Twisted bilayer graphene revisited: Minimal two-band model for low-energy bands

An accurate description of the low-energy electronic bands in twisted bilayer graphene (tBLG) is of great interest due to their relation to correlated electron phases such as superconductivity and Mott-insulator behavior at half-filling. The paradigmatic model of Bistritzer and MacDonald [Proc. Natl. Acad. Sci. USA 108, 12233 (2011)], based on the moiré pattern formed by tBLG, predicts the existence of “magic angles” at which the Fermi velocity of the low-energy bands goes to zero, and the bands themselves become dispersionless. Here, we reexamine the low-energy bands of tBLG from the ab initio electronic structure perspective, motivated by features related to the atomic relaxation in the moiré pattern, namely, circular regions of AA stacking, triangular regions of AB/BA stacking and domain walls separating the latter. We find that the bands are never perfectly flat and the Fermi velocity never vanishes, but rather a “magic range” exists where the lower band becomes extremely flat and the Fermi velocity attains a nonzero minimum value. We propose a simple $(2+2)$-band model, comprised of two different pairs of orbitals, both on a honeycomb lattice: the first pair represents the low-energy bands with high localization at the AA sites, while the second pair represents highly dispersive bands associated with domain-wall states. This model gives an accurate description of the low-energy bands with few (13) parameters that are physically motivated and vary smoothly in the magic range. In addition, we derive an effective two-band Hamiltonian which also gives an accurate description of the low-energy bands. This minimal two-band model affords a connection to a Hubbard-like description of the occupancy of subbands and can be used a basis for exploring correlated states.

Figure 1

(a) Sketch of the local twisting and untwisting which occurs around the AA and AB/BA domain centers, respectively. The orange hexagons correspond to the AA regions, the green triangles to the AB/BA regions and the purple ellipses correspond to the DWs. $a_{\mathrm{sc}}$ is the moiré period. (b) The monolayer BZs, in red and blue, which are rotated with respect to one another by $\theta$. The high-symmetry points $\mathrm{\Gamma }$, K/K', and M/M' are shown. The moiré BZ is the hexagon whose edges are formed by joining the K and K' points of the monolayers. The magnified diagram shows the $\mathbf{k}$-point paths considered and their ${\mathcal{C}}_3$ rotation symmetries and mirror planes $\mathcal{M}$.

Figure 2

(Top) Low-energy bands of tBLG along the K-$\mathrm{\Gamma }$-M-K (solid) and K'-$\mathrm{\Gamma }$-M'-K' (dashed) paths, for a selection of twist angles near the magic angle. The special angles ${\theta }_1^{*}$ and ${\theta }_2^{*}$, which define the magic range, as well as the angle ${\theta }_0^{*}$, at which the bands are degenerate at $\mathrm{\Gamma }$, are shown. For the smallest and largest twist angles, the plots are shaded up to the half-filling lines of the lower bands. (Bottom) Fermi surfaces of the lower bands directly above, at half-filling.

Figure 3

Magic range from $\mathbf{k}·\mathbf{p}$ bands. (a) Energy eigenvalues of the upper (red) and lower (blue) low-energy bands at the $\mathrm{\Gamma }$ and M points as a function of twist angle. The twist angles ${\theta }_1^{*}$ and ${\theta }_2^{*}$ at which the $\mathrm{\Gamma }$ and M eigenvalues of the lower band are equal are marked, and the region between them is shaded, indicating the magic range. The twist angles ${\theta }_0^{*}$ and ${\theta }_{\mathrm{m}}^{*}$, at which the eigenvalues of both bands at $\mathrm{\Gamma }$ and M are degenerate, respectively, are also shown. (b) Fermi velocity $v_{\mathrm{K}}\left({\theta }^{*}\right)$ of the upper (red) and lower (blue) low-energy bands as a function of twist angle, as a percentage of the Fermi velocity of graphene, $v_{\mathrm{K}}^0={10}^6$ m/s [58]. The absolute value of the Fermi velocity of both bands reaches a minimum at ${\theta }_{\mathrm{m}}^{*}$.

Figure 4

(a) Illustration of the $s$ (red) and $p_x,p_y$ (blue) orbitals and (b) their linear combinations into the three $s{p}^2$ hybrid orbitals, labeled ${\varphi }_1,{\varphi }_2,{\varphi }_3$. (c) The ${\varphi }_0, {\varphi }_{+}, {\varphi }_{-}$ effective orbitals are shown on the right. For each orbital, the lobes of positive sign are shown in magenta, and the lobes of negative sign are shown as gray. (d) Wave-function magnitudes of the AA, DW, and AB/BA states, projected onto the A and B sublattices from the ten-band model in Ref. [20], for $\theta =0.9^{\circ }$.

Figure 5

(a) Triangular lattice with the lattice vectors ${\mathbf{a}}_1, {\mathbf{a}}_2$, and ${\mathbf{a}}_3$ (black arrows), and the vectors ${\mathbf{b}}_1, {\mathbf{b}}_2$, and ${\mathbf{b}}_3$ (red arrows). A total of four unit cells of the periodic lattice are shown. The central lattice site is decorated with one ${\varphi }_0$ orbital consisting of the three positive lobes of ${\varphi }_1,{\varphi }_2,{\varphi }_3$ for the A sublattice (orange), and a second orbital with the lobes rotated by $\pi$ for the B sublattice (light blue). The purple ellipses represent the DW states. (b) Re-arrangement of the orbitals so that the lobes ${\varphi }_1,{\varphi }_2,{\varphi }_3$ are located at different lattice sites. The centers of the orbitals attributed to the A and B sublattices are shown, which form a honeycomb lattice, as indicated by the dashed black lines. (c) Nearest neighbor hoppings included in the $(2+2)$-band Hamiltonian, Eq. (6). The red arrows indicate the first nearest neighbor hoppings, between opposite sublattices, connected by vectors ${\mathbf{b}}_i$. The green arrows indicate the second nearest neighbor hoppings, on the same sublattice, connected by vectors ${\mathbf{a}}_i$. The blue arrows indicate the third nearest neighbor hoppings, between opposite sublattices, connected by vectors $-2{\mathbf{b}}_i$.

Figure 6

Values of the hopping parameters $t_i^{*}, t_i^{▵}$, and $t_i^{\mathrm{int}}, i=0,1,2,\text{and}3$, that appear in ${\mathcal{H}}^{*}, {\mathcal{H}}^{▵}$, and ${\mathcal{H}}^{\mathrm{int}}$, respectively, as determined by fitting to the $\mathbf{k}·\mathbf{p}$ low-energy bands, for the range of twist angles $0.{95}^{\circ }\le {\theta }^{*}\le 1.2^{\circ }$. The shaded regions indicate the magic range.

Figure 7

Low-energy bands obtained by solving Eq. (10), setting $E=0$ in Eq. (11), using the fits to the $(2+2)$-band Hamiltonian in Fig. 6. The $\mathbf{k}·\mathbf{p}$ bands are shown in gray. The solid and dashed lines show the bands along the K-$\mathrm{\Gamma }$-M-K and K'-$\mathrm{\Gamma }$-M'-K' paths, respectively.