Spectroscopy of Wigner crystal polarons in an atomically thin semiconductor

Abstract

Strongly interacting electrons in two-dimensional systems can spontaneously break translational symmetry, forming a periodic Wigner crystal. Although these crystals have been realized in several platforms, experimental studies of their collective many-body excitations in the absence of a magnetic field remain an outstanding challenge. Here, we access this regime optically by uncovering Wigner crystal polarons: novel light-matter excitations arising from the dressing of excitons by collective excitations of the Wigner crystal. These hybrid quasiparticles manifest as new optical resonances in cryogenic reflectance spectra of a charge-tunable WSe$_2$ monolayer, appearing concurrently with previously identified exciton umklapp transitions. In contrast to the latter, the energies of Wigner crystal polarons are governed not only by the electronic lattice constant but also by their hybridization with attractive exciton-polarons, whose strength is controlled by electronic interactions. These novel many-body excitations provide an optical interface to the spin state of the Wigner crystal, which as we demonstrate, can be controlled both magnetically and optically. Our work establishes layered materials as a unique platform for exploring dynamical impurity dressing by strongly correlated electronic orders.

Figures (13)

• Figure 1: Optical signatures of a Wigner crystal in a WSe$_2$ monolayer. ( a,b) Schematic ( a) and optical micrograph ( b) of the charge-tunable WSe$_2$ monolayer device explored in the main text (see Methods Sec. 1 for the fabrication details). ( c) Reflectance contrast spectrum as a function of the electron density $n_\mathrm{e}$ at $B=0$ and $T=1.6$ K. ( d) Schematic illustrating the band structure of a WSe$_2$ monolayer as well as valley-selective optical selection rules. ( e) $n_\mathrm{e}$-evolution of the reflectance contrast differentiated with respect to the gate voltage and photon energy. Solid lines represent the fitted energies of exciton and both AP transitions, while dashed lines mark the corresponding umklapp and WP resonances originating from scattering off the Wigner crystal. ( f) Energies of X, AP$_\mathrm{T}$, and AP$_\mathrm{S}$ transitions as well as the corresponding umklapp and WP peaks extracted from the data in ( c,e) in the low electron density limit (see Methods Sec. 4 for details). ( g) Illustration of the back-folded exciton band structure, giving rise to a blueshifted-umklapp resonance (purple). ( h) $n_\mathrm{e}$-dependent energy splittings between each main peak and the corresponding umklapp transition. Each dataset is fitted with a linear dependence $\Delta E=h^2n_\mathrm{e}/\sqrt{3}m_\mathrm{X}+\Delta$ with fixed $m_\mathrm{X}\approx0.68m_0$ and different offset $\Delta$: 0 for U$_\mathrm{X}$, 6 meV for WP$_\mathrm{T}$, and 5 meV for WP$_\mathrm{S}$.
• Figure 2: Robustness of the Wigner crystal to thermal and quantum fluctuations. ( a) Electron-density evolutions of differentiated reflectance contrast spectra acquired at various temperatures (as indicated). Solid lines mark the fitted energies of main X, AP$_\mathrm{T}$, and AP$_\mathrm{S}$ resonances, while dashed lines represent expected positions of the umklapp peaks at $T=5$ K. ( b) Linecuts through the plots in ( a) (as well as similar datasets from different temperatures) showing temperature-dependent spectral profiles of the exciton umklapp (top) and Wigner crystal polarons (bottom) at $n_\mathrm{e}\approx3\cdot10^{11}\ \mathrm{cm}^{-2}$. To account for the temperature-induced bandgap shift, the energy axes of all plots are offset by the exciton energy $E_\mathrm{X}(T)$ at charge neutrality. The shaded areas mark a-few-meV-wide energy ranges around the respective umklapp resonances used for determining their spectral weight by averaging background-corrected $|d^2(\Delta R/R_0)/dV_\mathrm{TG}dE|$ signal (see Methods Sec. 8 for details). ( c) $T$--$n_\mathrm{e}$ phase diagrams of the Wigner crystal acquired using U$_\mathrm{X}$ (left), WP$_\mathrm{S}$ (middle), and WP$_\mathrm{T}$ (right). Each map shows averaged spectral weight of the umklapp resonance divided by the oscillator strength of the corresponding main optical transition. The domed-shaped phase boundary (identical for all plots; dashed line) is a guide to the eye.
• Figure 3: Theoretical description of the Wigner crystal polaron. ( a) Illustration of an attractive and Wigner crystal polaron formation. In the weak WC limit with a small gap $\Delta_\mathrm{WC}$ (bottom), the AP (WP) originates primarily from particle-hole dressing of a ground (Bragg-scattered) exciton with momentum $k\approx0$ ($k\approx k_\mathrm{W}$). For larger $\Delta_\mathrm{WC}$ (top), these two excitations hybridize due to strong attractive exciton-electron interactions, mediated by particle-hole excitations across the WC gap. The hybridization results in an enhancement of the WP spectral weight and an increase of its energy detuning from the AP. ( b) Cartoon showing the corresponding exciton umklapp scattering by repulsive exciton-electron interactions, where the energy splitting between the the main and umklapp branches is largely independent of $\Delta_\mathrm{WC}$. (c-e) Exciton spectral functions calculated using a spin-independent model within the Chevy approximation for a fixed $n_\mathrm{e}=5\cdot10^{11}\ \mathrm{cm}^{-2}$, and various WC gaps: ( e) $\Delta_\mathrm{WC}=0$; ( d) weak WC with a small gap where U$_\mathrm{X}$ appears at an energy detuning $\Delta E_{\mathrm{U}_\mathrm{X}} \approx \hbar^2k_\mathrm{W}^2/2m_\mathrm{X}$; ( c) strong WC with a large gap, where WP of sizable spectral weight appears at an energy splitting $\Delta E_{\mathrm{WP}} > \Delta E_{\mathrm{U}_\mathrm{X}}$. (f) The relative spectral weight of the WP compared to the AP calculated as a function of $\Delta_{\mathrm{WC}}$ for a fixed $n_\mathrm{e}=5\cdot10^{11}$ cm$^{-2}$. (g) Electron-density dependent energies of both singlet and triplet AP and WP transitions: points represent the experimental data from Fig. \ref{['fig:Fig1']}, while dashed lines mark the fitted energies of WP$_\mathrm{S}$ and WP$_\mathrm{T}$ resonances using a single-parameter effective model given by Eq. \ref{['eq:hybridization']}. For each WP branch, the energy is obtained by adding the experimentally-determined AP energy to the theoretically calculated AP-WP energy splitting at a given $n_\mathrm{e}$.
• Figure 3: Oscillator strengths of optical resonances based on TM fitting. ( a,b) Example reflectance contrast spectra measured for the main device at $T=1.6$ K, $B=0$, and two different electron densities (as indicated). The solid lines mark the fits of the AP spectral profiles with the TM model. ( c,d) Close-ups of the WP energy ranges (marked by dashed rectangles in a,b), showing background-corrected spectra together with the corresponding TM fits of the WP spectral profiles. ( e) Relative oscillator strengths of WP$_\mathrm{T}$ and WP$_\mathrm{S}$ resonances divided by the spectral weights of corresponding main AP resonances as a function of electron density. Note that determination of WP intensities is possible only when they are spectrally well-separated from X/AP resonances.
• Figure 4: Magnetic and optical control of the Wigner crystal spin state. ( a,h) Schematics illustrating two approaches for inducing sizable spin-valley polarization of the Wigner crystal: by applying a magnetic field ( a) or by illuminating the sample with circularly polarized light at $B=0$ ( h). ( b,c) Electron density evolutions of reflectance contrast ( b) and its second derivative ( c). The spectra were measured under linearly polarized excitation in two circular polarizations of detection at $B=5$ T and $T=1.6$ K. Solid lines show the fitted energies of the main optical transitions, while dashed lines mark expected positions of the umklapp resonances determined by fitting their splitting from the corresponding co-polarized main exciton transitions with $h^2n_\mathrm{e}/\sqrt{3}m_\mathrm{X}+\Delta$ for a fixed $m_\mathrm{X}=0.68m_0$ and different offsets $\Delta$. ( d,e) Linecuts through the maps in ( b,c) at $n_\mathrm{e}\approx3\cdot10^{11}\ \mathrm{cm}^{-2}$ showing polarization-resolved main resonances ( d) and the corresponding umklapp transitions ( e). The data in ( e) were binned along the energy axis. Solid lines show the fitted spectral profiles, while dashed ones mark the extracted peak energies (see Methods Sec. 4 for the details of the fitting procedures). ( f) Zeeman splittings between $\sigma^-$ and $\sigma^+$ polarized branches of the main excitons (left) and of the corresponding umklapp resonances (right) determined as a function of electron density at $B=5$ T. ( g) Calculated Zeeman splitting of the umklapp-scattered exciton as a function of exciton-electron interaction strength $V_\mathrm{X}$ (see Methods Sec. 13). The splitting is positive (negative) for attractive (repulsive) interactions, consistent with experimental results. ( i,j) Similar spectra as in ( d,e) at the same $n_\mathrm{e}$, but acquired at $B=0$ and two different polarizations of excitation: $\sigma^+$ and linear. The difference in intensities of triplet and singlet branches of AP/WP transitions demonstrate optical spin pumping of the WC (cf. Methods Sec. 9).