Modeling of electronic dynamics in twisted bilayer graphene

Abstract

We consider the problem of numerically computing the quantum dynamics of an electron in twisted bilayer graphene. The challenge is that atomic-scale models of the dynamics are aperiodic for generic twist angles because of the incommensurability of the layers. The Bistritzer-MacDonald PDE model, which is periodic with respect to the bilayer's moiré pattern, has recently been shown to rigorously describe these dynamics in a parameter regime. In this work, we first prove that the dynamics of the tight-binding model of incommensurate twisted bilayer graphene can be approximated by computations on finite domains. The main ingredient of this proof is a speed of propagation estimate proved using Combes-Thomas estimates. We then provide extensive numerical computations which clarify the range of validity of the Bistritzer-MacDonald model.

Figures (8)

• Figure 1: Left: The atomic structures of twisted bilayer graphene at twist angle $\theta=4^\circ$. The moiré lattice vectors are $\boldsymbol{a}_{m,1}$ and $\boldsymbol{a}_{m,2}$. Right: The modulus of one entry of the moiré interlayer potential $\left[T(\boldsymbol{r})\right]_{11}$. It is periodic over the moiré lattice vectors.
• Figure 1: The relative error for the truncated tight-binding model in TBG. Each data point is the average of several simulations with different initial conditions while keeping their norm constant. As it is not possible to compute the infinite system $\psi(t)$ directly, we compare the solutions $P_R^*\Psi(t)$ to $P_{R'}^*\Psi(t)$, where the reference value is $R' = 86.60 \text{ \AA}$. When the truncation radius $R$ increases, $P_R^*\Psi(t)$ converges exponentially.
• Figure 1: The Bravais lattice (black) and a Wigner-Seitz unit cell $\Gamma_{\boldsymbol{R}}$ of $\boldsymbol{R}$ (red). After a shift of the lattice we can always ensure $\boldsymbol{\tau}$ is in a unit cell that contains the origin (blue).
• Figure 2: The BM band structure of TBG at twist angle $\theta = 1.05 ^\circ$. $1.05^\circ$ is called a magic angle for TBG because there are a pair of almost flat moiré bands in the TBG band structure. Left: The flat bands (red) as well as two bands above the flat bands (blue and green) around a moiré $\boldsymbol{K}_m$ point. Right: The contour plot of the third band with two points $\boldsymbol{k}_1 = (0 , -0.02)^\top$ and $\boldsymbol{k}_2 = (0.01, -0.0275 )^\top$ in $k$-space, and an illustration of the gradient of energy $\boldsymbol{\nabla} E(\boldsymbol{k}_i)$ at these points.
• Figure 3: Left: The modulus of the wave-function for the BM model, the tight-binding model and the corrector for a wave-packet initial condition concentrated at $k_1$. Only one layer is presented, as the two layers have similar behaviour. The arrow represents the direction of $\boldsymbol{\nabla} E(\boldsymbol{k}_i)$. Right: The same figure for a wave-packet concentrated at $k_2$. Recovering physical units, the axes have units $\AA$, and $t=T$ represents time at $T \cdot\hbar \cdot\text{eV}^{-1} \approx T \times 6.6 \times 10^{-16} \text{s}$.

Theorems & Definitions (17)

• Lemma 2.2
• Proof 1
• Example 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.7: Exponential Decay of Resolvent
• Proof 2
• Proposition 2.8
• Proof 3
• Theorem 2.9: Truncation Estimate