Correlated electronic phases in twisted bilayer transition metal dichalcogenides

Atomically Thin Kagome-Structured Co9Te16 Achieved through Self-Intercalation and Its Flat Band Visualization

Kagome materials have recently garnered substantial attention due to the intrinsic flat band feature and the stimulated magnetic and spin-related many-body physics. In contrast to their bulk counterparts, two-dimensional (2D) kagome materials feature more distinct kagome bands, beneficial for exploring novel quantum phenomena. Herein, we report the direct synthesis of an ultrathin kagome-structured Co-telluride (Co9Te16) via a molecular beam epitaxy (MBE) route and clarify its formation mechanism from the Co-intercalation in the 1T-CoTe2 layers. More significantly, we unveil the flat band states in the ultrathin Co9Te16 and identify the real-space localization of the flat band states by in situ scanning tunneling microscopy/spectroscopy (STM/STS) combined with first-principles calculations. A ferrimagnetic order is also predicted in kagome-Co9Te16. This work should provide a novel route for the direct synthesis of ultrathin kagome materials via a metal self-intercalation route, which should shed light on the exploration of the intriguing flat band physics in the related systems.

Thermodynamics of resonating-valence-bond states toward the understanding of quantum spin liquid phenomena

A quantum spin liquid (QSL) is a state of matter in which spins do not exhibit magnetic order. In contrast to paramagnets, the spins in a QSL interact strongly, similar to conventional ordered magnets; however the thermodynamic stability of QSLs is rarely studied. Here, the thermodynamic properties of centered hexagon nanoclusters were investigated using the Hubbard model, which was solved using exact numerical diagonalization. The total spin, spin-spin correlation functions, local magnetic moments, charge and spin gaps, and magnetocaloric effect were analyzed for a half-filled band as a function of the ratio between the on-site Coulomb repulsion and electronic hopping (U/t). The centered hexagon nanocluster exhibited an antiferromagnetic (AFM) behavior with exotic magnetic ordering. Resonating-valence-bond (RVB) states were observed for intermediate values of U/t, in which short-range spin-spin correlation functions were suppressed to minimize spin frustration. The AFM order was examined in terms of the Néel-like temperature derived from the temperature dependence of the magnetic susceptibility. An interesting result is that the systems under external magnetic fields exhibited an inverse magnetocaloric effect, which was remarkable for intermediate values of U/t, where the RVB state was observed. Owing to the novel discovery of exotic magnetic ordering in triangular moiré patterns in twisted bilayer graphene (TBLG) systems, these results provide insights into the onset of magnetism and the possible spin liquid states in these graphene moiré materials.

Preservation of Topological Surface States in Millimeter-Scale Transferred Membranes

Ultrathin topological insulator membranes are building blocks of exotic quantum matter. However, traditional epitaxy of these materials does not facilitate stacking in arbitrary orders, while mechanical exfoliation from bulk crystals is also challenging due to the non-negligible interlayer coupling therein. Here we liberate millimeter-scale films of the topological insulator Bi2Se3, grown by molecular beam epitaxy, down to 3 quintuple layers. We characterize the preservation of the topological surface states and quantum well states in transferred Bi2Se3 films using angle-resolved photoemission spectroscopy. Leveraging the photon-energy-dependent surface sensitivity, the photoemission spectra taken with 6 and 21.2 eV photons reveal a transfer-induced migration of the topological surface states from the top to the inner layers. By establishing clear electronic structures of the transferred films and unveiling the wave function relocation of the topological surface states, our work lays the physics foundation crucial for the future fabrication of artificially stacked topological materials with single-layer precision.

Remote epitaxy of single-crystal rhombohedral WS2 bilayers

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

Moiré band renormalization due to lattice mismatch in bilayer graphene

We investigated the band renormalization caused by the compressive-strain-induced lattice mismatch in parallel AA stacked bilayer graphene using two complementary methods: the tight-binding approach and the low-energy continuum theory. While a large mismatch does not alter the low-energy bands, a small one reduces the bandwidth of the low-energy bands along with a decrease in the Fermi velocity. In the tiny-mismatch regime, the low-energy continuum theory reveals that the long-period moiré pattern extensively renormalizes the low-energy bands, resulting in a significant reduction of bandwidth. Meanwhile, the Fermi velocity exhibits an oscillatory behavior and approaches zero at specific mismatches. However, the resulting low-energy bands are not perfectly isolated flat, as seen in twisted bilayer graphene at magic angles. These findings provide a deeper understanding of moiré physics and offer valuable guidance for related experimental studies in creating moiré superlattices using two-dimensional van der Waals heterostructures.

Imaging moiré excited states with photocurrent tunnelling microscopy

Moiré superlattices provide a highly tuneable and versatile platform to explore novel quantum phases and exotic excited states ranging from correlated insulators to moiré excitons. Scanning tunnelling microscopy has played a key role in probing microscopic behaviours of the moiré correlated ground states at the atomic scale. However, imaging of quantum excited states in moiré heterostructures remains an outstanding challenge. Here we develop a photocurrent tunnelling microscopy technique that combines laser excitation and scanning tunnelling spectroscopy to directly visualize the electron and hole distribution within the photoexcited moiré exciton in twisted bilayer WS2. The tunnelling photocurrent alternates between positive and negative polarities at different locations within a single moiré unit cell. This alternating photocurrent originates from the in-plane charge transfer moiré exciton in twisted bilayer WS2, predicted by our GW-Bethe-Salpeter equation calculations, that emerges from the competition between the electron-hole Coulomb interaction and the moiré potential landscape. Our technique enables the exploration of photoexcited non-equilibrium moiré phenomena at the atomic scale.

Mesoscopic Stacking Reconfigurations in Stacked van der Waals Film

Mesoscopic-scale stacking reconfigurations are investigated when van der Waals (vdW) films are stacked. A method to visualize complicated stacking structures and mechanical distortions simultaneously in stacked atom-thick films using Raman spectroscopy is developed. In the rigid limit, it is found that the distortions originate from the transfer process, which can be understood through thin film mechanics with a large elastic property mismatch. In contrast, with atomic corrugations, the in-plane strain fields are more closely correlated with the stacking configuration, highlighting the impact of atomic reconstructions on the mesoscopic scale. It is discovered that the grain boundaries do not have a significant effect while the cracks are causing inhomogeneous strain in stacked polycrystalline films. This result contributes to understanding the local variation of emerging properties from moiré structures and advancing the reliability of stacked vdW material fabrication.

Mapping charge excitations in generalized Wigner crystals

Transition metal dichalcogenide-based moiré superlattices exhibit strong electron-electron correlations, thus giving rise to strongly correlated quantum phenomena such as generalized Wigner crystal states. Evidence of Wigner crystals in transition metal dichalcogenide moire superlattices has been widely reported from various optical spectroscopy and electrical conductivity measurements, while their microscopic nature has been limited to the basic lattice structure. Theoretical studies predict that unusual quasiparticle excitations across the correlated gap between upper and lower Hubbard bands can arise due to long-range Coulomb interactions in generalized Wigner crystal states. However, the microscopic proof of such quasiparticle excitations is challenging because of the low excitation energy of the Wigner crystal. Here we describe a scanning single-electron charging spectroscopy technique with nanometre spatial resolution and single-electron charge resolution that enables us to directly image electron and hole wavefunctions and to determine the thermodynamic gap of generalized Wigner crystal states in twisted WS2 moiré heterostructures. High-resolution scanning single-electron charging spectroscopy combines scanning tunnelling microscopy with a monolayer graphene sensing layer, thus enabling the generation of individual electron and hole quasiparticles in generalized Wigner crystals. We show that electron and hole quasiparticles have complementary wavefunction distributions and that thermodynamic gaps of ∼50 meV exist for the 1/3 and 2/3 generalized Wigner crystal states in twisted WS2.

Mapping twist-tuned multiband topology in bilayer WSe2

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

Excitonic Thermalization Bottleneck in Twisted TMD Heterostructures

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

Linear resistivity at van Hove singularities in twisted bilayer WSe2

Different mechanisms driving a linear temperature dependence of the resistivity ρ ∼ T at van Hove singularities (VHSs) or metal-insulator transitions when doping a Mott insulator are being debated intensively with competing theoretical proposals. We experimentally investigate this using the exceptional tunability of twisted bilayer (TB) WSe2 by tracking the parameter regions where linear-in-T resistivity is found in dependency of displacement fields, filling, and magnetic fields. We find that even when the VHSs are tuned rather far away from the half-filling point and the Mott insulating transition is absent, the T-linear resistivity persists at the VHSs. When doping away from the VHSs, the T-linear behavior quickly transitions into a Fermi liquid behavior with a T2 relation. No apparent dependency of the linear-in-T resistivity, besides a rather strong change of prefactor, is found when applying displacement fields as long as the filling is tuned to the VHSs, including D ∼ 0.28 V/nm where a high-order VHS is expected. Intriguingly, such non-Fermi liquid linear-in-T resistivity persists even when magnetic fields break the spin-degeneracy of the VHSs at which point two linear in T regions emerge, for each of the split VHSs separately. This points to a mechanism of enhanced scattering at generic VHSs rather than only at high-order VHSs or by a quantum critical point during a Mott transition. Our findings provide insights into the many-body consequences arising out of VHSs, especially the non-Fermi liquid behavior found in moiré materials.

Phase Engineering of Nanomaterials: Transition Metal Dichalcogenides

Crystal phase, a critical structural characteristic beyond the morphology, size, dimension, facet, etc., determines the physicochemical properties of nanomaterials. As a group of layered nanomaterials with polymorphs, transition metal dichalcogenides (TMDs) have attracted intensive research attention due to their phase-dependent properties. Therefore, great efforts have been devoted to the phase engineering of TMDs to synthesize TMDs with controlled phases, especially unconventional/metastable phases, for various applications in electronics, optoelectronics, catalysis, biomedicine, energy storage and conversion, and ferroelectrics. Considering the significant progress in the synthesis and applications of TMDs, we believe that a comprehensive review on the phase engineering of TMDs is critical to promote their fundamental studies and practical applications. This Review aims to provide a comprehensive introduction and discussion on the crystal structures, synthetic strategies, and phase-dependent properties and applications of TMDs. Finally, our perspectives on the challenges and opportunities in phase engineering of TMDs will also be discussed.

Evidence of the fractional quantum spin Hall effect in moiré MoTe2

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

Two-Dimensional Moiré Polaronic Electron Crystals

Two-dimensional moiré materials have emerged as the most versatile platform for realizing quantum phases of electrons. Here, we explore the stability origins of correlated states in WSe_{2}/WS_{2} moiré superlattices. We find that ultrafast electronic excitation leads to partial melting of the Mott states on timescales 5 times longer than predictions from the charge hopping integrals and that the melting rates are thermally activated, with activation energies of 18±3 and 13±2  meV for the one- and two-hole Mott states, respectively, suggesting significant electron-phonon coupling. A density functional theory calculation of the one-hole Mott state confirms polaron formation and yields a hole-polaron binding energy of 16 meV. These findings reveal a close interplay of electron-electron and electron-phonon interactions in stabilizing the polaronic Mott insulators at transition metal dichalcogenide moiré interfaces.

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

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

Identification and Structural Characterization of Twisted Atomically Thin Bilayer Materials by Deep Learning

Two-dimensional materials are expected to play an important role in next-generation electronics and optoelectronic devices. Recently, twisted bilayer graphene and transition metal dichalcogenides have attracted significant attention due to their unique physical properties and potential applications. In this study, we describe the use of optical microscopy to collect the color space of chemical vapor deposition (CVD) of molybdenum disulfide (MoS2) and the application of a semantic segmentation convolutional neural network (CNN) to accurately and rapidly identify thicknesses of MoS2 flakes. A second CNN model is trained to provide precise predictions on the twist angle of CVD-grown bilayer flakes. This model harnessed a data set comprising over 10,000 synthetic images, encompassing geometries spanning from hexagonal to triangular shapes. Subsequent validation of the deep learning predictions on twist angles was executed through the second harmonic generation and Raman spectroscopy. Our results introduce a scalable methodology for automated inspection of twisted atomically thin CVD-grown bilayers.

Mixed Stochastic-Deterministic Approach for Many-Body Perturbation Theory Calculations

We present an approach for GW calculations of quasiparticle energies with quasiquadratic scaling by approximating high-energy contributions to the Green's function in its Lehmann representation with effective stochastic vectors. The method is easy to implement without altering the GW code, converges rapidly with stochastic parameters, and treats systems of various dimensionality and screening response. Our calculations on a 5.75° twisted MoS_{2} bilayer show how large-scale GW methods include geometry relaxations and electronic correlations on an equal basis in structurally nontrivial materials.

Maximally localized Wannier functions, interaction models, and fractional quantum anomalous Hall effect in twisted bilayer MoTe2

We investigate the moiré band structures and the strong correlation effects in twisted bilayer MoTe[Formula: see text] for a wide range of twist angles, employing a combination of various techniques. Using large-scale first-principles calculations, we pinpoint realistic continuum modeling parameters, subsequently deriving the maximally localized Wannier functions for the top three moiré bands. Simplifying our model with reasonable assumptions, we obtain a minimal two-band model, encompassing Coulomb repulsion, correlated hopping, and spin exchange. Our minimal interaction models pave the way for further exploration of the rich many-body physics in twisted MoTe[Formula: see text]. Furthermore, we explore the phase diagrams of the system through Hartree-Fock approximation and exact diagonalization (ED). Our two-band ED analysis underscores significant band-mixing effects in this system, which enlarge the optimal twist angle for fractional quantum anomalous Hall states.

Metal-Insulator Transition in a Semiconductor Heterobilayer Model

Transition metal dichalcogenide superlattices provide an exciting new platform for exploring and understanding a variety of phases of matter. The moiré continuum Hamiltonian, of two-dimensional jellium in a modulating potential, provides a fundamental model for such systems. Accurate computations with this model are essential for interpreting experimental observations and making predictions for future explorations. In this work, we combine two complementary quantum Monte Carlo (QMC) methods, phaseless auxiliary field quantum Monte Carlo and fixed-phase diffusion Monte Carlo, to study the ground state of this Hamiltonian. We observe a metal-insulator transition between a paramagnet and a 120° Néel ordered state as the moiré potential depth and the interaction strength are varied. We find significant differences from existing results by Hartree-Fock and exact diagonalization studies. In addition, we benchmark density-functional theory, and suggest an optimal hybrid functional which best approximates our QMC results.

Field-Induced Hybridization of Moiré Excitons in MoSe_{2}/WS_{2} Heterobilayers

We study experimentally and theoretically the hybridization among intralayer and interlayer moiré excitons in a MoSe_{2}/WS_{2} heterostructure with antiparallel alignment. Using a dual-gate device and cryogenic white light reflectance and narrow-band laser modulation spectroscopy, we subject the moiré excitons in the MoSe_{2}/WS_{2} heterostack to a perpendicular electric field, monitor the field-induced dispersion and hybridization of intralayer and interlayer moiré exciton states, and induce a crossover from type I to type II band alignment. Moreover, we employ perpendicular magnetic fields to map out the dependence of the corresponding exciton Landé g factors on the electric field. Finally, we develop an effective theoretical model combining resonant and nonresonant contributions to moiré potentials to explain the observed phenomenology, and highlight the relevance of interlayer coupling for structures with close energetic band alignment as in MoSe_{2}/WS_{2}.

Transport Study of Charge-Carrier Scattering in Monolayer WSe_{2}

Employing flux-grown single crystal WSe_{2}, we report charge-carrier scattering behaviors measured in h-BN encapsulated monolayer field effect transistors. We observe a nonmonotonic change of transport mobility as a function of hole density in the degenerately doped sample, which can be explained by energy dependent scattering amplitude of strong defects calculated using the T-matrix approximation. Utilizing long mean-free path (>500  nm), we also demonstrate the high quality of our electronic devices by showing quantized conductance steps from an electrostatically defined quantum point contact, showing the potential for creating ultrahigh quality quantum optoelectronic devices based on atomically thin semiconductors.

Ultraviolet interlayer excitons in bilayer WSe2

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

Correlated insulator and Chern insulators in pentalayer rhombohedral-stacked graphene

Rhombohedral-stacked multilayer graphene hosts a pair of flat bands touching at zero energy, which should give rise to correlated electron phenomena that can be tuned further by an electric field. Moreover, when electron correlation breaks the isospin symmetry, the valley-dependent Berry phase at zero energy may give rise to topologically non-trivial states. Here we measure electron transport through hexagonal boron nitride-encapsulated pentalayer graphene down to 100 mK. We observed a correlated insulating state with resistance at the megaohm level or greater at charge density n = 0 and displacement field D = 0. Tight-binding calculations predict a metallic ground state under these conditions. By increasing D, we observed a Chern insulator state with C = -5 and two other states with C = -3 at a magnetic field of around 1 T. At high D and n, we observed isospin-polarized quarter- and half-metals. Hence, rhombohedral pentalayer graphene exhibits two different types of Fermi-surface instability, one driven by a pair of flat bands touching at zero energy, and one induced by the Stoner mechanism in a single flat band. Our results establish rhombohedral multilayer graphene as a suitable system for exploring intertwined electron correlation and topology phenomena in natural graphitic materials without the need for moiré superlattice engineering.

Optical properties and plasmons in moiré structures

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

Spin Polarons and Ferromagnetism in Doped Dilute Moiré-Mott Insulators

Moiré heterostructures of transition metal dichalcogenides (TMDs) exhibit Mott-insulating behavior both at half filling as well as at fractional fillings, where electronic degrees of freedom form self-organized Wigner crystal states. An open question concerns magnetic states obtained by lifting the pseudospin-1/2 degeneracy of these states at lowest temperatures. While at half filling virtual hopping is expected to induce (weak) antiferromagnetic exchange interactions, these are strongly suppressed when considering dilute filling fractions. We argue that, instead, a small concentration of doped electrons leads to the formation of spin polarons, inducing ferromagnetic order at experimentally relevant temperatures, consistent with recently observed ferromagnetic states in moiré TMD systems. We predict explicit signatures of polaron formation in the magnetization profile.

Interlayer-coupling-engineerable flat bands in twisted MoSi2N4bilayers

The two-dimensional layered semiconductor MoSi2N4, which has several advantages including high strength, excellent stability, high hole mobility, and high thermal conductivity, was recently successfully synthesized using chemical vapor deposition. Based on first-principles calculations, we investigate the effects of the twist angle and interlayer distance variation on the electronic properties of twisted bilayer MoSi2N4. The flat bands are absent for twisted bilayer MoSi2N4when the twist angleθis reduced to 3.89°. Taking twisted bilayer MoSi2N4withθof 5.09° as an example, we find that flat bands emerge as the interlayer distance decreases. As the interlayer distance can be effectively modulated by hydrostatic pressure, we propose hydrostatic pressure as a knob for tailoring the flat bands in twisted bilayer MoSi2N4. Our findings provide theoretical support for extending the applications of MoSi2N4in strong correlation physics and superconductivity.

Terahertz linear/non-linear anomalous Hall conductivity of moiré TMD hetero-nanoribbons as topological valleytronics materials

Twisted moiré van der Waals heterostructures hold promise to provide a robust quantum simulation platform for strongly correlated materials and realize elusive states of matter such as topological states in the laboratory. We demonstrated that the moiré bands of twisted transition metal dichalcogenide (TMD) hetero-nanoribbons exhibit non-trivial topological order due to the tendency of valence and conduction band states in K valleys to form giant band gaps when spin-orbit coupling (SOC) is taken into account. Among the features of twisted WS[Formula: see text]/MoS[Formula: see text] and WSe[Formula: see text]/MoSe[Formula: see text], we found that the heavy fermions associated with the topological flat bands and the presence of strongly correlated states, enhance anomalous Hall conductivity (AHC) away from the magic angle. By band analysis, we showed that the topmost conduction bands from the ± K-valleys are perfectly flat and carry a spin/valley Chern number. Moreover, we showed that the non-linear anomalous Hall effect in moiré TMD hetero-nanoribbons can be used to manipulate terahertz (THz) radiation. Our findings establish twisted heterostructures of group-VI TMD nanoribbons as a tunable platform for engineering topological valley quantum phases and THz non-linear Hall conductivity.

Twistronics in two-dimensional transition metal dichalcogenide (TMD)-based van der Waals interface

Transition metal dichalcogenides (TMD) based heterostructures have gained significant attention lately because of their distinct physical properties and potential uses in electronics and optoelectronics. In the present work, the effects of twist on the structural, electronic, and optical properties (such as the static dielectric constant, refractive index, extinction coefficient, and absorption coefficient) of vertically stacked TMD heterostructures, namely MoSe2/WSe2, WS2/WSe2, MoSe2/WS2 and MoS2/WSe2, have been systematically studied and a thorough comparison is done among these heterostructures. In addition, the absence of negative frequency in the phonon dispersion curve and a low formation energy confirm the structural and thermodynamical stability of all the proposed TMD heterostructures. The calculations are performed using first-principles-based density functional theory (DFT) method. Beautiful Moiré patterns are formed due to the relative rotation of the layers as a consequence of the superposition of the periodic structures of the TMDs on each other. Twist engineering allows the modulation of bandgaps and a phase change from direct to indirect band gap semiconductors as well. The high optical absorption in the visible range of spectrum makes these twisted heterostructures very promising candidates in photovoltaic applications.

Magnetism and metallicity in moiré transition metal dichalcogenides

The ability to control the properties of twisted bilayer transition metal dichalcogenides in situ makes them an ideal platform for investigating the interplay of strong correlations and geometric frustration. Of particular interest are the low energy scales, which make it possible to experimentally access both temperature and magnetic fields that are of the order of the bandwidth or the correlation scale. In this manuscript, we analyze the moiré Hubbard model, believed to describe the low energy physics of an important subclass of the twisted bilayer compounds. We establish its magnetic and the metal-insulator phase diagram for the full range of magnetic fields up to the fully spin-polarized state. We find a rich phase diagram including fully and partially polarized insulating and metallic phases of which we determine the interplay of magnetic order, Zeeman-field, and metallicity, and make connection to recent experiments.

Engineering Gapless Edge States from Antiferromagnetic Chern Homobilayer

We put forward that stacked Chern insulators with opposite chiralities offer a strategy to achieve gapless helical edge states in two dimensions. We employ the square lattice as an example and elucidate that the gapless chiral and helical edge states emerge in the monolayer and antiferromagnetically stacked bilayer, characterized by Chern number C = - 1 and spin Chern number C S = - 1 , respectively. Particularly, for a topological phase transition to the normal insulator in the stacked bilayer, a band gap closing and reopening procedure takes place accompanied by helical edge states disappearing, where the Chern insulating phase in the monolayer vanishes at the same time. Moreover, EuO is revealed as a suitable candidate for material realization. This work is not only valuable to the research of the quantum anomalous Hall effect but also offers a favorable platform to realize magnetic topologically insulating materials for spintronics applications.

Carbon Kagome nanotubes-quasi-one-dimensional nanostructures with flat bands

In recent years, a number of bulk materials and heterostructures have been explored due their connections with exotic materials phenomena emanating from flat band physics and strong electronic correlation. The possibility of realizing such fascinating material properties in simple realistic nanostructures is particularly exciting, especially as the investigation of exotic states of electronic matter in wire-like geometries is relatively unexplored in the literature. Motivated by these considerations, we introduce in this work carbon Kagome nanotubes (CKNTs)-a new allotrope of carbon formed by rolling up Kagome graphene, and investigate this material using specialized first principles calculations. We identify two principal varieties of CKNTs-armchair and zigzag, and find both varieties to be stable at room temperature, based on ab initio molecular dynamics simulations. CKNTs are metallic and feature dispersionless states (i.e., flat bands) near the Fermi level throughout their Brillouin zone, along with an associated singular peak in the electronic density of states. We calculate the mechanical and electronic response of CKNTs to torsional and axial strains, and show that CKNTs appear to be more mechanically compliant than conventional carbon nanotubes (CNTs). Additionally, we find that the electronic properties of CKNTs undergo significant electronic transitions-with emergent partial flat bands and tilted Dirac points-when twisted. We develop a relatively simple tight-binding model that can explain many of these electronic features. We also discuss possible routes for the synthesis of CKNTs. Overall, CKNTs appear to be unique and striking examples of realistic elemental quasi-one-dimensional materials that may display fascinating material properties due to strong electronic correlation. Distorted CKNTs may provide an interesting nanomaterial platform where flat band physics and chirality induced anomalous transport effects may be studied together.

Tuning commensurability in twisted van der Waals bilayers

Moiré superlattices based on van der Waals bilayers1-4 created at small twist angles lead to a long wavelength pattern with approximate translational symmetry. At large twist angles (θt), moiré patterns are, in general, incommensurate except for a few discrete angles. Here we show that large-angle twisted bilayers offer distinctly different platforms. More specifically, by using twisted tungsten diselenide bilayers, we create the incommensurate dodecagon quasicrystals at θt = 30° and the commensurate moiré crystals at θt = 21.8° and 38.2°. Valley-resolved scanning tunnelling spectroscopy shows disparate behaviours between moiré crystals (with translational symmetry) and quasicrystals (with broken translational symmetry). In particular, the K valley shows rich electronic structures exemplified by the formation of mini-gaps near the valence band maximum. These discoveries demonstrate that bilayers with large twist angles offer a design platform to explore moiré physics beyond those formed with small twist angles.

Kinkless Electronic Junction along 1D Electronic Channel Embedded in a Van Der Waals Layer

Here, the formation of type-I and type-II electronic junctions with or without any structural discontinuity along a well-defined 1 nm-wide 1D electronic channel within a van der Waals layer is reported. Scanning tunneling microscopy and spectroscopy techniques are employed to investigate the atomic and electronic structure along peculiar domain walls formed on the charge-density-wave phase of 1T-TaS2 . Distinct kinds of abrupt electronic junctions with discontinuities of the band gap along the domain walls are found, some of which even do not have any structural kinks and defects. Density-functional calculations reveal a novel mechanism of the electronic junction formation; they are formed by a kinked domain wall in the layer underneath through substantial electronic interlayer coupling. This work demonstrates that the interlayer electronic coupling can be an effective control knob over nanometer-scale electronic property of 2D atomic monolayers.

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.

Evidence for Exciton Crystals in a 2D Semiconductor Heterotrilayer

Two-dimensional (2D) transition metal dichalcogenides (TMDC) and their moiré interfaces have been demonstrated for correlated electron states, including Mott insulators and electron/hole crystals commensurate with moiré superlattices. Here we present spectroscopic evidence for ordered bosons─interlayer exciton crystals in a WSe2/MoSe2/WSe2 trilayer, where the enhanced Coulomb interactions over those in heterobilayers have been predicted to result in exciton ordering. Ordered interlayer excitons in the trilayer are characterized by negligible mobility and by sharper PL peaks persisting to an exciton density of nex ∼ 1012 cm-2, which is an order of magnitude higher than the corresponding limit in the heterobilayer. We present evidence for the predicted quadrupolar exciton crystal and its transitions to dipolar excitons either with increasing nex or by an applied electric field. These ordered interlayer excitons may serve as models for the exploration of quantum phase transitions and quantum coherent phenomena.

Quantifying Strain in Moiré Superlattice

In twisted van der Waals (vdW) bilayers, intrinsic strain associated with the moiré superlattice and unintentionally introduced uniaxial strain may be present simultaneously. Both strains are able to lift the degeneracy of the E2g phonon modes in Raman spectra. Because of the different rotation symmetry of the two types of strain, the corresponding Raman intensity exhibits a distinct polarization dependence. We compare a 2.5° twisted MoS2 bilayer, in which the maximal intrinsic moiré strain is anticipated, and a natural MoS2 bilayer with an intentionally introduced uniaxial strain. By analyzing the frequency shift of the E2g doublet and their polarization dependence, we can not only determine the direction of unintentional uniaxial strain in the twisted bilayer but also quantify both strain components. This simple strain characterization method based on far-field Raman spectra will facilitate the studies of electronic properties of moiré superlattices under the influence of combined intrinsic and external strains.

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

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

Moiré-Assisted Realization of Octahedral MoTe2 Monolayer

A current key challenge in 2D materials is the realization of emergent quantum phenomena in hetero structures via controlling the moiré potential created by the periodicity mismatch between adjacent layers, as highlighted by the discovery of superconductivity in twisted bilayer graphene. Generally, the lattice structure of the original host material remains unchanged even after the moiré superlattice is formed. However, much less attention is paid for the possibility that the moiré potential can also modify the original crystal structure itself. Here, it is demonstrated that octahedral MoTe2 which is unstable in bulk is stabilized in a commensurate MoTe2 /graphene hetero-bilayer due to the moiré potential created between the two layers. It is found that the reconstruction of electronic states via the moiré potential is responsible for this stabilization, as evidenced by the energy-gap opening at the Fermi level observed by angle-resolved photoemission and scanning tunneling spectroscopies. The present results provide a fresh approach to realize novel 2D quantum phases by utilizing the moiré potential.

Mechanical, electronic, optical, piezoelectric and ferroic properties of strained graphene and other strained monolayers and multilayers: an update

This is an update of a previous review (Naumiset al2017Rep. Prog. Phys.80096501). Experimental and theoretical advances for straining graphene and other metallic, insulating, ferroelectric, ferroelastic, ferromagnetic and multiferroic 2D materials were considered. We surveyed (i) methods to induce valley and sublattice polarisation (P) in graphene, (ii) time-dependent strain and its impact on graphene's electronic properties, (iii) the role of local and global strain on superconductivity and other highly correlated and/or topological phases of graphene, (iv) inducing polarisationPon hexagonal boron nitride monolayers via strain, (v) modifying the optoelectronic properties of transition metal dichalcogenide monolayers through strain, (vi) ferroic 2D materials with intrinsic elastic (σ), electric (P) and magnetic (M) polarisation under strain, as well as incipient 2D multiferroics and (vii) moiré bilayers exhibiting flat electronic bands and exotic quantum phase diagrams, and other bilayer or few-layer systems exhibiting ferroic orders tunable by rotations and shear strain. The update features the experimental realisations of a tunable two-dimensional Quantum Spin Hall effect in germanene, of elemental 2D ferroelectric bismuth, and 2D multiferroic NiI2. The document was structured for a discussion of effects taking place in monolayers first, followed by discussions concerning bilayers and few-layers, and it represents an up-to-date overview of exciting and newest developments on the fast-paced field of 2D materials.

Stress and Curvature Effects in Layered 2D Ferroelectric CuInP2S6

Nanoscale ferroelectric 2D materials offer the opportunity to investigate curvature and strain effects on materials functionalities. Among these, CuInP2S6 (CIPS) has attracted tremendous research interest in recent years due to combination of room temperature ferroelectricity, scalability to a few layers thickness, and ferrielectric properties due to coexistence of 2 polar sublattices. Here, we explore the local curvature and strain effect on polarization in CIPS via piezoresponse force microscopy and spectroscopy. To explain the observed behaviors and decouple the curvature and strain effects in 2D CIPS, we introduce the finite element Landau-Ginzburg-Devonshire model, revealing strong changes in hysteresis characteristics in regions subjected to tensile and compressive strain. The piezoresponse force microscopy (PFM) results show that bending induces ferrielectric domains in CIPS, and the polarization-voltage hysteresis loops differ in bending and nonbending regions. These studies offer insights into the fabrication of curvature-engineered nanoelectronic devices.

Kinetic magnetism in triangular moiré materials

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

Non-Linear Optics at Twist Interfaces in h-BN/SiC Heterostructures

Understanding the emergent electronic structure in twisted atomically thin layers has led to the exciting field of twistronics. However, practical applications of such systems are challenging since the specific angular correlations between the layers must be precisely controlled and the layers have to be single crystalline with uniform atomic ordering. Here, an alternative, simple, and scalable approach is suggested, where nanocrystallinetwo-dimensional (2D) film on 3D substrates yields twisted-interface-dependent properties. Ultrawide-bandgap hexagonal boron nitride (h-BN) thin films are directly grown on high in-plane lattice mismatched wide-bandgap silicon carbide (4H-SiC) substrates to explore the twist-dependent structure-property correlations. Concurrently, nanocrystalline h-BN thin film shows strong non-linear second-harmonic generation and ultra-low cross-plane thermal conductivity at room temperature, which are attributed to the twisted domain edges between van der Waals stacked nanocrystals with random in-plane orientations. First-principles calculations based on time-dependent density functional theory manifest strong even-order optical nonlinearity in twisted h-BN layers. This work unveils that directly deposited 2D nanocrystalline thin film on 3D substrates could provide easily accessible twist-interfaces, therefore enabling a simple and scalable approach to utilize the 2D-twistronics integrated in 3D material devices for next-generation nanotechnology.

Room-temperature wavelike exciton transport in a van der Waals superatomic semiconductor

The transport of energy and information in semiconductors is limited by scattering between electronic carriers and lattice phonons, resulting in diffusive and lossy transport that curtails all semiconductor technologies. Using Re6Se8Cl2, a van der Waals (vdW) superatomic semiconductor, we demonstrate the formation of acoustic exciton-polarons, an electronic quasiparticle shielded from phonon scattering. We directly imaged polaron transport in Re6Se8Cl2 at room temperature, revealing quasi-ballistic, wavelike propagation sustained for a nanosecond and several micrometers. Shielded polaron transport leads to electronic energy propagation lengths orders of magnitude greater than in other vdW semiconductors, exceeding even silicon over a nanosecond. We propose that, counterintuitively, quasi-flat electronic bands and strong exciton-acoustic phonon coupling are together responsible for the transport properties of Re6Se8Cl2, establishing a path to ballistic room-temperature semiconductors.

Symmetry Engineering in Twisted Bilayer WTe2

The fabrication of artificial structures using a twisted van der Waals assembly has been a key technique for recent advancements in the research of two-dimensional (2D) materials. To date, various exotic phenomena have been observed thanks to the modified electron correlation or moiré structure controlled by the twist angle. However, the twisted van der Waals assembly has further potential to modulate the physical properties by controlling the symmetry. In this study, we fabricated twisted bilayer WTe2 and demonstrated that the twist angle successfully controls the spatial inversion symmetry and hence the spin splitting in the band structure. Our results reveal the further potential of a twisted van der Waals assembly, suggesting the feasibility of pursuing new physical phenomena in 2D materials based on the control of symmetry.

Strain-Induced Quasi-1D Channels in Twisted Moiré Lattices

We study the effects of strain in moiré systems composed of honeycomb lattices. We elucidate the formation of almost perfect one-dimensional moiré patterns in twisted bilayer systems. The formation of such patterns is a consequence of an interplay between twist and strain which gives rise to a collapse of the reciprocal space unit cell. As a criterion for such collapse we find a simple relation between the two quantities and the material specific Poisson ratio. The induced one-dimensional behavior is characterized by two, usually incommensurate, periodicities. Our results offer explanations for the complex patterns of one-dimensional channels observed in low angle twisted bilayer graphene systems and twisted bilayer dicalcogenides. Our findings can be applied to any hexagonal twisted moiré pattern and can be easily extended to other geometries.

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.

Thermally induced atomic reconstruction into fully commensurate structures of transition metal dichalcogenide layers

Twist angle between two-dimensional layers is a critical parameter that determines their interfacial properties, such as moiré excitons and interfacial ferro-electricity. To achieve better control over these properties for fundamental studies and various applications, considerable efforts have been made to manipulate twist angle. However, due to mechanical limitations and the inevitable formation of incommensurate regions, there remains a challenge in attaining perfect alignment of crystalline orientation. Here we report a thermally induced atomic reconstruction of randomly stacked transition metal dichalcogenide multilayers into fully commensurate heterostructures with zero twist angle by encapsulation annealing, regardless of twist angles of as-stacked samples and lattice mismatches. We also demonstrate the selective formation of R- and H-type fully commensurate phases with a seamless lateral junction using chemical vapour-deposited transition metal dichalcogenides. The resulting fully commensurate phases exhibit strong photoluminescence enhancement of the interlayer excitons, even at room temperature, due to their commensurate structure with aligned momentum coordinates. Our work not only demonstrates a way to fabricate zero-twisted, two-dimensional bilayers with R- and H-type configurations, but also provides a platform for studying their unexplored properties.

Phase Engineering of 2D Materials

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

Pressure tuning of minibands in MoS2/WSe2 heterostructures revealed by moiré phonons

Moiré superlattices of two-dimensional heterostructures arose as a new platform to investigate emergent behaviour in quantum solids with unprecedented tunability. To glean insights into the physics of these systems, it is paramount to discover new probes of the moiré potential and moiré minibands, as well as their dependence on external tuning parameters. Hydrostatic pressure is a powerful control parameter, since it allows to continuously and reversibly enhance the moiré potential. Here we use high pressure to tune the minibands in a rotationally aligned MoS2/WSe2 moiré heterostructure, and show that their evolution can be probed via moiré phonons. The latter are Raman-inactive phonons from the individual layers that are activated by the moiré potential. Moiré phonons manifest themselves as satellite Raman peaks arising exclusively from the heterostructure region, increasing in intensity and frequency under applied pressure. Further theoretical analysis reveals that their scattering rate is directly connected to the moiré potential strength. By comparing the experimental and calculated pressure-induced enhancement, we obtain numerical estimates for the moiré potential amplitude and its pressure dependence. The present work establishes moiré phonons as a sensitive probe of the moiré potential as well as the electronic structures of moiré systems.