Spatiotemporal dynamics of moiré excitons in van der Waals heterostructures

Technical Summary: Spatiotemporal Dynamics of Moiré Excitons in van der Waals Heterostructures

Problem Statement
Transition metal dichalcogenide (TMD) heterostructures offer a tunable platform for studying many-particle phenomena, particularly through the formation of interlayer excitons with permanent out-of-plane dipole moments. A central mechanism in these systems is the moiré potential, arising from lattice mismatch or twist angles, which reshapes the energy landscape into a complex, non-parabolic band structure with mini-Brillouin zones (mBZ). While recent experiments have enabled the observation and control of interlayer excitons in these moiré-patterned systems, a comprehensive microscopic theoretical framework is lacking. Existing approaches typically treat energy relaxation and spatial diffusion as decoupled processes or rely on simplified assumptions that fail to capture the coupled dynamics of excitons exhibiting non-trivial band structures, efficient phonon-mediated relaxation, and spatial localization. Consequently, the interplay between energy relaxation and real-space diffusion in moiré systems remains unexplored at a microscopic level.

Methodology
The authors develop a predictive, material-specific many-body model to track exciton dynamics across time, space, and momentum. The approach is based on an equation-of-motion formalism transformed into the Wigner representation, resulting in a Boltzmann transport equation for moiré excitons.

• Model Scope: The study focuses on the low excitation regime where exciton-exciton interactions are negligible. It accounts for the full two-dimensional momentum-dependent band structure, acknowledging that moiré-modified bands are non-parabolic.
• Simulation Technique: To manage the high dimensionality of the problem, the authors employ a Monte Carlo algorithm to solve the Boltzmann transport equation in both momentum and real space.
• System: The model is applied to a twisted, hBN-encapsulated WSe$_2$–MoSe$_2$ heterostructure. The study focuses on intermediate twist angles (3°–6°), where the moiré potential significantly modifies the band structure without completely trapping excitons (unlike the ~1° regime where bands become flat and group velocities vanish).
• Initial Conditions: Simulations initialize an exciton distribution with a Gaussian spatial profile (1 µm standard deviation) and a uniform energy distribution of approximately 60 meV ("hot" excitons).

Key Results
The study reveals a counterintuitive regime of exciton transport where flat bands, typically associated with immobile excitons, significantly enhance diffusion under specific conditions.

• Temperature Dependence: At low temperatures (e.g., 10 K), the diffusion coefficient ($D$) is predicted to be significantly enhanced compared to higher temperatures (e.g., 70 K). For a 3° twist angle, $D$ increases from 1.4 cm$^2$/s at 70 K to 6 cm$^2$/s at 10 K. This low-temperature value is more than double the value expected for a standard Boltzmann distribution.
• Mechanism of Enhancement: The enhanced propagation arises from a "relaxation bottleneck." At low temperatures, the mismatch between the interband energy gap and dominant optical phonon energies prevents excitons from fully relaxing to the ground state via phonon emission. Consequently, excitons remain trapped in relatively flat regions of the dispersion landscape but accumulate in higher-energy states.
• Role of Band Structure: Although the bands are flat along certain paths, the thermal population extends into more dispersive regions of the moiré Brillouin zone. This allows excitons to access states with higher group velocities. The interplay between the bottleneck effect (creating "hot" excitons) and the directional extension of the population into dispersive regions leads to a larger effective group velocity and, thus, enhanced diffusion.
• Twist Angle Dependence:
  • High Temperatures (>50–60 K): The diffusion coefficient increases monotonically with the twist angle, approaching the behavior of interlayer excitons with parabolic bands.
  • Low Temperatures (<50 K): For the smallest angle studied (3°), the diffusion coefficient decreases with increasing temperature. This is attributed to the band gap lying within the thermally populated region (40–70 K); the absence of available states in this window prevents the group velocity from compensating for the temperature-induced decrease in scattering time.
  • Intermediate Angles (>3°): A non-monotonic temperature dependence is observed, resulting from a competition between thermal occupation favoring higher-velocity states and enhanced phonon scattering reducing the scattering time.

Significance and Claims
The paper claims to provide the first microscopic theoretical framework that captures the coupled dynamics of momentum-space thermalization and real-space diffusion in moiré systems. The primary contribution is the revelation that flat bands do not necessarily inhibit exciton transport; instead, under low-temperature conditions, they can facilitate enhanced propagation through a bottleneck-induced accumulation of high-energy excitons.

The authors state that these insights lay the foundation for next-generation moiré-based optoelectronic and quantum technologies. Specifically, the work suggests that exciton transport can be controlled via "twist-angle engineering" and temperature tuning. This control over exciton flow is identified as crucial for potential applications in excitonic circuits, energy funneling, and diffusion-mediated light emission. The developed framework is noted to be applicable to a broader class of moiré systems, including those with lattice mismatches rather than just twist angles.