Moiré disorder effect in twisted bilayer graphene

We theoretically study the electronic structure of magic-angle twisted bilayer graphene with disordered moiré patterns. By using an extended continuum model incorporating nonuniform lattice distortion, we find that the local density of states of the flat band is hardly broadened, but splits into upper and lower subbands in most places. The spatial dependence of the splitting energy is almost exclusively determined by the local value of the effective vector potential induced by heterostrain, whereas the variation of local twist angle and local moiré period give relatively minor effects on the electronic structure. We explain the exclusive dependence on the local vector potential by a pseudo Landau level picture for the magic-angle flat band, and we obtain an analytic expression of the splitting energy as a function of the strain amplitude.

Figure 1

Moiré patterns of magic-angle TBG $(\theta =1.{05}^{\circ })$ with random nonuniform distortion of $\epsilon =0$, 0.0006, 0.0012, and 0.0018, where the characteristic wave length is $\lambda =7L_M$, and the supercell size (big parallelogram) is $n_{\mathrm{SM}}=8$. The bright region represents local AA stack and the dark region represents AB/BA stack. The red dots are the AA spots of the nondistorted TBG for reference.

Figure 2

Brillouin zones of (a) a nondistorted TBG and (b) a distorted TBG. Blue and orange hexagons on the left represent the first Brillouin zone of graphene layer 1 and 2 (twisted by $\mp \theta /2$), respectively, and red arrows are the displacement vectors from the layer 2's $K_{+}$ point to layer 1's. A green hexagon on the right side is the moiré Brillouin zone.

Figure 3

Band structure and the DOS of uniformly distorted magic-angle TBGs with different types of strain components, ${\epsilon }_{+},{\epsilon }_{-},{\epsilon }_{xy},\mathrm{\Omega }$. Different colors represent different amplitudes of strain. Horizontal lines in the right panels (DOS) indicate energies of the split levels in the pseudo Landau level picture.

Figure 4

(a) Moiré pattern of a disordered magic-angle TBG with $\epsilon =0.0004,\lambda =7L_{\mathrm{M}}$. The distortion is observed as slight shifts of AA points (yellow spots) relative to the regular red dots. (b) LDOS along line $X{X}^{\prime }$ [defined by a broken line in (a)]. (c) (Black, solid) LDOS at the points of $p_1,p_2,p_3$ in (a). (Red, dashed) LDOS at the AA point of the corresponding uniform TBG with the strain tensors fixed to the local value. (d) The spatial distribution of the splitting energy $\mathrm{\Delta }E$, or the energy distance between the two LDOS peaks. A hexagonal tile corresponds to a single moiré unit cell, and its color represents $\mathrm{\Delta }E$ at the center of the hexagon (the AA point). (e) A contour plot of the interlayer difference of the strain-induced vector potential, $ev\left|\mathit{A}\right(\mathit{r}\left)\right|$. (f) A scattered plot of $\mathrm{\Delta }E$ and $ev\left|\mathit{A}\right|$ (averaged in every moiré unit cell).

Figure 5

Plots similar to Figs. 4–4 calculated for different characteristic wave lengths $\lambda =5L_M,3L_M,L_M$.

Figure 6

The total DOS of disordered magic-angle TBGs with different distortion amplitudes $\epsilon$. For each curve, we take an average over different random configurations. Broken lines are the distribution function $D\left(\right|\mathit{A}\left|\right)$ with horizontal axis scaled by $E=0.7ev\left|\mathit{A}\right|$.