Moiré potential impedes interlayer exciton diffusion in van der Waals heterostructures

Controlled interlayer exciton ionization in an electrostatic trap in atomically thin heterostructures

Atomically thin semiconductor heterostructures provide a two-dimensional (2D) device platform for creating high densities of cold, controllable excitons. Interlayer excitons (IEs), bound electrons and holes localized to separate 2D quantum well layers, have permanent out-of-plane dipole moments and long lifetimes, allowing their spatial distribution to be tuned on demand. Here, we employ electrostatic gates to trap IEs and control their density. By electrically modulating the IE Stark shift, electron-hole pair concentrations above 2 × 1012 cm-2 can be achieved. At this high IE density, we observe an exponentially increasing linewidth broadening indicative of an IE ionization transition, independent of the trap depth. This runaway threshold remains constant at low temperatures, but increases above 20 K, consistent with the quantum dissociation of a degenerate IE gas. Our demonstration of the IE ionization in a tunable electrostatic trap represents an important step towards the realization of dipolar exciton condensates in solid-state optoelectronic devices.

Room Temperature, Twist Angle Independent, Momentum Direct Interlayer Excitons in van der Waals Heterostructures with Wide Spectral Tunability

Interlayer excitons (IXs) in van der Waals heterostructures with static out of plane dipole moment and long lifetime show promise in the development of exciton based optoelectronic devices and the exploration of many body physics. However, these IXs are not always observed, as the emission is very sensitive to lattice mismatch and twist angle between the constituent materials. Moreover, their emission intensity is very weak compared to that of corresponding intralayer excitons at room temperature. Here we report the room-temperature realization of twist angle independent momentum direct IX in the heterostructures of bulk PbI2 and bilayer WS2. Momentum conserving transitions combined with the large band offsets between the constituent materials enable intense IX emission at room temperature. A long lifetime (∼100 ns), noticeable Stark shift, and tunability of IX emission from 1.70 to 1.45 eV by varying the number of WS2 layers make these heterostructures promising to develop room temperature exciton based optoelectronic devices.

Spatial Filtering of Interlayer Exciton Ground State in WSe2/MoS2 Heterobilayer

Long-life interlayer excitons (IXs) in transition metal dichalcogenide (TMD) heterostructure are promising for realizing excitonic condensates at high temperatures. Critical to this objective is to separate the IX ground state (the lowest energy of IX state) emission from other states' emissions. Filtering the IX ground state is also essential in uncovering the dynamics of correlated excitonic states, such as the excitonic Mott insulator. Here, we show that the IX ground state in the WSe2/MoS2 heterobilayer can be separated from other states by its spatial profile. The emissions from different moiré IX modes are identified by their different energies and spatial distributions, which fits well with the rate-diffusion model for cascading emission. Our results show spatial filtering of the ground state mode and enrich the toolbox to realize correlated states at elevated temperatures.

Anomalous Interlayer Exciton Diffusion in WS2/WSe2 Moiré Heterostructure

Stacking van der Waals crystals allows for the on-demand creation of a periodic potential landscape to tailor the transport of quasiparticle excitations. We investigate the diffusion of photoexcited electron-hole pairs, or excitons, at the interface of WS2/WSe2 van der Waals heterostructure over a wide range of temperatures. We observe the appearance of distinct interlayer excitons for parallel and antiparallel stacking and track their diffusion through spatially and temporally resolved photoluminescence spectroscopy from 30 to 250 K. While the measured exciton diffusivity decreases with temperature, it surprisingly plateaus below 90 K. Our observations cannot be explained by classical models like hopping in the moiré potential. A combination of ab initio theory and molecular dynamics simulations suggests that low-energy phonons arising from the mismatched lattices of moiré heterostructures, also known as phasons, play a key role in describing and understanding this anomalous behavior of exciton diffusion. Our observations indicate that the moiré potential landscape is dynamic down to very low temperatures and that the phason modes can enable efficient transport of energy in the form of excitons.

Moiré superlattices in twisted two-dimensional halide perovskites

Moiré superlattices have emerged as a new platform for studying strongly correlated quantum phenomena, but these systems have been largely limited to van der Waals layer two-dimensional materials. Here we introduce moiré superlattices leveraging ultrathin, ligand-free halide perovskites, facilitated by ionic interactions. Square moiré superlattices with varying periodic lengths are clearly visualized through high-resolution transmission electron microscopy. Twist-angle-dependent transient photoluminescence microscopy and electrical characterizations indicate the emergence of localized bright excitons and trapped charge carriers near a twist angle of ~10°. The localized excitons are accompanied by enhanced exciton emission, attributed to an increased oscillator strength by a theoretically predicted flat band. This research showcases the promise of two-dimensional perovskites as unique room-temperature moiré materials.

Exciton Localization Modulated by Ultradeep Moiré Potential in Twisted Bilayer γ-Graphdiyne

Twisted moiré superlattice is featured with its moiré potential energy, the depth of which renders an effective approach to strengthening the exciton-exciton interaction and exciton localization toward high-performance quantum photonic devices. However, it remains as a long-standing challenge to further push the limit of moiré potential depth. Herein, owing to the pz orbital induced band edge states enabled by the unique sp-C in bilayer γ-graphdiyne (GDY), an ultradeep moiré potential of ∼289 meV is yielded. After being twisted into the hole-to-hole layer stacking configuration, the interlayer coupling is substantially intensified to augment the lattice potential of bilayer GDY up to 475%. The presence of lateral constrained moiré potential shifts the spatial distribution of electrons and holes in excitons from the regular alternating mode to their respective separated and localized mode. According to the well-established wave function distribution of electrons contained in excitons, the AA-stacked site is identified to serve for exciton localization. This work extends the materials systems available for moiré superlattice design further to serial carbon allotropes featured with benzene ring-alkyne chain coupling, unlocking tremendous potential for twistronic-based quantum device applications.

Electrical Control and Transport of Tightly Bound Interlayer Excitons in a MoSe_{2}/hBN/MoSe_{2} Heterostructure

Controlling interlayer excitons in Van der Waals heterostructures holds promise for exploring Bose-Einstein condensates and developing novel optoelectronic applications, such as excitonic integrated circuits. Despite intensive studies, several key fundamental properties of interlayer excitons, such as their binding energies and interactions with charges, remain not well understood. Here we report the formation of momentum-direct interlayer excitons in a high-quality MoSe_{2}/hBN/MoSe_{2} heterostructure under an electric field, characterized by bright photoluminescence (PL) emission with high quantum yield and a narrow linewidth of less than 4 meV. These interlayer excitons show electrically tunable emission energy spanning ∼180  meV through the Stark effect, and exhibit a sizable binding energy of ∼81  meV in the intrinsic regime, along with trion binding energies of a few millielectronvolts. Remarkably, we demonstrate the long-range transport of interlayer excitons with a characteristic diffusion length exceeding 10  μm, which can be attributed, in part, to their dipolar repulsive interactions. Spatially and polarization-resolved spectroscopic studies reveal rich exciton physics in the system, such as valley polarization, local trapping, and the possible existence of dark interlayer excitons. The formation and transport of tightly bound interlayer excitons with narrow linewidth, coupled with the ability to electrically manipulate their properties, open exciting new avenues for exploring quantum many-body physics, including excitonic condensate and superfluidity, and for developing novel optoelectronic devices, such as exciton and photon routers.

Ultralow Auger-Assisted Interlayer Exciton Annihilation in WS2/WSe2 Moiré Heterobilayers

Transition metal dichalcogenide (TMD) heterobilayers have emerged as a promising platform for exploring solid-state quantum simulators and many-body quantum phenomena. Their type II band alignment, combined with the moiré superlattice, inevitably leads to nontrivial exciton interactions and dynamics. Here, we unveil the distinct Auger annihilation processes for delocalized interlayer excitons in WS2/WSe2 moiré heterobilayers. By fitting the characteristic efficiency droop and bimolecular recombination rate, we quantitatively determine an ultralow Auger coefficient of 1.3 × 10-5 cm2 s-1, which is >100-fold smaller than that of excitons in TMD monolayers. In addition, we reveal selective exciton upconversion into the WSe2 layer, which highlights the significance of intralayer electron Coulomb interactions in dictating the microscopic scattering pathways. The distinct Auger processes arising from spatial electron-hole separation have important implications for TMD heterobilayers while endowing interlayer excitons and their strongly correlated states with unique layer degrees of freedom.

Twisted van der Waals Quantum Materials: Fundamentals, Tunability, and Applications

Twisted van der Waals (vdW) quantum materials have emerged as a rapidly developing field of two-dimensional (2D) semiconductors. These materials establish a new central research area and provide a promising platform for studying quantum phenomena and investigating the engineering of novel optoelectronic properties such as single photon emission, nonlinear optical response, magnon physics, and topological superconductivity. These captivating electronic and optical properties result from, and can be tailored by, the interlayer coupling using moiré patterns formed by vertically stacking atomic layers with controlled angle misorientation or lattice mismatch. Their outstanding properties and the high degree of tunability position them as compelling building blocks for both compact quantum-enabled devices and classical optoelectronics. This paper offers a comprehensive review of recent advancements in the understanding and manipulation of twisted van der Waals structures and presents a survey of the state-of-the-art research on moiré superlattices, encompassing interdisciplinary interests. It delves into fundamental theories, synthesis and fabrication, and visualization techniques, and the wide range of novel physical phenomena exhibited by these structures, with a focus on their potential for practical device integration in applications ranging from quantum information to biosensors, and including classical optoelectronics such as modulators, light emitting diodes, lasers, and photodetectors. It highlights the unique ability of moiré superlattices to connect multiple disciplines, covering chemistry, electronics, optics, photonics, magnetism, topological and quantum physics. This comprehensive review provides a valuable resource for researchers interested in moiré superlattices, shedding light on their fundamental characteristics and their potential for transformative applications in various fields.

Simultaneously Regulated Highly Polarized and Long-Lived Valley Excitons in WSe2/GaN Heterostructures

Interlayer excitons, with prolonged lifetimes and tunability, hold potential for advanced optoelectronics. Previous research on the interlayer excitons has been dominated by two-dimensional heterostructures. Here, we construct WSe2/GaN composite heterostructures, in which the doping concentration of GaN and the twist angle of bilayer WSe2 are employed as two ingredients for the manipulation of exciton behaviors and polarizations. The exciton energies in monolayer WSe2/GaN can be regulated continuously by the doping levels of the GaN substrate, and a remarkable increase in the valley polarizations is achieved. Especially in a heterostructure with 4°-twisted bilayer WSe2, a maximum polarization of 38.9% with a long lifetime is achieved for the interlayer exciton. Theoretical calculations reveal that the large polarization and long lifetime are attributed to the high exciton binding energy and large spin flipping energy during depolarization in bilayer WSe2/GaN. This work introduces a distinctive member of the interlayer exciton with a high degree of polarization and a long lifetime.

Unraveling the strain tuning mechanism of interlayer excitons in WSe2/MoSe2heterostructure

Atomically thin transition metal dichalcogenides (TMDs) exhibit rich excitonic physics, due to reduced dielectric screening and strong Coulomb interactions. Especially, some attractive topics in modern condensed matter physics, such as correlated insulator, superconductivity, topological excitons bands, are recently reported in stacking two monolayer (ML) TMDs. Here, we clearly reveal the tuning mechanism of tensile strain on interlayer excitons (IEXs) and intralayer excitons (IAXs) in WSe2/MoSe2heterostructure (HS) at low temperature. We utilize the cryogenic tensile strain platform to stretch the HS, and measure by micro-photoluminescence (μ-PL). The PL peaks redshifts of IEXs and IAXs in WSe2/MoSe2HS under tensile strain are well observed. The first-principles calculations by using density functional theory reveals the PL peaks redshifts of IEXs and IAXs origin from bandgap shrinkage. The calculation results also show the Mo-4d states dominating conduction band minimum shifts of the ML MoSe2plays a dominant role in the redshifts of IEXs. This work provides new insights into understanding the tuning mechanism of tensile strain on IEXs and IAXs in two-dimensional (2D) HS, and paves a way to the development of flexible optoelectronic devices based on 2D materials.

Nonlinear and Negative Effective Diffusivity of Interlayer Excitons in Moiré-Free Heterobilayers

Interlayer exciton diffusion is studied in atomically reconstructed MoSe_{2}/WSe_{2} heterobilayers with suppressed disorder. Local atomic registry is confirmed by characteristic optical absorption, circularly polarized photoluminescence, and g-factor measurements. Using transient microscopy we observe propagation properties of interlayer excitons that are independent from trapping at moiré- or disorder-induced local potentials. Confirmed by characteristic temperature dependence for free particles, linear diffusion coefficients of interlayer excitons at liquid helium temperature and low excitation densities are almost 1000 times higher than in previous observations. We further show that exciton-exciton repulsion and annihilation contribute nearly equally to nonlinear propagation by disentangling the two processes in the experiment and simulations. Finally, we demonstrate effective shrinking of the light emission area over time across several hundreds of picoseconds at the transition from exciton- to the plasma-dominated regimes. Supported by microscopic calculations for band gap renormalization to identify the Mott threshold, this indicates transient crossing between rapidly expanding, short-lived electron-hole plasma and slower, long-lived exciton populations.

Direct observation of single-atom defects in monolayer two-dimensional materials by using electron ptychography at 200 kV acceleration voltage

Electron ptychography has emerged as a popular technology for high-resolution imaging by combining the high coherence of electron sources with the ultra-fast scanning electron coil. However, the limitations of conventional pixelated detectors, including poor dynamic range and slow data readout speeds, have posed restrictions in the past on conducting electron ptychography experiments. We used the Gatan STELA pixelated detector to capture sequential diffraction data of monolayer two-dimensional (2D) materials for ptychographic reconstruction. By using the pixelated detector and electron ptychography, we demonstrate the observation of the radiation damage at atomic resolution in Transition Metal Dichalcogenides (TMDs).

Strongly Localized Moiré Exciton in Twisted Homobilayers

Artificially molding exciton flux is the cornerstone for developing promising excitonic devices. In the emerging hetero/homobilayers, the spatial separated charges prolong exciton lifetimes and create out-plane dipoles, facilitating electrically control exciton flux on a large scale, and the nanoscale periodic moiré potentials arising from twist-angle or/and lattice mismatch can substantially alter exciton dynamics, which are mainly proved in the heterostructures. However, the spatially indirect excitons dynamics in homobilayers without lattice mismatch remain elusive. Here the nonequilibrium dynamics of indirect exciton in homobilayers are systematically investigated. The homobilayers with slightly twist-angle can induce a deep moiré potential (>50 meV) in the energy landscape of indirect excitons, resulting in a strongly localized moiré excitons insulating the transport dynamics from phonons and disorder. These findings provide insights into the exciton dynamics and many-body physics in moiré superlattices modulated energy landscape, with implications for designing excitonic devices operating at room temperature.

Electrostatic moiré potential from twisted hexagonal boron nitride layers

Moiré superlattices host a rich variety of correlated electronic phases. However, the moiré potential is fixed by interlayer coupling, and it is dependent on the nature of carriers and valleys. In contrast, it has been predicted that twisted hexagonal boron nitride (hBN) layers can impose a periodic electrostatic potential capable of engineering the properties of adjacent functional layers. Here, we show that this potential is described by a theory of electric polarization originating from the interfacial charge redistribution, validated by its dependence on supercell sizes and distance from the twisted interfaces. This enables controllability of the potential depth and profile by controlling the twist angles between the two interfaces. Employing this approach, we further demonstrate how the electrostatic potential from a twisted hBN substrate impedes exciton diffusion in semiconductor monolayers, suggesting opportunities for engineering the properties of adjacent functional layers using the surface potential of a twisted hBN substrate.

Influence of Atomic Relaxations on the Moiré Flat Band Wave Functions in Antiparallel Twisted Bilayer WS2

Twisting bilayers of transition metal dichalcogenides gives rise to a moiré potential resulting in flat bands with localized wave functions and enhanced correlation effects. In this work, scanning tunneling microscopy is used to image a WS2 bilayer twisted approximately 3° off the antiparallel alignment. Scanning tunneling spectroscopy reveals localized states in the vicinity of the valence band onset, which is observed to occur first in regions with S-on-S Bernal stacking. In contrast, density functional theory calculations on twisted bilayers that have been relaxed in vacuum predict the highest-lying flat valence band to be localized in regions of AA' stacking. However, agreement with experiment is recovered when the calculations are performed on bilayers in which the atomic displacements from the unrelaxed positions have been reduced, reflecting the influence of the substrate and finite temperature. This demonstrates the delicate interplay of atomic relaxations and the electronic structure of twisted bilayer materials.

Recent progress of exciton transport in two-dimensional semiconductors

Spatial manipulation of excitonic quasiparticles, such as neutral excitons, charged excitons, and interlayer excitons, in two-dimensional semiconductors offers unique capabilities for a broad range of optoelectronic applications, encompassing photovoltaics, exciton-integrated circuits, and quantum light-emitting systems. Nonetheless, their practical implementation is significantly restricted by the absence of electrical controllability for neutral excitons, short lifetime of charged excitons, and low exciton funneling efficiency at room temperature, which remain a challenge in exciton transport. In this comprehensive review, we present the latest advancements in controlling exciton currents by harnessing the advanced techniques and the unique properties of various excitonic quasiparticles. We primarily focus on four distinct control parameters inducing the exciton current: electric fields, strain gradients, surface plasmon polaritons, and photonic cavities. For each approach, the underlying principles are introduced in conjunction with its progression through recent studies, gradually expanding their accessibility, efficiency, and functionality. Finally, we outline the prevailing challenges to fully harness the potential of excitonic quasiparticles and implement practical exciton-based optoelectronic devices.

Probing the interlayer excitation dynamics in WS2/WSe2 heterostructures with broadly tunable pump and probe energies

van der Waals heterostructures based on transition metal dichalcogenides (TMDs) provide a fascinating platform for exploring new physical phenomena and novel optoelectronic functionalities. Revealing the energy-dependence of photocarrier population dynamics in heterostructures is key for developing optoelectronic or valleytronic devices. Here, the broadband transient dynamics of interlayer excitation of a nearly-aligned WS2/WSe2 heterostructure is investigated by using energy-dependent pump-probe spectroscopy at cryogenic temperatures. Interestingly, WS2/WSe2 interlayer excitation, herein comprising a mixture of intra- and inter-layer excitons, exhibits largely constant lifetimes of a few hundred picoseconds across a broad energy range, in stark contrast to the salient energy-dependent dynamics of intralayer excitons in monolayer WSe2. While the PL emission of the WS2/WSe2 heterostructure is found to be strongly affected by electrostatic doping, the lifetimes of interlayer excitation show negligible changes. Our work elaborates the signatures of ultrafast dynamics introduced by intra- and interlayer co-existing excitonic species and enriches the understanding of interlayer couplings in van der Waals heterostructures.

Tuning and exploiting interlayer coupling in two-dimensional van der Waals heterostructures

Two-dimensional (2D) layered materials can stack into new material systems, with van der Waals (vdW) interaction between the adjacent constituent layers. This stacking process of 2D atomic layers creates a new degree of freedom-interlayer interface between two adjacent layers-that can be independently studied and tuned from the intralayer degree of freedom. In such heterostructures (HSs), the physical properties are largely determined by the vdW interaction between the individual layers,i.e.interlayer coupling, which can be effectively tuned by a number of means. In this review, we summarize and discuss a number of such approaches, including stacking order, electric field, intercalation, and pressure, with both their experimental demonstrations and theoretical predictions. A comprehensive overview of the modulation on structural, optical, electrical, and magnetic properties by these four approaches are also presented. We conclude this review by discussing several prospective research directions in 2D HSs field, including fundamental physics study, property tuning techniques, and future applications.

Observation of quadrupolar and dipolar excitons in a semiconductor heterotrilayer

Van der Waals (vdW) materials have opened up many avenues for discovery through layer assembly, as epitomized by interlayer dipolar excitons that exhibit electrically tunable luminescence, lasing and exciton condensation. Extending interlayer excitons to more vdW layers, however, raises fundamental questions concerning coherence within excitons and coupling between moiré superlattices at multiple interfaces. Here, by assembling angle-aligned WSe2/WS2/WSe2 heterotrilayers, we demonstrate the emergence of quadrupolar excitons. We confirm the exciton's quadrupolar nature by the decrease in its energy of 12 meV from coherent hole tunnelling between the two outer layers, its tunable static dipole moment under an external electric field and the reduced exciton-exciton interactions. At high exciton density, we also see signatures of a phase of oppositely aligned dipolar excitons, consistent with a staggered dipolar phase predicted to be driven by attractive dipolar interactions. Our demonstration paves the way for discovering emergent exciton orderings for three vdW layers and beyond.

Twist Angle-Dependent Intervalley Charge Carrier Transfer and Recombination in Bilayer WS2

A twist angle at a van der Waals junction provides a handle to tune its optoelectronic properties for a variety of applications, and a comprehensive understanding of how the twist modulates electronic structure, interlayer coupling, and carrier dynamics is needed. We employ time-dependent density functional theory and nonadiabatic molecular dynamics to elucidate angle-dependent intervalley carrier transfer and recombination in bilayer WS2. Repulsion between S atoms in twisted configurations weakens interlayer coupling, increases the interlayer distance, and softens layer breathing modes. Twisting has a minor influence on K valleys while it lowers Γ valleys and raises Q valleys because their wave functions are delocalized between layers. Consequently, the reduced energy gaps between the K and Γ valleys accelerate the hole transfer in the twisted structures. Intervalley electron transfer proceeds nearly an order of magnitude faster than hole transfer. The more localized wave functions at K than Q values and larger bandgaps result in smaller nonadiabatic couplings for intervalley recombination, making it 3-4 times slower in twisted than high-symmetry structures. B2g breathing, E2g in-plane, and A1g out-of-plane modes are most active during intervalley carrier transfer and recombination. The faster intervalley transfer and extended carrier lifetimes in twisted junctions are favorable for optoelectronic device performance.

Dynamics of Moiré Trion and Its Valley Polarization in a Microfabricated WSe2/MoSe2 Heterobilayer

The moiré potential, induced by stacking two monolayer semiconductors with slightly different lattice mismatches, acts as periodic quantum confinement for optically generated excitons, resulting in spatially ordered zero-dimensional quantum systems. However, there are limitations to exploring intrinsic optical properties of moiré excitons due to ensemble emissions and broadened emissions from many peaks caused by the inhomogeneity of the moiré potential. In this study, we proposed a microfabrication technique based on focused Ga⁺ ion beams, which enables us to control the number of peaks originating from the moiré potential and thus explore unknown moiré optical characteristics of WSe₂/MoSe₂ heterobilayer. By taking advantage of this approach, we reveal emissions from a single moiré exciton and charged moiré exciton (trion) under electrostatic doping conditions. We show the momentum dark moiré trion state above the bright trion state with a splitting energy of approximately 4 meV and clarify that the dynamics are determined by the initial trion population in the bright state. Furthermore, the degree of negative circularly polarized emissions and their valley dynamics of moiré trions are dominated by a very long valley relaxation process lasting ∼700 ns. Our findings on microfabricated heterobilayer could be viewed as an extension of our groundbreaking efforts in the field of quantum optics application using moiré superlattices.

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.

Bosonic Delocalization of Dipolar Moiré Excitons

In superlattices of twisted semiconductor monolayers, tunable moiré potentials emerge, trapping excitons into periodic arrays. In particular, spatially separated interlayer excitons are subject to a deep potential landscape and they exhibit a permanent dipole providing a unique opportunity to study interacting bosonic lattices. Recent experiments have demonstrated density-dependent transport properties of moiré excitons, which could play a key role for technological applications. However, the intriguing interplay between exciton-exciton interactions and moiré trapping has not been well understood yet. In this work, we develop a microscopic theory of interacting excitons in external potentials allowing us to tackle this highly challenging problem. We find that interactions between moiré excitons lead to a delocalization at intermediate densities, and we show how this transition can be tuned via twist angle and temperature. The delocalization is accompanied by a modification of optical moiré resonances, which gradually merge into a single free exciton peak.

Fermi Pressure and Coulomb Repulsion Driven Rapid Hot Plasma Expansion in a van der Waals Heterostructure

Transition metal dichalcogenide heterostructures provide a versatile platform to explore electronic and excitonic phases. As the excitation density exceeds the critical Mott density, interlayer excitons are ionized into an electron-hole plasma phase. The transport of the highly non-equilibrium plasma is relevant for high-power optoelectronic devices but has not been carefully investigated previously. Here, we employ spatially resolved pump-probe microscopy to investigate the spatial-temporal dynamics of interlayer excitons and hot-plasma phase in a MoSe₂/WSe₂ twisted bilayer. At the excitation density of ∼10¹⁴ cm^-2, well exceeding the Mott density, we find a surprisingly rapid initial expansion of hot plasma to a few microns away from the excitation source within ∼0.2 ps. Microscopic theory reveals that this rapid expansion is mainly driven by Fermi pressure and Coulomb repulsion, while the hot carrier effect has only a minor effect in the plasma phase.

Lattice Reconstruction in MoSe2-WSe2 Heterobilayers Synthesized by Chemical Vapor Deposition

Vertical van der Waals heterostructures of semiconducting transition metal dichalcogenides realize moiré systems with rich correlated electron phases and moiré exciton phenomena. For material combinations with small lattice mismatch and twist angles as in MoSe₂-WSe₂, however, lattice reconstruction eliminates the canonical moiré pattern and instead gives rise to arrays of periodically reconstructed nanoscale domains and mesoscopically extended areas of one atomic registry. Here, we elucidate the role of atomic reconstruction in MoSe₂-WSe₂ heterostructures synthesized by chemical vapor deposition. With complementary imaging down to the atomic scale, simulations, and optical spectroscopy methods, we identify the coexistence of moiré-type cores and extended moiré-free regions in heterostacks with parallel and antiparallel alignment. Our work highlights the potential of chemical vapor deposition for applications requiring laterally extended heterosystems of one atomic registry or exciton-confining heterostack arrays.

Complex Strain Scapes in Reconstructed Transition-Metal Dichalcogenide Moiré Superlattices

We investigate the intrinsic strain associated with the coupling of twisted MoS₂/MoSe₂ heterobilayers by combining experiments and molecular dynamics simulations. Our study reveals that small twist angles (between 0 and 2°) give rise to considerable atomic reconstructions, large moiré periodicities, and high levels of local strain (with an average value of ∼1%). Moreover, the formation of moiré superlattices is assisted by specific reconstructions of stacking domains. This process leads to a complex strain distribution characterized by a combined deformation state of uniaxial, biaxial, and shear components. Lattice reconstruction is hindered with larger twist angles (>10°) that produce moiré patterns of small periodicity and negligible strains. Polarization-dependent Raman experiments also evidence the presence of an intricate strain distribution in heterobilayers with near-0° twist angles through the splitting of the E_2g¹ mode of the top (MoS₂) layer due to atomic reconstruction. Detailed analyses of moiré patterns measured by AFM unveil varying degrees of anisotropy in the moiré superlattices due to the heterostrain induced during the stacking of monolayers.

Electrical control of hybrid exciton transport in a van der Waals heterostructure

Interactions between out-of-plane dipoles in bosonic gases enable the long-range propagation of excitons. The lack of direct control over collective dipolar properties has so far limited the degrees of tunability and the microscopic understanding of exciton transport. In this work we modulate the layer hybridization and interplay between many-body interactions of excitons in a van der Waals heterostructure with an applied vertical electric field. By performing spatiotemporally resolved measurements supported by microscopic theory, we uncover the dipole-dependent properties and transport of excitons with different degrees of hybridization. Moreover, we find constant emission quantum yields of the transporting species as a function of excitation power with radiative decay mechanisms dominating over nonradiative ones, a fundamental requirement for efficient excitonic devices. Our findings provide a complete picture of the many-body effects in the transport of dilute exciton gases, and have crucial implications for studying emerging states of matter such as Bose-Einstein condensation and optoelectronic applications based on exciton propagation.

Layer-Structured Anisotropic Metal Chalcogenides: Recent Advances in Synthesis, Modulation, and Applications

The unique electronic and catalytic properties emerging from low symmetry anisotropic (1D and 2D) metal chalcogenides (MCs) have generated tremendous interest for use in next generation electronics, optoelectronics, electrochemical energy storage devices, and chemical sensing devices. Despite many proof-of-concept demonstrations so far, the full potential of anisotropic chalcogenides has yet to be investigated. This article provides a comprehensive overview of the recent progress made in the synthesis, mechanistic understanding, property modulation strategies, and applications of the anisotropic chalcogenides. It begins with an introduction to the basic crystal structures, and then the unique physical and chemical properties of 1D and 2D MCs. Controlled synthetic routes for anisotropic MC crystals are summarized with example advances in the solution-phase synthesis, vapor-phase synthesis, and exfoliation. Several important approaches to modulate dimensions, phases, compositions, defects, and heterostructures of anisotropic MCs are discussed. Recent significant advances in applications are highlighted for electronics, optoelectronic devices, catalysts, batteries, supercapacitors, sensing platforms, and thermoelectric devices. The article ends with prospects for future opportunities and challenges to be addressed in the academic research and practical engineering of anisotropic MCs.

Surface acoustic wave induced phenomena in two-dimensional materials

Surface acoustic wave (SAW)-matter interaction provides a fascinating key for inducing and manipulating novel phenomena and functionalities in two-dimensional (2D) materials. The dynamic strain field and piezo-electric field associated with propagating SAWs determine the coherent manipulation and transduction between 2D excitons and phonons. Over the past decade, many intriguing acoustic-induced effects, including the acousto-electric effect, acousto-galvanic effect, acoustic Stark effect, acoustic Hall effect and acoustic exciton transport, have been reported experimentally. However, many more phenomena, such as the valley acousto-electric effect, valley acousto-electric Hall effect and acoustic spin Hall effect, were only theoretically proposed, the experimental verification of which are yet to be achieved. In this minireview, we attempt to overview the recent breakthrough of SAW-induced phenomena covering acoustic charge transport, acoustic exciton transport and modulation, and coherent acoustic phonons. Perspectives on the opportunities of the proposed SAW-induced phenomena, as well as open experimental challenges, are also discussed, attempting to offer some guidelines for experimentalists and theorists to explore the desired exotic properties and boost practical applications of 2D materials.

Thickness- and Twist-Angle-Dependent Interlayer Excitons in Metal Monochalcogenide Heterostructures

Interlayer excitons, or bound electron-hole pairs whose constituent quasiparticles are located in distinct stacked semiconducting layers, are being intensively studied in heterobilayers of two-dimensional semiconductors. They owe their existence to an intrinsic type-II band alignment between both layers that convert these into p-n junctions. Here, we unveil a pronounced interlayer exciton (IX) in heterobilayers of metal monochalcogenides, namely, γ-InSe on ε-GaSe, whose pronounced emission is adjustable just by varying their thicknesses given their number of layers dependent direct band gaps. Time-dependent photoluminescense spectroscopy unveils considerably longer interlayer exciton lifetimes with respect to intralayer ones, thus confirming their nature. The linear Stark effect yields a bound electron-hole pair whose separation d is just (3.6 ± 0.1) Å with d being very close to d_Se = 3.4 Å which is the calculated interfacial Se separation. The envelope of IX is twist-angle-dependent and describable by superimposed emissions that are nearly equally spaced in energy, as if quantized due to localization induced by the small moiré periodicity. These heterostacks are characterized by extremely flat interfacial valence bands making them prime candidates for the observation of magnetism or other correlated electronic phases upon carrier doping.

Enhanced interactions of interlayer excitons in free-standing heterobilayers

Strong, long-range dipole-dipole interactions between interlayer excitons (IXs) can lead to new multiparticle correlation regimes^1,2, which drive the system into distinct quantum and classical phases^2-5, including dipolar liquids, crystals and superfluids. Both repulsive and attractive dipole-dipole interactions have been theoretically predicted between IXs in a semiconductor bilayer^2,6-8, but only repulsive interactions have been reported experimentally so far^3,9-16. This study investigated free-standing, twisted (51°, 53°, 45°) tungsten diselenide/tungsten disulfide (WSe₂/WS₂) heterobilayers, in which we observed a transition in the nature of dipolar interactions among IXs, from repulsive to attractive. This was caused by quantum-exchange-correlation effects, leading to the appearance of a robust interlayer biexciton phase (formed by two IXs), which has been theoretically predicted^6-8 but never observed before in experiments. The reduced dielectric screening in a free-standing heterobilayer not only resulted in a much higher formation efficiency of IXs, but also led to strongly enhanced dipole-dipole interactions, which enabled us to observe the many-body correlations of pristine IXs at the two-dimensional quantum limit. In addition, we firstly observed several emission peaks from moiré-trapped IXs at room temperature in a well-aligned, free-standing WSe₂/WS₂ heterobilayer. Our findings open avenues for exploring new quantum phases with potential for applications in non-linear optics.

Interlayer Exciton Diode and Transistor

Controlling the flow of charge neutral interlayer exciton (IX) quasiparticles can potentially lead to low loss excitonic circuits. Here, we report unidirectional transport of IXs along nanoscale electrostatically defined channels in an MoSe₂-WSe₂ heterostructure. These results are enabled by a lithographically defined triangular etch in a graphene gate to create a potential energy "slide". By performing spatially and temporally resolved photoluminescence measurements, we measure smoothly varying IX energy along the structure and high speed exciton flow with a drift velocity up to 2 × 10⁶ cm/s, an order of magnitude larger than previous experiments. Furthermore, exciton flow can be controlled by saturating exciton population in the channel using a second laser pulse, demonstrating an optically gated excitonic transistor. Our work paves the way toward low loss excitonic circuits, the study of bosonic transport in one-dimensional channels, and custom potential energy landscapes for excitons in van der Waals heterostructures.

Twist Angle Tuning of Moiré Exciton Polaritons in van der Waals Heterostructures

Twisted atomically thin semiconductors are characterized by moiré excitons. Their optical signatures and selection rules are well understood. However, their hybridization with photons in the strong coupling regime for heterostructures integrated in an optical cavity has not been the focus of research yet. Here, we combine an excitonic density matrix formalism with a Hopfield approach to provide microscopic insights into moiré exciton polaritons. In particular, we show that exciton-light coupling, polariton energy, and even the number of polariton branches can be controlled via the twist angle. We find that these new hybrid light-exciton states become delocalized relative to the constituent excitons due to the mixing with light and higher-energy excitons. The system can be interpreted as a natural quantum metamaterial with a periodicity that can be engineered via the twist angle. Our study presents a significant advance in microscopic understanding and control of moiré exciton polaritons in twisted atomically thin semiconductors.

Interlayer Excitons in Transition Metal Dichalcogenide Semiconductors for 2D Optoelectronics

Optoelectronic materials that allow on-chip integrated light signal emitting, routing, modulation, and detection are crucial for the development of high-speed and high-throughput optical communication and computing technologies. Interlayer excitons in 2D van der Waals heterostructures, where electrons and holes are bounded by Coulomb interaction but spatially localized in different 2D layers, have recently attracted intense attention for their enticing properties and huge potential in device applications. Here, a general view of these 2D-confined hydrogen-like bosonic particles and the state-of-the-art developments with respect to the frontier concepts and prototypes is presented. Staggered type-II band alignment enables expansion of the interlayer direct bandgap from the intrinsic visible in monolayers up to the near- or even mid-infrared spectrum. Owing to large exciton binding energy, together with ultralong lifetime, room-temperature exciton devices and observation of quantum behaviors are demonstrated. With the rapid advances, it can be anticipated that future studies of interlayer excitons will not only allow the construction of all-exciton information processing circuits but will also continue to enrich the panoply of ideas on quantum phenomena.

Twist Angle-Dependent Interlayer Exciton in MoS2 Bilayers Revealed by Room-Temperature Reflectance

In 2H stacking bilayer MoS2, the exciton with an interlayer nature has been evidenced due to the hybridization of hole states among both layers. The transition energy of this interlayer exciton is located between the A and B excitons. In this work, we investigate the evolution of optical properties in stacking MoS2 bilayers with the twisted angles ranging from 0° to 60°, especially focusing on the interlayer exciton. The clear modulations of the exciton responses are observed by the room-temperature reflectance. The interlayer exciton transition is observed in the artificial stacking bilayer MoS2 with the twisted angle around 60°. We found that the interlayer exciton is very sensitive to the twisted angle. Once the stacking angle deviates the 2H stacking by a few degrees, the interlayer transition is quenched. This is due to the bilayer symmetry and interlayer coupling of this unique system.

Theory of Excitons in Atomically Thin Semiconductors: Tight-Binding Approach

Atomically thin semiconductors from the transition metal dichalcogenide family are materials in which the optical response is dominated by strongly bound excitonic complexes. Here, we present a theory of excitons in two-dimensional semiconductors using a tight-binding model of the electronic structure. In the first part, we review extensive literature on 2D van der Waals materials, with particular focus on their optical response from both experimental and theoretical points of view. In the second part, we discuss our ab initio calculations of the electronic structure of MoS2, representative of a wide class of materials, and review our minimal tight-binding model, which reproduces low-energy physics around the Fermi level and, at the same time, allows for the understanding of their electronic structure. Next, we describe how electron-hole pair excitations from the mean-field-level ground state are constructed. The electron-electron interactions mix the electron-hole pair excitations, resulting in excitonic wave functions and energies obtained by solving the Bethe-Salpeter equation. This is enabled by the efficient computation of the Coulomb matrix elements optimized for two-dimensional crystals. Next, we discuss non-local screening in various geometries usually used in experiments. We conclude with a discussion of the fine structure and excited excitonic spectra. In particular, we discuss the effect of band nesting on the exciton fine structure; Coulomb interactions; and the topology of the wave functions, screening and dielectric environment. Finally, we follow by adding another layer and discuss excitons in heterostructures built from two-dimensional semiconductors.

Scalable Moiré Lattice with Oriented TMD Monolayers

Moiré lattice in artificially stacked monolayers of two-dimensional (2D) materials effectively modulates the electronic structures of materials, which is widely highlighted. Formation of the electronic Moiré superlattice promises the prospect of uniformity among different moiré cells across the lattice, enabling a new platform for novel properties, such as unconventional superconductivity, and scalable quantum emitters. Recently, epitaxial growth of the monolayer transition metal dichalcogenide (TMD) is achieved on the sapphire substrate by chemical vapor deposition (CVD) to realize scalable growth of highly-oriented monolayers. However, fabrication of the scalable Moiré lattice remains challenging due to the lack of essential manipulation of the well-aligned monolayers for clean interface quality and precise twisting angle control. Here, scalable and highly-oriented monolayers of TMD are realized on the sapphire substrates by using the customized CVD process. Controlled growth of the epitaxial monolayers is achieved by promoting the rotation of the nuclei-like domains in the initial growth stage, enabling aligned domains for further grain growth in the steady-state stage. A full coverage and distribution of the highly-oriented domains are verified by second-harmonic generation (SHG) microscopy. By developing the method for clean monolayer manipulation, hetero-stacked bilayer (epi-WS₂/epi-MoS₂) is fabricated with the specific angular alignment of the two major oriented monolayers at the edge direction of 0°/ ± 60°. On account of the optimization for scalable Moiré lattice with a high-quality interface, the observation of interlayer exciton at low temperature illustrates the feasibility of scalable Moiré superlattice based on the oriented monolayers.

Excitons in semiconductor moiré superlattices

Semiconductor moiré superlattices represent a rapidly developing area of engineered photonic materials and a new platform to explore correlated electron states and quantum simulation. In this Review, we briefly introduce early experiments that identified new exciton resonances in transition metal dichalcogenide heterobilayers and discuss several topics including two types of transition metal dichalcogenide moiré superlattice, new optical selection rules, early evidence of moiré excitons, and how the resonant energy, dynamics and diffusion properties of moiré excitons can be controlled via the twist angle. To interpret optical spectra, it is important to measure the energy modulation within a moiré supercell. In this context, we describe a few scanning tunnelling microscopy experiments that measure the moiré potential landscape directly. Finally, we review a few recent experiments that applied excitonic optical spectroscopy to probe correlated electron phenomena in transition metal dichalcogenide moiré superlattices.

Excitonic transport driven by repulsive dipolar interaction in a van der Waals heterostructure

Dipolar bosonic gases are currently the focus of intensive research due to their interesting many-body physics in the quantum regime. Their experimental embodiments range from Rydberg atoms to GaAs double quantum wells and van der Waals heterostructures built from transition metal dichalcogenides. Although quantum gases are very dilute, mutual interactions between particles could lead to exotic many-body phenomena such as Bose-Einstein condensation and high-temperature superfluidity. Here, we report the effect of repulsive dipolar interactions on the dynamics of interlayer excitons in the dilute regime. By using spatial and time-resolved photoluminescence imaging, we observe the dynamics of exciton transport, enabling a direct estimation of the exciton mobility. The presence of interactions significantly modifies the diffusive transport of excitons, effectively acting as a source of drift force and enhancing the diffusion coefficient by one order of magnitude. The repulsive dipolar interactions combined with the electrical control of interlayer excitons opens up appealing new perspectives for excitonic devices.

An ultrafast photodetector driven by interlayer exciton dissociation in a van der Waals heterostructure

Ultrafast photodetectors based on two-dimensional materials suffer from low responsivities and high dark currents. Interlayer exciton dissociation in type-II vertical heterojunctions of transition metal dichalcogenides is a viable mechanism for achieving higher responsivities with picosecond response times. Here, we propose a novel device concept based on these structures, with potential for self-powered photodetector applications characterized by an unprecedented trade-off between speed and responsivity with zero dark current. In order to assess the realistic performance to be expected in the proposed device, we have purposely devised a simulation approach able to provide a detailed investigation of the physics at play, while showing excellent predictive capabilities when compared with experiments on interlayer exciton transport available in the literature. The proposed high-performance photodetectors with tunable responsivities are at reach with available fabrication techniques and could help in paving the way towards monolithically integrated artificial neural networks for ultrafast machine vision in speed sensitive applications.

Strong interaction between interlayer excitons and correlated electrons in WSe2/WS2 moiré superlattice

Heterobilayers of transition metal dichalcogenides (TMDCs) can form a moiré superlattice with flat minibands, which enables strong electron interaction and leads to various fascinating correlated states. These heterobilayers also host interlayer excitons in a type-II band alignment, in which optically excited electrons and holes reside on different layers but remain bound by the Coulomb interaction. Here we explore the unique setting of interlayer excitons interacting with strongly correlated electrons, and we show that the photoluminescence (PL) of interlayer excitons sensitively signals the onset of various correlated insulating states as the band filling is varied. When the system is in one of such states, the PL of interlayer excitons is relatively amplified at increased optical excitation power due to reduced mobility, and the valley polarization of interlayer excitons is enhanced. The moiré superlattice of the TMDC heterobilayer presents an exciting platform to engineer interlayer excitons through the periodic correlated electron states.

A roadmap for interlayer excitons

Interlayer excitons in van der Waals heterostructures have tunable electron-hole separation in both real space and momentum space, enabling unprecedented control over excitonic properties to be exploited in a wide array of future applications ranging from exciton condensation to valleytronic and optoelectronic devices.

Interlayer exciton formation, relaxation, and transport in TMD van der Waals heterostructures

Van der Waals (vdW) heterostructures based on transition metal dichalcogenides (TMDs) generally possess a type-II band alignment that facilitates the formation of interlayer excitons between constituent monolayers. Manipulation of the interlayer excitons in TMD vdW heterostructures holds great promise for the development of excitonic integrated circuits that serve as the counterpart of electronic integrated circuits, which allows the photons and excitons to transform into each other and thus bridges optical communication and signal processing at the integrated circuit. As a consequence, numerous studies have been carried out to obtain deep insight into the physical properties of interlayer excitons, including revealing their ultrafast formation, long population recombination lifetimes, and intriguing spin-valley dynamics. These outstanding properties ensure interlayer excitons with good transport characteristics, and may pave the way for their potential applications in efficient excitonic devices based on TMD vdW heterostructures. At present, a systematic and comprehensive overview of interlayer exciton formation, relaxation, transport, and potential applications is still lacking. In this review, we give a comprehensive description and discussion of these frontier topics for interlayer excitons in TMD vdW heterostructures to provide valuable guidance for researchers in this field.

Diffusivity Reveals Three Distinct Phases of Interlayer Excitons in MoSe_{2}/WSe_{2} Heterobilayers

Charge separated interlayer excitons in transition metal dichalcogenide heterobilayers are being explored for moiré exciton lattices and exciton condensates. The presence of permanent dipole moments and the poorly screened Coulomb interaction make many-body interactions particularly strong for interlayer excitons. Here we reveal two distinct phase transitions for interlayer excitons in the MoSe_{2}/WSe_{2} heterobilayer using time and spatially resolved photoluminescence imaging: from trapped excitons in the moiré potential to the modestly mobile exciton gas as exciton density increases to n_{ex}∼10^{11}  cm^{-2} and from the exciton gas to the highly mobile charge separated electron-hole plasma for n_{ex}>10^{12}  cm^{-2}. The latter is the Mott transition and is confirmed in photoconductivity measurements. These findings set fundamental limits for achieving quantum states of interlayer excitons.

Twist Angle-Dependent Interlayer Exciton Lifetimes in van der Waals Heterostructures

In van der Waals (vdW) heterostructures formed by stacking two monolayers of transition metal dichalcogenides, multiple exciton resonances with highly tunable properties are formed and subject to both vertical and lateral confinement. We investigate how a unique control knob, the twist angle between the two monolayers, can be used to control the exciton dynamics. We observe that the interlayer exciton lifetimes in MoSe_{2}/WSe_{2} twisted bilayers (TBLs) change by one order of magnitude when the twist angle is varied from 1° to 3.5°. Using a low-energy continuum model, we theoretically separate two leading mechanisms that influence interlayer exciton radiative lifetimes. The shift to indirect transitions in the momentum space with an increasing twist angle and the energy modulation from the moiré potential both have a significant impact on interlayer exciton lifetimes. We further predict distinct temperature dependence of interlayer exciton lifetimes in TBLs with different twist angles, which is partially validated by experiments. While many recent studies have highlighted how the twist angle in a vdW TBL can be used to engineer the ground states and quantum phases due to many-body interaction, our studies explore its role in controlling the dynamics of optically excited states, thus, expanding the conceptual applications of "twistronics".

Tunable Phases of Moiré Excitons in van der Waals Heterostructures

Stacking monolayers of transition metal dichalcogenides into a heterostructure with a finite twist-angle gives rise to artificial moiré superlattices with a tunable periodicity. As a consequence, excitons experience a periodic potential, which can be exploited to tailor optoelectronic properties of these materials. Whereas recent experimental studies have confirmed twist-angle-dependent optical spectra, the microscopic origin of moiré exciton resonances has not been fully clarified yet. Here, we combine first-principles calculations with the excitonic density matrix formalism to study transitions between different moiré exciton phases and their impact on optical properties of the twisted MoSe₂/WSe₂ heterostructure. At angles smaller than 2°, we find flat, moiré-trapped states for inter- and intralayer excitons. This moiré exciton phase changes into completely delocalized states at 3°. We predict a linear and quadratic twist-angle dependence of excitonic resonances for the moiré-trapped and delocalized exciton phases, respectively.

Mechanisms and Applications of Steady-State Photoluminescence Spectroscopy in Two-Dimensional Transition-Metal Dichalcogenides

Two-dimensional (2D) transition-metal dichalcogenide (TMD) semiconductors exhibit many important structural and optoelectronic properties, such as strong light-matter interactions, direct bandgaps tunable from visible to near-infrared regions, flexibility and atomic thickness, quantum-confinement effects, valley polarization possibilities, and so on. Therefore, they are regarded as a very promising class of materials for next-generation state-of-the-art nano/micro optoelectronic devices. To explore different applications and device structures based on 2D TMDs, intrinsic material properties, their relationships, and evolutions with fabrication parameters need to be deeply understood, very often through a combination of various characterization techniques. Among them, steady-state photoluminescence (PL) spectroscopy has been extensively employed. This class of techniques is fast, contactless, and nondestructive and can provide very high spatial resolution. Therefore, it can be used to obtain optoelectronic properties from samples of various sizes (from microns to centimeters) during the fabrication process without complex sample preparation. In this article, the mechanism and applications of steady-state PL spectroscopy in 2D TMDs are reviewed. The first part of this review details the physics of PL phenomena in semiconductors and common techniques to acquire and analyze PL spectra. The second part introduces various applications of PL spectroscopy in 2D TMDs. Finally, a broader perspective is discussed to highlight some limitations and untapped opportunities of PL spectroscopy in characterizing 2D TMDs.