Abstract
When the Coulomb repulsion between electrons dominates over their kinetic energy, electrons in twodimensional systems are predicted to spontaneously break continuoustranslation symmetry and form a quantum crystal^{1}. Efforts to observe^{2,3,4,5,6,7,8,9,10,11,12} this elusive state of matter, termed a Wigner crystal, in twodimensional extended systems have primarily focused on conductivity measurements on electrons confined to a single Landau level at high magnetic fields. Here we use optical spectroscopy to demonstrate that electrons in a monolayer semiconductor with density lower than 3 × 10^{11} per centimetre squared form a Wigner crystal. The combination of a high electron effective mass and reduced dielectric screening enables us to observe electronic charge order even in the absence of a moiré potential or an external magnetic field. The interactions between a resonantly injected exciton and electrons arranged in a periodic lattice modify the exciton bandstructure so that an umklapp resonance arises in the optical reflection spectrum, heralding the presence of charge order^{13}. Our findings demonstrate that chargetunable transition metal dichalcogenide monolayers^{14} enable the investigation of previously uncharted territory for manybody physics where interaction energy dominates over kinetic energy.
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The data that support the findings of this study are available at the ETH Research Collection (https://doi.org/10.3929/ethzb000478739).
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Acknowledgements
C.K. and A.I. thank M. Knap for discussions. This work was supported by the Swiss National Science Foundation (SNSF) under grant no. 200021178909/1. K.W. and T.T. acknowledge support from the Elemental Strategy Initiative conducted by MEXT, Japan, A3 Foresight by JSPS and CREST (grant no. JPMJCR15F3) and JST. P.E.D., I.E. and E.D. were supported by HarvardMIT CUA, AFOSRMURI: Photonic Quantum Matter award FA95501610323, Harvard Quantum Initiative.
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Contributions
T.S. and A.I. conceived the experiments. T.S. carried out the measurements and analysed the data. M.K. and Y.S. helped with the experimental setup and the data analysis. K.W. and T.T. grew the hBN crystals. A.P., P.B. and X.L. fabricated the samples. P.E.D., C.K. and I.E. did the theoretical modelling and carried out the calculations under the guidance of E.D. T.S., P.E.D., E.D. and A.I. wrote the manuscript. T.S., E.D. and A.I. supervised the project.
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Extended data figures and tables
Extended Data Fig. 1 Device structure and the experimental setup.
a, Illustration of the structure of the device investigated in the main text. b, Simplified schematic of the experimental setup used for magnetooptical, mKtemperature measurements. FLG, fewlayer graphene; NA, numerical aperture.
Extended Data Fig. 2 Normalization and differentiation of the reflectance data.
a, Reflectance spectra acquired for the main device at T = 80 mK at two different spots: one in the MoSe_{2} monolayer region (blue) and one off the MoSe_{2} monolayer (red). Both spectra were obtained at charge neutrality (V_{t} = −1 V) and in the absence of the magnetic field. b, The reflectance contrast spectrum R_{c} ≡ ΔR/R_{0} determined based on the two spectra from a. c, d, Colour plots showing zerofield topgate voltage evolutions of the derivative of reflectance contrast \({R}_{{\rm{c}}}^{{\prime} }={\rm{d}}{R}_{{\rm{c}}}/{\rm{d}}{V}_{{\rm{t}}}\) with respect to the V_{t} (dashed lines mark the exciton and umklapp energies). The left panel presents a derivative evaluated numerically using standard, symmetric difference quotient method as \({R}_{{\rm{c}}}^{{\prime} }({V}_{{\rm{t}},n})=[{R}_{{\rm{c}}}({V}_{{\rm{t}},n+1}){R}_{{\rm{c}}}({V}_{{\rm{t}},n1})]/[{V}_{{\rm{t}},n+1}{V}_{{\rm{t}},n1}]\). The right panel shows a derivative of the same data obtained using the other method, in which dR_{c}/dV_{t} is computed as a difference quotient between symmetric data points separated not by two, but by four gatevoltage steps, that is, \({R}_{{\rm{c}}}^{{\prime} }({V}_{{\rm{t}},n})=[{R}_{{\rm{c}}}({V}_{{\rm{t}},n+2}){R}_{{\rm{c}}}({V}_{{\rm{t}},n2})]/[{V}_{{\rm{t}},n+2}{V}_{{\rm{t}},n2}]\). This method was used to plot the gatevoltage derivatives in Figs. 1–3.
Extended Data Fig. 3 Fitting the exciton and umklapp peaks with dispersive Lorentzian spectral profiles.
a, Colour map showing topgate voltage evolution of the reflectance contrast R_{c} spectra measured at T = 80 mK for the main device, at zero magnetic field, for low electron doping densities, and in the spectral vicinity of the exciton peak. b, Similar map presenting evolution of the reflectance contrast R_{c} upon subtraction of the fitted exciton spectral profile R_{X} as well as a smooth Gaussianlike background B (independent of the voltage). The dashed lines in both panels indicate the energies of the exciton and umklapp peaks. c, d, Crosssections through the maps in a, b at V_{t} = 0.8 V showing, respectively, bare and backgroundcorrected reflectance contrast spectra. The solid lines indicate the fits to the experimental data with dispersive Lorentzian spectral profiles, based on which we determined the exciton and umklapp energies (marked by vertical dashed lines). In the case of the umklapp peak, the fitting was carried out only in the energy region covered by the data points shown in green, in order to avoid spurious contribution around the energy of the main exciton (of about 1,635.5 meV at the selected V_{t} = 0.8 V) that originates from the residual of the exciton resonance fitting.
Extended Data Fig. 4 Calibration of the electron density for the dilution refrigerator experiments based on the Shubnikov–de Haas oscillations in the exciton linewidth.
a, Colour map presenting σ^{−}polarized reflectance contrast spectra measured at T = 80 mK for the main device as a function of the topgate voltage at B = 16 T. b, Gatevoltage dependence of the exciton linewidth extracted from the data in a by fitting the exciton resonance with a dispersive Lorentzian spectral profile. Dashed lines in both panels indicate the positions of integer filling factors. c, Gate voltages V_{t}(ν, B) corresponding to the positions of the exciton linewidth minima extracted from the reflectance measurements carried out at different magnetic fields. All of the presented data points were obtained in the regime where the Fermi level does not exceed the valley Zeeman splitting in the conduction band. Solid lines represent the fit of the data points with a set of linear dependencies corresponding to subsequent integer filling factors, which form a Landau level fan chart.
Extended Data Fig. 5 Comparison of the zerofield Wigner crystal signatures at T = 80 mK and T = 4 K.
a, b, Colour maps showing zerofield topgate voltage evolution of the derivative of reflectance contrast spectra with respect to V_{t} measured in the dilution refrigerator for the main device at two different temperatures: T = 80 mK (a) and T = 4 K (b). Black dashed lines mark the fitted energy E_{X} of the exciton resonance, and the green lines indicate the expected position of the umklapp peak E_{X} + ΔE_{U} for \(\Delta {E}_{{\rm{U}}}={h}^{2}{n}_{{\rm{e}}}/\sqrt{3}{m}_{{\rm{X}}}\) corresponding to a triangular Wigner crystal and m_{X} = 1.1m_{e}.
Extended Data Fig. 6 Melting of the Wigner crystal at elevated temperatures.
a, Colour maps displaying backgate voltage evolution of the derivative of reflectance contrast spectra with respect to V_{b} measured in the VTIbased setup at zero magnetic field and different temperatures of the main device (as indicated). In each map, the black dashed lines mark the fitted energy E_{X} of the main exciton peak, while the green line in the map at T = 4.5 K represents the expected position of the umklapp resonance \({E}_{{\rm{X}}}+{h}^{2}{n}_{{\rm{e}}}/\sqrt{3}{m}_{{\rm{X}}}\) corresponding to a triangular Wigner crystal and m_{X} = 1.3m_{e}. b–d, Crosssections through the maps in a (as well as through a similar map measured at T = 15.2 K) at V_{b} = −0.6 V (b), V_{b} = −0.45 V (c), and V_{b} = −0.35 V (d) corresponding to electron densities n_{e}/10^{11} cm^{−2} of, respectively, 1.2, 1.6 and 1.9. e, Relative amplitude of the umklapp transition determined as a function of the temperature as the average value of d(ΔR/R_{0})/dV_{b} in a 0.8meVwide energy window around the umklapp energy (marked by shaded regions in b–d) at different electron densities (as indicated). The error bar for each data point is estimated as a standard error (that is, standard deviation of the mean) of all points within the aforementioned energy window for a given n_{e} and T. The solid lines represent the fits of the temperature dependence for each n_{e} with a linear decrease a(T_{m} − T) + b at T < T_{m} and a flat background level b at T > T_{m}, with a melting temperatures T_{m} = 11 ± 1 K of the Wigner crystal being found to be the same for all investigated values of n_{e} within the experimental uncertainty. f, The same crosssections as in c (determined at V_{b} = −0.45 V) together with the solid lines representing the fits of the umklapp lineshapes at different temperatures with a phenomenological Lorentzian profile overlaid on a (fixed) polynomial background. g, Temperaturedependent relative amplitude of the umklapp peak at n_{e} ≈ 1.6 × 10^{11} cm^{−2} extracted based on the Lorentzian fits from f (as well as based on similar fits of the data taken at different temperatures; the error bars correspond to standard errors increased by a temperatureindependent term that is introduced to account for systematic uncertainty stemming from determination of the background profile). Solid line indicates a linear fit to the data points with nonzero amplitude corresponding to the same Wigner crystal melting temperature of T_{m} = 11 ± 1 K.
Extended Data Fig. 7 Analysis of disorderinduced broadening of the umklapp peak.
a, b, Colour maps showing topgate voltage evolution of the differentiated reflectance contrast d(ΔR/R_{0})/dE (a) and the bare reflectance contrast ΔR/R_{0} (b) acquired for the main device at B = 14 T, T = 80 mK, and in σ^{−} polarization of detection (note that these results were obtained in a different measurement than that shown in Fig. 3e, f). Grey horizontal dashed lines mark the voltages corresponding to filling factors of ν = 0, 0.34, and 1. White (black) dashed (dotted) line indicates the fitted energy E_{X}(σ^{−}) (E_{X}(σ^{+})) of the σ^{−}polarized (σ^{+}polarized) main exciton resonance. The green dashed line represents the expected energy of the umklapp peak determined as \({E}_{{\rm{X}}}({\sigma }^{+})+\Delta {E}_{{\rm{U}}}^{+}\), where \(\Delta {E}_{{\rm{U}}}^{+}={h}^{2}{n}_{{\rm{e}}}/\sqrt{3}{m}_{{\rm{X}}}+s{k}_{{\rm{W}}}\sqrt{{(s{k}_{{\rm{W}}})}^{2}+{({\varDelta }_{{\rm{Z}}}/2)}^{2}}+{\varDelta }_{{\rm{Z}}}/2\) represents the umklapp–main exciton detuning calculated within the frame the model from Supplementary Information section 3 using the exciton mass of m_{X} = 1.3m_{e}, the strength of the electron–hole exchange sK = 300 meV (with K representing the valley momentum), and experimentally determined main exciton Zeeman splitting Δ_{Z} = 3.6 meV. c, d, Crosssections through the maps in a, b at V_{t} = 0.45 V corresponding to the electron density n_{e} ≈ 1.16 × 10^{11} cm^{−2} and Landau level filling factor ν ≈ 0.34. The solid line in c shows the fit of the experimental data with a sum of two differentiated dispersive Lorentzian spectral profiles \({R}_{{\rm{X}}}^{{\prime} }(E)+{R}_{{\rm{U}}}^{{\prime} }(E)\) representing the main and umklapp resonances. The solid line in d in turn depicts the fit of the main exciton resonance in the bare reflectance spectrum with a transfermatrix model, allowing us to extract the nonradiative broadening of the exciton \({\gamma }_{{\rm{X}}}^{{\rm{nrad}}}=0.78\,{\rm{meV}}\). e, The fits of the differentiated reflectance spectrum at ν ≈ 0.34 with the sum of two differentiated dispersive Lorentzians \({R}_{{\rm{X}}}^{{\prime} }(E)+{R}_{{\rm{U}}}^{{\prime} }(E)\) corresponding to different correlation lengths of the Wigner crystal (as indicated). For each fit the parameters of the main exciton resonance (that is, amplitude A_{X}, linewidth γ_{X}, energy E_{X} and phase φ_{X}) as well as the phase φ_{0} and the energy E_{U} of the umklapp peak are fixed at the values obtained from the fit in c. The linewidth and amplitude of the umklapp resonance are assumed to be given by \({\gamma }_{{\rm{U}}}={\gamma }_{{\rm{X}}}^{{\rm{nrad}}}+{\gamma }_{{\rm{U}}}^{{\rm{dis}}}\) and A_{U} = α_{U}A_{X}, where the disorderinduced broadening \({\gamma }_{{\rm{U}}}^{{\rm{dis}}}\) and the relative amplitude α_{U} are fitted based on the predictions of the theoretical model from Supplementary Information section 6 describing the umklapp lineshape for a Wigner crystal with a given correlation length. For all fits α_{U} is assumed to be smaller than 5%. f, The coefficient of determination R^{2} evaluated for the fits corresponding to different Wigner crystal correlation lengths based on the data points in a 2meVwide energy window around the umklapp resonance (marked by shaded region). The dashed line indicates the value of R^{2} obtained in the same way for an unconstrained fit in c.
Extended Data Fig. 8 Observation of zeromagneticfield Wigner crystal signatures for the second device.
a, Colour map showing reflectance contrast spectra measured as a function of the topgate voltage V_{t} for the second device. The data were acquired at T = 4 K and in the absence of the magnetic field. b, Gatevoltage evolution of the derivative of the spectra from a with respect to V_{t}. Black dashed lines in both panels indicate the energy of the exciton peak E_{X} obtained by fitting its spectral profile with dispersive Lorentzian lineshape. Green lines mark the expected position E_{X} + ΔE_{U} of the umklapp peak, where \(\Delta {E}_{{\rm{U}}}={h}^{2}{n}_{{\rm{e}}}/\sqrt{3}{m}_{{\rm{X}}}\) is computed under an assumption of a triangular Wigner crystal and for the value of exciton mass m_{X} = 1.3m_{e}.
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Smoleński, T., Dolgirev, P.E., Kuhlenkamp, C. et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor. Nature 595, 53–57 (2021). https://doi.org/10.1038/s41586021035904
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DOI: https://doi.org/10.1038/s41586021035904
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