Electronic crystals in layered materials

Abstract

In modern two-dimensional (2D) materials, such as graphene-based systems and atomically thin transition-metal dichalcogenides, the interplay of strong electronic correlations, tunable moiré superlattices, and nontrivial band topology has given rise to rich phase diagrams and collective phenomena. Among the novel phases that have been realized, electronic crystals—states of matter in which itinerant electrons spontaneously crystallize—play a particularly prominent role. In this Review, we summarize the current status of electron crystallization in van der Waals heterostructures, with emphasis on the experimental platforms and measurement techniques that enable their study. We also highlight open questions and outline future directions that may elucidate more of the fascinating properties of electronic crystals.

Introduction

At sufficiently low carrier densities, a homogeneous electron system can spontaneously “freeze” into an ordered electronic “Wigner” crystal (WC), driven by the dominance of electron-electron interactions over kinetic energy. First predicted by Eugene Wigner nearly a century ago¹, these charge-ordered states are a paradigm of strongly correlated electronic states of matter (Fig. 1a).

Fig. 1. Electronic phase diagram: Properties and sensing.

a Schematic phase diagram of the idealized 2DEG as a function of 1/r_s (tuned by varying the electron density) and temperature, highlighting open questions. b Optical detection of an umklapp scattering peak in the reflection contrast measurements on monolayer MoSe₂, indicating the formation of a WC; adapted with permission from ref. ². c STM imaging of the WC and stripe phases in a bilayer graphene system at B = 13.95, 6, and 4 T (from left to right); adapted with permission from ref. ¹⁸. d–g New features of electron crystal formation that can be probed with 2D materials: d disorder effects captured by STM imaging on bilayer MoSe₂, adapted with permission from ref. ⁸; e multilayer crystals, as observed in bilayer MoSe₂ (adapted with permission from ref. ⁷); f spin/valley magnetism, recently studied in monolayer MoSe₂ (adapted with permission from ref. ³); g the interplay between electron crystallization and band topology, as studied in a moiré pentalayer graphene system (adapted with permission from ref. ²⁰); C is the Chern number.

While the possibility of electronic crystallization in metallic systems was a theoretical curiosity in Wigner’s time, it has become clear that electron crystals play a prominent role in the phase diagrams of modern two-dimensional (2D) quantum materials based on van der Waals (vdW) heterostructures, including graphene-based systems and atomically-thin transition-metal dichalcogenides (TMDs). Experimental realizations of electron crystals in 2D materials to date include single^2–6 and bilayer^7–9 TMDs, twisted TMDs with moiré superlattices^10–17, crystalline graphene multilayers¹⁸ and moiré graphene^19,20. These remarkable discoveries, enabled by advances in device fabrication and novel sensing and detection techniques (Fig. 1b, c), have led to a resurgence of interest in electron crystals. Moreover, the prominence of these phases suggests that their detailed characterization is crucial for a comprehensive understanding of the phase diagrams of charge-tunable vdW devices. These systems are also especially promising for addressing fundamental questions associated with electron crystallization, such as the effects of quenched disorder, their spin and valley magnetism, crystallization in multilayer systems, and interplay with band topology (see Fig. 1d–g).

The purpose of this article is to review recent advances in electron crystallization in layered 2D materials. The first section, “Wigner Crystals in layered materials”, summarizes the fundamental features of 2D WCs and explains why modern 2D platforms have proved especially fruitful for their study. In Section “Sensing Methods”, we describe the experimental detection methods special to 2D materials that have been important in the discovery and study of electron crystals. Finally, in the last section, “Outlook”, we conclude with a discussion of more exotic types of electron crystals that are presently the subject of active theoretical and experimental investigation.

Wigner Crystals in layered materials

WCs in an ideal setting

In the simplest situation, the low-carrier density state of a 2D semiconductor in the effective mass approximation is described by the homogeneous 2D electron gas (2DEG). The important energy scales are the Fermi energy E_F = ℏ²/m^*a² (We assume an isotropic, two-fold (spin or valley) degenerate electronic energy band.) and the Coulomb energy V_C = e²/4πϵ₀ϵ_ra, where m^* is the effective mass, a is interparticle distance (defined in relation to the 2D electron density n = 1/πa²), and ϵ_r is the dielectric constant of the environment in which the 2D layer is embedded. The single dimensionless parameter that determines the ground state properties of the system is $r_s\equiv V_C/E_F=a/a_B^{*}$, where $a_B^{*}=4\pi {ϵ}_0{ϵ}_r{\hslash }^2/m^{*}{e}^2$ is the effective Bohr radius of the semiconductor. Implicit in this description is a charge-neutralizing background, which is provided by the gate electrodes in the 2D materials under consideration.

In the high-density limit r_s → 0, the kinetic energy dominates over the interaction energy, and the important physics is the metallic screening of the long-range Coulomb interaction, which is adequately captured within the random-phase approximation (Another interesting possibility at high density is that of Kohn-Luttinger superconductivity²¹.). In the opposite limit of low density, r_s → ∞, the electron-electron interactions dominate over the kinetic energy and the electrons order into a triangular WC lattice to minimize their mutual Coulomb repulsion²². Quantum Monte Carlo calculations place the transition between the homogeneous liquid and WC phase at r_s ≈ 37^23–27, which is in good agreement with recent experiments^2,3,28.

However, even in this simplest 2DEG realization, there remain a number of important open questions concerning the ground state phase diagram. As r_s → ∞, residual exchange couplings between the electrons localized on the WC lattice sites lead to a ferromagnetic ground state, as can be deduced from semiclassical considerations^29–32. However, as the system approaches the melting transition with decreasing r_s, larger multiparticle ring exchange processes become important, leading to highly frustrated and complex exchange dynamics^25,27,33,34. Despite both theoretical and experimental progress, the question of the magnetic ground state of the WC near the melting point is not settled. Another central question concerns the role of intermediate phases between the WC and the homogeneous electron liquid. A direct first-order transition is forbidden due to the Coulomb energy penalty in the associated two-phase coexistence region³⁵, which has led to the proposal of intermediate mixed or micro-emulsion phases of electron liquid and crystal^35–37, a proposal that has recently received experimental support³. The distinct possibility of an intermediate metallic electron crystal phase, arising as a self-doping instability of the WC, was also recently investigated³⁸. The situation is summarized in the schematic phase diagram in Fig. 1a.

As will be elaborated upon below, the phase diagram is significantly enriched in 2D vdW materials, where additional axes—including disorder, layer number, and band topology—lead to new possibilities for electron crystallization.

2D materials platforms

Charge-tunable vdW heterostructures assembled by combining individual layers of various 2D materials are especially well-suited to the study of WC phases. In the case of graphene-based systems, the highly adjustable electronic bands—which can be significantly flattened using magnetic fields, displacement fields in the case of multilayers with rhombohedral stacking configuration^39,40, or by forming narrow bands in moiré systems—lead to situations where interaction effects are dominant. In TMD systems, electronic crystallization can occur spontaneously even in the translationally-invariant (within the effective mass approximation) monolayer limit owing to relatively large effective masses and reduced dielectric screening, which combine to give high critical densities for WC formation (see Fig. 2a, b and Table 1). Crucially, the elevated electron densities imply that the electronic state is less susceptible to effects from sample disorder (to be discussed in more detail below). This situation may be compared with the 2DEGs historically realized in semiconductor quantum wells, which need much lower densities to access the WC regime and correspondingly require extremely clean samples (see Table 1).

Fig. 2. Properties of two-dimensional semiconductors.

a 2D materials can feature electronic bands with relatively large effective mass. b The reduced dimensionality leads to strong Coulomb interactions among charged carriers. This, for instance, results in strong correlation effects and tightly bound excitons with large oscillator strengths. c In a prototypical device structure, the 2D heterostructures under study are encapsulated in dielectric 2D layers such as hBN and gated with top and bottom electrodes, allowing independent control over carrier density and electric field. The heterostructure may consist of a monolayer, an electrostatically coupled bilayer, or bilayers in direct contact hosting a moiré superlattice. The corresponding electronic and/or excitonic potential is shown on the right of the panel. These configurations exhibit distinct electronic and excitonic properties: for instance, a monolayer preserves continuous symmetry (in the effective mass approximation), while the presence of a moiré superlattice leads to discrete translation symmetry. The bilayer system lies in between, capable of exhibiting either continuous or discrete symmetry depending on the specific configuration, such as interlayer coupling strength.

Table 1.

Representative material parameters for electrons (e) and holes in (h) in various TMDs, compared with a few quantum well systems

	m*/me	ϵr	nc (1011cm−2)	μ (cm2/Vs)
MoSe2 (e)	0.7 ref. 3	4.5	2.0	3000 ref. 168
biMoSe2 (h)	1.26 ref. 8	2.6 ref. 8	20	?
WSe2 (h)	0.45 ref. 76	4.5	0.83	80,000 ref. 76
WS2 (h)	0.35 ref. 169	4.5	0.5	2,000 ref. 170
AlAs (e)	0.46 ref. 171	10 ref. 171	0.18	2.4 × 106 ref. 172
ZnO (e)	0.3 ref. 28	8.5 ref. 28	0.10	6 × 105 ref. 28
GaAs (e)	0.067 ref. 171	13 ref. 171	0.002	57 × 106 ref. 173

Here m^*/m_e is the band mass in units of the free electron mass; ϵ_r is an effective dielectric constant of the semiconductor environment, the precise value of which depends on details of the heterostructure and, unless otherwise specified, we have put ϵ_r ≈ 4.5 as a representative value for different TMDs¹⁷⁴; n_c is the theoretically estimated critical density for WC formation corresponding to r_s = 37; μ is the highest reported mobility in the corresponding material.

In addition to favorable band structures and material parameters, the high tunability of 2D materials offers routes to enhance the stability of electron crystals, while enabling engineered crystal structures with behaviors inaccessible in single monolayers. The most direct example of this is stacking 2D materials to form multilayer structures, where the interlayer distances, tunneling, and twist angle between layers can all be controlled experimentally. The simplest case is a (non-twisted) bilayer of two Coulomb-coupled 2DEGs. By varying the ratio of the interlayer spacing d to the interparticle spacing a, new WC geometries can be realized^7,41,42 (see also Section “Outlook”), and the crystal stability is enhanced due to the commensurate locking of the two layers^7,41,43 (Fig. 1e). Utilizing optical probes, the emergence of bilayer WCs has been confirmed in bilayer MoSe₂ (ref. ⁷).

When the 2D layers are twisted relative to each other, the resulting moiré patterns create a tunable external periodic potential experienced by the electrons (a similar effect can also arise in lattice-mismatched layers, or lattice mismatch with a substrate). The geometry, strength, and filling (the number of electrons per moiré unit cell) of the external potential serve as control parameters that can be used to stabilize the formation of electronic crystals and influence their properties. In the case of TMD heterobilayers, several groups have observed the formation of electronic crystals at certain rational fillings of the moiré superlattice in twisted WSe₂/WS₂^10–17. Similar observations have recently been made in pentalayer graphene^19,20. These moiré electron crystals—or “generalized Wigner crystals”⁴⁴—exhibit properties distinct from those formed in the absence of the moiré potential. For instance, the presence of the potential leads to the formation of new types of crystals beyond the simple triangular lattice, such as stripe^13,14 and honeycomb¹⁴ lattices at 1/2 and 2/3 filling of a triangular moiré lattice, respectively.

Yet another interesting feature of electron crystallization in 2D materials is the interplay of strong electronic correlations with the topology of the host electronic band. Recent experiments on both crystalline and moiré graphene multilayers^19,20,40,45 have provided evidence for the formation of topological electron crystals at zero magnetic field, where the crystallization appears to coexist with a quantized Hall conductance. Intriguingly, the presence/absence of the quantized Hall effect has been shown to be tunable by external displacement and magnetic fields.

Real materials: deviations from the ideal 2DEG

In real materials, the intricate interplay between electron-electron interactions and disorder plays a critical role in determining the properties of the WC, as well as the electron liquid to WC transition. Sufficiently strong disorder will induce Anderson localization, where electron wavefunctions become localized due to interference from fluctuations in the disorder potential. Although Anderson localization also creates an incompressible state, it differs fundamentally from a WC, which is characterized by strong positional correlations and a periodic electron arrangement.

Experimentally, the amount of disorder in actual TMD devices depends on various factors, including synthesis methods, growth conditions, dielectric environment, and the particular fabrication procedures employed for making vdW heterostructures. For example, in TMDs prepared via mechanical exfoliation or chemical vapor deposition (CVD), defect densities can range from 10¹² to above 10¹³ cm⁻² (ref. ⁴⁶). In comparison, high-quality graphene samples can have disorder densities as low as 10⁹ cm⁻² (refs. ^47–49). The flux growth method has shown promise in reducing both charged and isovalent defect densities in TMD samples to below 10¹⁰ and 10¹¹ cm⁻², respectively^50,51 (see Fig. 3a). This suggests that such samples could exhibit charge disorder well below the critical densities for WC formation, making them particularly promising for studying WCs in the clean limit. In addition to minimizing intrinsic disorder in the TMDs themselves, controlling the surrounding environment is crucial. For instance, encapsulating TMDs in hexagonal boron nitride (hBN) or suspending them has been shown to effectively suppress extrinsic disorder^52,53.

Fig. 3. Influence of disorder on electronic crystal phases in 2D materials.

a Topographic images of CVT-grown (left) and flux-grown (right) WSe₂ crystals, visualizing appreciable difference in disorder density; adapted with permission from ref. ⁴⁶. b STM image of an electronic Wigner crystal in hole-doped MoSe₂ bilayer (left) and a topography image of the same, undoped MoSe₂ flake (right), enabling direct visualization of defect-induced deformations of electronic lattice; adapted with permission from ref. ⁸. c Temperature- and d electron-density dependent conductivity of high-mobility electron-doped WSe₂ monolayer, showing clear signature of metal-to-insulator transition; adapted with permission from ref. ⁷⁶. e Elastic deformation of a WC in the presence of weak, short-range attractive (blue) and repulsive (red) impurities. f Schematic showing nucleation of a local Wigner “crystallite” by a charged impurity in the liquid phase of the 2DEG (the expected 2k_F Friedel oscillations set in far from the impurity). g Energy competition between the homogeneous electron liquid and crystal; weak disorder may tend to locally stabilize the crystal over the liquid phase owing to the gain in elastic energy.

While strong disorder leads to Anderson localization, a small amount of disorder can (locally) enhance the stability of the WC due to the energy gain from impurity pinning, lowering the critical r_s and increasing its melting temperature^54–56. Pictorially, one may imagine a pristine WC that becomes a “glassy” localized state pinned by the disorder, while retaining short-range order^31,32. Electron localization is reinforced by the combined effects of Coulomb interactions and impurity potentials, effectively stabilizing the WC phase; see Fig. 3e–g. The properties of such pinned WCs are well-studied^57–60, and the emergence of an additional “pinning resonance” in optical absorption measurements has been a hallmark signature of the WC phase in GaAs-based 2DEGs^61–66. Experimentally, a decreased critical r_s and elevated critical temperatures have been found in MoSe₂ monolayers^2,3 and bilayers⁸, suggesting an important role of the disorder.

Intriguingly, recent scanning tunneling microscopy (STM) experiments on bilayer MoSe₂ have directly imaged the disordered WC and the melting transition^8,9, allowing detailed investigations of the interplay between disorder and correlations. By directly comparing STM images with quantum Monte Carlo simulation, it was found that short-range disorder (arising from isovalent defects) acts to stabilize an “amorphous” WC to a significantly higher density⁹. In this context, an important direction concerns the connection between the WC transition and well-documented anomalies in the transport of strongly correlated 2DEGs⁶⁷, most notable of which is the metal-insulator transition. Local imaging, combined with recent progress in contacting TMD monolayers for transport studies (see Fig. 3c, d), promises to shed light on these fundamental questions.

In practice, the WCs realized in present 2D materials likely do not fit neatly into either the categories of “strong” or “weak disorder” described above⁶⁸, and careful analysis, along with reasonable quantitative criteria, is thus required to distinguish between the WC and Anderson (or Efros-Shklovskii⁶⁹) insulator phases. This is especially true when considering more subtle phenomena such as the nature of the transition between the liquid and WC phases and possible two-phase coexistence⁷⁰. We also note that the availability of local probes for studying 2D materials (as elaborated upon below) means that disorder does not immediately preclude the possibility of studying WC physics.

Beyond quenched disorder, there are other perturbations to the idealized clean 2DEG that are especially relevant to the WCs realized in layered 2D materials. These include screening of the Coulomb interaction by nearby gate electrodes—which render the Coulomb interactions short-range at separations larger than the distance to the gate and increase the critical r_s⁷¹—and electron-atomic phonon coupling, which has recently been argued to be more important for the properties of WCs realized in 2D materials than in conventional quantum well systems⁷².

Sensing methods

The hallmark of electronic crystals is the presence of long-range charge order. Even though this order is conceptually identical to that of atoms in regular crystals, the electronic case is notoriously difficult to probe. This is primarily due to the much lower densities involved, which render regular crystallography techniques, such as X-ray diffraction, inefficient for the detection of electronic crystals. Early experimental efforts thus relied on indirect signatures of WCs: instead of probing the charge order, they focused on the compressibility. As discussed in the previous section, the WCs in real materials are incompressible due to pinning of the electron lattice by disorder. For this reason, the WC acts as an insulator, and its formation thus manifests as an increased resistivity in DC or low-frequency transport experiments^73–75. Another consequence of disorder pinning is the emergence of a pinning-mode resonance in high-frequency AC transport experiments, which corresponds to collective oscillations of an electronic crystal about the disorder potential minima^61–66.

While these techniques have served as workhorses for WC explorations in conventional materials (e.g., GaAs), their application in the context of vdW heterostructures turned out to be more challenging. First, due to limited lateral extension and spatial inhomogeneities originating from the mechanical stacking process, standard transport experiments often prove unsuccessful, as they inherently average electronic properties over the entire sample. Owing to difficulties in making high-quality electrical contacts to semiconducting 2D materials, such measurements of low-density electronic crystals have been mostly limited to graphene-based systems, although there has been recent progress in this direction^76–78, including very recent THz spectroscopy experiments of the AC conductivity of a WC in a TMD monolayer⁷⁹. Nevertheless, from this perspective, local sensing techniques offering in-situ selection of investigated spatial areas are clearly advantageous. This includes both optical and scanning probe methods that are discussed below.

Optical spectroscopy of crystalline electronic phases in 2D materials

With its sub-micron spatial resolution, confocal spectroscopy provides a unique compromise between experimental complexities and local access to electronic phases. TMD-based systems are particularly well-suited for this approach thanks to their strong exciton binding energies, which are enhanced with respect to conventional materials for exactly the same reasons that are responsible for more prominent electronic correlations: weak dielectric screening and relatively large effective masses. This renders the excitons in these materials as robust bosonic impurities for sensing electronic phases even at relatively large carrier densities ~ 10¹² cm⁻² (refs. ^80,81). In the presence of charge carriers, the excitons form trions that give rise to repulsive and attractive Fermi polarons (AP), which dominate the luminescence and absorption spectra.

The most straightforward utilization of these optical excitations is local compressibility sensing (Fig. 4a). Experimentally, whenever the electron system in a TMD-based material becomes incompressible, the optical resonances display a sharp change in their intensity, energy, or linewidth. This applies not only to charge-ordered phases (Fig. 4b–d) such as WCs ⁷, Mott states ⁸², or Mott-Wigner crystals ⁸³, but also to other gapped states, such as integer quantum Hall liquids ^84–86. Excitonic sensing can also be operated non-locally to optically detect electrical conductivity ¹⁰. Moreover, it remains efficient in the case of multilayer electronic phases, even if they are composed of layers of the same material separated by insulating spacers. This is due to natural strain variations in vdW heterostructures, which typically result in resolvable differences between energies of excitons in various layers, thus enabling layer-selective spectroscopic readout of electronic compressibility using the excitonic transition originating from a specific layer. Such a method has recently allowed for probing the formation of bilayer Mott states in the system of MoSe₂ monolayers with a monolayer hBN spacer ⁸⁷ (Fig. 4c).

Fig. 4. Optical sensing of charge-ordered electronic phases.

a Cartoon illustrating local compressibility sensing with excitons residing in the same layer as the electron system. Spectroscopic signatures of the formation of incompressible phases in different bilayer systems: b Mott state in an angle-aligned WSe₂/WS₂ bilayer, adapted with permission from ref. ⁸²; c Mott state in a twisted MoSe₂ bilayer with a monolayer hBN spacer, adapted with permission from ref. ⁸⁷; d WC in a translationally-invariant MoSe₂ bilayer with 3-monolayer-thick hBN spacer, adapted with permission from ref. ⁷. e Illustration of remote optical sensing of electronic compressibility with Rydberg excitons in an adjacent layer. Drastic changes in spectral positions and weights of Rydberg excitons in the sensing layer originating from the formation of incompressible Mott and Mott-Wigner phases in: f a proximal WSe₂/WS₂ bilayer hosting a moiré potential (adapted with permission from ref. ¹¹) and g a MoSe₂ monolayer with remotely imprinted electrostatic potential (adapted with permission from ref. ⁹³). h Schematic showing Bragg-umklapp scattering of excitons off the electron crystal, allowing for optical sensing of long-range charge order. i The interaction between excitons and the ordered electronic lattice folds back excitonic bands, giving rise to new umklapp optical resonances, whose energy offset from the main exciton transition is directly determined by the lattice constant of the electronic crystal. Experimental Bragg-umklapp signatures of various charged-ordered phases: j a WC in a MoSe₂ monolayer (adapted with permission from ref. ²); k a Mott state in a twisted MoSe₂ bilayer with a monolayer hBN spacer (adapted with permission from ref. ⁹⁶); l a Mott-Wigner crystal in a MoSe₂ monolayer subjected to electrostatically-defined moiré potential due to proximal twisted hBN monolayer (adapted with permission from ref. ⁹⁷).

Another unique aspect of optical experiments is that they provide a direct interface to the spin state of correlated electrons. Owing to spin-valley locking in TMD-based systems, the circular polarization of optical excitations is determined by the spin polarization of resident electrons⁸⁸. This enables direct optical sensing of electronic magnetism, which has recently been exploited to probe the evolution of the magnetic susceptibility across the solid-to-liquid electron phase transition in a MoSe₂ monolayer³, shedding light on the evolution of exchange interactions across the WC melting transition. A similar technique has also been employed to study the collective magnetism of charge-ordered electronic phases in moiré bilayers^11,89–91.

In addition to the above optical sensing schemes involving electrons and excitons from the same layer, the electronic correlations in a given layer can be also probed using excitons residing in an adjacent layer. For example, by probing the onset of doping in this sensor layer, it is possible to optically detect the chemical potential of a remote electron system⁹². When the sensor layer remains charge-neutral, its Rydberg excitonic states can serve as sensitive optical probes of incompressible electronic phases (Fig. 4e) not only in proximal TMD bilayers systems^11,93 (Fig. 4f, g), but also in layered materials that are otherwise optically-inaccessible, such as graphene^94,95.

Despite their versatility, the above methods cannot be employed for detecting a crystalline electronic lattice directly. Much more powerful in this regard is Bragg-umklapp spectroscopy. At the heart of this approach is the periodic potential experienced by the exciton interacting with an electronic crystal (Fig. 4h). This allows for an optically-inactive exciton of finite momentum to get folded back to the light cone, provided that its momentum matches the reciprocal WC lattice vector k_W (Fig. 4i). This gives rise to a new umklapp resonance in the absorption spectrum, which is blueshifted by the kinetic energy of Bragg-scattered excitons ${\hslash }^2{k}_W^2/2m_X^{*}$. This resonance not only serves as a direct signature of long-range crystalline order, but also as a quantitative probe of the underlying lattice constant determining k_W. This method has been exploited both for sensing charge-ordered phases, including WCs in translationally-invariant monolayer² (Fig. 4j) and bilayer MoSe₂ systems ⁷, as well as Mott and Mott-Wigner states in TMD bilayers hosting moiré potentials ^96,97 (Fig. 4k, l). In the latter case, the key prerequisite for applicability of the method is the lack of periodic potential for excitons in the absence of electrons, which is realized in systems where conduction and valence bands experience moiré potentials of identical periodicity, such as twisted TMD bilayers with a monolayer hBN spacer^87,96 or TMD monolayers interfaced with twisted hBN bilayer⁹⁷.

Local probe sensing of crystalline electronic phases in 2D materials

To access WCs on length scales shorter than the optical wavelength, scanning probe sensing becomes the technique of choice. In this method, a sharp tip is brought in close proximity to a device, thereby not only achieving outstanding spatial resolution but also allowing to probe electronic properties without the need for high-quality electrical contacts. Among the multitude of local probe schemes (see ref. ⁹⁸ for a comprehensive recent review on their usage in 2D materials), those applicable for the spectroscopy of charge-ordered phases can be classified into two main categories depending on the physical observable they give access to.

The first one includes techniques such as microwave impedance microscopy (MIM) (Fig. 5a) or scanning electron transistor (SET) spectroscopy (Fig. 5b) that enable local measurements of macroscopic quantities like electronic conductivity or compressibility on length scales ~100 nm. These techniques, although not able to directly reveal the periodic electronic order, allow for distinguishing insulating and metallic phases, and have been successfully employed to sense the formation of Mott states, Mott–Wigner crystals¹² (Fig. 5c) or B-field-induced Wigner crystals^99,100 (Fig. 5d) in various vdW structures.

Fig. 5. Local probes of charge-ordered electronic phases.

a MIM and b SET sensing of electronic compressibility with ~100 nm spatial resolution. Example local signatures of the formation of incompressible electronic phases in c WSe₂/WS₂ bilayers using MIM (adapted with permission from ref. ¹²) and d magic-angle twisted bilayer graphene using SET (adapted with permission from ref. ⁹⁹). e STM probing of charge ordering with spatial resolution of single nanometers. Visualizations of real space lattices of various electronic phases in different systems: f Mott states in WSe₂/WS₂ bilayers (adapted with permission from ref. ¹⁴); g Wigner molecules in hole-doped twisted WS₂ bilayers (adapted with permission from ref. ¹⁰¹); h WCs in hole-doped natural MoSe₂ bilayers (adapted with permission from ref. ⁸); i electronic crystals in graphene subjected to strong magnetic fields (adapted with permission from ref. ¹⁸).

The second, more direct scheme, involves scanning tunneling microscopy (STM) (Fig. 5e) that offers spatial resolution at the level of single nanometers and uniquely enables visualization of the real-space structure of electronic crystals. In one of the first STM applications to electron crystals, the moiré bilayer hosting a generalized WC was not exposed directly to the STM tip, but covered with a thin insulating spacer and a doped graphene layer¹⁴. In this configuration, the tunneling current between the tip and graphene layer is spatially modulated due to interlayer interactions with the proximal WC, enabling direct mapping of the corresponding charge distribution with a spatial resolution limited by the hBN spacer thickness¹⁴ (Fig. 5f). More recently, this scheme has been further improved to enable direct sensing of WCs without the need for an intermediate metallic layer: by carefully balancing the band edge and vacuum energy levels in the tip and the probed layer, the tip-induced perturbation of the electron system was reduced to an extent that allowed for non-invasive sensing. This has enabled imaging¹⁰¹ the formation of Wigner molecules^102–106 at high filling factors of a deep moiré potential (Fig. 5g), quantum melting of the hole WC in bilayer MoSe₂ (ref. ⁸) (Fig. 5h), and the competition between the WC and fractional quantum Hall states in graphene subject to a Landau-quantizing magnetic field¹⁸ (Fig. 5i).

Outlook

More exotic electron crystals

The versatility of layered vdW materials opens up entirely new avenues for studying electron crystallization. Even in the single-layer limit, intriguing possibilities remain beyond those discussed above. For example, interband screening in gapped monolayer graphene can strongly influence electronic charge-ordering¹⁰⁷, while multivalley systems with anisotropic effective mass tensors may host interesting forms of pseudo-spin order (associated with the valley degree of freedom) at elevated temperature scales¹⁰⁸.

The modularity of 2D materials also offers unprecedented opportunities for heterostructure design, where stacking layers with distinct properties to form coupled multilayers significantly expands the accessible phase space. In addition to stabilizing novel phases, these structures can provide access to the fascinating phenomenology inherent to electronic crystals, such as their magnetic properties and excitation spectra.

A particularly elegant example of this, which has been mentioned already, is the Coulomb-coupled bilayer WC with like charges in both layers. Here, two additional axes appear in the ground state phase diagram: the interlayer separation and the density imbalance between layers. For commensurate layer densities, interlayer interactions stabilize a variety of geometries, including square and rectangular lattices^7,41, which may be directly imaged using scanning probe techniques. Additionally, gate tuning of individual layer concentrations allows for controlled doping away from commensurate densities. In this situation, a sufficiently small density imbalance has been theoretically predicted to result in defect-doped (interstitial or vacancy) metallic electron crystal ground states¹⁰⁹, which may themselves have further ordering instabilities.

Probing and manipulating the magnetic (either spin or valley pseudo-spin) states of WCs is another promising direction. While theory predicts a ferromagnetic ground state as r_s → ∞ in the monolayer WC, in the experimentally relevant range of densities, the combination of relatively weak exchange interactions and a high degree of frustration suppresses magnetic ordering temperatures (see Section “Wigner Crystals in layered materials”). This makes experimental investigation of magnetic properties challenging, despite significant interest.

Bilayer WCs offer a promising platform to address these challenges, as their higher critical densities lead to stronger exchange interactions. Moreover, recent theoretical work suggests that varying the interlayer coupling in different bilayer lattice configurations can yield distinct magnetic ordering¹¹⁰—including ferromagnetic and multi-sublattice antiferromagnetic states (Fig. 6a)—opening new avenues for controllable magnetism. In twisted multilayers, moiré potentials significantly modify the magnetic properties of electron crystals^111–116. Beyond conventional exchange dynamics, quantum tunneling of WC defects has recently been predicted to lead to kinetic magnetism with significantly elevated energy scales^{38,89,112,117,118}.

Fig. 6. Outlook.

a Bilayer Wigner crystals are predicted to host a variety of structural and magnetic phases depending on the interlayer coupling (adapted with permission from ref. ¹¹⁰). b An electrostatically coupled electron-hole bilayer system can support exotic phases, including bilayer Wigner crystals, excitonic superfluids, and more exotic Wigner supersolids (adapted with permission from ref. ¹²¹). c A Wigner crystal can imprint a periodic electrostatic potential on another neighboring layer as a source to create new ordered phases. d A schematic phase diagram as a function of interaction strength and Berry curvature concentration, showing regions of Fermi liquid, conventional Wigner crystal, and anomalous Hall crystal phases (adapted with permission from ref. ¹³³). e Schematic showing dressing of an exciton into Wigner polarons by excitations of a Wigner crystal.

Another intriguing possibility involves oppositely charged 2D layers (electrons and holes) brought into close proximity^119,120. At large layer separations and for sufficiently dilute densities, carriers form interlocking triangular WCs; as the separation decreases, stronger interlayer interactions lead to the formation of bound electron-hole pairs with phase coherence; i.e., superfluidity. This transition has been predicted to proceed via an intermediate bosonic supersolid—a quantum phase that combines crystalline order with superfluidity^121,122 (Fig. 6b). How to precisely engineer the optimal interlayer and intralayer Coulomb interactions—and how to detect these phases experimentally—remains an open and exciting frontier. Very recently, umklapp signatures of exciton crystallization in a bilayer system composed of monolayer MoSe₂ and a WS₂/WSe₂ moiré superlattice have been reported¹²³.

WCs themselves can also act as sources of tunable periodic potentials for adjacent layers, similar to the remote imprinting of moiré potentials^93,124–126. By dynamically tuning the WC, one can imprint a reconfigurable periodic landscape onto adjacent layers (Fig. 6c). This tunable potential enables the stabilization of a variety of phases, including charge-ordered states, dipolar insulators^127,128, and states with fractal energy spectra such as Hofstadter butterflies¹²⁹.

Finally, the interplay between electron crystallization and the topology of the host electronic band presents a new direction for exploration. With regards to magnetism, recent semiclassical calculations have shown that Berry curvature of the host electronic band can modify the spin dynamics qualitatively, leading to chiral terms in the effective exchange Hamiltonian that may stabilize chiral spin-density wave or chiral spin liquid phases^130,131. Perhaps even more dramatic, the breaking of translation symmetry in a band with Berry curvature via electron crystallization can lead to back-folded bands with non-zero Chern numbers and corresponding quantized Hall responses. While related “Hall crystal” phases were proposed decades ago in quantum Hall systems¹³², recent experiments in both graphene/hBN moiré superlattices^19,40,45 and twisted bilayer-trilayer graphene²⁰ have observed a quantized Hall resistance and vanishing longitudinal resistance at zero magnetic field that appears to coexist with some form of electron crystallization. In these systems, a quantized Hall effect appears at both commensurate fillings of the moiré lattice^19,20,40 and, more surprisingly, over extended ranges of filling⁴⁵, suggestive of a continuous translation symmetry breaking. The temperature dependence of the resistances and the threshold current-voltage characteristics also resemble those of WCs. These “anomalous Hall crystals” (AHCs), featuring both charge order and nontrivial topology in the form of a quantized Hall response, are now the subject of active theoretical investigation^133–151 (Fig. 6d). One could also imagine stabilizing an AHC by placing it adjacent to a topologically trivial WC that serves as a tunable “pinning potential”, as described in the preceding paragraph.

How can it be observed

New experimental approaches to studying WCs promise significant advances in understanding and controlling these intriguing phases. Although transport studies have been challenging, future improvements in metal-2D semiconductor contacts¹⁵² could lead to a detailed investigation of the thermodynamic stability and nonlinear transport properties of WCs. Microwave spectroscopy techniques^153,154, which can characterize the vibrations of WCs^22,155, including their phonons and pinning/de-pinning dynamics, could provide critical information on their dynamic properties and stability.

Scanning probe measurements have proven powerful for studying WCs and can be further enhanced by incorporating additional capabilities. For instance, integrating optical or microwave methods with scanning probe microscopy^{100,156–158} not only enables nanoscale mapping of WC’s response in the high-frequency domain, but also provides a powerful approach to manipulate electrons within WCs through optical or spin resonances. Furthermore, the recent development of ref. ¹⁵⁹, where a twisted 2D junction serves as the tunneling junction in scanning tunneling microscopy (STM), may reveal critical insights into the properties of WCs in momentum space.

Optical methods could also enable the detection of collective dynamics and spin physics. The dressing of excitons by the Wigner crystal is expected to lead to the formation of Wigner polarons (Fig. 6e)^{4,5,160–162}. Recent experiments suggest that these Wigner polarons could be observable in optical measurements as additional features in reflectance^4–6. Although the exact nature of Wigner polarons is not fully established—for example, regarding the role of WC phonons—these results indicate that optical studies can probe polaron dynamics and provide a platform for exploring Bose-Fermi mixtures. Optical spectroscopy has also been used to probe phonons in a moiré electron crystal¹⁶³. Furthermore, owing to the locking between electronic spin orientation and helicity of excitonic resonances, by probing temporal correlations of photons interacting with a TMD layer, it might be possible to infer the dynamics of spin fluctuations of the WC electrons and study their evolution across the quantum melting transition. This could reveal the formation of spin-ordered WC at sufficiently low temperatures, either with ferro- or anti-ferromagnetic spin arrangement. In the latter case, umklapp spectroscopy can uniquely enable one to probe the geometry and size of spin sublattices, which will interact differently with excitons of a given circular polarization. This is expected to give rise to a fine structure of the umklapp resonance¹⁶⁴. Terahertz optics has also recently been proposed as a means to probe and even manipulate valley pseudospin order in WCs¹⁶⁵.

Additionally, ultrafast optical techniques offer the capability to dynamically manipulate and control WCs. Such ultrafast control enables the exploration of transient phenomena and non-equilibrium states, potentially leading to novel quantum phases and transitions that are inaccessible under equilibrium conditions^166,167.

Collectively, these advanced methodologies could significantly deepen our understanding of both the ground and excited states of electronic crystals, and also open exciting new avenues for manipulating correlated electron systems more broadly.

Author contributions

This review has been conceptualized and written jointly by Y.Z., I.E. and T.S.

Data availability

No datasets were generated or analyzed during the current study.

Competing interests

The authors declare no competing interests.