Time-evolution patterns of electrons in twisted bilayer graphene

We characterize the dynamics of electrons in twisted bilayer graphene by analyzing the time evolution of electron waves in an atomic lattice. We perform simulations based on a kernel polynomial technique using Chebyshev polynomials; this method does not requires any diagonalization of the system Hamiltonian. Our simulations reveal that interlayer electronic coupling induces an exchange of waves between the two graphene layers. This wave transfer manifests as oscillations of the layer-integrated probability densities as a function of time. For the bilayer case, it also causes a difference in the wavefront dynamics compared to monolayer graphene. The intralayer spreading of electron waves is irregular and progresses as a two-stage process. The first one, characterized by a well-defined wavefront, occurs in a short time—a wavefront forms instead during the second stage. The wavefront takes a hexagonlike shape with the vertices developing faster than the edges. Though the detail spreading form of waves depends on initial states, we observe localization of waves in specific regions of the moiré zone. To characterize the electron dynamics, we also analyze the time autocorrelation functions. We show that these quantities exhibit beating modulation when reducing interlayer coupling.

Figure 1

Spreading of the distribution of electron probability densities (in red) in the monolayer graphene taken at several time moments. The arrows denote the vectors of probability current density (in green). The blue lines denote the three mirror-symmetry planes ${\sigma }_v$ of the hexagonal lattice. We highlight the direction of the armchair and zigzag lines in red in the frame with $t=0.8\mathrm{fs}$.

Figure 2

The real and imaginary part of the autocorrelation function $C_i\left(t\right)=\langle i|\psi \left(t\right)\rangle$ calculated from three hopping models: the NN in red, the NNN in green, and the NNNN in blue. The red rectangle box identifies the typical oscillation pattern predicted by the NN model. The inset shows the continuous variation of the real part of $C_i\left(t\right)$ in a longer evolution time.

Figure 3

Neutrino-like oscillation of the layer-integrated probability density ${\mathcal{P}}_{\text{T/B}}\left(t\right)$ for the (a) AA- and (b) AB-stacking bilayer graphene. Data plotted for two values of the Slater-Koster parameter $V_{pp\sigma }^0=0.48\mathrm{eV}$ (the solid curves) and 0.12 eV (the dot-dashed curves). Only data in the interval of $(0,30)\mathrm{fs}$ are displayed to zoom in the oscillations. The vertical lines highlight the times ${\tau }_{\text{T}\to \text{B}}$ and $t=1.3\mathrm{fs}$ discussed in the text.

Figure 4

Spreading of the electron probability density for the AA-stacking configuration taken at several time moments. Lattice nodes in red (black) belong to the top (bottom) graphene layer. The blue lines in the frames at $t=1.3$ and $3.2\mathrm{fs}$ denote the three symmetrical mirrors ${\sigma }_v$ of the lattice.

Figure 5

Spreading of the electron probability density for the AB-stacking configuration taken at several time moments. The initial state $|i\rangle$ is set at the position of an A-atom of the top graphene layer, which is on top of a B-atom of the bottom layer. Lattice nodes in red (black) belong to the top (bottom) graphene layer. The blue lines in the frames at $t=1$ and $3.1\mathrm{fs}$ denote the three symmetrical mirrors ${\sigma }_v$ of the lattice.

Figure 6

Time autocorrelation functions $C_i\left(t\right)$ of the AA- (red) and AB-stacking (blue) bilayer graphene calculated using the NN model with $V_{pp\sigma }^0=0.48$ and 0.12 eV.

Figure 7

Layer-integrated probability densities ${\mathcal{P}}_{\text{T/B}}\left(t\right)$ in the top (solid lines) and bottom (dot-dashed lines) layers of three TBG configurations with the twist angles of $5^{\circ }$ (green), $2.5^{\circ }$ (blue), and $1^{\circ }$ (red). Panels (a) and (b) are for the cases in which the initial state $|2p_z\rangle$ is located at the central point of the ${\mathrm{AA}}_0$-like and AB-like region, respectively. The parameter $V_{pp\sigma }=0.48\mathrm{eV}$. The vertical lines highlight the time $t=1.5\mathrm{fs}$ discussed in the text.

Figure 8

Spreading of the electron probability density in the TBG configuration with $\theta =2.5^{\circ }$ taken at several time moments. The initial state $\left|\psi \right(0)\rangle =|i\rangle$ located in the center of the AA-like region in the moiré zone. Lattice nodes in red (black) belong to the top (bottom) layer. The blue hexagon denotes the moiré zone. The characters ${\mathrm{AA}}_i$ ($i=1,\cdots ,6$), ${\mathrm{AB}}_i$, and ${\mathrm{BA}}_i$ ($i=1,2,3$) remark the AA- and AB-like regions in the moiré zone.

Figure 9

Distribution of the probability densities in two TBG samples with $\theta =5^{\circ }$ (left panel) and $\theta =2.5^{\circ }$ (right panel) at large evolution times $t=27$ and $30\mathrm{fs}$.

Figure 10

Similar to Fig. 9 but with the initial state $\left|\psi \right(0)\rangle =|i\rangle$ located in the center of the AB-like region in the moiré zone. Lattice nodes in red (black) belong to the top (bottom) layer.

Figure 11

The real (a) and imaginary (b) parts of the time autocorrelation function $C\left(t\right)$ for three TBG configurations with the twist angles of $\theta =5^{\circ }$ (green), $2.5^{\circ }$ (blue), and $1^{\circ }$ (red).