Shear instability in twisted bilayer graphene

In twisted bilayer graphene (TBLG), extremely small deviations from the magic twist angle ${\theta }_m\approx 1.{08}^{\circ }$ change its electronic structure near the Fermi level drastically, causing a meV-wide flat band to appear or disappear. In view of such sensitivity to minute structural deformations, we investigate the combined effect of shear and atomic relaxation on the electronic structure. Using precise experimental data for monolayer and bilayer graphene as input in a simplified formalism for the electronic structure and elastic energy, we find TBLG near ${\theta }_m$ to be unstable with respect to global shear by the angle $\alpha \approx 0.{08}^{\circ }$. In TBLG, the effect of shear on the electronic structure is as important as that of atomic relaxation. Under optimum global shear, calculated ${\theta }_m$ is reduced by $0.{04}^{\circ }$ and agrees with the observed value.

Figure 1

Shear and twist in BLG. (a) Definition of shear and twist operations in BLG, initially a bilayer in AA stacking indicated by the honeycomb lattice. A rectangular segment of the top layer, with one side closing the angle $\beta$ with respect to the highlighted armchair direction, is indicated in yellow and surrounded by the dotted line. The segment is first sheared by the angle $\alpha$ and subsequently rotated by the angle $\theta$. (b) Schematic top view of a uniformly twisted and sheared monolayer graphene (MLG) on top of an undeformed MLG. The resulting relaxed moiré pattern contains regions of AA stacking, highlighted by the white circles, and those of AB or BA stacking. (c) Dark-field TEM image of bilayer graphene reproduced from Ref. [19]. (d) Schematic side view of a relaxed sheared and twisted BLG with locally varying stacking.

Figure 2

(a) Energy difference $\mathrm{\Delta }E$ caused by shearing the top layer of BLG, which had been twisted by ${\theta }_m=1.{08}^{\circ }$, by varying the shear angle $\alpha$ along the $\beta =0^{\circ }$ direction. Results for the BLG with unrelaxed atomic positions in planar, sheared monolayers are shown by the red dotted line and for the BLG with relaxed atomic positions by the black solid line. Contour plots of the local shift vector length $\delta =\left|\mathbf{\delta }\right|$ in the BLG structure, which had been twisted by ${\theta }_m=1.{08}^{\circ }$ and sheared by $\alpha =0.{08}^{\circ }$ along the $\beta =0^{\circ }$ direction, (b) in the absence and (c) in the presence of lattice relaxation.

Figure 3

Electronic density of states (DOS) in the BLG structure subject to the magic twist angle ${\theta }_m=1.{08}^{\circ }$. Results for the unsheared structure are shown in (a) and (c), and those for the top layer sheared by $\alpha =0.{08}^{\circ }$ along the $\beta =0^{\circ }$ direction are shown in (b) and (d). Results for the unrelaxed structure in (a) and (b) are compared to those for the relaxed structure in (c) and (d).

Figure 4

Contour plots of the local shift vector length $\delta =\left|\mathbf{\delta }\right|$ in the BLG structure, which had been twisted by $\theta =0.4^{\circ }$ and sheared by $\alpha =0.{08}^{\circ }$ along the $\beta =0^{\circ }$ direction, (a) in the absence and (b) in the presence of lattice relaxation.

Figure 5

Electronic density of states (DOS) in the BLG structure subject to the magic angle twist ${\theta }_m=1.{08}^{\circ }$. Results for the unsheared structure are shown in (a) and (c), and those for the top layer sheared by $\alpha =0.{08}^{\circ }$ along the $\beta =0^{\circ }$ direction are shown in (b) and (d). Results for the unrelaxed structure in (a) and (b) are compared to those for the relaxed structure in (c) and (d). The energy scale is extended in comparison to Fig. 3.

Figure 6

The width of the flat band in a relaxed twisted bilayer graphene structure as a function of the twist angle $\theta$. Data points for commensurate, relaxed structures in the vicinity of the observed value ${\theta }_m=1.{08}^{\circ }$ are presented by the black diamonds for the unsheared lattice and by the red circles for the structure with the top layer sheared by $\alpha \approx 0.{08}^{\circ }$ along the $\beta =0^{\circ }$ direction. The dotted and dashed lines are guides to the eye.