Continuum model of the twisted graphene bilayer

The continuum model of the twisted graphene bilayer [Lopes dos Santos, Peres, and Castro Neto, Phys. Rev. Lett. 99, 256802 (2007)] is extended to include all types of commensurate structures. The essential ingredient of the model, the Fourier components of the spatially modulated hopping amplitudes, can be calculated analytically for any type of commensurate structures in the low-twist-angle limit. We show that the Fourier components that could give rise to a gap in the sublattice exchange symmetric (SE-even) structures discussed by Mele [Phys. Rev. B 81, 161405 (2010)] vanish linearly with angle, whereas the amplitudes that saturate to finite values, as $\theta \to 0$, ensure that all low-angle structures share essentially the same physics. We extend our previous calculations beyond the validity of perturbation theory to discuss the disappearance of Dirac cone structure at angles below $\theta \lesssim 1$${}^{\circ }$.

Figure 1

(a) Geometry of the honeycomb lattice. (b) A shell of 12 Bravais lattice sites, their position related by the rotation and reflection symmetries of the hexagonal lattice.

Figure 2

The ${\tilde{t}}_{\perp }^{BA}\left(\mathbf{G}\right)/t_{\perp }$ as a function of $G^{\prime }L$: the dots are numerically calculated values for a $(m,r)=(10,1)$ structure, with $\theta =3.{15}^{\circ }$, and the red line is the integral of Eq. (19).

Figure 3

(a) $2(L/a)\sin (\theta /2)$ vs $\theta$. The various lines correspond to different values of $r$; the lower line corresponds to the structures with $r=1$. (b) A SE-even structure is almost periodic with the period of a structure with $r=1$; here, a $(m,r)=(7,3)$ is shown overlaid with the Wigner-Seitz cells of $(m,r)=(2,1);$ the black hexagon is the true unit cell of the structure.

Figure 4

Comparison of Fourier amplitudes of pairs of structures: each point has an $x$ coordinate ${\tilde{t}}_{\perp }^{\beta \alpha }\left(\mathbf{G}\right)$ for $\left[\right(m-1/3,r/3]$ and a $y$ coordinate ${\tilde{t}}_{\perp }^{\beta \alpha }\left(\mathbf{G}\right)$ for a SE-even structure $(m,r$). $\tilde{\mathbf{G}}=k{\tilde{\mathbf{G}}}_1+l{\tilde{\mathbf{G}}}_2$ and $\mathbf{G}=(2k-l){\mathbf{G}}_1+(k+l){\mathbf{G}}_2$. According to Eq. (26), these amplitudes should be equal. The line is $y=x$, not a fit. The inset has an expanded scale to include the dominant amplitudes ($k,l)=\{(0,0),(-1,0),(-1,-1)\}.$ The angles are in the range $2.5^{\circ }<\theta <7.3^{\circ }$.

Figure 5

$|{\tilde{t}}_{\perp }^{\beta \alpha }\left(\mathbf{G}\right)|/t_{\perp }$ for $\mathbf{G}=k{\mathbf{G}}_1+l{\mathbf{G}}_2$ for SE-even lattices, with different angles of rotation. For $(k,l)=(-1,1)$ or $(2,1)$, $\mathbf{G}$ is approximately equal to a reciprocal lattice vector of an $r=1$ lattice and, ${\tilde{t}}_{\perp }^{\beta \alpha }\left(\mathbf{G}\right)/t_{\perp }$is almost constant; but for $(k,l)=(1,1)$, which corresponds to $\mathbf{G}=\Delta \mathbf{K}$, the amplitude vanishes linearly with $\theta$.

Figure 6

Velocity renormalization, by perturbation theory in ${\tilde{t}}_{\perp }/\left(\hslash v_F\Delta K\right)$, and by the continuum model with numerical diagonalization of the secular equation, with a sufficient set of momentum values for convergence. The latter calculation deviates from the perturbative results as $\left(\hslash v_F\Delta K\right)$ becomes comparable to ${\tilde{t}}_{\perp }$, but is in quite good agreement with band-structure calculations (Refs. 28, 29 and 33).

Figure 7

Contour plot of the first positive energy band; the hexagon is the first Brillouin zone. (a) $\theta =1.{79}^{\circ }$, the cones are visible, but the saddle point is not located on the line joining the two Dirac cones; (b) $\theta =1.{20}^{\circ },$ the arc joining the cones through the saddle point has become a very flat valley, and the cones are no longer well defined.

Figure 8

Densities of states (DOS) for two angles: (a) for $\theta =1.{79}^{\circ }$, the cones are still well defined and the dispersion shows the usual linear dependence near zero energy; (b) for $\theta =1.2^{\circ }$, there appears to be a finite density of states near zero energy, making if difficult to define a Fermi velocity: the dispersion is no longer linear. The red line is the DOS for two uncoupled layers.

Figure 9

Densities of states (DOS) for two small angles: (a) for $\theta =1.{08}^{\circ }$, there is a very sharp central peak, with a barely resolved two-peak structure, corresponding to a flat band of states localized in $AA$ stacking regions of the unit cell; (b) for $\theta =0.{87}^{\circ }$, the central band is broader, and still displays the two-peak structure, although, at the meV resolution, there is a finite DOS between the peaks, precluding the unambiguous definition of a Fermi velocity.

Figure 10

Density plots of the local density of states, in the Wigner-Seitz cell, integrated for $\left|\epsilon \right|<20\mathrm{meV}$: (a) $\theta =1.{78}^{\circ },$ (b) $\theta =1.{08}^{\circ }$.

Figure 11

Band structures of pure $AA$ and pure $AB$ stacking bilayers.