Moiré semiconductors on the twisted bilayer dice lattice

We propose an effective lattice model for the moiré structure of the twisted bilayer dice lattice. In the chiral limit, we find that there are flat bands at the zero-energy level at any twist angle besides the magic ones and these flat bands are broadened by small perturbation away from the chiral limit. The flat bands contain both bands with zero Chern number which originate from the destructive interference of the states on the dice lattice and the topological nontrivial bands at the magic angle. The existence of the flat bands can be detected from the peak-splitting structure of the optical conductance at all angles, while the transition peaks do not split and only occur at magic angles in twisted bilayer graphene.

Figure 1

(a) Structure of the dice lattice. Blue, red, and black dots correspond to $A, B$, and $C$ sites, respectively. The lattice vectors are denoted by ${\mathit{a}}_2$ and ${\mathit{a}}_2$. (b) Bilayer dice lattice in $A-B$ (Bernal) stacking. The vector $\tau$ labels the position of the sublattice site in a unit cell.

Figure 2

(a) Moiré period structure of the TBD with unit lattice vectors ${\mathit{a}}_1^m=L_s(\frac{\sqrt{3}}{2},-\frac{1}{2})$ and ${\mathit{a}}_2^m=L_s(\frac{\sqrt{3}}{2},\frac{1}{2})$, where $L_s=\frac{a}{2sin(\theta /2)}$ is the size of the moiré unit cell. (b) Reciprocal space of the TBD with reciprocal vectors ${\mathit{b}}_1^m=\frac{4\pi }{\sqrt{3}{L}_s}(\frac{1}{2},-\frac{\sqrt{3}}{2})$ and ${\mathit{b}}_2^m=\frac{4\pi }{\sqrt{3}{L}_s}(\frac{1}{2},\frac{\sqrt{3}}{2})$. The coordinate $(l_1,l_2)$ means the vector $\mathbf{b}=l_1{\mathit{b}}_1^m+l_2{\mathit{b}}_2^m$. The red dashed line marks the first Brillouin zone with green dashed lines marking the reciprocal path for calculating the TBD band structure.

Figure 3

Band structure of the $A-B$ stacking untwisted bilayer dice lattice model. Here we set the reciprocal path the same as for graphene [55]; the lattice vector and coordinates of atoms in the unit cell are the same as in Fig. 1. The other parameters are $W_{A{B}^{\prime }}=W_{B{C}^{\prime }}=0.33\text{eV}$ and (a) $W_{A{C}^{\prime }}=0.33\text{eV}$ (the chiral symmetry is broken) and (b) $W_{A{C}^{\prime }}=0$ (the chiral symmetry is preserved).

Figure 4

Here $W_{A{B}^{\prime }}=W=0.11\text{eV}$. The band structure of TBG in the chiral limit is shown at twist angles (a) $\theta =0.5^{\circ }$, (c) $\theta =1.{08}^{\circ }$, and (e) $\theta =5^{\circ }$. The flat band only occurs at the magic angle $\theta =1.{08}^{\circ }$. The band structure of the TBD in the chiral limit is shown at twist angles (b) $\theta =0.5^{\circ }$, (d) $\theta =1.{08}^{\circ }$, and (f) $\theta =5^{\circ }$. Totally flat bands with a high level degeneracy appear at zero energy at all angles. The band structure of the TBD is shown (g) at magic angle $\theta =1.{08}^{\circ }$, with $W_{A{A}^{\prime }}=0\text{eV}$ and $W_{A{C}^{\prime }}=0.1\text{eV}$, and (h) at $\theta =5^{\circ }$, with $W_{A{A}^{\prime }}/W_{A{B}^{\prime }}=0.5\text{eV}$. In (g) and (h) the chiral symmetry is broken and the degeneracy is lifted. The momentum labels $1\to 2\to 3\to 4\to 1$ are defined in Fig. 2.

Figure 5

Landau-level structure of (a) TBG and (b) the TBD, with $u_0^{\prime }=0.01\text{eV}, u_0=0.1$, and $\mu =0.05\text{eV}$ (red dotted line), and optical conductance ${\sigma }_{xx}$ of (c) TBG and (d) the TBD. We choose the room temperature $T=300\text{K}$. The symbol $\pm$ in (c) represents the two flat bands near zero energy. For energy bands bigger (or smaller) than the zero energy with a typical Landau-level gap, we denote by $L=0^{\pm },\pm 1,\pm 2,...$. For example, $T_{\pm ,0^{+}}$ represents the contribution from two zero-energy bands to the first Landau level with positive energy denoted by $L=0^{+}$. For the flat bands near zero energy in (d), we use their energy value to denote, for example, $T_{-0.012,0^{+}}$, which represents the contribution from the flat band with energy $E=-0.012\text{eV}$ to $L=0^{+}$.