Lattice relaxation and energy band modulation in twisted bilayer graphene

We theoretically study the lattice relaxation in the twisted bilayer graphene (TBG) and its effect on the electronic band structure. We develop an effective continuum theory to describe the lattice relaxation in general TBGs and obtain the optimized structure to minimize the total energy. At small rotation angles $<2^{\circ }$, in particular, we find that the relaxed lattice drastically reduces the area of the AA stacking region and forms a triangular domain structure with alternating AB and BA stacking regions. We then investigate the effect of the domain formation on the electronic band structure. The most notable change from the nonrelaxed model is that an energy gap of up to 20 meV opens at the superlattice subband edges on the electron and hole sides. We also find that the lattice relaxation significantly enhances the Fermi velocity, which was strongly suppressed in the nonrelaxed model.

Figure 1

Twisted bilayer graphene at rotation angle $\theta =2.{65}^{\circ }$. The blue squares indicate the regions where the lattice structure locally resembles a regular stacking arrangement such as AA, AB, BA, and SP (see the text). Parallelogram is the moiré unit cell.

Figure 2

Brillouin zones of layer 1 (black hexagon), layer 2 (red dashed hexagon), and twisted bilayer graphene (blue hexagons) at $\theta =2.{65}^{\circ }$.

Figure 3

(a) One-dimensional moiré superlattice model. (b) Schematic picture of the relaxed structure, where the atoms are locked to the vertically aligned positions, leaving a domain boundary in the middle. (c) Interlayer binding energy per length as a function of the relative translation $\delta$ for commensurate double chains.

Figure 4

(a) Interlayer atomic shift $\delta \left(x\right)$ and (b) local binding energy $V\left[\delta \right(x\left)\right]$ plotted against the position $x$ in one-dimensional moiré superlattice with $\eta =$ 0, 0.3, 1, and 3.

Figure 5

Interlayer binding energy $V\left[\mathbf{\delta }\right]$ of TBG as a function of local atomic shift $\mathbf{\delta }\left(\mathbf{r}\right)$.

Figure 6

Dimensionless parameter $\eta$ as a function of rotation angle $\theta$.

Figure 7

Outermost panel: Distribution of the displacement vector ${\mathbf{u}}^{-}\left(\mathbf{r}\right)$ in the TBG of $\theta =1.{05}^{\circ }$. Right side panels: Local atomic structure near AA (AB or SP) stacked point before and after the relaxation. The small dashed circles in the center panel indicate the areas where the local structure is sampled.

Figure 8

Two-dimensional maps for (a) absolute value of the displacement vector ${\mathbf{u}}^{-}\left(\mathbf{r}\right)$, (b) the local binding energy $V\left[\mathbf{\delta }\right(\mathbf{r}\left)\right]$, and (c) strain-induced pseudomagnetic field $B_{\mathrm{eff}}\left(\mathbf{r}\right)$, calculated for TBGs with various rotation angles.

Figure 9

Band structure and density of state of relaxed (black solid lines) and nonrelaxed (red dashed lines) TBGs at various rotation angles. The energy gap is indicated by a pair of arrows in the DOS panel.

Figure 10

(a) Band gap between the lowest band and the first excited bands in the electron side (red solid line) and in the hole side (blue dashed), in relaxed TBGs against the rotation angle. (b) Band velocity at $K$ point as a function of $\theta$ for relaxed (black solid) and nonrelaxed (red dashed) TBGs.

Figure 11

Schematic of all q points within radius $4G_M$ in reciprocal space. The independent q points when $\theta >1^{\circ }$($\theta <1^{\circ }$) are marked by blue (green) triangles.