Excitons in moiré superlattices with disordered electrons

TL;DR

This work develops a comprehensive theory of moiré excitons in fractionally filled TMD moiré superlattices by extending the hydrogenic exciton model to include moiré-induced $s$-$p$ orbital mixing and solving in a Bloch framework with a basis of $|\alpha\rangle = \varphi_{\alpha}(\mathbf{r})$ states. The exciton problem is formulated with the center-of-mass coordinate $\mathbf{R}$ and relative coordinate $\mathbf{r}$, yielding a matrix eigenproblem in reciprocal space that captures inter-orbital couplings via $V_{\alpha,\beta}(\mathbf{G}-\mathbf{G}')$ and enables analysis of $1s$, $2s$, and $3s$ resonances under fractional fillings. The absorption spectrum is computed from the $K=0$ exciton states, with the oscillator strength determined by the $G=0$ component and the $s$-orbital content at $\mathbf{r}=0$, linking spectral features directly to moiré-induced $s$-$p$ hybridization. By incorporating defect-induced quasi-ordering and thermal fluctuations through large-$N$ supercells and classical Monte Carlo, the study shows that disorder and finite temperature can partially or fully suppress moiré exciton signatures, especially for higher Rydberg states, and reveals distinct melting behaviors across fractional fillings. The results align with experiments, provide a framework for probing correlated electron states with excitons, and suggest avenues for voltage-tunable THz devices based on controllable $2s$-$2p$ splittings in moiré devices.

Abstract

Moiré superlattices in transition metal dichalcogenides (TMDs) heterobilayers exhibit various correlated insulating states driven by long-range Coulomb interactions, and these states crucially alter exciton resonances, particularly at fractional fillings. We revisit a theoretical framework to investigate the doping dependence of exciton spectra by extending hydrogenic exciton wavefunctions, systematically analyzing how the 1$s$, 2$s$, and 3$s$ Rydberg states respond to moiré-induced mixing of $s$- and $p$-type orbitals. Notably, while the 1$s$ state remains relatively robust against doping, higher Rydberg excitons show strong redshifts and oscillator-strength quenching near specific fractional fillings. We incorporate both defect-induced quasi-ordering and thermal fluctuations to capture realistic device conditions, employing a large supercell approach. By selectively randomizing a subset of electrons or utilizing classical Monte Carlo simulations, we present direct calculations of exciton spectra under varying defect densities and temperatures. Our results emphasize how even moderate disorder or finite temperature can partially or completely suppress characteristic moiré exciton physics. Especially, we show how the 2$s$ exciton states respond to the phase transition in correlated electron states. This comprehensive picture not only clarifies recent experimental observations but also provides a framework to guide the design of moiré-based optoelectronic devices.

Figures (8)

• Figure 1: Schematic illustration of (a) the device structure and (b) representative electron configurations at various fractional fillings. The moiré superlattice (orange layer) is stacked above a separate “sensor” monolayer (green layer), and both layers are sandwiched between top and bottom hBN dielectrics. By tuning the gate voltages, one controls the fractional electron filling ($\nu$) in the moiré layer. For each fractional filling $\nu$, white circles mark empty moiré lattice sites, and black dots indicate occupied sites. Blue dashed polygons highlight unit cells of each specific filling.
• Figure 2: Center-of-mass probability distributions, $\rho(\mathbf{R})$, of the 2$s$ exciton at fractional fillings (a) $\nu=1/4$, (b) $\nu=1/3$, (c) $\nu=2/5$, and (d) $\nu=1/2$. The color scale indicates $\rho(\mathbf{R})$ from red (zero) to blue (maximum). These plots highlight how the 2$s$-like exciton wavefunction adapts to different ordered charge configurations in the moiré superlattice. For each fractional filling $\nu$, white and black circles mark empty and occupied moiré lattice sites, respectively. Dashed-line polygons represent unit cells of each specific filling.
• Figure 3: Comparison between experimental (from Ref. Xu2020) and theoretical exciton spectra. (a) Measured reflectance contrast ($R/R_{0}$) spectra over a broad energy range. (b)--(d) Zoomed-in views of the same experimental data focusing on the 3$s$, 2$s$, and 1$s$ exciton energy ranges, respectively. (e)--(g) Theoretically calculated absorption spectra (oscillator strength) for the 3$s$, 2$s$, and 1$s$ excitons, assuming an ordered electron arrangement in the moiré superlattice. In each panel, the color scale is adjusted to distinguish the intensity levels clearly for each exciton state. Note that the 3$s$ and 2$s$ resonances undergo pronounced redshifts, whereas the 1$s$ exciton remains robust.
• Figure 4: Schematic illustrations of different defect densities in a moiré lattice, highlighting how even a small number of displaced electrons can partially disrupt long-range order. For simplicity, a small section of the lattice is shown here, whereas our numerical calculations are performed on a $12 \times 12$ moiré supercell based on the unit cell of each filling factor. (a) Single-disorder: only one electron departs its original site (indicated by the red arrow and colored boxes), representing minimal quasi-ordering. (b) Five-disorder: five electrons have left their original sites, illustrating a moderate level of quasi-ordering. (c) Fully disordered: all electrons are randomly distributed without any long-range ordering. Black and white symbols represent filled and empty moiré site, respectively.
• Figure 5: Impact of defect-induced quasi-ordering on exciton absorption in moiré superlattices. Simulated absorption spectra for the (a)--(c) 2$s$ and (d)--(f) 1$s$ excitons under different levels of structural disorder. Here, panels (a) and (d) show single-disorder charge states; (b) and (e) correspond to a quasi-diorder state with five electrons displaced and (c) and (f) represent fully disordered electron distributions. Note that in panels (a)–(f), we show data only for filling factors $\nu$ from 1/7 to 1/2, as the corresponding spectra for $1-\nu$ are nearly identical to those for $\nu$. (g) Summary of the absorption quenching due to defect-induced quasi-ordering, plotted for multiple fractional fillings $\nu$. The horizontal axis indicates the total number of electrons displaced from their ideal moiré sites, and the vertical axis is the extracted absorption amplitude (a.u.). Circles correspond to the 1$s$ exciton and squares to the 2$s$ exciton. Different colors represent distinct filling factors from $\nu = 1/7$ to $\nu = 6/7$. As the fraction of displaced electrons grows, higher-Rydberg excitons (2$s$) quickly lose their strong resonance features, whereas the 1$s$ exciton remains relatively unaffected.