Low-energy theory for the graphene twist bilayer

The graphene twist bilayer represents the prototypical system for investigating the stacking degree of freedom in few-layer graphenes. The electronic structure of this system changes qualitatively as a function of angle, from a large-angle limit in which the two layers are essentially decoupled—with the exception of a 28-atom commensuration unit cell for which the layers are coupled on an energy scale of $\approx 8\mathrm{meV}$—to a small-angle strong-coupling limit. Despite sustained investigation, a fully satisfactory theory of the twist bilayer remains elusive. The outstanding problems are (i) to find a theoretically unified description of the large- and small-angle limits, and (ii) to demonstrate agreement between the low-energy effective Hamiltonian and, for instance, ab initio or tight-binding calculations. In this article, we develop a low-energy theory that in the large-angle limit reproduces the symmetry-derived Hamiltonians of Mele [Phys. Rev. B 81, 161405 (2010)], and in the small-angle limit shows almost perfect agreement with tight-binding calculations. The small-angle effective Hamiltonian is that of Bistritzer and MacDonald [Proc. Natl. Acad. Sci. (U.S.A.) 108, 12233 (2011)], but with the momentum scale $\mathrm{\Delta }K$, the difference of the momenta of the unrotated and rotated special points, replaced by a coupling momentum scale $g^{\left(c\right)}=\frac{8\pi }{\sqrt{3}a}sin\frac{\theta }{2}$. Using this small-angle Hamiltonian, we are able to determine the complete behavior as a function of angle, finding a complex small-angle clustering of van Hove singularities in the density of states (DOS) that after a “zero-mode” peak regime between $0.{90}^{\circ }<\theta <0.{15}^{\circ }$ limits $\theta <0.{05}^{\circ }$ to a DOS that is essentially that of a superposition DOS of all bilayer stacking possibilities. In this regime, the Dirac spectrum is entirely destroyed by hybridization for $-0.25<E<0.25\text{eV}$ with an average band velocity $\approx 0.3v_F^{\left(\text{SLG}\right)}$ (where SLG denotes single-layer graphene). We study the fermiology of the twist bilayer in this limit, finding remarkably structured constant energy surfaces with multiple Lifshitz transitions between $K$- and $\mathrm{\Gamma }$-centered Fermi sheets and a rich pseudospin texture.

Figure 1

Backfolding of single-layer graphene states to the bilayer Brillouin zone, where the $\mathbf{k}$ vector in the bilayer Brillouin zone to which basis states fold back has been set to ${\mathbf{K}}_B$. Also shown are the high-energy states that couple to $\mathbf{K}$ by the coupling momenta ${\mathbf{g}}_i^{\left(c\right)}$. For two of these (${\mathbf{g}}_1^{\left(c\right)}$ and ${\mathbf{g}}_3^{\left(c\right)}$), these high-energy states lie outside the second-layer (rotated) Brillouin zone, and umklapp vectors are required to backfold them. The three high-energy $\mathbf{k}$ vectors involved in the scattering paths used in constructing the large-angle continuum Hamiltonians are labeled as “$H$.”

Figure 2

Band structure of the two smallest commensurate twist bilayer structures: $\theta =38.{21}^{\circ }$ (top panel) and $\theta =21.{79}^{\circ }$ (bottom panel). For each of these twist angles, the supercell consists of 28 carbon atoms. The full dark (black) lines display the full two-center tight-binding approximation, and the light shaded (red) broken line displays that of the low-energy Hamiltonians Eq. (20) (top panel) and Eq. (18) (bottom panel).

Figure 3

Density of states (DOS) in the small-angle limit. Shown are calculations using the full two-center tight-binding scheme [dark shaded (red) full lines], along with the corresponding results generated using the low-energy Hamiltonian given by Eq. (26) [light shaded (green) full lines]. Shown are the DOS for twist angles from $0.{90}^{\circ }$ to $0.{03}^{\circ }$; note that below $0.{10}^{\circ }$ computational resources allowed only for calculation using the low-energy theory. For all cases shown, the DOS of both methods are seen to be in good agreement, particularly near the Dirac point at $E=0\mathrm{eV}$. The light (cyan) shaded regions indicate the energy region of the central peak from which the electron density in the first column of Fig. 5 is constructed. Similarly, the dark shaded (red) regions illustrate the states summed over in the negative energy neighbor peak, from which the electron density displayed in the second column of Fig. 5 is constructed. The black dashed line in the $\theta =0.{03}^{\circ }$ panel is a superposition DOS constructed from the DOS of all possible interlayer stacking vectors, showing that at this angle the system has effectively electronically segmented into separate stacking regions.

Figure 4

Band structure of the twist bilayer plotted on a standard $M-K-\mathrm{\Gamma }$ path through the Brillouin zone. Open circles are tight-binding results, and full lines show the result provided by the low-energy effective Hamiltonian, Eq. (26). The pronounced set of flat “moiré bands” seen near the Dirac point correspond to the peak in the density of states of the bilayer spectrum shown in Fig. 3.

Figure 5

Electron density formed by integrating ${\left|\mathrm{\Psi }\left(\mathbf{r}\right)\right|}^2$ over all states in the Dirac point peak indicated by the light shaded region in Fig. 3 (first column), the dark shaded region of the negative energy satellite peak (second column), and over all states in an energy window $-0.5<E<-0.41\text{eV}$ for $0.{90}^{\circ }$ and $0.{40}^{\circ }$ and $-0.40<E<-0.35\text{eV}$ for$0.{10}^{\circ }$ (third column). Shown is the electron density at rotation angles of $0.{90}^{\circ }$ (a), $0.{40}^{\circ }$ (b), and $0.{10}^{\circ }$ (c) obtained both from tight-binding calculations and the low-energy Hamiltonian of Eq. (26) as indicated.

Figure 6

Average band velocity as a function of twist angle of the bilayer and energy [panel (a)]. Valleys of reduced band velocity that arise from the interlayer coupling of the two Dirac cones may be seen, with such valleys directed toward the small-angle low-energy regime in which the average band velocity exhibits a complex $(E,\theta )$ dependence shown in panel (b). Panel (c) displays the average band velocity as a function of energy for two constant angles, and panel (d) shows the band velocity at the Dirac point (not averaged). In this panel, calculations using the low-energy Hamiltonian of this work [Eq. (26)] are shown as the solid black line, the low-energy Hamiltonian of Bistritzer et al. [8] is shown as the light dashed line, and full tight-binding calculations are shown both from this work (blue diamonds) and from Ref. [26] (black squares). The “magic angle” structure seen in panel (d) is to some extent washed out by convolving with a Gaussian, thus it is more difficult to detect in panel (b).

Figure 7

Fermi surfaces at twist angles of $\theta ={\text{8.0}}^{\circ }, {\text{3.0}}^{\circ }$, and ${\text{0.9}}^{\circ }$ for a range of Fermi energies between 10 and $200\mathrm{meV}$ at both $T=0$ (left panel) and $T=300\mathrm{K}$ (right panel). The color indicates the band velocity of the bands (presented as the ratio $v/v_F^{\left(\text{SLG}\right)}$) intersecting the constant energy surface. For the large twist angle of ${\text{8.0}}^{\circ }$, the fermiology is recognizably that of a Dirac cone; at smaller angles, the low-energy fermiology differs dramatically from the cone topology.

Figure 8

Pseudospin texture of the twisted bilayer in the large- ($\theta ={30}^{\circ }$), intermediate- ($\theta =3^{\circ }$), and small- ($\theta =0.9^{\circ }$) angle regimes. The Fermi energies of the Fermi surfaces are $0.9\mathrm{eV}$ for the large-angle case and $0.13\mathrm{eV}$ for the intermediate- and small-angle cases. The large-angle structure resembles the double vortex of a decoupled graphene bilayer, whereas in the intermediate-angle case a trigonally warped vortex is found. In the small-angle case, the pseudospin texture is seen to “flow” along the sixfold “arms” of the starlike Fermi surface. Note that the definitions of pseudospin up and pseudospin down differ between the two layers due to the rotation of the basis atoms of the second layer of the twist bilayer.