Correlated electronic phases in twisted bilayer transition metal dichalcogenides

Moiré bands in twisted trilayer black phosphorene: effects of pressure and electric field

Twist-induced moiré bands and accompanied correlated phenomena have been extensively investigated in twisted hexagonal lattices with weak interlayer coupling. However, the formation of moiré bands in strongly coupled layered materials and their controlled tuning remain largely unexplored. Here, we systematically study the moiré bands in twisted trilayer black phosphorene (TTbP) and the influences of pressure and electric field on them. Moiré states can form in various TTbPs even when the twist angle is larger than 16° similar to that of twisted bilayer bP. However, different TTbPs show different localization patterns depending on the twisting layer, leading to distinct dipolar behaviors. While these moiré states become quasi-one-dimensional (1D) as the twist angle decreases, external pressure causes the crossover of moiré states from quasi-1D to 0D with a dramatic change in localization areas and greatly reduced bandwidth. Interestingly, compared to twisted bilayer and pristine bP, TTbPs show a much larger electric-field induced Stark effect, controllable by either the twist angle or twist layer. Our work thus demonstrates TTbP as an attractive platform to explore moiré-controlled electronic and optical properties, as well as tunable optoelectronic applications.

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.

Tunable angle-dependent electrochemistry at twisted bilayer graphene with moiré flat bands

Tailoring electron transfer dynamics across solid-liquid interfaces is fundamental to the interconversion of electrical and chemical energy. Stacking atomically thin layers with a small azimuthal misorientation to produce moiré superlattices enables the controlled engineering of electronic band structures and the formation of extremely flat electronic bands. Here, we report a strong twist-angle dependence of heterogeneous charge transfer kinetics at twisted bilayer graphene electrodes with the greatest enhancement observed near the 'magic angle' (~1.1°). This effect is driven by the angle-dependent tuning of moiré-derived flat bands that modulate electron transfer processes with the solution-phase redox couple. Combined experimental and computational analysis reveals that the variation in electrochemical activity with moiré angle is controlled by a structural relaxation of the moiré superlattice at twist angles of <2°, and 'topological defect' AA stacking regions, where flat bands are localized, produce a large anomalous local electrochemical enhancement that cannot be accounted for by the elevated local density of states alone.

Terahertz Fingerprint of Monolayer Wigner Crystals

The strong Coulomb interaction in monolayer semiconductors represents a unique opportunity for the realization of Wigner crystals without external magnetic fields. In this work, we predict that the formation of monolayer Wigner crystals can be detected by their terahertz response spectrum, which exhibits a characteristic sequence of internal optical transitions. We apply the density matrix formalism to derive the internal quantum structure and the optical conductivity of the Wigner crystal and to microscopically analyze the multipeak shape of the obtained terahertz spectrum. Moreover, we predict a characteristic shift of the peak position as a function of charge density for different atomically thin materials and show how our results can be generalized to an arbitrary two-dimensional system.

Reproducibility in the fabrication and physics of moiré materials

Overlaying two atomic layers with a slight lattice mismatch or at a small rotation angle creates a moiré superlattice, which has properties that are markedly modified from (and at times entirely absent in) the 'parent' materials. Such moiré materials have progressed the study and engineering of strongly correlated phenomena and topological systems in reduced dimensions. The fundamental understanding of the electronic phases, such as superconductivity, requires a precise control of the challenging fabrication process, involving the rotational alignment of two atomically thin layers with an angular precision below 0.1 degrees. Here we review the essential properties of moiré materials and discuss their fabrication and physics from a reproducibility perspective.

Commensurate Stacking Phase Transitions in an Intercalated Transition Metal Dichalcogenide

Intercalation and stacking-order modulation are two active ways in manipulating the interlayer interaction of transition metal dichalcogenides (TMDCs), which lead to a variety of emergent phases and allow for engineering material properties. Herein, the growth of Pb-intercalated TMDCs-Pb(Ta_1+x Se₂ )₂ , the first 124-phase, is reported. Pb(Ta_1+x Se₂ )₂ exhibits a unique two-step first-order structural phase transition at around 230 K. The transitions are solely associated with the stacking degree of freedom, evolving from a high-temperature (high-T) phase with ABC stacking and R3m symmetry to an intermediate phase with AB stacking and P3m1, and finally to a low-temperature (low-T) phase again with R3msymmetry, but with ACB stacking. Each step involves a rigid slide of building blocks by a vector [1/3, 2/3, 0]. Intriguingly, gigantic lattice contractions occur at the transitions on warming. At low-T, bulk superconductivity with T_c  ≈ 1.8 K is observed. The underlying physics of the structural phase transitions are discussed from first-principle calculations. The symmetry analysis reveals topological nodal lines in the band structure. The results demonstrate the possibility of realizing higher-order metal-intercalated phases of TMDCs and advance the knowledge of polymorphic transitions, and may inspire stacking-order engineering in TMDCs and beyond.

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.

Tunable Orbital Ferromagnetism at Noninteger Filling of a Moiré Superlattice

The flat bands resulting from moiré superlattices exhibit fascinating correlated electron phenomena such as correlated insulators, ( Nature 2018, 556 (7699), 80-84), ( Nature Physics 2019, 15 (3), 237) superconductivity, ( Nature 2018, 556 (7699), 43-50), ( Nature 2019, 572 (7768), 215-219) and orbital magnetism. ( Science 2019, 365 (6453), 605-608), ( Nature 2020, 579 (7797), 56-61), ( Science 2020, 367 (6480), 900-903) Such magnetism has been observed only at particular integer multiples of n₀, the density corresponding to one electron per moiré superlattice unit cell. Here, we report the experimental observation of ferromagnetism at noninteger filling (NIF) of a flat Chern band in a ABC-TLG/hBN moiré superlattice. This state exhibits prominent ferromagnetic hysteresis behavior with large anomalous Hall resistivity in a broad region of densities centered in the valence miniband at n = -2.3n₀. We observe that, not only the magnitude of the anomalous Hall signal, but also the sign of the hysteretic ferromagnetic response can be modulated by tuning the carrier density and displacement field. Rotating the sample in a fixed magnetic field demonstrates that the ferromagnetism is highly anisotropic and likely purely orbital in character.

Strong Moiré Excitons in High-Angle Twisted Transition Metal Dichalcogenide Homobilayers with Robust Commensuration

The burgeoning field of twistronics, which concerns how changing the relative twist angles between two materials creates new optoelectronic properties, offers a novel platform for studying twist-angle dependent excitonic physics. Herein, by surveying a range of hexagonal phase transition metal dichalcogenides (TMD) twisted homobilayers, we find that 21.8 ± 1.0°-twisted (7a×7a) and 27.8 ± 1.0°-twisted (13a×13a) bilayers account for nearly 20% of the total population of twisted bilayers in solution-phase restacked bilayers and can be found also in chemical vapor deposition (CVD) samples. Examining the optical properties associated with these twisted angles, we found that 21.8 ± 1.0° twisted MoS₂ bilayers exhibit an intense moiré exciton peak in the photoluminescence (PL) spectra, originating from the refolded Brillouin zones. Our work suggests that commensurately twisted TMD homobilayers with short commensurate wavelengths can have interesting optoelectronic properties that are different from the small twist angle counterparts.

Developing Graphene-Based Moiré Heterostructures for Twistronics

Graphene-based moiré heterostructures are strongly correlated materials, and they are considered to be an effective platform to investigate the challenges of condensed matter physics. This is due to the distinct electronic properties that are unique to moiré superlattices and peculiar band structures. The increasing research on strongly correlated physics via graphene-based moiré heterostructures, especially unconventional superconductors, greatly promotes the development of condensed matter physics. Herein, the preparation methods of graphene-based moiré heterostructures on both in situ growth and assembling monolayer 2D materials are discussed. Methods to improve the quality of graphene and optimize the transfer process are presented to mitigate the limitations of low-quality graphene and damage caused by the transfer process during the fabrication of graphene-based moiré heterostructures. Then, the topological properties in various graphene-based moiré heterostructures are reviewed. Furthermore, recent advances regarding the factors that influence physical performances via a changing twist angle, the exertion of strain, and regulation of the dielectric environment are presented. Moreover, various unique physical properties in graphene-based moiré heterostructures are demonstrated. Finally, the challenges faced during the preparation and characterization of graphene-based moiré heterostructures are discussed. An outlook for the further development of moiré heterostructures is also presented.

Emergence and Tuning of Multiple Flat Bands in Twisted Bilayer γ-Graphyne

Using twisted bilayer γ-graphyne (TBGY) as an example, we show that it is possible to create multiple flat bands in carbon allotropes without the requirement of a specified magic angle. The origin of the flat bands can be understood by a simple two-level coupling model. The narrow bandwidth and strong localization of the flat band states might lead to strong correlation effects, which make TBGY a good platform for studying correlation physics. On the basis of the two-level coupling model, we further propose that the width and extent of localization of flat bands can be tuned by an energy mismatch ΔE between the two layers of TBGY, which can be realized by either applying a perpendicular electric field or introducing a heterostrain. This allows continuous modulation of TBGY from the strong-correlation regime to the medium- or weak-correlation regime, which could be utilized to study the quantum phase transition.

Coupling and Decoupling of Bilayer Graphene Monitored by Electron Energy Loss Spectroscopy

We studied the interlayer coupling and decoupling of bilayer graphene (BLG) using spatially resolved electron energy loss spectroscopy with a monochromated electron source. We correlated the twist-angle-dependent energy band hybridization with Moiré superlattices and the corresponding optical absorption peaks. The optical absorption peak originates from the excitonic transition between the hybridized van Hove singularities (vHSs), which shifts systematically with the twist angle. We then proved that the BLG decouples when a monolayer of metal chloride is intercalated in its van der Waals gap and results in the elimination of the vHS peak.

Nickel particle-enabled width-controlled growth of bilayer molybdenum disulfide nanoribbons

Transition metal dichalcogenides exhibit a variety of electronic behaviors depending on the number of layers and width. Therefore, developing facile methods for their controllable synthesis is of central importance. We found that nickel nanoparticles promote both heterogeneous nucleation of the first layer of molybdenum disulfide and simultaneously catalyzes homoepitaxial tip growth of a second layer via a vapor-liquid-solid (VLS) mechanism, resulting in bilayer nanoribbons with width controlled by the nanoparticle diameter. Simulations further confirm the VLS growth mechanism toward nanoribbons and its orders of magnitude higher growth speed compared to the conventional noncatalytic growth of flakes. Width-dependent Coulomb blockade oscillation observed in the transfer characteristics of the nanoribbons at temperatures up to 60 K evidences the value of this proposed synthesis strategy for future nanoelectronics.

SU(4) Chiral Spin Liquid, Exciton Supersolid, and Electric Detection in Moiré Bilayers

We propose a moiré bilayer as a platform where exotic quantum phases can be stabilized and electrically detected. Moiré bilayers consist of two separate moiré superlattice layers coupled through the interlayer Coulomb repulsion. In the small distance limit, an SU(4) spin can be formed by combining layer pseudospin and the real spin. As a concrete example, we study an SU(4) spin model on triangular lattice in the fundamental representation. By tuning a three-site ring exchange term K∼(t^{3}/U^{2}), we find the SU(4) symmetric crystallized phase and an SU(4)_{1} chiral spin liquid at the balanced filling. We also predict two different exciton supersolid phases with interlayer coherence at imbalanced filling under displacement field. Especially, the system can simulate an SU(2) Bose-Einstein condensation by injecting interlayer excitons into the magnetically ordered Mott insulator at the layer polarized limit. Smoking gun evidences of these phases can be obtained by measuring the pseudospin transport in the counterflow channel.

Interlayer electronic coupling on demand in a 2D magnetic semiconductor

When monolayers of two-dimensional (2D) materials are stacked into van der Waals structures, interlayer electronic coupling can introduce entirely new properties, as exemplified by recent discoveries of moiré bands that host highly correlated electronic states and quantum dot-like interlayer exciton lattices. Here we show the magnetic control of interlayer electronic coupling, as manifested in tunable excitonic transitions, in an A-type antiferromagnetic 2D semiconductor CrSBr. Excitonic transitions in bilayers and above can be drastically changed when the magnetic order is switched from the layered antiferromagnetic ground state to a field-induced ferromagnetic state, an effect attributed to the spin-allowed interlayer hybridization of electron and hole orbitals in the latter, as revealed by Green's function-Bethe-Salpeter equation (GW-BSE) calculations. Our work uncovers a magnetic approach to engineer electronic and excitonic effects in layered magnetic semiconductors.

Ultrathin Twisted Germanium Sulfide van der Waals Nanowires by Bismuth Catalyzed Vapor-Liquid-Solid Growth

1D nanowires of 2D layered crystals are emerging nanostructures synthesized by combining van der Waals (vdW) epitaxy and vapor-liquid-solid (VLS) growth. Nanowires of the group IV monochalcogenide germanium sulfide (GeS) are of particular interest for twistronics due to axial screw dislocations giving rise to Eshelby twist and precision interlayer twist at helical vdW interfaces. Ultrathin vdW nanowires have not been realized, and it is not clear if confining layered crystals into extremely thin wires is even possible. If axial screw dislocations are still stable, ultrathin vdW nanowires can reach large twists and should display significant quantum confinement. Here it is shown that VLS growth over Bi catalysts yields vdW nanowires down to ≈15 nm diameter while maintaining tens of µm length. Combined electron microscopy and diffraction demonstrate that ultrathin GeS nanowires crystallize in the orthorhombic bulk structure but can realize nonequilibrium stacking that may lead to 1D ferroelectricity. Ultrathin nanowires carry screw dislocations, remain chiral, and achieve very high twist rates. Whenever the dislocation extends to the nanowire tip, it continues into the Bi catalyst. Eshelby twist analysis demonstrates that the ultrathin nanowires follow continuum predictions. Cathodoluminescence on individual nanowires, finally, shows pronounced emission blue shifts consistent with quantum confinement.

Quantum anomalous Hall effect from intertwined moiré bands

Electron correlation and topology are two central threads of modern condensed matter physics. Semiconductor moiré materials provide a highly tuneable platform for studies of electron correlation^1-12. Correlation-driven phenomena, including the Mott insulator^2-5, generalized Wigner crystals^2,6,9, stripe phases¹⁰ and continuous Mott transition^11,12, have been demonstrated. However, non-trivial band topology has remained unclear. Here we report the observation of a quantum anomalous Hall effect in AB-stacked MoTe₂ /WSe₂ moiré heterobilayers. Unlike in the AA-stacked heterobilayers¹¹, an out-of-plane electric field not only controls the bandwidth but also the band topology by intertwining moiré bands centred at different layers. At half band filling, corresponding to one particle per moiré unit cell, we observe quantized Hall resistance, h/e² (with h and e denoting the Planck's constant and electron charge, respectively), and vanishing longitudinal resistance at zero magnetic field. The electric-field-induced topological phase transition from a Mott insulator to a quantum anomalous Hall insulator precedes an insulator-to-metal transition. Contrary to most known topological phase transitions¹³, it is not accompanied by a bulk charge gap closure. Our study paves the way for discovery of emergent phenomena arising from the combined influence of strong correlation and topology in semiconductor moiré materials.

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.

Magic in twisted transition metal dichalcogenide bilayers

The long-wavelength moiré superlattices in twisted 2D structures have emerged as a highly tunable platform for strongly correlated electron physics. We study the moiré bands in twisted transition metal dichalcogenide homobilayers, focusing on WSe2, at small twist angles using a combination of first principles density functional theory, continuum modeling, and Hartree-Fock approximation. We reveal the rich physics at small twist angles θ < 4∘, and identify a particular magic angle at which the top valence moiré band achieves almost perfect flatness. In the vicinity of this magic angle, we predict the realization of a generalized Kane-Mele model with a topological flat band, interaction-driven Haldane insulator, and Mott insulators at the filling of one hole per moiré unit cell. The combination of flat dispersion and uniformity of Berry curvature near the magic angle holds promise for realizing fractional quantum anomalous Hall effect at fractional filling. We also identify twist angles favorable for quantum spin Hall insulators and interaction-induced quantum anomalous Hall insulators at other integer fillings.

Moiré trions in MoSe2/WSe2 heterobilayers

Transition metal dichalcogenide moiré bilayers with spatially periodic potentials have emerged as a highly tunable platform for studying both electronic^1-6 and excitonic^4,7-13 phenomena. The power of these systems lies in the combination of strong Coulomb interactions with the capability of controlling the charge number in a moiré potential trap. Electronically, exotic charge orders at both integer and fractional fillings have been discovered^2,5. However, the impact of charging effects on excitons trapped in moiré potentials is poorly understood. Here, we report the observation of moiré trions and their doping-dependent photoluminescence polarization in H-stacked MoSe₂/WSe₂ heterobilayers. We find that as moiré traps are filled with either electrons or holes, new sets of interlayer exciton photoluminescence peaks with narrow linewidths emerge about 7 meV below the energy of the neutral moiré excitons. Circularly polarized photoluminescence reveals switching from co-circular to cross-circular polarizations as moiré excitons go from being negatively charged and neutral to positively charged. This switching results from the competition between valley-flip and spin-flip energy relaxation pathways of photo-excited electrons during interlayer trion formation. Our results offer a starting point for engineering both bosonic and fermionic many-body effects based on moiré excitons¹⁴.

Excitons and emergent quantum phenomena in stacked 2D semiconductors

The design and control of material interfaces is a foundational approach to realize technologically useful effects and engineer material properties. This is especially true for two-dimensional (2D) materials, where van der Waals stacking allows disparate materials to be freely stacked together to form highly customizable interfaces. This has underpinned a recent wave of discoveries based on excitons in stacked double layers of transition metal dichalcogenides (TMDs), the archetypal family of 2D semiconductors. In such double-layer structures, the elegant interplay of charge, spin and moiré superlattice structure with many-body effects gives rise to diverse excitonic phenomena and correlated physics. Here we review some of the recent discoveries that highlight the versatility of TMD double layers to explore quantum optics and many-body effects. We identify outstanding challenges in the field and present a roadmap for unlocking the full potential of excitonic physics in TMD double layers and beyond, such as incorporating newly discovered ferroelectric and magnetic materials to engineer symmetries and add a new level of control to these remarkable engineered materials.

Strain and Spin-Orbit Coupling Engineering in Twisted WS2/Graphene Heterobilayer

The strain in hybrid van der Waals heterostructures, made of two distinct two-dimensional van der Waals materials, offers an interesting handle on their corresponding electronic band structure. Such strain can be engineered by changing the relative crystallographic orientation between the constitutive monolayers, notably, the angular misorientation, also known as the “twist angle”. By combining angle-resolved photoemission spectroscopy with density functional theory calculations, we investigate here the band structure of the WS₂/graphene heterobilayer for various twist angles. Despite the relatively weak coupling between WS₂ and graphene, we demonstrate that the resulting strain quantitatively affects many electronic features of the WS₂ monolayers, including the spin-orbit coupling strength. In particular, we show that the WS₂ spin-orbit splitting of the valence band maximum at K can be tuned from 430 to 460 meV. Our findings open perspectives in controlling the band dispersion of van der Waals materials.

Studying 2D materials with advanced Raman spectroscopy: CARS, SRS and TERS

Raman spectroscopy has been established as a valuable tool to study and characterize two-dimensional (2D) systems, but it exhibits two drawbacks: a relatively weak signal response and a limited spatial resolution. Recently, advanced Raman spectroscopy techniques, such as coherent anti-Stokes spectroscopy (CARS), stimulated Raman scattering (SRS) and tip-enhanced Raman spectroscopy (TERS), have been shown to overcome these two limitations. In this article, we review how useful physical information can be retrieved from different 2D materials using these three advanced Raman spectroscopy and imaging techniques, discussing results on graphene, hexagonal boron-nitride, and transition metal di- and mono-chalcogenides, thus providing perspectives for future work in this early-stage field of research, including similar studies on unexplored 2D systems and open questions.

Electrically tunable Feshbach resonances in twisted bilayer semiconductors

[Figure: see text].

Correlated Insulating States and Transport Signature of Superconductivity in Twisted Trilayer Graphene Superlattices

Layers of two-dimensional materials stacked with a small twist angle give rise to beating periodic patterns on a scale much larger than the original lattice, referred to as a "moiré superlattice." Here, we demonstrate a higher-order "moiré of moiré" superlattice in twisted trilayer graphene with two consecutive small twist angles. We report correlated insulating states near the half filling of the moiré of moiré superlattice at an extremely low carrier density (∼10^{10}  cm^{-2}), near which we also report a zero-resistance transport behavior typically expected in a 2D superconductor. The full-occupancy (ν=-4 and ν=4) states are semimetallic and gapless, distinct from the twisted bilayer systems.

Charge-order-enhanced capacitance in semiconductor moiré superlattices

Van der Waals moiré materials have emerged as a highly controllable platform to study electronic correlation phenomena^1-17. Robust correlated insulating states have recently been discovered at both integer and fractional filling factors of semiconductor moiré systems^10-17. In this study we explored the thermodynamic properties of these states by measuring the gate capacitance of MoSe₂/WS₂ moiré superlattices. We observed a series of incompressible states for filling factors 0-8 and anomalously large capacitance in the intervening compressible regions. The anomalously large capacitance, which was nearly 60% above the device's geometrical capacitance, was most pronounced at small filling factors, below the melting temperature of the charge-ordered states, and for small sample-gate separation. It is a manifestation of the device-geometry-dependent Coulomb interaction between electrons and phase mixing of the charge-ordered states. Based on these results, we were able to extract the thermodynamic gap of the correlated insulating states and the device's electronic entropy and specific heat capacity. Our findings establish capacitance as a powerful probe of the correlated states in semiconductor moiré systems and demonstrate control of these states via sample-gate coupling.

Carrier transport theory for twisted bilayer graphene in the metallic regime

Understanding the normal-metal state transport in twisted bilayer graphene near magic angle is of fundamental importance as it provides insights into the mechanisms responsible for the observed strongly correlated insulating and superconducting phases. Here we provide a rigorous theory for phonon-dominated transport in twisted bilayer graphene describing its unusual signatures in the resistivity (including the variation with electron density, temperature, and twist angle) showing good quantitative agreement with recent experiments. We contrast this with the alternative Planckian dissipation mechanism that we show is incompatible with available experimental data. An accurate treatment of the electron-phonon scattering requires us to go well beyond the usual treatment, including both intraband and interband processes, considering the finite-temperature dynamical screening of the electron-phonon matrix element, and going beyond the linear Dirac dispersion. In addition to explaining the observations in currently available experimental data, we make concrete predictions that can be tested in ongoing experiments.

Anisotropic Disorder and Thermal Stability of Silicane

Atomically thin silicon nanosheets (SiNSs), such as silicane, have potential for next-generation computing paradigms, such as integrated photonics, owing to their efficient photoluminescence emission and complementary-metal-oxide-semiconductor (CMOS) compatibility. To be considered as a viable material for next-generation photonics, the SiNSs must retain their structural and optical properties at operating temperatures. However, the intersheet disorder of SiNSs and their nanoscale structure makes structural characterization difficult. Here, we use synchrotron X-ray diffraction and atomic pair distribution function (PDF) analysis to characterize the anisotropic disorder within SiNSs, demonstrating they exhibit disorder within the intersheet spacing, but have little translational or rotational disorder among adjacent SiNSs. Furthermore, we identify changes in their structural, chemical, and optical properties after being heated in an inert atmosphere up to 475 °C. We characterized changes of the annealed SiNSs using synchrotron-based total X-ray scattering, infrared spectroscopy, X-ray photoelectron spectroscopy, scanning electron microscopy, electron paramagnetic resonance, absorbance, photoluminescence, and excited-state lifetime. We find that the silicon framework is robust, with an onset of amorphization at ∼300 °C, which is well above the required operating temperatures of photonic devices. Above ∼300 °C, we demonstrate that the SiNSs begin to coalesce while keeping their translational alignment to yield amorphous silicon nanosheets. In addition, our DFT results provide information on the structure, energetics, band structures, and vibrational properties of 11 distinct oxygen-containing SiNSs. Overall, these results provide critical information for the implementation of atomically thin silicon nanosheets in next-generation CMOS-compatible integrated photonic devices.

Unconventional van der Waals heterostructures beyond stacking

Two-dimensional crystals provide exceptional opportunities for integrating dissimilar materials and forming interfaces where distinct properties and phenomena emerge. To date, research has focused on two basic heterostructure types: vertical van der Waals stacks and laterally joined monolayer crystals with in-plane line interfaces. Much more diverse architectures and interface configurations can be realized in the few-layer and multilayer regime, and if mechanical stacking and single-layer growth are replaced by processes taking advantage of self-organization, conversions between polymorphs, phase separation, strain effects, and shaping into the third dimension. Here, we highlight such opportunities for engineering heterostructures, focusing on group IV chalcogenides, a class of layered semiconductors that lend themselves exceptionally well for exploring novel van der Waals architectures, as well as advanced methods including in situ microscopy during growth and nanometer-scale probes of light-matter interactions. The chosen examples point to fruitful future directions and inspire innovative developments to create unconventional van der Waals heterostructures beyond stacking.

Realization of nearly dispersionless bands with strong orbital anisotropy from destructive interference in twisted bilayer MoS2

Recently, the twist angle between adjacent sheets of stacked van der Waals materials emerged as a new knob to engineer correlated states of matter in two-dimensional heterostructures in a controlled manner, giving rise to emergent phenomena such as superconductivity or correlated insulating states. Here, we use an ab initio based approach to characterize the electronic properties of twisted bilayer MoS₂. We report that, in marked contrast to twisted bilayer graphene, slightly hole-doped MoS₂ realizes a strongly asymmetric p_x-p_y Hubbard model on the honeycomb lattice, with two almost entirely dispersionless bands emerging due to destructive interference. The origin of these dispersionless bands, is similar to that of the flat bands in the prototypical Lieb or Kagome lattices and co-exists with the general band flattening at small twist angle due to the moiré interference. We study the collective behavior of twisted bilayer MoS₂ in the presence of interactions, and characterize an array of different magnetic and orbitally-ordered correlated phases, which may be susceptible to quantum fluctuations giving rise to exotic, purely quantum, states of matter.

Twisted van der Waals systems are known to host flat electronic bands, originating from moire potential. Here, the authors predict from purely geometric considerations a new type of nearly dispersionless bands in twisted bilayer MoS₂, resulting from destructive interference between effective lattice hopping matrix elements.

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.

Engineering Three-Dimensional Moiré Flat Bands

Twisting two adjacent layers of van der Waals materials with respect to each other can lead to flat two-dimensional electronic bands which enables a wealth of physical phenomena. Here, we generalize this concept of so-called moiré flat bands to engineer flat bands in all three spatial dimensions controlled by the twist angle. The basic concept is to stack the material such that the large spatial moiré interference patterns are spatially shifted from one twisted layer to the next. We exemplify the general concept by considering graphitic systems, boron nitride, and WSe2, but the approach is applicable to any two-dimensional van der Waals material. For hexagonal boron nitride, we develop an ab initio fitted tight binding model that captures the corresponding three-dimensional low-energy electronic structure. We outline that interesting three-dimensional correlated phases of matter can be induced and controlled following this route, including quantum magnets and unconventional superconducting states.

Continuous Mott transition in semiconductor moiré superlattices

The evolution of a Landau Fermi liquid into a non-magnetic Mott insulator with increasing electronic interactions is one of the most puzzling quantum phase transitions in physics^1-6. The vicinity of the transition is believed to host exotic states of matter such as quantum spin liquids^4-7, exciton condensates⁸ and unconventional superconductivity¹. Semiconductor moiré materials realize a highly controllable Hubbard model simulator on a triangular lattice^9-22, providing a unique opportunity to drive a metal-insulator transition (MIT) via continuous tuning of the electronic interactions. Here, by electrically tuning the effective interaction strength in MoTe₂/WSe₂ moiré superlattices, we observe a continuous MIT at a fixed filling of one electron per unit cell. The existence of quantum criticality is supported by the scaling collapse of the resistance, a continuously vanishing charge gap as the critical point is approached from the insulating side, and a diverging quasiparticle effective mass from the metallic side. We also observe a smooth evolution of the magnetic susceptibility across the MIT and no evidence of long-range magnetic order down to ~5% of the Curie-Weiss temperature. This signals an abundance of low-energy spinful excitations on the insulating side that is further corroborated by the Pomeranchuk effect observed on the metallic side. Our results are consistent with the universal critical theory of a continuous Mott transition in two dimensions^4,23.

Capturing 3D atomic defects and phonon localization at the 2D heterostructure interface

The three-dimensional (3D) local atomic structures and crystal defects at the interfaces of heterostructures control their electronic, magnetic, optical, catalytic, and topological quantum properties but have thus far eluded any direct experimental determination. Here, we use atomic electron tomography to determine the 3D local atomic positions at the interface of a MoS2-WSe2 heterojunction with picometer precision and correlate 3D atomic defects with localized vibrational properties at the epitaxial interface. We observe point defects, bond distortion, and atomic-scale ripples and measure the full 3D strain tensor at the heterointerface. By using the experimental 3D atomic coordinates as direct input to first-principles calculations, we reveal new phonon modes localized at the interface, which are corroborated by spatially resolved electron energy-loss spectroscopy. We expect that this work will pave the way for correlating structure-property relationships of a wide range of heterostructure interfaces at the single-atom level.

Quantum criticality in twisted transition metal dichalcogenides

Near the boundary between ordered and disordered quantum phases, several experiments have demonstrated metallic behaviour that defies the Landau Fermi paradigm^1-5. In moiré heterostructures, gate-tuneable insulating phases driven by electronic correlations have been recently discovered^6-23. Here, we use transport measurements to characterize metal-insulator transitions (MITs) in twisted WSe₂ near half filling of the first moiré subband. We find that the MIT as a function of both density and displacement field is continuous. At the metal-insulator boundary, the resistivity displays strange metal behaviour at low temperatures, with dissipation comparable to that at the Planckian limit. Further into the metallic phase, Fermi liquid behaviour is recovered at low temperature, and this evolves into a quantum critical fan at intermediate temperatures, before eventually reaching an anomalous saturated regime near room temperature. An analysis of the residual resistivity indicates the presence of strong quantum fluctuations in the insulating phase. These results establish twisted WSe₂ as a new platform to study doping and bandwidth-controlled metal-insulator quantum phase transitions on the triangular lattice.

Visualizing electron localization of WS2/WSe2 moiré superlattices in momentum space

The search for materials with flat electronic bands continues due to their potential to drive strong correlation and symmetry breaking orders. Electronic moirés formed in van der Waals heterostructures have proved to be an ideal platform. However, there is no holistic experimental picture for how superlattices modify electronic structure. By combining spatially resolved angle-resolved photoemission spectroscopy with optical spectroscopy, we report the first direct evidence of how strongly correlated phases evolve from a weakly interacting regime in a transition metal dichalcogenide superlattice. By comparing short and long wave vector moirés, we find that the electronic structure evolves into a highly localized regime with increasingly flat bands and renormalized effective mass. The flattening is accompanied by the opening of a large gap in the spectral function and splitting of the exciton peaks. These results advance our understanding of emerging phases in moiré superlattices and point to the importance of interlayer physics.

Spin-textured Chern bands in AB-stacked transition metal dichalcogenide bilayers

While transition-metal dichalcogenide (TMD)-based moiré materials have been shown to host various correlated electronic phenomena, topological states have not been experimentally observed until now [T. Li et al., Quantum anomalous Hall effect from intertwined moiré bands. arXiv [Preprint] (2021). https://arxiv.org/abs/2107.01796 (Accessed 5 July 2021)]. In this work, using first-principle calculations and continuum modeling, we reveal the displacement field-induced topological moiré bands in AB-stacked TMD heterobilayer [Formula: see text]/[Formula: see text] Valley-contrasting Chern bands with nontrivial spin texture are formed from interlayer hybridization between [Formula: see text] and [Formula: see text] bands of nominally opposite spins. Our study establishes a recipe for creating topological bands in AB-stacked TMD bilayers in general, which provides a highly tunable platform for realizing quantum-spin Hall and interaction-induced quantum anomalous Hall effects.

Interaction-Driven Filling-Induced Metal-Insulator Transitions in 2D Moiré Lattices

Using a realistic band structure for twisted WSe_{2} materials, we develop a theory for the interaction-driven correlated insulators to conducting metals transitions through the tuning of the filling factor around commensurate fractional fillings of the moiré unit cell in the 2D honeycomb lattice, focusing on the dominant half-filled Mott insulating state, which exists for both long- and short-range interactions. We find metallic states slightly away from half-filling, as have recently been observed experimentally. We discuss the stabilities and the magnetic properties of the resulting insulating and metallic phases, and comment on their experimental signatures. We also discuss the nature of the correlated insulator states at the rational fractional fillings.

Twisting of 2D Kagomé Sheets in Layered Intermetallics

Chemical bonding in 2D layered materials and van der Waals solids is central to understanding and harnessing their unique electronic, magnetic, optical, thermal, and superconducting properties. Here, we report the discovery of spontaneous, bidirectional, bilayer twisting (twist angle ∼4.5°) in the metallic kagomé MgCo₆Ge₆ at T = 100(2) K via X-ray diffraction measurements, enabled by the preparation of single crystals by the Laser Bridgman method. Despite the appearance of static twisting on cooling from T ∼300 to 100 K, no evidence for a phase transition was found in physical property measurements. Combined with the presence of an Einstein phonon mode contribution in the specific heat, this implies that the twisting exists at all temperatures but is thermally fluctuating at room temperature. Crystal Orbital Hamilton Population analysis demonstrates that the cooperative twisting between layers stabilizes the Co-kagomé network when coupled to strongly bonded and rigid (Ge₂) dimers that connect adjacent layers. Further modeling of the displacive disorder in the crystal structure shows the presence of a second, Mg-deficient, stacking sequence. This alternative stacking sequence also exhibits interlayer twisting, but with a different pattern, consistent with the change in electron count due to the removal of Mg. Magnetization, resistivity, and low-temperature specific heat measurements are all consistent with a Pauli paramagnetic, strongly correlated metal. Our results provide crucial insight into how chemical concepts lead to interesting electronic structures and behaviors in layered materials.

WS2 moiré superlattices derived from mechanical flexibility for hydrogen evolution reaction

The discovery of moiré superlattices (MSLs) opened an era in the research of ‘twistronics’. Engineering MSLs and realizing unique emergent properties are key challenges. Herein, we demonstrate an effective synthetic strategy to fabricate MSLs based on mechanical flexibility of WS₂ nanobelts by a facile one-step hydrothermal method. Unlike previous MSLs typically created through stacking monolayers together with complicated method, WS₂ MSLs reported here could be obtained directly during synthesis of nanobelts driven by the mechanical instability. Emergent properties are found including superior conductivity, special superaerophobicity and superhydrophilicity, and strongly enhanced electro-catalytic activity when we apply ‘twistronics’ to the field of catalytic hydrogen production. Theoretical calculations show that such excellent catalytic performance could be attributed to a closer to thermoneutral hydrogen adsorption free energy value of twisted bilayers active sites. Our findings provide an exciting opportunity to design advanced WS₂ catalysts through moiré superlattice engineering based on mechanical flexibility.

Expanding the available materials with moiré superlattices is interesting but also challenging. Here the authors use a one-step hydrothermal approach to synthesis WS₂ moiré superlattices with high catalytic activity for hydrogen evolution reaction

A simple fabrication strategy for orientationally accurate twisted heterostructures

Van der Waals (vdW) heterostructure is a type of metamaterial where multiple layers of 2D materials are vertically aligned at controlled misorientation. The relative rotation in between the adjacent layers, or the twist angle between them plays a crucial role in changing the electronic band structure of the superlattice. The assembly of multi-layers of precisely twisted two dimensional layered materials requires knowledge of the atomic structure at the edge of the flake. It may be artificially created by the 'tear and stack' process. Otherwise, the crystallographic orientation needs to be determined through invasive processes such as transmission electron microscopy or scanning tunneling microscopy, and via second-harmonic generation (SHG). Here, we demonstrate a simple and elegant transfer protocol using only an optical microscope as a edge identifier tool through which, controlled transfer of twisted homobilayer and heterobilayer transition metal dichalcogenides is performed with close to 100% yield. The fabricated twisted vdW heterostructures have been characterized by SHG, Raman spectroscopy and photoluminiscence spectroscopy, confirming the desired twist angle within ∼0.5° accuracy. The presented method is reliable, quick and prevents the use of invasive tools which is desirable for reproducible device functionalities.

Direct Visualization and Manipulation of Stacking Orders in Few-Layer Graphene by Dynamic Atomic Force Microscopy

Stacking order plays a central role in governing a wide range of properties in layered two-dimensional materials. In the case of few-layer graphene, there are two common stacking configurations: ABA and ABC stacking, which have been proven to exhibit dramatically different electronic properties. However, the controllable characterization and manipulation between them remain a great challenge. Here, we report that ABA- and ABC-stacked domains can be directly visualized in phase imaging by tapping-mode atomic force microscopy with much higher spatial resolution than conventional optical spectroscopy. The contrasting phase is caused by the different energy dissipation by the tip-sample interaction. We further demonstrate controllable manipulation on the ABA/ABC domain walls by means of propagating stress transverse waves generated by the tapping of tip. Our results offer a reliable strategy for direct imaging and precise control of the atomic structures in few-layer graphene, which can be extended to other two-dimensional materials.

Phonon renormalization in reconstructed MoS2 moiré superlattices

In moiré crystals formed by stacking van der Waals materials, surprisingly diverse correlated electronic phases and optical properties can be realized by a subtle change in the twist angle. Here, we discover that phonon spectra are also renormalized in MoS₂ twisted bilayers, adding an insight to moiré physics. Over a range of small twist angles, the phonon spectra evolve rapidly owing to ultra-strong coupling between different phonon modes and atomic reconstructions of the moiré pattern. We develop a low-energy continuum model for phonons that overcomes the outstanding challenge of calculating the properties of large moiré supercells and successfully captures the essential experimental observations. Remarkably, simple optical spectroscopy experiments can provide information on strain and lattice distortions in moiré crystals with nanometre-size supercells. The model promotes a comprehensive and unified understanding of the structural, optical and electronic properties of moiré superlattices.

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.

Pauli-limit violation and re-entrant superconductivity in moiré graphene

Moiré quantum matter has emerged as a materials platform in which correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics^1-5. However, direct evidence of unconventional pairing remains to be experimentally demonstrated. Here we show that magic-angle twisted trilayer graphene exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large (2-3 times) violation of the Pauli limit for conventional spin-singlet superconductors^6,7. This is an unexpected observation for a system that is not predicted to have strong spin-orbit coupling. The Pauli-limit violation is observed over the entire superconducting phase, which indicates that it is not related to a possible pseudogap phase with large superconducting amplitude pairing. Notably, we observe re-entrant superconductivity at large magnetic fields, which is present over a narrower range of carrier densities and displacement fields. These findings suggest that the superconductivity in magic-angle twisted trilayer graphene is likely to be driven by a mechanism that results in non-spin-singlet Cooper pairs, and that the external magnetic field can cause transitions between phases with potentially different order parameters. Our results demonstrate the richness of moiré superconductivity and could lead to the design of next-generation exotic quantum matter.

Superconductivity in an extreme strange metal

Some of the highest-transition-temperature superconductors across various materials classes exhibit linear-in-temperature 'strange metal' or 'Planckian' electrical resistivities in their normal state. It is thus believed by many that this behavior holds the key to unlock the secrets of high-temperature superconductivity. However, these materials typically display complex phase diagrams governed by various competing energy scales, making an unambiguous identification of the physics at play difficult. Here we use electrical resistivity measurements into the micro-Kelvin regime to discover superconductivity condensing out of an extreme strange metal state-with linear resistivity over 3.5 orders of magnitude in temperature. We propose that the Cooper pairing is mediated by the modes associated with a recently evidenced dynamical charge localization-delocalization transition, a mechanism that may well be pertinent also in other strange metal superconductors.

Excitonic and Valley-Polarization Signatures of Fractional Correlated Electronic Phases in a WSe_{2}/WS_{2} Moiré Superlattice

We have measured the reflectance contrast, photoluminescence, and valley polarization of a WSe_{2}/WS_{2} heterobilayer moiré superlattice at gate-tunable charge density. We observe absorption modulation of three intralayer moiré excitons at filling factors ν=1/3 and 2/3. We also observe luminescence modulation of interlayer trions at around a dozen fractional filling factors, including ν=-3/2, 1/4, 1/3, 2/5, 2/3, 6/7, 5/3. Remarkably, the valley polarization of interlayer trions is suppressed at some fractional fillings. These results demonstrate that electron crystallization can modulate the absorption, emission, and valley dynamics of the excitonic states in a moiré superlattice.

Imaging moiré flat bands in three-dimensional reconstructed WSe2/WS2 superlattices

Moiré superlattices in transition metal dichalcogenide (TMD) heterostructures can host novel correlated quantum phenomena due to the interplay of narrow moiré flat bands and strong, long-range Coulomb interactions^1-9. However, microscopic knowledge of the atomically reconstructed moiré superlattice and resulting flat bands is still lacking, which is critical for fundamental understanding and control of the correlated moiré phenomena. Here we quantitatively study the moiré flat bands in three-dimensional (3D) reconstructed WSe₂/WS₂ moiré superlattices by comparing scanning tunnelling spectroscopy (STS) of high-quality exfoliated TMD heterostructure devices with ab initio simulations of TMD moiré superlattices. A strong 3D buckling reconstruction accompanied by large in-plane strain redistribution is identified in our WSe₂/WS₂ moiré heterostructures. STS imaging demonstrates that this results in a remarkably narrow and highly localized K-point moiré flat band at the valence band edge of the heterostructure. A series of moiré flat bands are observed at different energies that exhibit varying degrees of localization. Our observations contradict previous simplified theoretical models but agree quantitatively with ab initio simulations that fully capture the 3D structural reconstruction. Our results reveal that the strain redistribution and 3D buckling in TMD heterostructures dominate the effective moiré potential and the corresponding moiré flat bands at the Brillouin zone K points.