Stacking Domains and Dislocation Networks in Marginally Twisted Bilayers of Transition Metal Dichalcogenides

Tunnel junctions based on interfacial two dimensional ferroelectrics

Van der Waals heterostructures have opened new opportunities to develop atomically thin (opto)electronic devices with a wide range of functionalities. The recent focus on manipulating the interlayer twist angle has led to the observation of out-of-plane room temperature ferroelectricity in twisted rhombohedral bilayers of transition metal dichalcogenides. Here we explore the switching behaviour of sliding ferroelectricity using scanning probe microscopy domain mapping and tunnelling transport measurements. We observe well-pronounced ambipolar switching behaviour in ferroelectric tunnelling junctions with composite ferroelectric/non-polar insulator barriers and support our experimental results with complementary theoretical modelling. Furthermore, we show that the switching behaviour is strongly influenced by the underlying domain structure, allowing the fabrication of diverse ferroelectric tunnelling junction devices with various functionalities. We show that to observe the polarisation reversal, at least one partial dislocation must be present in the device area. This behaviour is drastically different from that of conventional ferroelectric materials, and its understanding is an important milestone for the future development of optoelectronic devices based on sliding ferroelectricity.

Polarization-driven band topology evolution in twisted MoTe2 and WSe2

Motivated by recent experimental observations of opposite Chern numbers in R-type twisted MoTe2 and WSe2 homobilayers, we perform large-scale density-functional-theory calculations with machine learning force fields to investigate moiré band topology across a range of twist angles in both materials. We find that the Chern numbers of the moiré frontier bands change sign as a function of twist angle, and this change is driven by the competition between moiré ferroelectricity and piezoelectricity. Our large-scale calculations, enabled by machine learning methods, reveal crucial insights into interactions across different scales in twisted bilayer systems. The interplay between atomic-level relaxation effects and moiré-scale electrostatic potential variation opens new avenues for the design of intertwined topological and correlated states, including the possibility of mimicking higher Landau level physics in the absence of magnetic field.

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.

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.

Competition of Moiré Network Sites to Form Electronic Quantum Dots in Reconstructed MoX2/WX2 Heterostructures

Twisted bilayers of two-dimensional semiconductors offer a versatile platform for engineering quantum states for charge carriers using moiré superlattice effects. Among the systems of recent interest are twistronic MoX2/WX2 heterostructures (X = Se or S), which undergo reconstruction into preferential stacking domains and highly strained domain wall networks, determining the electron/hole localization across moiré superlattices. Here, we present a catalogue of options for the formation of self-organized quantum dots and wires in lattice-reconstructed marginally twisted MoX2/WX2 bilayers with a relative lattice mismatch δ ≪ 1 for twist angles ranging from perfect alignment to θ ∼ 1°. On the basis of multiscale modeling taking into account twirling of domain wall networks, we analyze bilayers with both parallel and antiparallel orientations of their unit cells and describe crossovers between different positioning of band edges for electrons and holes across moiré superlattices when θ < δ and θ > δ.

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.

Bright and Dark Quadrupolar Excitons in the WSe_{2}/MoSe_{2}/WSe_{2} Heterotrilayer

Transition metal dichalcogenide heterostructures have been extensively studied as a platform for investigating exciton physics. While heterobilayers such as WSe_{2}/MoSe_{2} have received significant attention, there has been comparatively less research on heterotrilayers, which may offer new excitonic species and phases, as well as unique physical properties. In this Letter, we present theoretical and experimental investigations on the emission properties of quadrupolar excitons (QXs), a newly predicted type of exciton, in a WSe_{2}/MoSe_{2}/WSe_{2} heterotrilayer device. Our findings reveal that the optical brightness or darkness of QXs is determined by horizontal mirror symmetry and valley and spin selection rules. Additionally, the emission intensity and energy of both bright and dark QXs can be adjusted by applying an out-of-plane electric field, due to changes in hole distribution and the Stark effect. These results not only provide experimental evidence for the existence of QXs in heterotrilayers but also uncover their novel properties, which have the potential to drive the development of new exciton-based applications.

Twirling and Spontaneous Symmetry Breaking of Domain Wall Networks in Lattice-Reconstructed Heterostructures of Two-Dimensional Materials

Lattice relaxation in twistronic bilayers with close lattice parameters and almost perfect crystallographic alignment of the layers results in the transformation of the moiré pattern into a sequence of preferential stacking domains and domain wall networks. Here, we show that reconstructed moiré superlattices of the perfectly aligned heterobilayers of same chalcogen transition metal dichalcogenides have broken-symmetry structures featuring twisted nodes ("twirls") of domain wall networks. The analysis of twist-angle dependence of strain characteristics for the broken-symmetry structures shows that the formation of twirl reduces the amount of hydrostatic strain around the nodes, potentially weakening their influence on the band edge energies of electrons and holes.

Visualizing moiré ferroelectricity via plasmons and nano-photocurrent in graphene/twisted-WSe2 structures

Ferroelectricity, a spontaneous and reversible electric polarization, is found in certain classes of van der Waals (vdW) materials. The discovery of ferroelectricity in twisted vdW layers provides new opportunities to engineer spatially dependent electric and optical properties associated with the configuration of moiré superlattice domains and the network of domain walls. Here, we employ near-field infrared nano-imaging and nano-photocurrent measurements to study ferroelectricity in minimally twisted WSe2. The ferroelectric domains are visualized through the imaging of the plasmonic response in a graphene monolayer adjacent to the moiré WSe2 bilayers. Specifically, we find that the ferroelectric polarization in moiré domains is imprinted on the plasmonic response of the graphene. Complementary nano-photocurrent measurements demonstrate that the optoelectronic properties of graphene are also modulated by the proximal ferroelectric domains. Our approach represents an alternative strategy for studying moiré ferroelectricity at native length scales and opens promising prospects for (opto)electronic devices.

Ferroelectric Switching at Symmetry-Broken Interfaces by Local Control of Dislocations Networks

Semiconducting ferroelectric materials with low energy polarization switching offer a platform for next-generation electronics such as ferroelectric field-effect transistors. Recently discovered interfacial ferroelectricity in bilayers of transition metal dichalcogenide films provides an opportunity to combine the potential of semiconducting ferroelectrics with the design flexibility of 2D material devices. Here, local control of ferroelectric domains in a marginally twisted WS2 bilayer is demonstrated with a scanning tunneling microscope at room temperature, and their observed reversible evolution is understood using a string-like model of the domain wall network (DWN). Two characteristic regimes of DWN evolution are identified: (i) elastic bending of partial screw dislocations separating smaller domains with twin stackings due to mutual sliding of monolayers at domain boundaries and (ii) merging of primary domain walls into perfect screw dislocations, which become the seeds for the recovery of the initial domain structure upon reversing electric field. These results open the possibility to achieve full control over atomically thin semiconducting ferroelectric domains using local electric fields, which is a critical step towards their technological use.

Shear Strain-Induced Two-Dimensional Slip Avalanches in Rhombohedral MoS2

Slip avalanches are ubiquitous phenomena occurring in three-dimensional materials under shear strain, and their study contributes immensely to our understanding of plastic deformation, fragmentation, and earthquakes. So far, little is known about the role of shear strain in two-dimensional (2D) materials. Here we show some evidence of 2D slip avalanches in exfoliated rhombohedral MoS2, triggered by shear strain near the threshold level. Utilizing interfacial polarization in 3R-MoS2, we directly probe the stacking order in multilayer flakes and discover a wide variety of polarization domains with sizes following a power-law distribution. These findings suggest that slip avalanches can occur during the exfoliation of 2D materials, and the stacking orders can be changed via shear strain. Our observation has far-reaching implications for the development of new materials and technologies, where precise control over the atomic structure of these materials is essential for optimizing their properties as well as for our understanding of fundamental physical phenomena.

Operando electron microscopy investigation of polar domain dynamics in twisted van der Waals homobilayers

Conventional antiferroelectric materials with atomic-scale anti-aligned dipoles undergo a transition to a ferroelectric (FE) phase under strong electric fields. The moiré superlattice formed in the twisted stacks of van der Waals crystals exhibits polar domains alternating in moiré length with anti-aligned dipoles. In this moiré domain antiferroelectic (MDAF) arrangement, the distribution of electric dipoles is distinguished from that of two-dimensional FEs, suggesting dissimilar domain dynamics. Here we performed an operando transmission electron microscopy investigation on twisted bilayer WSe2 to observe the polar domain dynamics in real time. We find that the topological protection, provided by the domain wall network, prevents the MDAF-to-FE transition. As one decreases the twist angle, however, this transition occurs as the domain wall network disappears. Exploiting stroboscopic operando transmission electron microscopy on the FE phase, we measure a maximum domain wall velocity of 300 μm s-1. Domain wall pinnings by various disorders limit the domain wall velocity and cause Barkhausen noises in the polarization hysteresis loop. Atomic-scale analysis of the pinning disorders provides structural insight on how to improve the switching speed of van der Waals FEs.

Excitons in mesoscopically reconstructed moiré heterostructures

Moiré effects in vertical stacks of two-dimensional crystals give rise to new quantum materials with rich transport and optical phenomena that originate from modulations of atomic registries within moiré supercells. Due to finite elasticity, however, the superlattices can transform from moiré-type to periodically reconstructed patterns. Here we expand the notion of such nanoscale lattice reconstruction to the mesoscopic scale of laterally extended samples and demonstrate rich consequences in optical studies of excitons in MoSe₂-WSe₂ heterostructures with parallel and antiparallel alignments. Our results provide a unified perspective on moiré excitons in near-commensurate semiconductor heterostructures with small twist angles by identifying domains with exciton properties of distinct effective dimensionality, and establish mesoscopic reconstruction as a compelling feature of real samples and devices with inherent finite size effects and disorder. Generalized to stacks of other two-dimensional materials, this notion of mesoscale domain formation with emergent topological defects and percolation networks will instructively expand the understanding of fundamental electronic, optical and magnetic properties of van der Waals heterostructures.

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.

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.

Signatures of Electric Field and Layer Separation Effects on the Spin-Valley Physics of MoSe2/WSe2 Heterobilayers: From Energy Bands to Dipolar Excitons

Multilayered van der Waals heterostructures based on transition metal dichalcogenides are suitable platforms on which to study interlayer (dipolar) excitons, in which electrons and holes are localized in different layers. Interestingly, these excitonic complexes exhibit pronounced valley Zeeman signatures, but how their spin-valley physics can be further altered due to external parameters-such as electric field and interlayer separation-remains largely unexplored. Here, we perform a systematic analysis of the spin-valley physics in MoSe2/WSe2 heterobilayers under the influence of an external electric field and changes of the interlayer separation. In particular, we analyze the spin (Sz) and orbital (Lz) degrees of freedom, and the symmetry properties of the relevant band edges (at K, Q, and Γ points) of high-symmetry stackings at 0° (R-type) and 60° (H-type) angles-the important building blocks present in moiré or atomically reconstructed structures. We reveal distinct hybridization signatures on the spin and the orbital degrees of freedom of low-energy bands, due to the wave function mixing between the layers, which are stacking-dependent, and can be further modified by electric field and interlayer distance variation. We find that H-type stackings favor large changes in the g-factors as a function of the electric field, e.g., from -5 to 3 in the valence bands of the Hhh stacking, because of the opposite orientation of Sz and Lz of the individual monolayers. For the low-energy dipolar excitons (direct and indirect in k-space), we quantify the electric dipole moments and polarizabilities, reflecting the layer delocalization of the constituent bands. Furthermore, our results show that direct dipolar excitons carry a robust valley Zeeman effect nearly independent of the electric field, but tunable by the interlayer distance, which can be rendered experimentally accessible via applied external pressure. For the momentum-indirect dipolar excitons, our symmetry analysis indicates that phonon-mediated optical processes can easily take place. In particular, for the indirect excitons with conduction bands at the Q point for H-type stackings, we find marked variations of the valley Zeeman (∼4) as a function of the electric field, which notably stands out from the other dipolar exciton species. Our analysis suggests that stronger signatures of the coupled spin-valley physics are favored in H-type stackings, which can be experimentally investigated in samples with twist angle close to 60°. In summary, our study provides fundamental microscopic insights into the spin-valley physics of van der Waals heterostructures, which are relevant to understanding the valley Zeeman splitting of dipolar excitonic complexes, and also intralayer excitons.

Moiré Potential, Lattice Relaxation, and Layer Polarization in Marginally Twisted MoS2 Bilayers

Artificially twisted heterostructures of semiconducting transition-metal dichalcogenides (TMDs) offer unprecedented control over their electronic and optical properties via the spatial modulation of interlayer interactions and structural reconstruction. Here we study twisted MoS₂ bilayers in a wide range of twist angles near 0° using scanning tunneling microscopy/spectroscopy. We investigate the twist angle dependence of the moiré pattern, which is dominated by lattice reconstruction for small angles (<2°), leading to large triangular domains with rhombohedral stacking. Local spectroscopy measurements reveal a large moiré-potential strength of 100-200 meV for angles <3°. In reconstructed regions, we see a bias-dependent asymmetry between neighboring triangular domains, which we relate to the vertical polarization that is intrinsic to rhombohedral stacked TMDs. This viewpoint is further supported by spectroscopy maps and ambient piezoresponse measurements. Our results provide a microscopic perspective of this new class of interfacial ferroelectrics and can offer clues for designing novel heterostructures that harness this effect.

Spatially Dependent Electronic Structures and Excitons in a Marginally Twisted Moiré Superlattice of Spiral WS2

Twisted two-dimensional transition metal dichalcogenide (TMD) moiré superlattices provide an additional degree of freedom to engineer electronic and optical properties. Nevertheless, controllable synthesis of marginally twisted homo TMD moiré superlattices is still a challenge. Here, physical vapor deposition grown spiral WS₂ nanosheets are demonstrated to be a marginally twisted moiré superlattice using scanning tunneling microscopy and spectroscopy. Periodic moiré superlattices are found on the third layer (3L) and 4L of the spiral WS₂ nanosheet owing to the marginally twisted alignment between two neighboring layers, resulting in a highly localized flat band near the valence band maximum. Their bandgap depends on atomic stacking configurations, which gives a good interpretation for split moiré excitons using photoluminescence at 77 K. This work can benefit the development of twisted homo TMD moiré superlattices and could promote the profound research of twisted TMDs in the prospective field, such as strongly correlated physics and twistronics.

Spatially Resolved Polarization Manipulation of Ferroelectricity in Twisted hBN

Robust room-temperature interfacial ferroelectricity has been formed in the 2D limit by simply twisting two atomic layers of non-ferroelectric hexagonal boron nitride (hBN). A thorough understanding of this newly discovered ferroelectric system is required. Here, twisted hBN is used as a tunneling junction and it is studied at the nanometer scale using conductive atomic force microscopy. Three properties unique to this system are discovered. First, the polarization dependence of the tunneling resistance contrasts with the conventional theory. Second, the ferroelectric domains can be controlled using mechanical stress, highlighting the original meaning of the emergent "slidetronics". Third, ferroelectric hysteresis is highly spatially dependent. The hysteresis is symmetric at the domain walls. A few nanometers away, the hysteresis shifts completely to the positive or negative side, depending on the original polarization. These findings reveal the unconventional ferroelectricity in this 2D system.

Bond-Orbital-Resolved Piezoelectricity in Sp2-Hybridized Monolayer Semiconductors

Sp²-hybridized monolayer semiconductors (e.g., planar group III-V and IV-IV binary compounds) with inversion symmetry breaking (ISB) display piezoelectricity governed by their σ- and π-bond electrons. Here, we studied their bond-orbital-resolved electronic piezoelectricity (i.e., the σ- and π-piezoelectricity). We formulated a tight-binding piezoelectric model to reveal the different variations of σ- and π-piezoelectricity with the ISB strength (Δ). As Δ varied from positive to negative, the former decreased continuously, but the latter increased piecewise and jumped at Δ=0 due to the criticality of the π-electrons' ground-state geometry near this quantum phase-transition point. This led to a piezoelectricity predominated by the π-electrons for a small |Δ|. By constructing an analytical model, we clarified the microscopic mechanisms underlying the anomalous π-piezoelectricity and its subtle relations with the valley Hall effect. The validation of our models was justified by applying them to the typical sp² monolayers including hexagonal silicon carbide, Boron-X (X = N, P, As, Ab), and a BN-doped graphene superlattice.

Moiré modulation of charge density waves

Here we investigate how charge density waves (CDWs), inherent to a monolayer, are effected by creating twisted van der Waals structures. Homobilayers of metallic transition metal dichalcogenides (TMDs), at small twist angles where there is significant atomic reconstruction, are utilised as an example to investigate the interplay between the moiré domain structure and CDWs of different periods. For3×3CDWs, there is no geometric constraint to prevent the CDWs from propagating throughout the moiré structure. Whereas for2×2CDWs, to ensure the CDWs in each layer have the most favourable interactions in the domains, the CDW phase must be destroyed in the connecting domain walls. For3×3CDWs with twist angles close to 180^∘, moiré-scale triangular structures can form; while close to 0^∘, moiré-scale dimer domains occur. The star-of-David CDW (13×13) is found to host CDWs in the domains only, since there is one low energy stacking configuration, similar to the2×2CDWs. These predictions are offered for experimental verification in twisted bilayer metallic TMDs which host CDWs, and we hope this will stimulate further research on the interplay between the moiré superlattice and CDW phases intrinsic to the comprising 2D materials.

Morphology Deformation and Giant Electronic Band Modulation in Long-Wavelength WS2 Moiré Superlattices

As a lattice interference effect, moiré superlattices feature a magnification effect that they respond sensitively to both the extrinsic mechanical perturbations and intrinsic atomic reconstructions. Here, using scanning tunneling microscopy and spectroscopy, we observe that long-wavelength WS₂ superlattices are reconstructed into various moiré morphologies, ranging from regular hexagons to heavily deformed ones. We show that a dedicated interplay between the extrinsic nonuniform heterostrain and the intrinsic atomic reconstruction is responsible for this interesting moiré structure evolution. Importantly, the interplay between these two factors also introduces a local inhomogeneous intralayer strain within a moiré. Contrary to the commonly reported electronic modulation that occurred at the valence band edge due to interlayer hybridization, we find that this local intralayer strain induces a strong modulation at K point of the conduction band, reaching up to 300 meV in the heavily deformed moiré. Our microscopic explorations provide valuable information in understanding the intriguing physics in TMD moirés.

Tunable Strong Magnetic Anisotropy in Two-Dimensional van der Waals Antiferromagnets

We show that the anisotropic energy of a 2D antiferromagnet is greatly enhanced via stacking on a magnetic substrate layer, arising from the sublattice-dependent interlayer magnetic interaction that defines an effective anisotropic energy. Interestingly, this effective energy couples strongly with the interlayer stacking order and the magnetic order of the substrate layer, providing unique mechanical and magnetic means to control the antiferromagnetic order. These two types of control methods distinctly affect the sublattice magnetization dynamics, with a change in the ratio of sublattice precession amplitudes in the former and its chirality in the latter. In moiré superlattices formed by a relative twist or strain between the layers, the coupling with stacking order introduces a landscape of effective anisotropic energy across the moiré, which can be utilized to create nonuniform antiferromagnetic textures featuring periodically localized low-energy magnons.

Interfacial ferroelectricity in marginally twisted 2D semiconductors

Twisted heterostructures of two-dimensional crystals offer almost unlimited scope for the design of new metamaterials. Here we demonstrate a room temperature ferroelectric semiconductor that is assembled using mono- or few-layer MoS₂. These van der Waals heterostructures feature broken inversion symmetry, which, together with the asymmetry of atomic arrangement at the interface of two 2D crystals, enables ferroelectric domains with alternating out-of-plane polarization arranged into a twist-controlled network. The last can be moved by applying out-of-plane electrical fields, as visualized in situ using channelling contrast electron microscopy. The observed interfacial charge transfer, movement of domain walls and their bending rigidity agree well with theoretical calculations. Furthermore, we demonstrate proof-of-principle field-effect transistors, where the channel resistance exhibits a pronounced hysteresis governed by pinning of ferroelectric domain walls. Our results show a potential avenue towards room temperature electronic and optoelectronic semiconductor devices with built-in ferroelectric memory functions.

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.

A Scalable Network Model for Electrically Tunable Ferroelectric Domain Structure in Twistronic Bilayers of Two-Dimensional Semiconductors

Moiré structures in small-angle-twisted bilayers of two-dimensional (2D) semiconductors with a broken-symmetry interface form arrays of ferroelectric (FE) domains with periodically alternating out-of-plane polarization. Here, we propose a network theory for the tunability of such FE domain structure by applying an electric field perpendicular to the 2D crystal. Using multiscale analysis, we derive a fully parametrized string-theory-like description of the domain wall network (DWN) and show that it undergoes a qualitative change, after the arcs of partial dislocation (PD) like domain walls merge (near the network nodes) into streaks of perfect screw dislocations (PSD), which happens at a threshold displacement field dependent on the DWN period.

Valley-Polarized Quantum Anomalous Hall State in Moiré MoTe_{2}/WSe_{2} Heterobilayers

Moiré heterobilayer transition metal dichalcogenides (TMDs) emerge as an ideal system for simulating the single-band Hubbard model and interesting correlated phases have been observed in these systems. Nevertheless, the moiré bands in heterobilayer TMDs were believed to be topologically trivial. Recently, it was reported that both a quantum valley Hall insulating state at filling ν=2 (two holes per moiré unit cell) and a valley-polarized quantum anomalous Hall state at filling ν=1 were observed in AB stacked moiré MoTe_{2}/WSe_{2} heterobilayers. However, how the topologically nontrivial states emerge is not known. In this Letter, we propose that the pseudomagnetic fields induced by lattice relaxation in moiré MoTe_{2}/WSe_{2} heterobilayers could naturally give rise to moiré bands with finite Chern numbers. We show that a time-reversal invariant quantum valley Hall insulator is formed at full filling ν=2, when two moiré bands with opposite Chern numbers are filled. At half filling ν=1, the Coulomb interaction lifts the valley degeneracy and results in a valley-polarized quantum anomalous Hall state, as observed in the experiment. Our theory identifies a new way to achieve topologically nontrivial states in heterobilayer TMD materials.

Moiré-Enabled Topological Superconductivity

The search for artificial topological superconductivity has been limited by the stringent conditions required for its emergence. As exemplified by the recent discoveries of various correlated electronic states in twisted van der Waals materials, moiré patterns can act as a powerful knob to create artificial electronic structures. Here, we demonstrate that a moiré pattern between a van der Waals superconductor and a monolayer ferromagnet creates a periodic potential modulation that enables the realization of a topological superconducting state that would not be accessible in the absence of the moiré. The magnetic moiré pattern gives rise to Yu-Shiba-Rusinov minibands and periodic modulation of the Majorana edge modes that we detect using low-temperature scanning tunneling microscopy (STM) and spectroscopy (STS). Moiré patterns and, more broadly, periodic potential modulations are powerful tools to overcome the conventional constraints for realizing and controlling topological superconductivity.

Chiral Phonons in Moiré Superlattices

We discover chiral phonons at the lowest energy bands in moiré superlattices. The moiré chiral phonons we uncover are the collective excitations of the stacking domains. Their origin is uniquely attributed to the stacking configurations whose interlayer binding energy breaks the C_2z symmetry on the moiré length scale. Within elastic theory, we use a general symmetry analysis to provide a complete classification of van der Waals heterostructures in respect to hosting moiré chiral phonons and show the calculation for twisted MoS₂ as an example. We present a low-energy effective model to qualitatively understand the moiré chiral phonons and show that it captures the essential physics remarkably well. Our result potentially opens up new possibilities in phononic twistronics as the moiré chiral phonons have high tunability, moiré scale wavelengths, excitation energies in only a few meV and can possibly be mechanically excited.

Electronic Structure and Stacking Arrangement of Tungsten Disulfide at the Gold Contact

There is an intensive effort to control the nature of attractive interactions between ultrathin semiconductors and metals and to understand its impact on the electronic properties at the junction. Here, we present a photoelectron spectroscopy study on the interface between WS₂ films and gold, with a focus on the occupied electronic states near the Brillouin zone center (i.e., the Γ point). To delineate the spectra of WS₂ supported on crystalline Au from the suspended WS₂, we employ a microscopy approach and a tailored sample structure, in which the WS₂/Au junction forms a semi-epitaxial relationship and is adjacent to suspended WS₂ regions. The photoelectron spectra, as a function of WS₂ thickness, display the expected splitting of the highest occupied states at the Γ point. In multilayer WS₂, we discovered variations in the electronic states that spatially align with the crystalline grains of underlying Au. Corroborated by density functional theory calculations, we attribute the electronic structure variations to stacking variations within the WS₂ films. We propose that strong interactions exerted by Au grains cause slippage of the interfacing WS₂ layer with respect to the rest of the WS₂ film. Our findings illustrate that the electronic properties of transition metal dichalcogenides, and more generally 2D layered materials, are physically altered by the interactions with the interfacing materials, in addition to the electron screening and defects that have been widely considered.

Lattice reconstruction induced multiple ultra-flat bands in twisted bilayer WSe2

Moiré superlattices in van der Waals heterostructures provide a tunable platform to study emergent properties that are absent in the natural crystal form. Twisted bilayer transition metal dichalcogenides (TB-TMDs) can host moiré flat bands over a wide range of twist angles. For twist angle close to 60°, it was predicted that TB-TMDs undergo a lattice reconstruction which causes the formation of ultra-flat bands. Here, by using scanning tunneling microscopy and spectroscopy, we show the emergence of multiple ultra-flat bands in twisted bilayer WSe₂ when the twist angle is within 3° of 60°. The ultra-flat bands are manifested as narrow tunneling conductance peaks with estimated bandwidth less than 10 meV, which is only a fraction of the estimated on-site Coulomb repulsion energy. The number of these ultra-flat bands and spatial distribution of the wavefunctions match well with the theoretical predictions, strongly evidencing that the observed ultra-flat bands are induced by lattice reconstruction. Our work provides a foundation for further study of the exotic correlated phases in TB-TMDs.

It was predicted that lattice reconstruction can lead to the emergence of multiple ultra-flat electronic bands in twisted bilayer transition metal dichalcogenides. Here, by using scanning tunneling microscopy and spectroscopy, the authors demonstrate such bands in twisted bilayer WSe₂.

Layer-dependent interface reconstruction and strain modulation in twisted WSe2

Twistronics has emerged as one of the most attractive playgrounds for manipulating the interfacial structures and electronic properties of two-dimensional materials. However, the layer-dependent lattice reconstruction and resulted strain distribution in marginally twisted transition metal dichalcogenides still remain elusive. Here we report a systematic study by both electron diffraction quantification and atomic-resolution imaging on the interface reconstruction of twisted WSe₂, which shows a strong dependence on the constituent layer numbers and twist angles. The competition between the interlayer interaction, which varies with local atomic configurations, and the intralayer elastic deformation, related to the layer thickness, leads to rich superlattice motifs and strain modulation patterns, i.e. triangular for odd and kagome-like textures for even layer numbers, against the rigid stacking moiré model. The strain effects of small twist angles are further demonstrated by electrical transport measurements, manifesting intriguing conducting states at low temperatures beyond the flat band features of large twist angles. Our work not only provides a comprehensive understanding of layer-dependent twist structures, but also may shed light on the future design of twistronic devices.

Large-Scale Mapping of Moiré Superlattices by Hyperspectral Raman Imaging

Moiré superlattices can induce correlated-electronic phases in twisted van der Waals materials: strongly correlated quantum phenomena emerge, such as superconductivity and the Mott-insulating state. However, moiré superlattices produced through artificial stacking can be quite inhomogeneous, which hampers the development of a clear correlation between the moiré period and the emerging electrical and optical properties. Here, it is demonstrated in twisted-bilayer transition-metal dichalcogenides that low-frequency Raman scattering can be utilized not only to detect atomic reconstruction, but also to map out the inhomogeneity of the moiré lattice over large areas. The method is established based on the finding that both the interlayer-breathing mode and moiré phonons are highly susceptible to the moiré period and provide characteristic fingerprints. Hyperspectral Raman imaging visualizes microscopic domains of a 5° twisted-bilayer sample with an effective twist-angle resolution of about 0.1°. This ambient methodology can be conveniently implemented to characterize and preselect high-quality areas of samples for subsequent device fabrication, and for transport and optical experiments.

Weak ferroelectric charge transfer in layer-asymmetric bilayers of 2D semiconductors

In bilayers of two-dimensional semiconductors with stacking arrangements which lack inversion symmetry charge transfer between the layers due to layer-asymmetric interband hybridisation can generate a potential difference between the layers. We analyse bilayers of transition metal dichalcogenides (TMDs)—in particular, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {WSe}_2$$\end{document}WSe2—for which we find a substantial stacking-dependent charge transfer, and InSe, for which the charge transfer is found to be negligibly small. The information obtained about TMDs is then used to map potentials generated by the interlayer charge transfer across the moiré superlattice in twistronic bilayers.

Moiré Patterns in 2D Materials: A Review

Quantum materials have attracted much attention in recent years due to their exotic and incredible properties. Among them, van der Waals materials stand out due to their weak interlayer coupling, providing easy access to manipulating electrical and optical properties. Many fascinating electrical, optical, and magnetic properties have been reported in the moiré superlattices, such as unconventional superconductivity, photonic dispersion engineering, and ferromagnetism. In this review, we summarize the methods to prepare moiré superlattices in the van der Waals materials and focus on the current discoveries of moiré pattern-modified electrical properties, recent findings of atomic reconstruction, as well as some possible future directions in this field.

Moiré excitons in MoSe2-WSe2 heterobilayers and heterotrilayers

Layered two-dimensional materials exhibit rich transport and optical phenomena in twisted or lattice-incommensurate heterostructures with spatial variations of interlayer hybridization arising from moiré interference effects. Here, we report experimental and theoretical studies of excitons in twisted heterobilayers and heterotrilayers of transition metal dichalcogenides. Using MoSe₂-WSe₂ stacks as representative realizations of twisted van der Waals bilayer and trilayer heterostructures, we observe contrasting optical signatures and interpret them in the theoretical framework of interlayer moiré excitons in different spin and valley configurations. We conclude that the photoluminescence of MoSe₂-WSe₂ heterobilayer is consistent with joint contributions from radiatively decaying valley-direct interlayer excitons and phonon-assisted emission from momentum-indirect reservoirs that reside in spatially distinct regions of moiré supercells, whereas the heterotrilayer emission is entirely due to momentum-dark interlayer excitons of hybrid-layer valleys. Our results highlight the profound role of interlayer hybridization for transition metal dichalcogenide heterostacks and other realizations of multi-layered semiconductor van der Waals heterostructures.

Γ valley transition metal dichalcogenide moiré bands

The valence band maxima of most group VI transition metal dichalcogenide thin films remain at the Γ point all of the way from bulk to bilayer. In this paper, we develop a continuum theory of the moiré minibands that are formed in the valence bands of Γ-valley homobilayers by a small relative twist. Our effective theory is benchmarked against large-scale ab initio electronic structure calculations that account for lattice relaxation. As a consequence of an emergent [Formula: see text] symmetry, we find that low-energy Γ-valley moiré holes differ qualitatively from their K-valley counterparts addressed previously; in energetic order, the first three bands realize 1) a single-orbital model on a honeycomb lattice, 2) a two-orbital model on a honeycomb lattice, and 3) a single-orbital model on a kagome lattice.

Moiré metrology of energy landscapes in van der Waals heterostructures

The emerging field of twistronics, which harnesses the twist angle between two-dimensional materials, represents a promising route for the design of quantum materials, as the twist-angle-induced superlattices offer means to control topology and strong correlations. At the small twist limit, and particularly under strain, as atomic relaxation prevails, the emergent moiré superlattice encodes elusive insights into the local interlayer interaction. Here we introduce moiré metrology as a combined experiment-theory framework to probe the stacking energy landscape of bilayer structures at the 0.1 meV/atom scale, outperforming the gold-standard of quantum chemistry. Through studying the shapes of moiré domains with numerous nano-imaging techniques, and correlating with multi-scale modelling, we assess and refine first-principle models for the interlayer interaction. We document the prowess of moiré metrology for three representative twisted systems: bilayer graphene, double bilayer graphene and H-stacked MoSe2/WSe2. Moiré metrology establishes sought after experimental benchmarks for interlayer interaction, thus enabling accurate modelling of twisted multilayers.

Layer Pseudospin Dynamics and Genuine Non-Abelian Berry Phase in Inhomogeneously Strained Moiré Pattern

Periodicity of long wavelength moiré patterns is very often destroyed by the inhomogeneous strain introduced in fabrications of van der Waals layered structures. We present a framework to describe massive Dirac fermions in such distorted moiré pattern of transition metal dichalcogenides homobilayers, accounting for the dynamics of layer pseudospin. In decoupled bilayers, we show two causes of in-plane layer pseudospin precession: By the coupling of layer antisymmetric strain to valley magnetic moment; and by the Aharonov-Bohm effect in the SU(2) gauge potential for the case of R-type bilayer under antisymmetric strain and H-type under symmetric strain. With interlayer coupling in the moiré, its interplay with strain manifests as a non-Abelian gauge field. We show a genuine non-Abelian Aharonov-Bohm effect in such field, where the evolution operators for different loops are noncommutative. This provides an exciting platform to explore non-Abelian gauge field effects on electron, with remarkable tunability of the field by strain and interlayer bias.

Bending moduli for forty-four select atomic monolayers from first principles

We calculate bending moduli along the principal directions for forty-four select atomic monolayers using ab initio density functional theory (DFT). Specifically, considering representative materials from each of Groups IV, III-V, V monolayers, Group IV monochalcogenides, transition metal trichalcogenides, transition metal dichalcogenides and Group III monochalcogenides, we utilize the recently developed Cyclic DFT method to calculate the bending moduli in the practically relevant but previously intractable low-curvature limit. We find that the moduli generally increase with thickness of the monolayer, while spanning three orders of magnitude between the different materials. In addition, structures with a rectangular lattice are prone to a higher degree of anisotropy relative to those with a honeycomb lattice. Exceptions to these trends are generally a consequence of unusually strong/weak bonding and/or significant structural relxation related effects.

Broken mirror symmetry in excitonic response of reconstructed domains in twisted MoSe2/MoSe2 bilayers

Van der Waals heterostructures obtained via stacking and twisting have been used to create moiré superlattices¹, enabling new optical and electronic properties in solid-state systems. Moiré lattices in twisted bilayers of transition metal dichalcogenides (TMDs) result in exciton trapping^2-5, host Mott insulating and superconducting states⁶ and act as unique Hubbard systems^7-9 whose correlated electronic states can be detected and manipulated optically. Structurally, these twisted heterostructures feature atomic reconstruction and domain formation^10-14. However, due to the nanoscale size of moiré domains, the effects of atomic reconstruction on the electronic and excitonic properties have not been systematically investigated. Here we use near-0°-twist-angle MoSe₂/MoSe₂ bilayers with large rhombohedral AB/BA domains¹⁵ to directly probe the excitonic properties of individual domains with far-field optics. We show that this system features broken mirror/inversion symmetry, with the AB and BA domains supporting interlayer excitons with out-of-plane electric dipole moments in opposite directions. The dipole orientation of ground-state Γ-K interlayer excitons can be flipped with electric fields, while higher-energy K-K interlayer excitons undergo field-asymmetric hybridization with intralayer K-K excitons. Our study reveals the impact of crystal symmetry on TMD excitons and points to new avenues for realizing topologically non-trivial systems^16,17, exotic metasurfaces¹⁸, collective excitonic phases¹⁹ and quantum emitter arrays^20,21 via domain-pattern engineering.

Atomic reconstruction in twisted bilayers of transition metal dichalcogenides

Van der Waals heterostructures form a unique class of layered artificial solids in which physical properties can be manipulated through controlled composition, order and relative rotation of adjacent atomic planes. Here we use atomic-resolution transmission electron microscopy to reveal the lattice reconstruction in twisted bilayers of the transition metal dichalcogenides, MoS₂ and WS₂. For twisted 3R bilayers, a tessellated pattern of mirror-reflected triangular 3R domains emerges, separated by a network of partial dislocations for twist angles θ < 2°. The electronic properties of these 3R domains, featuring layer-polarized conduction-band states caused by lack of both inversion and mirror symmetry, appear to be qualitatively different from those of 2H transition metal dichalcogenides. For twisted 2H bilayers, stable 2H domains dominate, with nuclei of a second metastable phase. This appears as a kagome-like pattern at θ ≈ 2°, transitioning at θ → 0 to a hexagonal array of screw dislocations separating large-area 2H domains. Tunnelling measurements show that such reconstruction creates strong piezoelectric textures, opening a new avenue for engineering of 2D material properties.