A tunable bilayer Hubbard model in twisted WSe2

Hidden states and dynamics of fractional fillings in twisted MoTe2 bilayers

The fractional quantum anomalous Hall (FQAH) effect was recently discovered in twisted MoTe2 (tMoTe2) bilayers1-4. Experiments so far have revealed Chern insulators from hole doping at ν = -1, -2/3, -3/5 and -4/7 (per moiré unit cell)1-6. In parallel, theories predict that, between v = -1 and -3, there exist exotic quantum phases7-15, such as the coveted fractional topological insulators, fractional quantum spin Hall (FQSH) states and non-Abelian fractional states. Here we use transient optical spectroscopy16,17 on tMoTe2 to reveal nearly 20 hidden states at fractional fillings that are absent in static optical sensing or transport measurements. A pump pulse selectively excites charge across the correlated or pseudogaps, leading to the disordering (melting) of correlated states18. A probe pulse detects the subsequent melting and recovery dynamics by means of exciton and trion sensing1,3,19-21. Besides the known states, we observe further fractional fillings between ν = 0 and -1 and a large number of states on the electron doping side (ν > 0). Most importantly, we observe new states at fractional fillings of the Chern bands at ν = -4/3, -3/2, -5/3, -7/3, -5/2 and -8/3. These states are potential candidates for the predicted exotic topological phases7-15. Moreover, we show that melting of correlated states occurs on two distinct timescales, 2-4 ps and 180-270 ps, attributed to electronic and phonon mechanisms, respectively. We discuss the differing dynamics of the electron-doped and hole-doped states from the distinct moiré conduction and valence bands.

Correlated states controlled by a tunable van Hove singularity in moiré WSe2 bilayers

Twisted transition metal dichalcogenide (TMD) bilayers have enabled the discovery of superconductivity, ferromagnetism, correlated insulators, and a series of new topological phases of matter. However, the connection between these electronic phases of matter and the underlying band structure singularities has remained largely unexplored. Here, combining magnetic circular dichroism and exciton sensing measurements, we investigate the influence of a van Hove singularity (vHS) on the correlated phases in bilayer WSe2 with twist angle between 2 and 3 degrees. By tuning the vHS across the Fermi level using electric and magnetic fields, we observe Stoner ferromagnetism below moiré lattice filling one and Chern insulators at filling one. The experimental observations are supported by the continuum model band structure calculations. Our results highlight the prospect of engineering electronic phases of matter in moiré materials by tunable van Hove singularities.

Valley charge-transfer insulator in twisted double bilayer WSe2

In flat-band systems, emergent physics can be substantially modified by the presence of another nearby electronic band. For example, a Mott˘Hubbard insulator can turn into a charge transfer insulator if other electronic states enter between the upper and lower Hubbard bands. Here, we introduce twisted double bilayer (TDB) WSe2, with twist angles near 60°, as a controllable platform in which the K-valley band can be tuned to close vicinity of the Γ-valley moiré flat band. At half-filling, correlations split the Γ-valley flat band into upper and lower Hubbard bands and a charge-transfer insulator forms between the Γ-valley upper Hubbard band and K-valley band. Using gate control, we continuously move the K-valley band across the Γ-valley Hubbard bands, and observe a tunable charge-transfer insulator gap and subsequently a continuous phase transition to a metal. The tuning of Mott˘Hubbard to charge-transfer insulator establishes valley degree of freedom as a suitable knob for transitions between exotic correlated phases.

Two-Dimensional Transition Metal Dichalcogenides: A Theory and Simulation Perspective

Two-dimensional transition metal dichalcogenides (2D TMDs) are a promising class of functional materials for fundamental physics explorations and applications in next-generation electronics, catalysis, quantum technologies, and energy-related fields. Theory and simulations have played a pivotal role in recent advancements, from understanding physical properties and discovering new materials to elucidating synthesis processes and designing novel devices. The key has been developments in ab initio theory, deep learning, molecular dynamics, high-throughput computations, and multiscale methods. This review focuses on how theory and simulations have contributed to recent progress in 2D TMDs research, particularly in understanding properties of twisted moiré-based TMDs, predicting exotic quantum phases in TMD monolayers and heterostructures, understanding nucleation and growth processes in TMD synthesis, and comprehending electron transport and characteristics of different contacts in potential devices based on TMD heterostructures. The notable achievements provided by theory and simulations are highlighted, along with the challenges that need to be addressed. Although 2D TMDs have demonstrated potential and prototype devices have been created, we conclude by highlighting research areas that demand the most attention and how theory and simulation might address them and aid in attaining the true potential of 2D TMDs toward commercial device realizations.

Time-domain signatures of distinct correlated insulators in a moiré superlattice

Among expanding discoveries of quantum phases in moiré superlattices, correlated insulators stand out as both the most stable and most commonly observed. Despite the central importance of these states in moiré physics, little is known about their underlying nature. Here, we use pump-probe spectroscopy to show distinct time-domain signatures of correlated insulators at fillings of one (ν = -1) and two (ν = -2) holes per moiré unit cell in the angle-aligned WSe2/WS2 system. Following photo-doping, we find that the disordering time of the ν = -1 state is independent of excitation density (nex), as expected from the characteristic phonon response time associated with a polaronic state. In contrast, the disordering time of the ν = -2 state scales with 1 / n ex , in agreement with plasmonic screening from free holons and doublons. These states display disparate reordering behavior dominated either by first order (ν = -1) or second order (ν = -2) recombination, suggesting the presence of Hubbard excitons and free carrier-like holons/doublons, respectively. Our work delineates the roles of electron-phonon (e-ph) versus electron-electron (e-e) interactions in correlated insulators on the moiré landscape and establishes non-equilibrium responses as mechanistic signatures for distinguishing and discovering quantum phases.

Magnetochiral charge pumping due to charge trapping and skin effect in chirality-induced spin selectivity

Chirality-induced spin selectivity (CISS) generates giant spin polarization in transport through chiral molecules, paving the way for novel spintronic devices and enantiomer separation. Unlike conventional transport, CISS magnetoresistance (MR) violates Onsager's reciprocal relation, exhibiting significant resistance changes when reversing electrode magnetization at zero bias. However, its underlying mechanism remains unresolved. In this work, we propose that CISS MR originates from charge trapping that modifies the electron tunneling barrier and circumvents Onsager's relation, distinct from previous spin polarization-based models. Charge trapping is governed by the non-Hermitian skin effect, where dissipation leads to exponential wavefunction localization at the ferromagnet-chiral molecule interface. Reversing magnetization or chirality alters the localization direction, changing the occupation of impurity/defect states in the molecule (i.e., charge trapping) - a phenomenon we term magnetochiral charge pumping. Our theory explains why CISS MR can far exceed the ferromagnet spin polarization and why chiral molecules violate the reciprocal relation but chiral metals do not. Furthermore, it predicts exotic phenomena beyond the conventional CISS framework, including asymmetric MR induced by magnetic fields alone (without ferromagnetic electrodes), as confirmed by recent experiments. This work offers a deeper understanding of CISS and opens avenues for controlling electrostatic interactions in chemical and biological systems through the magnetochiral charge pumping.

Scanning Electron Microscopy Imaging of Twist Domains in Transition Metal Dichalcogenide Heterostructures

Twisted two-dimensional (2D) material heterostructures provide an exciting platform for investigating fundamental physical phenomena. Many of the most interesting behaviors emerge at small twist angles, where the materials reconstruct to form areas of perfectly stacked crystals separated by partial dislocations. However, understanding the properties of these systems is often impossible without correlative imaging of their local reconstructed domain configuration, which exhibits random variations due to disorder and contamination. In particular, visualization of the local domain configuration allows determination of the local twist angle and, hence, the local lattice strain. Here, we demonstrate a simple and widely accessible route to visualize domains in the as-produced twisted transition metal dichalcogenide (TMD) heterostructures using electron channeling contrast imaging (ECCI) in scanning electron microscopy (SEM). This nondestructive approach is compatible with conventional substrates and allows domains to be visualized even when sealed beneath an encapsulation layer. Complementary theoretical calculations reveal how a combination of elastic and inelastic scattering leads to contrast inversions at the specified detector scattering angles and sample tilts. We demonstrate that optimal domain contrast is therefore achieved by maximizing signal collection while avoiding contrast inversion conditions.

Enhancing Resonant Second-Harmonic Generation in Bilayer WSe2 by Layer-Dependent Exciton-Polaron Effect

Two-dimensional (2D) materials serve as exceptional platforms for controlled second-harmonic generation (SHG). Current approaches to SHG control often depend on nonresonant conditions or symmetry breaking via single-gate control. Here, we employ dual-gate bilayer WSe2 to demonstrate an SHG enhancement concept that leverages strong exciton resonance and a layer-dependent exciton-polaron effect. By selectively localizing injected holes within one layer, we induce exciton-polaron states in the hole-filled layer while maintaining normal exciton states in the charge-neutral layer. The distinct resonant conditions of these layers effectively break interlayer inversion symmetry, thereby promoting resonant SHG. This method achieves a remarkable 40-fold enhancement of SHG at minimal electric field, equivalent to conditions near the dielectric-breakdown threshold but using only ∼3% of the critical breakdown field. Our results highlight SHG sensitivity to carrier density and type, offering a new tool for manipulating SHG and probing quantum states in 2D excitonic systems.

Realizing one-dimensional moiré chains with strong electron localization in two-dimensional twisted bilayer WSe2

Two-dimensional (2D) moiré systems based on twisted bilayer graphene and transition metal dichalcogenides provide a promising platform to investigate emergent phenomena driven by strong electron-electron interactions in partially filled flat bands. A natural question arises: Is it possible to expand the 2D correlated moiré physics to one-dimensional (1D) that electron-electron correlation is expected to be further enhanced? This requires selectively doping of 1D moiré chain, which seems to be not within the grasp of today's technology. Therefore, an experimental demonstration of the 1D moiré chain with partially filled electronic states remains absent. Here, we show that we can introduce 1D boundaries, separating two regions with different twist angles, in twisted bilayer WSe2 (tWSe2) by using scanning tunneling microscopy (STM) and demonstrate that the electronic states of 1D moiré sites along the boundaries can be selectively filled. The strong localized charge states of correlated moiré electrons in the 1D moiré chain can be directly imaged and manipulated by combining a back-gate voltage with the STM bias voltage. Our results open the door for realizing new correlated electronic states of the 1D moiré chain in 2D systems.

Low-Defect-Density Monolayer MoS2 Wafer by Oxygen-Assisted Growth-Repair Strategy

Atomic chalcogen vacancy is the most commonly observed defect category in two dimensional (2D) transition-metal dichalcogenides, which can be detrimental to the intrinsic properties and device performance. Here a low-defect density, high-uniform, wafer-scale single crystal epitaxial technology by in situ oxygen-incorporated "growth-repair" strategy is reported. For the first time, the oxygen-repairing efficiency on MoS2 monolayers at atomic scale is quantitatively evaluated. The sulfur defect density is greatly reduced from (2.71 ± 0.65) × 1013 down to (4.28 ± 0.27) × 1012 cm-2, which is one order of magnitude lower than reported as-grown MoS2. Such prominent defect deduction is owing to the kinetically more favorable configuration of oxygen substitution and an increase in sulfur vacancy formation energy around oxygen-incorporated sites by the first-principle calculations. Furthermore, the sulfur vacancies induced donor defect states is largely eliminated confirmed by quenched defect-related emission. The devices exhibit improved carrier mobility by more than three times up to 65.2 cm2 V-1 s-1 and lower Schottky barrier height reduced by half (less than 20 meV), originating from the suppressed Fermi-level pinning effect from disorder-induced gap state. The work provides an effective route toward engineering the intrinsic defect density and electronic states through modulating synthesis kinetics of 2D materials.

Twist-Controlled Ferroelectricity and Emergent Multiferroicity in WSe2 Bilayers

Recently, researchers have been investigating artificial ferroelectricity, which arises when inversion symmetry is broken in certain R-stacked, i.e., zero-degree twisted, van der Waals (vdW) bilayers. Here, the study reports the twist-controlled ferroelectricity in tungsten diselenide (WSe2) bilayers. The findings show noticeable room temperature ferroelectricity that decreases with twist angle within the range 0° < θ < 3°, and disappears completely for θ ≥ 4°. This variation aligns with moiré length scale-controlled ferroelectric dynamics (0° < θ < 3°), while loss beyond 4° may relate to twist-controlled commensurate to non-commensurate transitions. This twist-controlled ferroelectricity serves as a spectroscopic tool for detecting transitions between commensurate and incommensurate moiré patterns. At 5.5 K, 3° twisted WSe2 exhibits ferroelectric and correlation-driven ferromagnetic ordering, indicating twist-controlled multiferroic behavior. The study offers insights into twist-controlled coexisting ferro-ordering and serves as valuable spectroscopic tools.

Opto-twistronic Hall effect in a three-dimensional spiral lattice

Studies of moiré systems have explained the effect of superlattice modulations on their properties, demonstrating new correlated phases1. However, most experimental studies have focused on a few layers in two-dimensional systems. Extending twistronics to three dimensions, in which the twist extends into the third dimension, remains underexplored because of the challenges associated with the manual stacking of layers. Here we study three-dimensional twistronics using a self-assembled twisted spiral superlattice of multilayered WS2. Our findings show an opto-twistronic Hall effect driven by structural chirality and coherence length, modulated by the moiré potential of the spiral superlattice. This is an experimental manifestation of the noncommutative geometry of the system. We observe enhanced light-matter interactions and an altered dependence of the Hall coefficient on photon momentum. Our model suggests contributions from higher-order quantum geometric quantities to this observation, providing opportunities for designing quantum-materials-based optoelectronic lattices with large nonlinearities.

Multiferroicity and Topology in Twisted Transition Metal Dichalcogenides

Van der Waals heterostructures have recently emerged as an exciting platform for investigating the effects of strong electronic correlations, including various forms of magnetic or electrical orders. Here, we perform an unbiased exact diagonalization study of the effects of interactions on topological flat bands of twisted transition metal dichalcogenides (TMDs) at odd integer fillings. For hole-filling ν_{h}=1, we find that the Chern insulator phase, expected from interaction-induced spin-valley polarization of the bare bands, is quite fragile, and gives way to spontaneous multiferroic order-coexisting ferroelectricity and ferromagnetism, in the presence of long-range Coulomb repulsion. We provide a simple real-space picture to understand the phase diagram as a function of interaction range and strength. Our findings establish twisted TMDs as a novel and highly tunable platform for multiferroicity, and we outline a potential route towards electrical control of magnetism in the multiferroic phase.

Remote epitaxy of single-crystal rhombohedral WS2 bilayers

Compared to transition metal dichalcogenide (TMD) monolayers, rhombohedral-stacked (R-stacked) TMD bilayers exhibit remarkable electrical performance, enhanced nonlinear optical response, giant piezo-photovoltaic effect and intrinsic interfacial ferroelectricity. However, from a thermodynamics perspective, the formation energies of R-stacked and hexagonal-stacked (H-stacked) TMD bilayers are nearly identical, leading to mixed stacking of both H- and R-stacked bilayers in epitaxial films. Here, we report the remote epitaxy of centimetre-scale single-crystal R-stacked WS2 bilayer films on sapphire substrates. The bilayer growth is realized by a high flux feeding of the tungsten source at high temperature on substrates. The R-stacked configuration is achieved by the symmetry breaking in a-plane sapphire, where the influence of atomic steps passes through the lower TMD layer and controls the R-stacking of the upper layer. The as-grown R-stacked bilayers show up-to-30-fold enhancements in carrier mobility (34 cm2V-1s-1), nearly doubled circular helicity (61%) and interfacial ferroelectricity, in contrast to monolayer films. Our work reveals a growth mechanism to obtain stacking-controlled bilayer TMD single crystals, and promotes large-scale applications of R-stacked TMD.

Strong-Coupling Phases of Trions and Excitons in Electron-Hole Bilayers at Commensurate Densities

We introduce density imbalanced electron-hole bilayers at a commensurate 2:1 density ratio as a platform for realizing novel phases of electrons, excitons, and trions. Through the independently tunable carrier densities and interlayer spacing, competition between kinetic energy, intralayer repulsion, and interlayer attraction yields a rich phase diagram. By a combination of theoretical analysis and numerical calculation, we find a variety of strong-coupling phases in different parameter regions, including quantum crystals of electrons, excitons, and trions. We also propose an "electron-exciton supersolid" phase that features electron crystallization and exciton superfluidity simultaneously. The material realization and experimental signature of these phases are discussed in the context of semiconductor transition metal dichalcogenide bilayers.

Resonant Band Hybridization in Alloyed Transition Metal Dichalcogenide Heterobilayers

Bandstructure engineering using alloying is widely utilized for achieving optimized performance in modern semiconductor devices. While alloying has been studied in monolayer transition metal dichalcogenides, its application in van der Waals heterostructures built from atomically thin layers is largely unexplored. Here, heterobilayers made from monolayers of WSe2 (or MoSe2) and MoxW1 - xSe2 alloy are fabricated and nontrivial tuning of the resultant bandstructure is observed as a function of concentration x. This evolution is monitored by measuring the energy of photoluminescence (PL) of the interlayer exciton (IX) composed of an electron and hole residing in different monolayers. In MoxW1 - xSe2/WSe2, a strong IX energy shift of ≈100 meV is observed for x varied from 1 to 0.6. However, for x < 0.6 this shift saturates and the IX PL energy asymptotically approaches that of the indirect bandgap in bilayer WSe2. This observation is theoretically interpreted as the strong variation of the conduction band K valley for x > 0.6, with IX PL arising from the K - K transition, while for x < 0.6, the bandstructure hybridization becomes prevalent leading to the dominating momentum-indirect K - Q transition. This bandstructure hybridization is accompanied with strong modification of IX PL dynamics and nonlinear exciton properties. This work provides foundation for bandstructure engineering in van der Waals heterostructures highlighting the importance of hybridization effects and opening a way to devices with accurately tailored electronic properties.

Mapping twist-tuned multiband topology in bilayer WSe2

Semiconductor moiré superlattices have been shown to host a wide array of interaction-driven ground states. However, twisted homobilayers have been difficult to study in the limit of large moiré wavelengths, where interactions are most dominant. In this study, we conducted local electronic compressibility measurements of twisted bilayer WSe2 (tWSe2) at small twist angles. We demonstrated multiple topological bands that host a series of Chern insulators at zero magnetic field near a "magic angle" around 1.23°. Using a locally applied electric field, we induced a topological quantum-phase transition at one hole per moiré unit cell. Our work establishes the topological phase diagram of a generalized Kane-Mele-Hubbard model in tWSe2, demonstrating a tunable platform for strongly correlated topological phases.

Excitonic Thermalization Bottleneck in Twisted TMD Heterostructures

Twisted van der Waals heterostructures show intriguing interface exciton physics, including hybridization effects and emergence of moiré potentials. Recent experiments have revealed that moiré-trapped excitons exhibit remarkable dynamics, where excited states show lifetimes that are several orders of magnitude longer than in monolayers. The origin of this behavior is still under debate. Based on a microscopic many-particle approach, we investigate the phonon-driven relaxation cascade of nonequilibrium moiré excitons in the exemplary MoSe2-WSe2 heterostructure. We track exciton relaxation pathways across different moiré mini-bands and identify the phonon-scattering channels assisting the spatial redistribution of excitons into low-energy pockets of the moiré potential. We unravel a phonon bottleneck in the flat band structure at low twist angles preventing excitons from fully thermalizing into the lowest state, explaining the measured enhanced emission intensity and lifetime of excited moiré excitons. Overall, our work provides important insights into exciton relaxation dynamics in flat-band exciton materials.

Evidence of the fractional quantum spin Hall effect in moiré MoTe2

Quantum spin Hall (QSH) insulators are two-dimensional electronic materials that have a bulk band gap similar to an ordinary insulator but have topologically protected pairs of edge modes of opposite chiralities1-6. So far, experimental studies have found only integer QSH insulators with counter-propagating up-spins and down-spins at each edge leading to a quantized conductance G0 = e2/h (with e and h denoting the electron charge and Planck's constant, respectively)7-14. Here we report transport evidence of a fractional QSH insulator in 2.1° twisted bilayer MoTe2, which supports spin-Sz conservation and flat spin-contrasting Chern bands15,16. At filling factor ν = 3 of the moiré valence bands, each edge contributes a conductance3 2 G 0 with zero anomalous Hall conductivity. The state is probably a time-reversal pair of the even-denominator 3/2-fractional Chern insulators. Furthermore, at ν = 2, 4 and 6, we observe a single, double and triple QSH insulator with each edge contributing a conductance G0, 2G0 and 3G0, respectively. Our results open up the possibility of realizing time-reversal symmetric non-abelian anyons and other unexpected topological phases in highly tunable moiré materials17-19.

Stacking engineering in layered homostructures: transitioning from 2D to 3D architectures

Artificial materials, characterized by their distinctive properties and customized functionalities, occupy a central role in a wide range of applications including electronics, spintronics, optoelectronics, catalysis, and energy storage. The emergence of atomically thin two-dimensional (2D) materials has driven the creation of artificial heterostructures, harnessing the potential of combining various 2D building blocks with complementary properties through the art of stacking engineering. The promising outcomes achieved for heterostructures have spurred an inquisitive exploration of homostructures, where identical 2D layers are precisely stacked. This perspective primarily focuses on the field of stacking engineering within layered homostructures, where precise control over translational or rotational degrees of freedom between vertically stacked planes or layers is paramount. In particular, we provide an overview of recent advancements in the stacking engineering applied to 2D homostructures. Additionally, we will shed light on research endeavors venturing into three-dimensional (3D) structures, which allow us to proactively address the limitations associated with artificial 2D homostructures. We anticipate that the breakthroughs in stacking engineering in 3D materials will provide valuable insights into the mechanisms governing stacking effects. Such advancements have the potential to unlock the full capability of artificial layered homostructures, propelling the future development of materials, physics, and device applications.

Ultraviolet interlayer excitons in bilayer WSe2

Interlayer excitons in van der Waals heterostructures are fascinating for applications like exciton condensation, excitonic devices and moiré-induced quantum emitters. The study of these charge-transfer states has almost exclusively focused on band edges, limiting the spectral region to the near-infrared regime. Here we explore the above-gap analogues of interlayer excitons in bilayer WSe2 and identify both neutral and charged species emitting in the ultraviolet. Even though the transitions occur far above the band edge, the states remain metastable, exhibiting linewidths as narrow as 1.8 meV. These interlayer high-lying excitations have switchable dipole orientations and hence show prominent Stark splitting. The positive and negative interlayer high-lying trions exhibit significant binding energies of 20-30 meV, allowing for a broad tunability of transitions via electric fields and electrostatic doping. The Stark splitting of these trions serves as a highly accurate, built-in sensor for measuring interlayer electric field strengths, which are exceedingly difficult to quantify otherwise. Such excitonic complexes are further sensitive to the interlayer twist angle and offer opportunities to explore emergent moiré physics under electrical control. Our findings more than double the accessible energy range for applications based on interlayer excitons.

Remote imprinting of moiré lattices

Two-dimensional moiré materials are formed by overlaying two layered crystals with small differences in orientation or/and lattice constant, where their direct coupling generates moiré potentials. Moiré materials have emerged as a platform for the discovery of new physics and device concepts, but while moiré materials are highly tunable, once formed, moiré lattices cannot be easily altered. Here we demonstrate the electrostatic imprinting of moiré lattices onto a target monolayer semiconductor. The moiré potential-created by a lattice of electrons that is supported by a Mott insulator state in a remote MoSe2/WS2 moiré bilayer-imprints a moiré potential that generates flat bands and correlated insulating states in the target monolayer and can be turned on/off by gate tuning the doping density of the moiré bilayer. Additionally, we studied the interplay between the electrostatic and structural relaxation contributions to moiré imprinting. Our results demonstrate a pathway towards gate control of moiré lattices.

Hexagonal Boron Nitride Slab Waveguides for Enhanced Spectroscopy of Encapsulated 2D Materials

The layered insulator hexagonal boron nitride (hBN) is a critical substrate that brings out the exceptional intrinsic properties of two-dimensional (2D) materials such as graphene and transition metal dichalcogenides (TMDs). In this work, the authors demonstrate how hBN slabs tuned to the correct thickness act as optical waveguides, enabling direct optical coupling of light emission from encapsulated layers into waveguide modes. Molybdenum selenide (MoSe2 ) and tungsten selenide (WSe2 ) are integrated within hBN-based waveguides and demonstrate direct coupling of photoluminescence emitted by in-plane and out-of-plane transition dipoles (bright and dark excitons) to slab waveguide modes. Fourier plane imaging of waveguided photoluminescence from MoSe2 demonstrates that dry etched hBN edges are an effective out-coupler of waveguided light without the need for oil-immersion optics. Gated photoluminescence of WSe2 demonstrates the ability of hBN waveguides to collect light emitted by out-of-plane dark excitons.Numerical simulations explore the parameters of dipole placement and slab thickness, elucidating the critical design parameters and serving as a guide for novel devices implementing hBN slab waveguides. The results provide a direct route for waveguide-based interrogation of layered materials, as well as a way to integrate layered materials into future photonic devices at arbitrary positions whilst maintaining their intrinsic properties.

Optical properties and plasmons in moiré structures

The discoveries of numerous exciting phenomena in twisted bilayer graphene (TBG) are stimulating significant investigations on moiré structures that possess a tunable moiré potential. Optical response can provide insights into the electronic structures and transport phenomena of non-twisted and twisted moiré structures. In this article, we review both experimental and theoretical studies of optical properties such as optical conductivity, dielectric function, non-linear optical response, and plasmons in moiré structures composed of graphene, hexagonal boron nitride (hBN), and/or transition metal dichalcogenides. Firstly, a comprehensive introduction to the widely employed methodology on optical properties is presented. After, moiré potential induced optical conductivity and plasmons in non-twisted structures are reviewed, such as single layer graphene-hBN, bilayer graphene-hBN and graphene-metal moiré heterostructures. Next, recent investigations of twist-angle dependent optical response and plasmons are addressed in twisted moiré structures. Additionally, we discuss how optical properties and plasmons could contribute to the understanding of the many-body effects and superconductivity observed in moiré structures.

Recent Strategies for the Synthesis of Phase-Pure Ultrathin 1T/1T' Transition Metal Dichalcogenide Nanosheets

The past few decades have witnessed a notable increase in transition metal dichalcogenide (TMD) related research not only because of the large family of TMD candidates but also because of the various polytypes that arise from the monolayer configuration and layer stacking order. The peculiar physicochemical properties of TMD nanosheets enable an enormous range of applications from fundamental science to industrial technologies based on the preparation of high-quality TMDs. For polymorphic TMDs, the 1T/1T' phase is particularly intriguing because of the enriched density of states, and thus facilitates fruitful chemistry. Herein, we comprehensively discuss the most recent strategies for direct synthesis of phase-pure 1T/1T' TMD nanosheets such as mechanical exfoliation, chemical vapor deposition, wet chemical synthesis, atomic layer deposition, and more. We also review frequently adopted methods for phase engineering in TMD nanosheets ranging from chemical doping and alloying, to charge injection, and irradiation with optical or charged particle beams. Prior to the synthesis methods, we discuss the configuration of TMDs as well as the characterization tools mostly used in experiments. Finally, we discuss the current challenges and opportunities as well as emphasize the promising fields for the future development.

Giant Correlated Gap and Possible Room-Temperature Correlated States in Twisted Bilayer MoS_{2}

Moiré superlattices have emerged as an exciting condensed-matter quantum simulator for exploring the exotic physics of strong electronic correlations. Notable progress has been witnessed, but such correlated states are achievable usually at low temperatures. Here, we report evidence of possible room-temperature correlated electronic states and layer-hybridized SU(4) model simulator in AB-stacked MoS_{2} homobilayer moiré superlattices. Correlated insulating states at moiré band filling factors v=1, 2, 3 are unambiguously established in twisted bilayer MoS_{2}. Remarkably, the correlated electronic state at v=1 shows a giant correlated gap of ∼126  meV and may persist up to a record-high critical temperature over 285 K. The realization of a possible room-temperature correlated state with a large correlated gap in twisted bilayer MoS_{2} can be understood as the cooperation effects of the stacking-specific atomic reconstruction and the resonantly enhanced interlayer hybridization, which largely amplify the moiré superlattice effects on electronic correlations. Furthermore, extreme large nonlinear Hall responses up to room temperature are uncovered near correlated electronic states, demonstrating the quantum geometry of moiré flat conduction band.

Artificial Atoms, Wigner Molecules, and an Emergent Kagome Lattice in Semiconductor Moiré Superlattices

Semiconductor moiré superlattices comprise an array of artificial atoms and provide a highly tunable platform for exploring novel electronic phases. We introduce a theoretical framework for studying moiré quantum matter that treats intra-moiré-atom interactions exactly and is controlled in the limit of large moiré period. We reveal an abundance of new physics arising from strong electron interactions when there are multiple electrons within a moiré unit cell. In particular, at filling factor n=3, the Coulomb interaction within each three-electron moiré atom leads to a three-lobed "Wigner molecule." When their size is comparable to the moiré period, the Wigner molecules form an emergent Kagome lattice. Our Letter identifies two universal length scales characterizing the kinetic and interaction energies in moiré materials and demonstrates a rich phase diagram due to their interplay.

Kinetic magnetism in triangular moiré materials

Magnetic properties of materials ranging from conventional ferromagnetic metals to strongly correlated materials such as cuprates originate from Coulomb exchange interactions. The existence of alternate mechanisms for magnetism that could naturally facilitate electrical control has been discussed theoretically1-7, but an experimental demonstration8 in an extended system has been missing. Here we investigate MoSe2/WS2 van der Waals heterostructures in the vicinity of Mott insulator states of electrons forming a frustrated triangular lattice and observe direct evidence of magnetic correlations originating from a kinetic mechanism. By directly measuring electronic magnetization through the strength of the polarization-selective attractive polaron resonance9,10, we find that when the Mott state is electron-doped, the system exhibits ferromagnetic correlations in agreement with the Nagaoka mechanism.

Phase Engineering of 2D Materials

Polymorphic 2D materials allow structural and electronic phase engineering, which can be used to realize energy-efficient, cost-effective, and scalable device applications. The phase engineering covers not only conventional structural and metal-insulator transitions but also magnetic states, strongly correlated band structures, and topological phases in rich 2D materials. The methods used for the local phase engineering of 2D materials include various optical, geometrical, and chemical processes as well as traditional thermodynamic approaches. In this Review, we survey the precise manipulation of local phases and phase patterning of 2D materials, particularly with ideal and versatile phase interfaces for electronic and energy device applications. Polymorphic 2D materials and diverse quantum materials with their layered, vertical, and lateral geometries are discussed with an emphasis on the role and use of their phase interfaces. Various phase interfaces have demonstrated superior and unique performance in electronic and energy devices. The phase patterning leads to novel homo- and heterojunction structures of 2D materials with low-dimensional phase boundaries, which highlights their potential for technological breakthroughs in future electronic, quantum, and energy devices. Accordingly, we encourage researchers to investigate and exploit phase patterning in emerging 2D materials.

Thermodynamic evidence of fractional Chern insulator in moiré MoTe2

Chern insulators, which are the lattice analogues of the quantum Hall states, can potentially manifest high-temperature topological orders at zero magnetic field to enable next-generation topological quantum devices1-3. Until now, integer Chern insulators have been experimentally demonstrated in several systems at zero magnetic field3-8, whereas fractional Chern insulators have been reported in only graphene-based systems under a finite magnetic field9,10. The emergence of semiconductor moiré materials11, which support tunable topological flat bands12,13, provides an opportunity to realize fractional Chern insulators13-16. Here we report thermodynamic evidence of both integer and fractional Chern insulators at zero magnetic field in small-angle twisted bilayer MoTe2 by combining the local electronic compressibility and magneto-optical measurements. At hole filling factor ν = 1 and 2/3, the system is incompressible and spontaneously breaks time-reversal symmetry. We show that they are integer and fractional Chern insulators, respectively, from the dispersion of the state in the filling factor with an applied magnetic field. We further demonstrate electric-field-tuned topological phase transitions involving the Chern insulators. Our findings pave the way for the demonstration of quantized fractional Hall conductance and anyonic excitation and braiding17 in semiconductor moiré materials.

Composite Fermi Liquid at Zero Magnetic Field in Twisted MoTe_{2}

The pursuit of exotic phases of matter outside of the extreme conditions of a quantizing magnetic field is a long-standing quest of solid state physics. Recent experiments have observed spontaneous valley polarization and fractional Chern insulators in zero magnetic field in twisted bilayers of MoTe_{2}, at partial filling of the topological valence band (ν=-2/3 and -3/5). We study the topological valence band at half filling, using exact diagonalization and density matrix renormalization group calculations. We discover a composite Fermi liquid (CFL) phase even at zero magnetic field that covers a large portion of the phase diagram near twist angle ∼3.6°. The CFL is a non-Fermi liquid phase with metallic behavior despite the absence of Landau quasiparticles. We discuss experimental implications including the competition between the CFL and a Fermi liquid, which can be tuned with a displacement field. The topological valence band has excellent quantum geometry over a wide range of twist angles and a small bandwidth that is, remarkably, reduced by interactions. These key properties stabilize the exotic zero field quantum Hall phases. Finally, we present an optical signature involving "extinguished" optical responses that detects Chern bands with ideal quantum geometry.

Orbitally Controlled Quantum Hall States in Decoupled Two-Bilayer Graphene Sheets

The authors report on integer and fractional quantum Hall states in a stack of two twisted Bernal bilayer graphene sheets. By exploiting the momentum mismatch in reciprocal space, the single-particle tunneling between both bilayers is suppressed. Since the bilayers are spatially separated by only 0.34 nm, the stack benefits from strong interlayer Coulombic interactions. These interactions can cause the formation of a Bose-Einstein condensate. Indeed, such a condensate is observed for half-filling in each bilayer sheet. However, only when the partially filled levels have orbital index 1. It is absent for partially filled levels with orbital index 0. This discrepancy is tentatively attributed to the role of skyrmion/anti-skyrmion pair excitations and the dependence of the energy of these excitations on the orbital index. The application of asymmetric top and bottom gate voltages enables to influence the orbital nature of the electronic states of the graphene bilayers at the chemical potential and to navigate in orbital mixed space. The latter hosts an even denominator fractional quantum Hall state at total filling of -3/2. These observations suggest a unique edge reconstruction involving both electrons and chiral p-wave composite fermions.

Flat Γ Moiré Bands in Twisted Bilayer WSe_{2}

The recent observation of correlated phases in transition metal dichalcogenide moiré systems at integer and fractional filling promises new insight into metal-insulator transitions and the unusual states of matter that can emerge near such transitions. Here, we combine real- and momentum-space mapping techniques to study moiré superlattice effects in 57.4° twisted WSe_{2} (tWSe_{2}). Our data reveal a split-off flat band that derives from the monolayer Γ states. Using advanced data analysis, we directly quantify the moiré potential from our data. We further demonstrate that the global valence band maximum in tWSe_{2} is close in energy to this flat band but derives from the monolayer K states which show weaker superlattice effects. These results constrain theoretical models and open the perspective that Γ-valley flat bands might be involved in the correlated physics of twisted WSe_{2}.

Electrically tunable dipolar interactions between layer-hybridized excitons

Transition-metal dichalcogenide bilayers exhibit a rich exciton landscape including layer-hybridized excitons, i.e. excitons which are of partly intra- and interlayer nature. In this work, we study hybrid exciton-exciton interactions in naturally stacked WSe2 homobilayers. In these materials, the exciton landscape is electrically tunable such that the low-energy states can be rendered more or less interlayer-like depending on the strength of the external electric field. Based on a microscopic and material-specific many-particle theory, we reveal two intriguing interaction regimes: a low-dipole regime at small electric fields and a high-dipole regime at larger fields, involving interactions between hybrid excitons with a substantially different intra- and interlayer composition in the two regimes. While the low-dipole regime is characterized by weak inter-excitonic interactions between intralayer-like excitons, the high-dipole regime involves mostly interlayer-like excitons which display a strong dipole-dipole repulsion and give rise to large spectral blue-shifts and a highly anomalous diffusion. Overall, our microscopic study sheds light on the remarkable electrical tunability of hybrid exciton-exciton interactions in atomically thin semiconductors and can guide future experimental studies in this growing field of research.

Programming correlated magnetic states with gate-controlled moiré geometry

The ability to control the underlying lattice geometry of a system may enable transitions between emergent quantum ground states. Here, we report in-situ gate switching between honeycomb and triangular lattice geometries of an electron many-body Hamiltonian in R-stacked MoTe₂ moiré bilayers, resulting in switchable magnetic exchange interactions. At zero electric field, we observe a correlated ferromagnetic insulator near one hole per moiré unit cell with a widely tunable Curie temperature up to 14K. Applying an electric field switches the system into a half-filled triangular lattice with antiferromagnetic interactions; further doping this layer-polarized superlattice tunes the antiferromagnetic exchange interaction back to ferromagnetic. Our work demonstrates R-stacked MoTe₂ moirés to be a laboratory for engineering correlated states with nontrivial topology.

Tunable spin and valley excitations of correlated insulators in Γ-valley moiré bands

Moiré superlattices formed from transition metal dichalcogenides support a variety of quantum electronic phases that are highly tunable using applied electromagnetic fields. While the valley degree of freedom affects optoelectronic properties in the constituent transition metal dichalcogenides, it has yet to be fully explored in moiré systems. Here we establish twisted double-bilayer WSe₂ as an experimental platform to study electronic correlations within Γ-valley moiré bands. Through local and global electronic compressibility measurements, we identify charge-ordered phases at multiple integer and fractional moiré fillings. By measuring the magnetic field dependence of their energy gaps and the chemical potential upon doping, we reveal spin-polarized ground states with spin-polaron quasiparticle excitations. In addition, an applied displacement field induces a metal-insulator transition driven by tuning between Γ- and K-valley moiré bands. Our results demonstrate control over the spin and valley character of the correlated ground and excited states in this system.

Rotational and dilational reconstruction in transition metal dichalcogenide moiré bilayers

Lattice reconstruction and corresponding strain accumulation plays a key role in defining the electronic structure of two-dimensional moiré superlattices, including those of transition metal dichalcogenides (TMDs). Imaging of TMD moirés has so far provided a qualitative understanding of this relaxation process in terms of interlayer stacking energy, while models of the underlying deformation mechanisms have relied on simulations. Here, we use interferometric four-dimensional scanning transmission electron microscopy to quantitatively map the mechanical deformations through which reconstruction occurs in small-angle twisted bilayer MoS₂ and WSe₂/MoS₂ heterobilayers. We provide direct evidence that local rotations govern relaxation for twisted homobilayers, while local dilations are prominent in heterobilayers possessing a sufficiently large lattice mismatch. Encapsulation of the moiré layers in hBN further localizes and enhances these in-plane reconstruction pathways by suppressing out-of-plane corrugation. We also find that extrinsic uniaxial heterostrain, which introduces a lattice constant difference in twisted homobilayers, leads to accumulation and redistribution of reconstruction strain, demonstrating another route to modify the moiré potential.

Thickness Mapping and Layer Number Identification of Exfoliated van der Waals Materials by Fourier Imaging Micro-Ellipsometry

As performance of van der Waals heterostructure devices is governed by the nanoscale thicknesses and homogeneity of their constituent mono- to few-layer flakes, accurate mapping of these properties with high lateral resolution becomes imperative. Spectroscopic ellipsometry is a promising optical technique for such atomically thin-film characterization due to its simplicity, noninvasive nature and high accuracy. However, the effective use of standard ellipsometry methods on exfoliated micron-scale flakes is inhibited by their tens-of-microns lateral resolution or slow data acquisition. In this work, we demonstrate a Fourier imaging spectroscopic micro-ellipsometry method with sub-5 μm lateral resolution and three orders-of-magnitude faster data acquisition than similar-resolution ellipsometers. Simultaneous recording of spectroscopic ellipsometry information at multiple angles results in a highly sensitive system, which is used for performing angstrom-level accurate and consistent thickness mapping on exfoliated mono-, bi- and trilayers of graphene, hexagonal boron nitride (hBN) and transition metal dichalcogenide (MoS₂, WS₂, MoSe₂, WSe₂) flakes. The system can successfully identify highly transparent monolayer hBN, a challenging proposition for other characterization tools. The optical microscope integrated ellipsometer can also map minute thickness variations over a micron-scale flake, revealing its lateral inhomogeneity. The prospect of adding standard optical elements to augment generic optical imaging and spectroscopy setups with accurate in situ ellipsometric mapping capability presents potential opportunities for investigation of exfoliated 2D materials.

Layer-dependent correlated phases in WSe2/MoS2 moiré superlattice

Electron correlation plays an essential role in the macroscopic quantum phenomena in the moiré heterostructure, such as antiferromagnetism and correlated insulating phases. Unlike the phenomena where the interaction involves only electrons in one layer, the interaction of distinct phases in two or more layers represents a new horizon forward, such as the one in the Kondo lattice model. Here, using interlayer excitons as a probe, we show that the interlayer interactions in heterobilayers of tungsten diselenide and molybdenum disulfide (WSe₂/MoS₂) can be electrically switched on and off, resulting in a layer-dependent correlated phase diagram, including single-layer, layer-selective, excitonic-insulator and layer-hybridized regions. We demonstrate that these correlated phases affect the interlayer exciton non-radiative decay pathways. These results reveal the role of strong correlation on interlayer exciton dynamics and pave the way for studying the layer-resolved strong correlation behaviour in moiré heterostructures.

Substrate engineering for wafer-scale two-dimensional material growth: strategies, mechanisms, and perspectives

The fabrication of wafer-scale two-dimensional (2D) materials is a prerequisite and important step for their industrial applications. Chemical vapor deposition (CVD) is the most promising approach to produce high-quality films in a scalable way. Recent breakthroughs in the epitaxy of wafer-scale single-crystalline graphene, hexagonal boron nitride, and transition-metal dichalcogenides highlight the pivotal roles of substrate engineering by lattice orientation, surface steps, and energy considerations. This review focuses on the existing strategies and underlying mechanisms, and discusses future directions in epitaxial substrate engineering to deliver wafer-scale 2D materials for integrated electronics and photonics.

Exciton density waves in Coulomb-coupled dual moiré lattices

Strongly correlated bosons in a lattice are a platform that can realize rich bosonic states of matter and quantum phase transitions¹. While strongly correlated bosons in a lattice have been studied in cold-atom experiments^2-4, their realization in a solid-state system has remained challenging⁵. Here we trap interlayer excitons-bosons composed of bound electron-hole pairs, in a lattice provided by an angle-aligned WS₂/bilayer WSe₂/WS₂ multilayer. The heterostructure supports Coulomb-coupled triangular moiré lattices of nearly identical period at the top and bottom interfaces. We observe correlated insulating states when the combined electron filling factor of the two lattices, with arbitrary partitions, equals [Formula: see text] and [Formula: see text]. These states can be interpreted as exciton density waves in a Bose-Fermi mixture of excitons and holes^6,7. Because of the strong repulsive interactions between the constituents, the holes form robust generalized Wigner crystals^8-11, which restrict the exciton fluid to channels that spontaneously break the translational symmetry of the lattice. Our results demonstrate that Coulomb-coupled moiré lattices are fertile ground for correlated many-boson phenomena^12,13.

Switching between Mott-Hubbard and Hund Physics in Moiré Quantum Simulators

Mott-Hubbard and Hund electron correlations have been realized thus far in separate classes of materials. Here, we show that a single moiré homobilayer encompasses both kinds of physics in a controllable manner. We develop a microscopic multiband model that we solve by dynamical mean-field theory to nonperturbatively address the local many-body correlations. We demonstrate how tuning with twist angle, dielectric screening, and hole density allows us to switch between Mott-Hubbard and Hund correlated states in a twisted WSe₂ bilayer. The underlying mechanism is based on controlling Coulomb-interaction-driven orbital polarization and the energetics of concomitant local singlet and triplet spin configurations. From a comparison to recent experimental transport data, we find signatures of a filling-controlled transition from a triplet charge-transfer insulator to a Hund-Mott metal. Our finding establishes twisted transition-metal dichalcogenides as a tunable platform for exotic phases of quantum matter emerging from large local spin moments.

Semiconductor moiré materials

Moiré materials have emerged as a platform for exploring the physics of strong electronic correlations and non-trivial band topology. Here we review the recent progress in semiconductor moiré materials, with a particular focus on transition metal dichalcogenides. Following a brief overview of the general features in this class of materials, we discuss recent theoretical and experimental studies on Hubbard physics, Kane-Mele-Hubbard physics and equilibrium moiré excitons. We also comment on the future opportunities and challenges in the studies of transition metal dichalcogenide and other semiconductor moiré materials.