Optical Signatures of Periodic Charge Distribution in a Mott-like Correlated Insulator State

The elementary optical excitations in two-dimensional semiconductors hosting itinerant electrons are attractive and repulsive polarons—excitons that are dynamically screened by electrons. Exciton polarons have hitherto been studied in translationally invariant degenerate Fermi systems. Here, we show that periodic distribution of electrons breaks the excitonic translational invariance and leads to a direct optical signature in the exciton-polaron spectrum. Specifically, we demonstrate that new optical resonances appear due to spatially modulated interactions between excitons and electrons in an incompressible Mott-like correlated state. Our observations demonstrate that resonant optical spectroscopy provides an invaluable tool for studying strongly correlated states, such as Wigner crystals and density waves, where exciton-electron interactions are modified by the emergence of charge order.

Popular Summary

When some 2D crystals are stacked with a small twist angle, their energy landscapes change to create a periodic patchwork of “valleys” in which electrons can localize. Such systems offer a way to explore emergent phenomena—such as superconductivity—that arise from strong correlations among electrons. Recently, researchers reported observations of a correlated insulating state in one such system, a bilayer 2D semiconductor based on transition-metal dichalcogenides (TMDs). Here, we report direct evidence of how electrons organize themselves in this state.

To probe the underlying charge distribution, we use excitons—bound pairs of an electron and an electron hole. For excitons, any periodic distribution of charge acts like a diffraction grating, scattering high-momentum excitons to zero-momentum states, which manifest as an additional optical resonance. Using optical reflectance spectroscopy, we find that this resonance appears when the TMD heterostructure takes on properties of a Mott insulator; that is, just one electron fills every well, and the 2D layer insulates when conventional theory predicts it should conduct. Furthermore, this new resonance is unchanged by an external magnetic field, indicating its origin as high-momentum excitons. Since these observed behaviors are consistent with theoretical models that predict that excitons are scattered by charges periodically distributed in a Mott-like correlated state, we conclude that is how the system’s electrons are arranged.

This work furthers our goal of using optics to verify periodic ordering of charges on length scales much smaller than optical wavelengths. Using this scheme, we recently confirmed the existence of a Wigner crystal, a system of spontaneously crystallized electrons.

Figure 1

Moiré lattice in ${\mathrm{MoS}\mathrm{e}}_2/\mathrm{hBN}/{\mathrm{MoS}\mathrm{e}}_2$ and basic characterization. (a) Schematic picture of the device structure. The left bottom picture shows a moiré lattice. (b) Schematic picture of the potential for excitons created by electrons trapped in a moiré lattice. (c) Differential reflectance spectrum for each charge configuration. “i” and “n” indicate neutral and electron doped charge configurations and are shown in the order of (top, bottom). (d) Color map of layer contrasted differential reflectance signal [$\mathrm{\Delta }R/R_0(1.6379\mathrm{eV})-\mathrm{\Delta }R/R_0(1.6311\mathrm{eV})$].

Figure 2

Umklapp exciton resonances at $\nu =1$. (a),(b) $V_{\mu }$ dependence of differential reflectance differentiated with respect to energy $E$ at $V_E=0.24\mathrm{V}$ (a) and $V_E=-1.98\mathrm{V}$ (b). (c) $V_E$ dependence of differential reflectance spectra at $\nu =1$. The cyan, yellow, and magenta dashed lines correspond to $V_E=-1.98$, $-0.80$, and 0.24 V, respectively. The scale of the color bars is logarithmic for $|d(\mathrm{\Delta }R/R_0)/dE|>{10}^2\mathrm{e}{\mathrm{V}}^{-1}$ and linear for $|d(\mathrm{\Delta }R/R_0)/dE|<{10}^2\mathrm{e}{\mathrm{V}}^{-1}$.

Figure 3

Umklapp exciton resonances at $\nu =2$. (a) $V_{\mu }$ dependence of differential reflectance differentiated with respect to energy $E$ at $V_E=-0.80\mathrm{V}$. (b) $V_E$ dependence of differential reflectance spectra at $\nu =2$. The cyan, yellow, and magenta dashed lines correspond to $V_E=-1.98$, $-0.80$, and 0.24 V, respectively. The scale of the color bars is logarithmic for $|d(\mathrm{\Delta }R/R_0)/dE|>{10}^2\mathrm{e}{\mathrm{V}}^{-1}$ and linear for $|d(\mathrm{\Delta }R/R_0)/dE|<{10}^2\mathrm{e}{\mathrm{V}}^{-1}$.

Figure 4

Exciton band structure. Spectrum of the effective band Hamiltonian of Eq. (3) for parameters given in the main text and $J=300\mathrm{meV}$. (a) Bare dispersion of mobile excitons. Strong intervalley electron-hole exchange coupling splits the linearly polarized exciton modes and yields two branches with linear (green line) and parabolic (blue line) dispersion. For simplicity, we do not show that the degeneracy of the excitonic branches extends throughout the light cone. The exciton valley pseudospin is shown as blue (green) arrows for the parabolic (linear) dispersion. Periodic potentials allow states connected by reciprocal lattice vectors to mix, yielding higher bands (transparent lines), as shown in an extended zone scheme. Higher bands of the linearly dispersing mode decouple due to their large energy detuning. (b) Exciton bands along a path in a moiré Brillouin zone. The oscillator strength of each state is indicated by the color bar, which is saturated for all blue lines. As photons carry almost vanishing momentum, only modes close to the $\mathrm{\Gamma }$ point may obtain finite oscillator strength (we artificially extend the momentum range of the bright states for better visibility). While most states remain dark, only a single umklapp band per polarization obtains sizable oscillator strength. Left: dispersion in the absence of a magnetic field. We find an energy splitting between the umklapp state and the main resonance of 3 meV. This additional bright state carries 1.2% of the oscillator strength of the bare exciton. The oscillator strength of higher bands is significantly suppressed due to the larger energy splitting, which reduces the coupling to light. Right: exciton dispersion for $B_z=7\mathrm{T}$. While the main resonance splits significantly, the umklapp peaks are only marginally affected by the magnetic field.

Figure 5

Exciton and umklapp exciton resonances at $\nu =1$ under magnetic field $B_z=7\mathrm{T}$. (a),(b) $V_{\mu }$ dependence of differential reflectance differentiated with respect to energy $E$ in ${\sigma }_{+}$ (a) and ${\sigma }_{-}$ (b) polarization. (c),(d) Line cuts of the reflectance spectrum (c) and its energy differentiation (d) at $\nu =1$, $V_E=0.61\mathrm{V}$.