Electrons Surf Phason Waves in Moiré Bilayers

We investigate the effect of thermal fluctuations on the atomic and electronic structure of a twisted MoSe2/WSe2 heterobilayer using a combination of classical molecular dynamics and ab initio density functional theory calculations. Our calculations reveal that thermally excited phason modes give rise to an almost rigid motion of the moiré lattice. Electrons and holes in low-energy states are localized in specific stacking regions of the moiré unit cell and follow the thermal motion of these regions. In other words, charge carriers surf phason waves that are excited at finite temperatures. We also show that such surfing survives in the presence of a substrate and frozen potential. This effect has potential implications for the design of charge and exciton transport devices based on moiré materials.


Structural relaxations
The moiré patterns are relaxed using the LAMMPS package with the Stillinger-Weber [S2], and Kolmogorov-Crespi [S3] potentials to capture the intralayer and interlayer interactions of the twisted bilayer of WSe 2 , respectively. The Kolmogorov-Crespi parameters used in this work can correctly reproduce the interlayer binding energy landscape, obtained using density functional theory. The atomic relaxations produced using these parameters are in excellent agreement with relaxations performed using density functional theory. We relax the atoms within a fixed simulation box with the force tolerance of 10 −5 eV/Å for any atom along any direction.

Molecular dynamics simulations
We equilibrate the moiré material under periodic boundary conditions in the canonical ensemble at several temperatures for a nano-second using a Noseé-Hoover thermostat. We track the dynamics of moié sites using the micro-canonical ensemble for several nanoseconds.
To extract the speed of the moiré sites, we have used a 3 × 3 × 1 moiré supercell. Additional molecular dynamics simulations were performed for smaller twist angles (1 • and 59 • ). Similar movements of moiré sites were observed. To simulate different initial conditions, we used a 2 Langevin thermostat with different increments in temperature within a loop up to 150 K before equilibration. The surfing speeds for all these different initial conditions are obtained for 3.14 • moiré 3 × 3 × 1 supercell to be 31, 40, 16 nm/ns.

Inclusion of the substrate and frozen potential
We use hexagonal Boron Nitride (hBN) as a substrate for the MoSe 2 /WSe 2 twisted bilayer. The intralayer interactions were described using a Tersoff potential [S4] and the interlayer interactions were captured using a Kolmogorov-Crespi potential [S5]. The lattice constant for hBN was 2.516Å and 24 × 24 unit cells were added to the twisted bilayer. As long as the dynamics of the hBN and the twisted bilayer are allowed at finite T , we find the moiré sites move. However, if we freeze the movements of hBN in all directions, the moiré sites stop free movement but show large displacements around a mean position. The frozen hBN atoms in such a scenario act as a frozen potential which pin the phason motion.

Electronic structure calculations
We use a double-ζ plus polarization basis for the expansion of wavefunctions. For all the electronic structure calculations we use the Γ point in the moiré Brillouin zone to obtain the converged ground state charge density. A large vacuum spacing of 20Å is used in the out-of-plane direction for all the density functional theory (DFT) calculations. All the electronic structure calculations at finite temperature are performed using a snapshot of the moiré unit cell from the classical molecular dynamics simulations. For computing the averaged electronic band gap at 150 K, we used 6 snapshots from our molecular dynamics simulations. Note that all the electronic structure calculations were performed on the moiré unit cell (i.e., 1 × 1 × 1 moiré supercell).

Frozen potential
The attached video (Surfingindisorder.mp4 ) shows the atomic motion (obtained from a molecular dynamics simulation) in the presence of frozen potential as described in the main text at a high temperature of 1200 K. We find that surfing survives even in the presence of frozen potential. Also, we present the same dynamics (Surfingindisordersideview.mp4 ) from a side view focusing on the atomistic details to emphasize the moiré magnification that happens at the moiré scale.

Long time scale with different initial conditions
The movements of the moiré sites over a large time window for three different initial conditions are presented as movies Movementlargewindow1.mp4, Movementlargewindow2.mp4, Movementlargewindow3.mp4. Each movie was created with 8 ns long trajectories. Moreover, the direction changes when sufficiently long simulations are carried out (here we use a simulation time of 8 ns while in the original manuscript, the simulation time was less than 1 ns). The situation is similar to a drunkard's walk where the drunkard might initially take a couple of steps in a specific direction. In such a situation, even though there's no preferred direction for the steps taken, after n steps the walker traverses a distance proportional to √ n.    The surfing direction is marked with an arrow.

X: TEMPERATURE-GRADIENT TO BREAK ISOTROPY
To demonstrate this, we have created a temperature gradient as shown in Figure S10.
We have computed the mean-square-displacements of the moiré sites with a 3 × 3 × 1 and a 7 × 7 × 1 moiré supercell with T h = 175 K and T l = 125 K and found it to be linear in time, indicating diffusive motion. We have used Langevin thermostats to maintain the temperature in hot and cold regions [S6]. The average temperature of the system is kept at 150 K.