle ($\theta <2^{\circ }$).

Figure 9

(a) The farthest distance $d_{\mathrm{far}}$ and the nearest distances $d_{\mathrm{near}}$ between the graphene layers in tBLGs, as functions of the twist angle $\theta$. The solid and dashed lines are from vdW-DFT and LDA-DFT calculations, respectively. The interlayer distances of purely AA or AB stacking bilayers are also shown by horizontal lines. (b) The band dispersions of a tBLG with $\theta =7.{34}^{\circ }$, obtained by vdW-DFT. The dashed line is the corresponding dispersion of the SLG. The Fermi levels are set to be 0.