Imaging Generalized Wigner Crystal States in a WSe2/WS2 Moir\'e Superlattice

The Wigner crystal state, first predicted by Eugene Wigner in 1934, has fascinated condensed matter physicists for nearly 90 years2-14. Studies of two-dimensional (2D) electron gases first revealed signatures of the Wigner crystal in electrical transport measurements at high magnetic fields2-4. More recently optical spectroscopy has provided evidence of generalized Wigner crystal states in transition metal dichalcogenide (TMDC) moir\'e superlattices. Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Scanning tunneling microscopy (STM) in principle has sufficient spatial resolution to image a Wigner crystal, but conventional STM measurements can potentially alter fragile Wigner crystal states in the process of measurement. Here we demonstrate real-space imaging of 2D Wigner crystals in WSe2/WS2 moir\'e heterostructures using a novel non-invasive STM spectroscopy technique. We employ a graphene sensing layer in close proximity to the WSe2/WS2 moir\'e superlattice for Wigner crystal imaging, where local STM tunneling current into the graphene sensing layer is modulated by the underlying electron lattice of the Wigner crystal in the WSe2/WS2 heterostructure. Our measurement directly visualizes different lattice configurations associated with Wigner crystal states at fractional electron fillings of n = 1/3, 1/2, and 2/3, where n is the electron number per site. The n=1/3 and n=2/3 Wigner crystals are observed to exhibit a triangle and a honeycomb lattice, respectively, in order to minimize nearest-neighbor occupations. The n = 1/2 state, on the other hand, spontaneously breaks the original C3 symmetry and forms a stripe structure in real space. Our study lays a solid foundation toward the fundamental understanding of rich Wigner crystal states in WSe2/WS2 moir\'e heterostructures.

2 Abstract: The Wigner crystal state, first predicted by Eugene Wigner in 1934 1 , has fascinated condensed matter physicists for nearly 90 years [2][3][4][5][6][7][8][9][10][11][12][13][14] . Studies of two-dimensional (2D) electron gases first revealed signatures of the Wigner crystal in electrical transport measurements at high magnetic fields [2][3][4] . More recently optical spectroscopy has provided evidence of generalized Wigner crystal states in transition metal dichalcogenide (TMDC) moiré superlattices [6][7][8][9] . Direct observation of the 2D Wigner crystal lattice in real space, however, has remained an outstanding challenge. Scanning tunneling microscopy (STM) in principle has sufficient spatial resolution to image a Wigner crystal, but conventional STM measurements can potentially alter fragile Wigner crystal states in the process of measurement. Here we demonstrate real-space imaging of 2D Wigner crystals in WSe2/WS2 moiré heterostructures using a novel non-invasive STM spectroscopy technique. We employ a graphene sensing layer in close proximity to the The long pursuit of Wigner crystals 2-10 has motivated the study of 2D electron gases at high magnetic field where electron kinetic energy is quenched by degenerate Landau levels 15,16 and has led to the discovery of new quantum hall states 17,18 . Electrical transport signatures of Wigner crystal states have been reported in extremely clean GaAs/AlGaAs quantum wells 2,3 as well as graphene 4 at sufficiently low doping and high magnetic field. Signs of Wigner crystallization have also been detected for electrons trapped at the surface of liquid helium [11][12][13][14] . Recently, the discovery of moiré flat minibands in van der Waals heterostructures has opened a new route to realize Wigner crystal states at zero magnetic field. Several optical and conductance measurements have provided evidence of rich generalized Wigner crystal states in different TMDC moiré superlattices [6][7][8][9] . Direct observation of the real-space electron lattice in 2D, however, has remained challenging experimentally.
Real-space imaging of 2D Wigner crystals requires a measurement technique that satisfies several stringent requirements. It must (1) have sufficient spatial resolution to resolve the electron lattice, (2) have sufficient sensitivity to detect the presence of single electrons in the lattice, (3) be adequately non-invasive to not destroy fragile Wigner crystal states. The last two requirements conflict with each other since strong coupling to the Wigner crystal is required for high sensitivity, whereas weak coupling is required to avoid strongly perturbing fragile states.
For example, conventional STM measurements have excellent spatial resolution and charge sensitivity but can be highly invasive since inevitable tip-gating effects at finite tip bias can destroy the delicate electron lattice of the Wigner crystal. In this work, we utilize a novel STM measurement scheme that strikes a balance between these two contradictory requirements, thus enabling real-space imaging of the n=2/3, n=1/2, and n=1/3 2D Wigner crystal states in WSe2/WS2 moiré heterostructures.
Our new STM scheme employs a specially designed van der Waals heterostructure as illustrated in Fig. 1a (see Methods for the sample fabrication details). It integrates a gated WSe2/WS2 moiré heterostructure and a top graphene monolayer sensing layer that are separated by a hexagonal boron nitride (hBN) layer with a thickness dt = 5nm, chosen to be smaller than the moiré lattice constant (LM = 8nm). This separation is small enough that the STM tip and graphene sensing layer can efficiently couple to individual moiré electrons in the WSe2/WS2 superlattice, but it is large enough that the tip and graphene layer remain non-invasive with respect to the delicate Wigner crystal states. STM tunneling current into the graphene sensing layer can be modulated by the charge states of different moiré sites in the WSe2/WS2 superlattice through local Coulomb blockade effects 19 . This technique allows us to detect the local charge distribution in the WSe2/WS2 heterostructure and to image the embedded Wigner crystal lattice. We implement dual gates in our van der Waals heterostructure devices ( Fig. 1a)  For VTG = 0, the Fermi level is within the band gap for the WSe2/WS2 heterostructure (see illustration in Fig. 1d). In this case, tuning VBG dopes charge carriers exclusively into the graphene layer. Fig. 1e shows a 2D plot of the STM differential conductivity (dI/dV) spectra of This inelastic tunneling gap causes the graphene CNP curve to abruptly shift as it shifts over the zero-bias region 22,24 .
We are able to dope electrons into the WSe2/WS2 heterostructure by applying a positive VTG such that the Fermi level of the WSe2/WS2 heterostructure lies near the conduction band edge (see illustration in Fig. 1f). Here we choose VTG ~ 0.5V so that the WSe2/WS2 heterostructure can be electron doped while the graphene sensing layer remains close to charge neutral. The reason for doing this is that the resulting small density of states for graphene provides the highest sensitivity for imaging Wigner crystal states in the moiré superlattice.
Charge neutral graphene also has less of a screening effect on the moiré electron-electron interactions due to the long screening length of Dirac electrons at the CNP 25 . Fig. 1g shows the resulting dI/dV tunneling spectra into the graphene sensing layer as a function of VBG at a fixed VTG = 0.53V. This panel corresponds to the same {VBG, Vbias} phase space outlined by the dashed white box in Fig.1e, but for nonzero VTG.    The n=1/2 generalized Wigner crystal state is predicted to be highly degenerate, with multiple electron lattice configurations having the same energy in the case of only nearestneighbor interactions 7 . The spontaneous broken symmetry of the n = 1/2 state might therefore be governed by higher-order effects, such as next-nearest-neighbor interactions and/or accidental strain in the lattice. Experimentally we here found that the n = 1/2 state is more fragile than the n=1/3 and n=2/3 states. A well-defined generalized Wigner crystal stripe phase is present only in a very narrow parameter space of Vbias and VBG. The n = 1/2 state is also more sensitive to local inhomogeneity, as reflected by the disordering of the stripe electron lattice near the right edge of the image in Fig. 2i as compared to the more defect-free n=2/3 (Fig. 2c) and n=1/3 (Fig. 2e) states. Further studies of the generalized Wigner crystal electron lattice at n = 1/2 could potentially lead to a better understanding of the competition of different quantum phases controlled by long-range Coulomb interactions.
We last discuss the imaging mechanism underlying the dI/dV mapping of generalized To distinguish between these two mechanisms we have systematically examined how the dI/dV maps evolve as Vbias is changed. Figs. 3a-e show dI/dV maps of the n = 2/3 Wigner crystal state as Vbias is increased from 130mV to 190mV in 15mV steps. No FFT filtering was performed on these images. The honeycomb electron lattice associated with the n = 2/3 generalized Wigner crystal state is not so clearly seen in the map at Vbias = 130 mV (Fig. 3a), but emerges when Vbias is increased to 145 mV (Fig. 3b). The dominant features are the bright dots centered on the AB1 stacking sites. These features expand with increased Vbias (Fig. 3c) and ultimately form ring-like features (Figs. 3d,e). Such behavior (i.e. expanding rings with increased tip bias) is characteristic of tip-induced electrical discharging rings 19,31-34 and occurs because electrical discharging for larger tip-electron distances requires larger tip biases. This indicates that mechanism (2) discussed above is the dominant contrast mechanism for imaging Wigner crystal states in our dI/dV maps. The STM tip locally discharges the moiré electron localized at the AB1 site closest to the tip apex once Vbias is large enough and the tip-electron distance is short enough. This enables discharge features centered around filled AB1 sites to be observed in dI/dV maps of the graphene sensing layer.
In conclusion, we have developed a new STM imaging technique that combines high spatial resolution and sensitivity with minimal perturbation to the probed electronic system. This  Zoom-in image of the red dashed box in (b). The red rhombus labels a primitive cell. Peaks correspond to AA stacking regions and the two inequivalent low points correspond to distinct AB stacking regions (denoted AB1 and AB2). d. Schematic of the heterostructure band alignment and Fermi levels for VTG = 0 and VBG > 0. At zero VTG, the Fermi level of the WSe2/WS2 heterostructure is located in the band gap. e. VBG-dependent dI/dV spectra measured on the graphene sensing layer over an AA stacking site for VTG = 0. The dispersive feature marked by the white dotted curve shows the evolution of the graphene charge neutral point (CNP) induced by electrostatic doping from VBG. The persistent gap near Vbias = 0 arises from an inelastic tunneling gap that exists at all gate voltages. Due to this inelastic tunneling gap, the graphene CNP curve shows an abrupt shift as it shifts over the zero-bias region. The tip height was set by the following parameters: Vbias = -300mV and I = 100 pA. f. Schematic of the heterostructure band alignment and Fermi levels for VTG > 0 and VBG > 0. Application of an appropriate positive VTG allows the Fermi level of the WSe2/WS2 heterostructure to be lifted into the conduction band. g. VBG-dependent dI/dV spectra measured on the graphene sensing layer over an AA stacking site for VTG = 0.53V. This is a zoom-in of the electron doped regime corresponding to the phase space denoted by the white dashed box in e. The graphene is hole-doped in the region below the horizontal dashed line (VBG < 7V), and the WSe2/WS2 is electron-doped in the region above it (VBG > 7V). The vertical white dash curve indicates the expected movement of the graphene CNP for a non-interacting picture. Significant electron doping of the graphene layer takes place at n = 1/3, 1/2, 2/3, and 1 (denoted by horizontal black dashed lines in (g)). The tip height was set by the following parameters: Vbias = -200mV and I = 100 pA. h. Vertical line-cut through the VBG-dependent dI/dV spectra in (g) at Vbias = 0.1V shows peaks at n=1, 2/3, 1/2, and 1/3.

Notes
The authors declare no financial competing interests. if not further specified.

Data availability
The data supporting the findings of this study are included in the main text and in the Supplementary Information files, and are also available from the corresponding authors upon reasonable request. We observe that the Dirac point (characterized by a dip in the LDOS and marked by grey arrows)

Individual dI/dV spectra
shifts from positive bias to negative bias voltage with increased VBG, corresponding to a change from the graphene hole-doped regime to the graphene electron-doped regime. The small tunneling current close to zero bias voltage is due to reduced tunneling from graphene electrons at the K and K' points of the graphene band structure 1 (phonon-induced inelastic tunneling additionally causes rises in tunnel current that can result in gap-like features in graphene dI/dV spectra 1 ).  b we display typical dI/dV spectra at n = 1/3, 1/2, 2/3, and 1 for correlated states (red) as well as for three other filling factors that lack correlated states (black). In each panel the dI/dV spectra are shifted vertically for clarity. The spectra indicate that the graphene sensing layer is more electron doped when the moiré heterostructure is in a correlated insulator state.
2. Moiré site dependence of the dI/dV spectra

Raw images and FFT filtering of the generalized Wigner crystal states
The dI/dV images of the generalized Wigner crystal states shown in Fig. 2 of the main text are FFT filtered to suppress the periodic feature associated with the moiré superlattice. Figure S3 shows the unfiltered raw images of the generalized Wigner crystal states and the process of the FFT filtering. shows the corresponding filtered FFT image (same as Fig. 2f in the main text).
Similar filtering procedures are performed for the n = 1/3 and n = 1/2 Wigner crystal states. Figure S3e and S3f show the raw real space and FFT images, respectively, of the n=1/3 Wigner crystal state, and the corresponding FFT filtered images are shown in Fig. S3g and S3h. Figure S3i and S3j show the raw real space and FFT images, respectively, of the n=1/2 Wigner crystal state, and the corresponding FFT filtered images are shown in Fig. S3k and S3l. Figure S3. Raw images and FFT filtering of the dI/dV maps for the generalized Wigner crystal states. a. Raw dI/dV map of the n = 2/3 state. b. FFT image of (a). c. Real space dI/dV map after FFT filtering of (a). In the filtering process, we removed the Fourier components within the six red circles indicated in (b). This FFT filtering suppresses the periodic feature associated with the moiré superlattice. d. FFT image of (c). e. Raw dI/dV map of the n = 1/3 state. f. FFT image of (e). g. Real space dI/dV map after FFT filtering of (e). The Fourier components within the red circles shown in (f) have been filtered out. h. FFT image of (g). i. Raw dI/dV map of the n = 1/2 state. j. FFT image of (i). k. Real space dI/dV map after FFT filtering of (i). The Fourier components within the red circles shown in (j) have been filtered out. l. FFT image of (k).