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# Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure

## Abstract

One of the first theoretically predicted manifestations of strong interactions in many-electron systems was the Wigner crystal1,2,3, in which electrons crystallize into a regular lattice. The crystal can melt via either thermal or quantum fluctuations4. Quantum melting of the Wigner crystal is predicted to produce exotic intermediate phases5,6 and quantum magnetism7,8 because of the intricate interplay of Coulomb interactions and kinetic energy. However, studying two-dimensional Wigner crystals in the quantum regime has often required a strong magnetic field9,10,11 or a moiré superlattice potential12,13,14,15, thus limiting access to the full phase diagram of the interacting electron liquid. Here we report the observation of bilayer Wigner crystals without magnetic fields or moiré potentials in an atomically thin transition metal dichalcogenide heterostructure, which consists of two MoSe2 monolayers separated by hexagonal boron nitride. We observe optical signatures of robust correlated insulating states at symmetric (1:1) and asymmetric (3:1, 4:1 and 7:1) electron doping of the two MoSe2 layers at cryogenic temperatures. We attribute these features to bilayer Wigner crystals composed of two interlocked commensurate triangular electron lattices, stabilized by inter-layer interaction16. The Wigner crystal phases are remarkably stable, and undergo quantum and thermal melting transitions at electron densities of up to 6 × 1012 per square centimetre and at temperatures of up to about 40 kelvin. Our results demonstrate that an atomically thin heterostructure is a highly tunable platform for realizing many-body electronic states and probing their liquid–solid and magnetic quantum phase transitions4,5,6,7,8,17.

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## Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

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## Acknowledgements

We acknowledge support from the DoD Vannevar Bush Faculty Fellowship (N00014-16-1-2825 for H.P., N00014-18-1-2877 for P.K.), NSF CUA (PHY-1125846 for H.P., E.D. and M.D.L.), Samsung Electronics (for H.P. and P.K.), NSF (PHY-1506284 for H.P. and M.D.L., DGE-1745303 for E.B., DMR-2038011 for Y.W.), AFOSR MURI (FA9550-17-1-0002), ARL (W911NF1520067 for H.P. and M.D.L.), DOE (DE-SC0020115 for H.P. and M.D.L.), AFOSR (FA9550-21-1-0216 for E.D.) and BME’s TKP 2020 Nanotechnology grant (G.Z.). Device fabrication was carried out at the Harvard Center for Nanoscale Systems.  Y.W. acknowledges Frontera computing system at the Texas Advanced Computing Center for geometric optimization of crystal structures. I.E. and E.D. acknowledge resources of the National Energy Research Scientific Computing Center (NERSC) for phase transition calculations. NERSC is a US Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231.

## Author information

Authors

### Contributions

H.P. and Y.Z. conceived the project. Y.Z., J.S. and E.B. fabricated the samples, and designed and performed the experiments. I.E., Y.W., G.Z. and E.D. developed the theoretical model, and Y.Z., J.S., E.B., I.E. and Y.W. analysed the data. G.S. assisted with optical measurements, G.S. and R.J.G. assisted with sample fabrication, and H.H. grew the MoSe2 crystals. T.T. and K.W. provided hBN samples. Y.Z., J.S., E.B., I.E., Y.W., G.S., E.D. and H.P. wrote the manuscript with extensive input from the other authors. H.P., E.D., P.K. and M.D.L. supervised the project.

### Corresponding authors

Correspondence to Eugene Demler or Hongkun Park.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Peer review information Nature thanks the anonymous reviewers for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Extended data figures and tables

### Extended Data Fig. 1 Optical spectroscopic characterization of device D1 at T = 4 K.

a, A spatial map of integrated PL intensity at (0 V, 0 V). The PL intensity from the MoSe2/hBN/MoSe2, which is quite uniform throughout the entire region, is stronger than that from the monolayer regions, suggesting that both MoSe2 layers maintain their direct bandgap. b, Reflection contrast R/R0 at (0 V, 0 V). c, Representative PL spectra taken from the gated region of device D1 at 4 K. We find the standard deviation of the exciton energy across the gated region to be about 1 meV, much less than the exciton linewidth. The inset shows the spots where the PL spectra were collected, with the colour of the dot and of the spectrum curve matching each other. The dashed line encloses the bilayer region covered by both top and bottom gates. d, The linewidth of the neutral exciton X0 as a function of total electron density n, measured along nt:nb = 1:1 and 2:1, compared with a monolayer MoSe2. e, 2D maps of X0 linewidth as a function of Vtg/dtg and Vbg/dbg. Free charge carriers in MoSe2 can interact with X0 by introducing additional scattering, which leads to the broadening of X0 (refs. 20,21). The features I, II and IV are defined in the same way as in Fig. 2. Along the nt:nb = 1:1 and 4:1 features (I and II), we observe narrower X0 linewidth. This is further corroboration that the system is in insulating states along these density ratios. The blue, red and yellow dashed lines represent the linecuts where we extract data shown in the same colour in d. For large Vtg and Vbg (top right corner of the graph), the intensity of X0 becomes very weak and therefore the linewidth cannot be reliably extracted. f, 2D maps of integrated charged exciton emission (XT) as a function of Vtg/dtg and Vbg/dbg. The features I–IV are defined in the same way as in Fig. 2. We observe reduced XT emission along features I and II (Fig. 2b).

### Extended Data Fig. 2 Gate dependence of PL spectra from device D1 at 4 K.

ac, PL spectra as a function of Vbg with a fixed Vtg of 0 V (a), 2.5 V (b) and 5 V (c). df, PL spectra as a function of Vtg with a fixed Vbg of 0 V (d), 1.5 V (e) and 4 V (f). When the top (bottom) MoSe2 is intrinsic or highly doped, the application of Vbg (Vtg) mainly modifies the response of the bottom (top) layer. Here X0b and X0t represent the neutral exciton from the bottom and the top layer, respectively, while XTb and XTt represent the charged exciton from the bottom and the top layer, respectively. On the basis of such gate dependence, we can distinguish the contributions from the top versus the bottom layer when they have non-degenerate optical response (when the filling ratio is not 1:1). At intermediate gate voltage (b, e), however, the total PL of the neutral exciton is enhanced at a particular voltage configuration (denoted as I), corresponding to the feature I observed in Fig. 2.

### Extended Data Fig. 3 Gate dependence of reflectance spectra from device D1 at 4 K.

af, As for Extended Data Fig. 2 but for normalized reflectance, R/R0; the spectra are normalized to the background reflectance when both MoSe2 layers are highly doped with electrons.

### Extended Data Fig. 4 Optical characterization of device D1 at 100 mK.

a, b, 2D maps of integrated neutral exciton X0 PL (a) and the second derivative of X0 PL intensity with respect to the electric field (b), as a function of Vtg/dtg and Vbg/dbg. The density ratios can be determined from the troughs in b. We added guide lines I, II, III and V, which correspond to constant density ratio nt:nb of 1:1, 4:1, 7:1 and 3:1, respectively.

### Extended Data Fig. 5 Observation of bilayer Wigner crystals in another MoSe2/hBN/MoSe2 device, D2, at 4 K.

a, PL spectrum of D2 at zero gate bias. Inset, a microscope image of D2. In this device, the inter-layer hBN thickness is 1.6 nm. The white and yellow dashed lines indicate top and bottom monolayer MoSe2, respectively. Scale bar, 10 μm. b, c, 2D maps of neutral exciton X0 reflectance contrast R/R0 (b) and integrated X0 PL (c), as a function of Vtg/dtg and Vbg/dbg. The emerging insulating feature I corresponds to a density ratio of nt:nb = 1:1, with a lower critical density than found in device D1; we did not observe insulating states at other density ratios. Taken together, this suggests that bilayer Wigner crystals are less stable for this larger inter-layer separation. df, Gate-dependence of PL spectra from device D2 at 4 K with a fixed Vtg of 0 V (d), 0.8 V (e) and 2 V (f). gi, As df but with a fixed Vbg of 0 V (g), 0.725 V (h) and 2 V (i). Such gate-dependent behaviours are similar to those of D1.

### Extended Data Fig. 6 Optically detected resistance and capacitance measurements of device D1 at 4 K.

a, Schematic of the optically detected resistance and capacitance measurements of device D1 with a partial top gate and a global back gate (grey area at top and bottom, respectively). The green layers are hBN dielectrics. A d.c. bias is applied to the top and bottom gates (Vtg and Vbg, respectively) and a small a.c. bias ($$\Delta \tilde{V}$$) is superimposed on the top gate voltage. When the frequency of the a.c. voltage is high (typically a few kHz in our experiment), no charge can be injected from the metal contact into MoSe2, but there is charge redistribution between region 1 and region 2. The dynamics of such charge redistribution provide information on the quantum capacitance and resistance of the gated region, which can be probed by the reflectance change of excitons in the ungated region 2 (here we represent the incident probe light using the red arrow). b, A spatial map of integrated PL intensity showing the extent of the t-MoSe2 and b-MoSe2 layers as yellow and white dashed lines, respectively. The green circle represents the detection spot. The solid red lines show the outline of the partial top gates. c, Reflection spectra from the spot indicated by the green circle in b at (0 V, 0 V). The red dashed line represents the continuous wave laser centred at 1.637 eV for probing resistance and capacitance. d, A 2D map of neutral exciton reflectance contrast at 1.637 eV as a function of Vtg/dtg and Vbg/dbg. Reflectance contrast is not affected by the d.c. bias applied to the local top gate, Vtg, as expected. e, f, 2D maps of the reflectance contrast change ΔOC which is in-phase (e) and out-of-phase (f) with the a.c. modulation voltage. We observe a reduction in the in-phase component X and an increase in the out-of-phase component Y along the 1:1 feature. The magnitude of |ΔOC| is reduced along this feature. g, h, Extracted capacitance (g) and resistance (h) of region 1 based on the effective a.c. circuit model and equation presented in Methods. The coloured dashed lines indicate the bottom-gate voltages for the line profiles in Fig. 2d in the main text.

### Extended Data Fig. 7 Examples of procedures for estimating It(nt), Ib(nb) and δ.

a, b, The integrated PL intensity from X0, I0, as a function of Vbg (Vtg) while keeping Vtg (Vbg) at a large value. Because the X0 emission from the heavily doped layer can be omitted, these values can used to estimate It(nt) and Ib(nb): that is, I0(nt, nb+) = It(nt) + Ib(nb+) = It(nt), and I0(nt+, nb) = It(nt+) + Ib(nb) = Ib(nb). Here nt+ and nb+ denote the heavily electron-doped top and bottom layer, respectively. We note the value of I0 only begins to decrease above a finite gate voltage, which corresponds to the conduction band minimum. On the basis of this, we can estimate the carrier density using ΔVtgVbg), which is the applied top (bottom) gate voltage relative to the onset voltage corresponding to the conduction band edge. c, The value of I0 as function of Vbg while keeping Vtg at 2.5 V (yellow curve). To calculate δ, we first subtract It(2.5 V) from the raw I0 data (red curve). The relative enhancement of the red curve with respect to the Ib(Vbg) curve (blue) represents the enhancement due to bilayer Wigner crystal formation. d, The value of δ at Vtg = 2.5 V determined using this method. We note that near the band edge δ becomes less than zero because of the MoSe2 layer’s weak screening and the difficulties in estimating It(nt) or Ib(nb) when the layer is intrinsic.

### Extended Data Fig. 8 2D maps of integrated X0 PL intensity of device D1 as a function of Vtg/dtg and Vbg/dbg at various temperatures.

ai, 2D maps measured at temperatures (in K) of 4, 15, 17, 23, 30, 35, 40, 45 and 50, respectively. The insulating features become weaker and broader at higher temperatures and eventually disappear.

### Extended Data Fig. 9 2D maps of γ and Γ as a function of total density n and temperature T.

a, 2D map of Lindemann parameter γ. The dashed black and yellow line represents the contour of γ = 0.56, the phase boundary between the classical bilayer Wigner crystal and the electron liquid, estimated from the experimentally determined critical density at 4 K (see Supplementary Information for details). The white dashed line represents the density at which theory predicts the staggered triangular lattice structure becomes unstable, giving way to a sequence of structural transitions34. b, 2D map of Γ, which is the ratio of the average Coulomb energy to the thermally averaged kinetic energy per electron. The black dashed line represents Γ  = 17, also estimated from the experimentally determined critical density at 4 K. The black arrows along the x and y directions indicate respectively thermal melting and quantum melting of the bilayer Wigner crystal. We note that at finite temperatures Γ is not equivalent to the interaction parameter rs, which is the ratio of the Coulomb energy to the Fermi energy at 0 K (see Supplementary Information).

### Extended Data Fig. 10 Umklapp scattering from Wigner crystals measured in device D1 at 4 K.

a, The derivative of the reflectance contrast R/R0 with respect to photon energy E, as a function of bottom-layer carrier density nb, measured along nt:nb = 1 in device D1 at 4 K. In addition to the neutral exciton X0, an umklapp scattering peak may be seen, whose energy shifts linearly with carrier density. b, To further enhance the contrast of the features, we take the derivative of the reflectance contrast R/R0 with respect to Vbg; this clearly shows the evolution of both exciton X0 and its umklapp scattering. In the weak coupling regime, the exciton scattering will probably be dominated by the electrons from the same layer. In addition, for a bilayer Wigner crystal with the same density in each layer, the primitive unit cell is the same as that of each individual layer. On the basis of these considerations, we can estimate the exciton mass from the density dependence of the splitting between X0 and the umklapp peak. We estimate the exciton mass to be about 1.8me, which is in agreement with literature22, given the uncertainties in determining the slope of the energy shift and electron densities.

## Supplementary information

### Supplementary Information

This file contains the Supplementary Methods and Discussion and Supplementary Figures 1–7.

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Zhou, Y., Sung, J., Brutschea, E. et al. Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure. Nature 595, 48–52 (2021). https://doi.org/10.1038/s41586-021-03560-w

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