Abstract
Wigner predicted that when the Coulomb interactions between electrons become much stronger than their kinetic energy, electrons crystallize into a closely packed lattice1. A variety of two-dimensional systems have shown evidence for Wigner crystals2,3,4,5,6,7,8,9,10,11 (WCs). However, a spontaneously formed classical or quantum WC has never been directly visualized. Neither the identification of the WC symmetry nor direct investigation of its melting has been accomplished. Here we use high-resolution scanning tunnelling microscopy measurements to directly image a magnetic-field-induced electron WC in Bernal-stacked bilayer graphene and examine its structural properties as a function of electron density, magnetic field and temperature. At high fields and the lowest temperature, we observe a triangular lattice electron WC in the lowest Landau level. The WC possesses the expected lattice constant and is robust between filling factor ν ≈ 0.13 and ν ≈ 0.38 except near fillings where it competes with fractional quantum Hall states. Increasing the density or temperature results in the melting of the WC into a liquid phase that is isotropic but has a modulated structure characterized by the Bragg wavevector of the WC. At low magnetic fields, the WC unexpectedly transitions into an anisotropic stripe phase, which has been commonly anticipated to form in higher Landau levels. Analysis of individual lattice sites shows signatures that may be related to the quantum zero-point motion of electrons in the WC lattice.
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Data availability
Other data that support the findings of this study are available from the corresponding author upon request. Source data are provided with this paper for the main figures and Extended Data figures.
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Acknowledgements
We acknowledge fruitful discussions with D. Huse, S. Kivelson and M. Heiblum. This work was primarily supported by DOE-BES grant DE-FG02-07ER46419 and the EPiQS initiative grants GBMF9469 of the Gordon and Betty Moore Foundation to A.Y. Other support for the experimental infrastructure was provided by NSF-MRSEC through the Princeton Center for Complex Materials NSF-DMR-2011750, DMR-2312311, ARO MURI (W911NF-21-2-0147) and ONR N00012-21-1-2592. A.Y. acknowledges the hospitality of the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, where part of this work was carried out. M.P.Z. and T.W. were supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division, under contract no. DE-AC02-05CH11231, in the van der Waals Heterostructures Program (KCWF16).
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Y.-C.T., M.H., Y.H. and A.Y. devised the experiments; Y.-C.T., M.H. and Y.H. created the structures of the devices and carried out the STM measurements and data analysis. E.L., T.W. and M.P.Z. carried out the theoretical calculations. K.W. and T.T. provided the h-BN substrates. All authors contributed to the writing of the paper.
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Extended data figures and tables
Extended Data Fig. 1 Effects of the bias voltage VB on the Wigner crystal images.
The two Idc images show the WC imaged at the same area, same ν with different bias voltages. Left: VB = 5 mV; right: VB = −5 mV. Notice that the Idc > 0 for VB > 0, and Idc < 0 for VB < 0. With the bias voltage VB applied on the sample and the tip grounded, the electron is tunneled from tip (sample) to the sample (tip) for positive (negative) bias voltage VB. The observed suppression (enhancement) of current indicates the presence of a localized electron. When VB > 0, a larger energy penalty is required for tunneling into the site, therefore a suppression of the current for a fixed bias voltage image; whereas for VB < 0, since there’s an extra charge to contribute from the site, an enhancement of the current for a fixed bias voltage image is expected. However, a slight shift of the positions of the sites is detected in these images with reversed bias, along with slight shape changes. These observations could potentially be explained by the tip perturbation. For example, it could be tip gating effects which can slightly change the local filling factor ν, resulting in the slight shift. Other than potential local gating from tip, the spatial shift of the localized electron might also be explained as results of a small horizontal electric field E (in the graphene sample) from voltage Vt (sum of work function mismatch and bias voltage VB) which is different for measurements taken at positive/negative coulmb gap edges. Other possibility, such as the piezo drifting effect has been excluded.
Extended Data Fig. 2 Full data set (I) of electronic ground state imaging at B = 13.95 T.
The data are presented in the sequence in rows of δIdc, \(S({\boldsymbol{q}})\), and autocorrelation of δIdc. The filling factor ν of each set is noted on the top. The scale bars for δIdc, \(S({\boldsymbol{q}})\), autocorrelation of δIdc are 100 nm, 0.2 nm−1, 100 nm, respectively.
Extended Data Fig. 3 Full data set (II) of electronic ground state imaging at B = 13.95 T.
The data are presented in the sequence in rows of δIdc, \(S({\boldsymbol{q}})\), and autocorrelation of δIdc. The filling factor ν of each set is noted on the top. The scale bars for δIdc, \(S({\boldsymbol{q}})\), autocorrelation of δIdc are 100 nm, 0.2 nm−1, 100 nm, respectively.
Extended Data Fig. 4 Full data set (III) of electronic ground state imaging at B = 13.95 T.
The data are presented in the sequence in rows of δIdc, \(S({\boldsymbol{q}})\), and autocorrelation of δIdc. The filling factor ν of each set is noted on the top. The scale bars for δIdc, \(S({\boldsymbol{q}})\), autocorrelation of δIdc are 100 nm, 0.2 nm−1, 100 nm, respectively.
Extended Data Fig. 5 Phase diagram and the extracted \(|{\bf{q}}|\) near the ν = 1/3 FQH state at B = 13.95 T.
The solid line is the expected magnitude \(|{{\bf{q}}}_{{\rm{W}}{\rm{C}}}|\) of the Bragg peaks of the WC. The grey mask represents where the FQHs ν = 1/3 sets in. The data are presented in solid triangles (WC), hollow triangles (distorted WC), and hollow circles (liquid). As approaching the ν = 1/3 FQH state, the extracted \(|{\bf{q}}|\) starts deviating away from the expected \(|{{\bf{q}}}_{{\rm{W}}{\rm{C}}}|\), but still the values are very close to one another. And on both sides of the FQHs there are liquid states, signifying a solid to liquid/liquid to solid transition near ν = 1/3.
Extended Data Fig. 6 Filling factor ν dependence of the extracted \({\boldsymbol{|}}{\bf{q}}{\boldsymbol{|}}\) at B = 13.95 T.
The solid line is the expected magnitude \(|{{\bf{q}}}_{{\rm{W}}{\rm{C}}}|\) of the Bragg peaks of the WC. The dashed line is the expected magnitude \(|{{\bf{q}}}_{{\rm{F}}{\rm{L}}}|\) for Fermi liquid or composite Fermi liquid, if we assume the interference pattern originates from the scattering between two opposite points \(\pm {{\bf{k}}}_{{\rm{F}}}\) on the Fermi surface at zero magnetic field. The data are presented in filled red circle (WC), hollow red circles (distorted WC), and filled blue circles (liquid). Most points are below \(|{{\bf{q}}}_{{\rm{W}}{\rm{C}}}|\). Interestingly, deviations of \(|{{\bf{q}}}_{{\rm{W}}{\rm{C}}}|\) are towards the direction away from \(|{{\bf{q}}}_{{\rm{F}}{\rm{L}}}|=2|{{\bf{k}}}_{{\rm{F}}}|=2\sqrt{4{\rm{\pi }}n}\). The error bars in vertical direction represent the fitting error of determining \(|{\bf{q}}|\), and horizontal error bars denote the uncertainty of the filling factor ν determination (see Method).
Supplementary information
Supplementary Information
Supplementary Note 1, Supplementary Figs. 1–17 and Supplementary Tables 1–6.
Supplementary Video 1
Imaging of a WC. Filling factor ν dependence of δIdc and S(q) maps at magnetic field B = 13.95 T and T = 210 mK.
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Tsui, YC., He, M., Hu, Y. et al. Direct observation of a magnetic-field-induced Wigner crystal. Nature 628, 287–292 (2024). https://doi.org/10.1038/s41586-024-07212-7
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DOI: https://doi.org/10.1038/s41586-024-07212-7
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