Mapping the twist-angle disorder and Landau levels in magic-angle graphene

Universal scaling laws on the rotational energy landscape for twisted van der Waals bilayers

The emerging field of twistronics utilizes the interfacial twist angle between two-dimensional materials to design and explore unconventional electronic properties. However, recent investigations revealed that not every twist angle is stable. Understanding and predicting preferred twist angles are therefore of vital importance and have received considerable attention; however, general analytical theories that can feasibly address the stability of twist angles have not yet been developed. In this work, we reveal the existence of universal analytical scaling laws that delineate the interface rotational energy landscape, enabling the determination of both stable angles and interlayer rotational torque. The universality of our theoretical results is fundamentally based on the evolution of moiré geometry, which is applicable across many material interface systems. Our results not only unify experimental observations and literature atomistic simulations, but also provide new perspectives for the rational design of nanoscale rotation-tunable electronic devices. Our theories can potentially inspire a deeper understanding of moiré-correlated interface mechanics.

Terahertz photocurrent probe of quantum geometry and interactions in magic-angle twisted bilayer graphene

Moiré materials represent strongly interacting electron systems bridging topological and correlated physics. Despite notable advances, decoding wavefunction properties underlying the quantum geometry remains challenging. Here we utilize polarization-resolved photocurrent measurements to probe magic-angle twisted bilayer graphene, leveraging its sensitivity to the Berry connection that encompasses quantum 'textures' of electron wavefunctions. Using terahertz light resonant with optical transitions of its flat bands, we observe bulk photocurrents driven by broken symmetries and reveal the interplay between electron interactions and quantum geometry. We observe inversion-breaking gapped states undetectable through quantum transport, sharp changes in the polarization axes caused by interaction-induced band renormalization and recurring photocurrent patterns at integer filling factors of the moiré unit cell that track the evolution of quantum geometry through the cascade of phase transitions. The large and tunable terahertz response intrinsic to flat-band systems offers direct insights into the quantum geometry of interacting electrons and paves the way for innovative terahertz quantum technologies.

Synthetic Band Structure Engineering of Graphene Using Block Copolymer-Templated Dielectric Superlattices

Engineering the electronic band structure of two-dimensional (2D) materials by imposing spatially periodic superlattice (SL) potentials opens a pathway to unconventional electronics. Nanopatterning the gate electrode or surface dielectric near 2D crystals provides a powerful strategy for realizing electrostatically tunable "remote" SLs with flexibility in lattice design. Here, we demonstrate the effectiveness of block copolymer (BCP)-templated dielectric nanopatterns for fabricating etch-free high-grade metal oxide SLs. Alumina (AlOx) nanopatterns with hexagonal symmetry and a 38 nm SL wavelength are produced as a model material by directly converting a self-assembled BCP film via block-selective vapor phase infiltration. Despite micrometer-scale rotational disorder inherent to BCP self-assembly, electronic transport measurements of graphene reveal replica Dirac points at zero field and Hofstadter mini-gaps under finite magnetic fields. These results indicate the successful formation of remote SL potentials in graphene resulting from optimized AlOx nanopattern fabrication to achieve consistent lattice symmetry and periodicity at a macroscopic scale. The findings of this study, combined with the versatile, scalable, and cost-effective nature of BCP nanopatterning, highlight the potential of BCP-templated nanostructures for remote SL engineering in 2D crystals.

Magnetic Bloch states at integer flux quanta induced by super-moiré potential in graphene aligned with twisted boron nitride

Two-dimensional electron systems in both magnetic fields and periodic potentials are described by the Hofstadter butterfly, a fundamental problem of solid-state physics. While moiré systems provide a powerful method to realize this type of spectrum, previous experiments have been limited to fractional flux quanta regime, due to the difficulty of building ~ 50 nm periodic modulations. Here, we demonstrate a super-moiré strategy to overcome this challenge. By aligning monolayer graphene (G) with 1.0° twisted hexagonal boron nitride (t-hBN), a 63.2 nm bichromatic G/t-hBN super-moiré is constructed, made possible by exploiting the electrostatic nature of t-hBN potential. Under magnetic field B , magnetic Bloch states at ϕ / ϕ 0 = 1 - 9 are achieved and observed as integer Brown-Zak oscillations, expanding the flux quanta from fractions to integers. Theoretical analysis reproduces these experimental findings. This work opens promising avenues to study unexplored Hofstadter butterfly, explore emergent topological order at integer flux quanta and engineer long-wavelength periodic modulations.

Coulomb interactions and migrating Dirac cones imaged by local quantum oscillations in twisted graphene

Flat-band moiré graphene systems are a quintessential platform for investigating correlated phases of matter. Various interaction-driven ground states have been proposed, but despite extensive experimental effort, there has been little direct evidence that distinguishes between various phases, in particular near the charge neutrality point. Here we probe the fine details of the density of states and the effects of Coulomb interactions in alternating-twist trilayer graphene by imaging the local thermodynamic quantum oscillations with a nanoscale scanning superconducting quantum interference device. We find that the charging self-energy due to occupied electronic states is most important in explaining the high-carrier-density physics. At half-filling of the conduction flat band, we observe ferromagnetic-driven symmetry breaking, suggesting that it is the most robust mechanism in the hierarchy of phase transitions. Near charge neutrality, where exchange energy dominates over charging self-energy, we find a nematic semimetal ground state, which is theoretically favoured over gapped states in the presence of heterostrain. In this semimetallic phase, the flat-band Dirac cones migrate towards the mini-Brillouin zone centre, spontaneously breaking the threefold rotational symmetry. Our low-field local quantum oscillation technique can be used to explore the ground states of many strongly interacting van der Waals systems.

Electronic confinement induced quantum dot behavior in magic-angle twisted bilayer graphene

Magic-angle twisted bilayer graphene (TBLG) has emerged as a versatile platform to explore correlated electron phases driven primarily by low-energy flat bands in moiré superlattices. While techniques for controlling the twist angle between graphene layers have spurred rapid experimental progress, understanding the effects of doping inhomogeneity on electronic transport in correlated electron systems remains challenging. In this work, we investigate the interplay of confinement and doping inhomogeneity on the electrical transport properties of TBLG by leveraging device dimensions and twist angles. We show that reducing device dimensions can magnify disorder potentials caused by doping inhomogeneity, resulting in pronounced carrier confinement. This phenomenon is evident in charge transport measurements, where the Coulomb blockade effect is observed. Temperature-dependent measurements reveal a large variation in the activation gap across the device. These findings highlight the critical role of doping inhomogeneity in TBLG and its significant impact on the transport properties of the system.

Nanoscale Structure and Interfacial Electrochemical Reactivity of Moiré-Engineered Atomic Layers

ConspectusThe electronic properties of atomically thin van der Waals (vdW) materials can be precisely manipulated by vertically stacking them with a controlled offset (for example, a rotational offset─i.e., twist─between the layers, or a small difference in lattice constant) to generate moiré superlattices. In recent years, the application of this "twistronics" concept to interfacial electrochemistry has unveiled unique pathways for tailoring the electrochemical reactivity. This Account provides an overview of our work that leveraged a suite of structural characterization methods, such as interferometric four-dimensional scanning transmission electron microscopy, dark-field transmission electron microscopy, and scanning tunneling microscopy, along with nanoscale electrochemical measurement techniques, namely, scanning electrochemical cell microscopy (SECCM), to uncover and dissect the profound impact of electrode electronic structure, controlled by interlayer twist, on interfacial electron transfer kinetics. At the heart of our findings is the discovery that moiré engineering enables the isolation of thermodynamically unfavorable stacking configurations, or topological defects, that substantially increase the standard electron transfer rate constant at the solid-liquid interface beyond what has been measured on conventional, nontwisted two-dimensional (2D) materials. This enhancement in interfacial reactivity can be attributed to the localization of a high density of electronic states within these particular sites in the superlattice, a similar effect to that which occurs upon incorporation of physical defects or vacancies in an electrode material but instead using an atomically pristine surface with a highly tunable structure. Throughout our studies, understanding the nuances of the relationship between the preimposed moiré twist angle and the observed electron transfer kinetics has heavily relied on the interrogation of additional factors such as spontaneous superlattice reconstruction and three-dimensional localization of electronic states, illustrating the importance of combining electrochemical measurements with both nanoscale structural probes and theoretical modeling for designing and optimizing moiré-engineered electrodes. The insight afforded by our efforts in this space continues to deepen our understanding of the fundamental mechanisms governing electron transfer at electrochemical interfaces at large and also points to the revolutionary prospect of twistronics for advancing electrochemical technologies. While our electrochemical studies have, so far, focused largely on graphene-based moiré materials, we also offer a perspective on the promise of transition metal dichalcogenide (TMD)-based moirés as candidates for highly versatile (photo)electrode surfaces. Accordingly, we provide a discussion of our studies on the structural relaxation observed in moiré superlattices of TMDs, and we summarize our work combining SECCM with field-effect electrostatic gating of TMDs to deconvolute the influences of material conductivity and intrinsic electron transfer kinetics from the overall electrochemical response of a semiconducting 2D material. Overall, this body of work establishes a distinctive foundation for the design of a wide range of materials with tailored properties that can provide crucial insights into interfacial charge transfer chemistry─potentially serving as platforms for sensing, energy conversion, and electrocatalysis─in addition to the emergent exotic correlated electron physics that originally ignited intense interest in moiré twistronics.

Direct probing of energy gaps and bandwidth in gate-tunable flat band graphene systems

Moiré systems featuring flat electronic bands exhibit a vast landscape of emergent exotic quantum states, making them one of the resourceful platforms in condensed matter physics. Tuning these systems via twist angle and the electric field greatly enhances our comprehension of their strongly correlated ground states. Here, we report a technique to investigate the nuanced intricacies of band structures in dual-gated multilayer graphene systems. We utilize the Landau levels of a decoupled monolayer graphene to extract the electric field-dependent bilayer graphene charge neutrality point gap. Then, we extend this method to analyze the evolution of the band gap and the flat bandwidth in twisted mono-bilayer graphene. The band gap maximizes at the same displacement field where the flat bandwidth minimizes, concomitant with the emergence of a strongly correlated phase. Moreover, we extract integer and fractional quantum Hall gaps to further demonstrate the strength of this method. Our technique paves the way for improving the understanding of electronic band structures in versatile flat band systems.

Superfluid stiffness of twisted trilayer graphene superconductors

The robustness of the macroscopic quantum nature of a superconductor can be characterized by the superfluid stiffness, ρs, a quantity that describes the energy required to vary the phase of the macroscopic quantum wavefunction. In unconventional superconductors, such as cuprates, the low-temperature behaviour of ρs markedly differs from that of conventional superconductors owing to quasiparticle excitations from gapless points (nodes) in momentum space. Intensive research on the recently discovered magic-angle twisted graphene family has revealed, in addition to superconducting states, strongly correlated electronic states associated with spontaneously broken symmetries, inviting the study of ρs to uncover the potentially unconventional nature of its superconductivity. Here we report the measurement of ρs in magic-angle twisted trilayer graphene (TTG), revealing unconventional nodal-gap superconductivity. Utilizing radio-frequency reflectometry techniques to measure the kinetic inductive response of superconducting TTG coupled to a microwave resonator, we find a linear temperature dependence of ρs at low temperatures and nonlinear Meissner effects in the current-bias dependence, both indicating nodal structures in the superconducting order parameter. Furthermore, the doping dependence shows a linear correlation between the zero-temperature ρs and the superconducting transition temperature Tc, reminiscent of Uemura's relation in cuprates, suggesting phase-coherence-limited superconductivity. Our results provide strong evidence for nodal superconductivity in TTG and put strong constraints on the mechanisms of these graphene-based superconductors.

Low-Frequency Resistance Noise in Near-Magic-Angle Twisted Bilayer Graphene

The low-frequency resistance fluctuations, or noise, in electrical resistance not only set a performance benchmark in devices but also form a sensitive tool to probe nontrivial electronic phases and band structures in solids. Here, we report the measurement of such noise in the electrical resistance in twisted bilayer graphene (tBLG), where the layers are misoriented close to the magic angle (θ ∼ 1°). At high temperatures (T ≳ 60-70 K), the power spectral density (PSD) of the fluctuation inside the low-energy moiré bands is predominantly ∝1/f, where f is the frequency, being generally lowest close to the magic angle, and can be well-explained within the conventional McWhorter model of the '1/f noise' with trap-assisted density-mobility fluctuations. At low T (≲10 K), the measured noise exhibits a strong two-level random telegraphic signal (RTS), especially close to the moiré gap, which exhibits a ∝1/f2-like PSD that can be attributed to poorly screened resonances of the Fermi energy to specific bands of defects in the encapsulating boron nitride (hBN) layers. The low-T noise within the moiré band exhibits a series of minima at the integral as well as half-integral fillings, which align with the frequently observed van Hove singularities in the density-of-states driven by strong Coulomb interaction. Apart from providing a comprehensive account of the origin and the magnitude of noise in tBLG, our experiment also reveals noise to be significantly more sensitive to the underlying interaction effects in tBLG than the conventional time-averaged transport.

Tunable Topological Transitions Probed by the Quantum Hall Effect in Twisted Double Bilayer Graphene

The moiré system provides a tunable platform for investigating exotic quantum phases. Particularly, the displacement field D is crucial for tuning the electronic structures and topological properties of twisted double bilayer graphene (TDBG). Here, we present a series of D-tunable topological transitions by the evolution of quantum Hall phases (QHPs) in the valence bands of TDBG. As D increases, we observe the alternating emergence of two distinct quantum Hall regions originating from full-filling and half-filling, which we attribute to the D-induced Lifshitz transition. Moreover, we delve into the remote valence bands of TDBG and observe a transition in the sequence of Landau levels upon the application of D, shifting from 8N + 4 to 8N. This observation, combined with theoretical calculations, unveils an alteration in the Berry phase. Our findings highlight the TDBG as an exemplary platform for understanding the origin of the topological transitions in the graphene-based moiré systems.

Infrared Spectroscopy for Diagnosing Superlattice Minibands in Twisted Bilayer Graphene near the Magic Angle

Twisted bilayer graphene (TBG) represents a highly tunable, strongly correlated electron system. However, understanding the single-particle band structure alone has been challenging due to a lack of spectroscopic measurements over a broad energy range. Here, we probe the band structure of TBG around the magic angle using infrared spectroscopy and reveal spectral features that originate from interband transitions. In combination with quantum transport, we connect spectral features over a broad energy range (10-700 meV) and track their evolution with the twist angle. We compare our data with calculations of the band structures obtained via the continuum model and find good agreement only when considering a variation of interlayer/intralayer tunneling parameters with the twist angle. Our analysis suggests that the magic angle also shifts due to lattice relaxation and is better defined for a wide angular range of 0.9-1.1°. Additionally, our measurements offer an optical fingerprint of the magic angle for screening heterostructures before nanofabrication.

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

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

Electron Correlation in 2D Periodic Systems from Periodic Bootstrap Embedding

Given the growing significance of 2D materials in various optoelectronic applications, it is imperative to have simulation tools that can accurately and efficiently describe electron correlation effects in these systems. Here, we show that the recently developed bootstrap embedding (BE) accurately predicts electron correlation energies and structural properties for 2D systems. Without explicit dependence on the reciprocal space sum (k-points) in the correlation calculation, our proof-of-concept calculations shows that BE can typically recover ∼99.5% of the total minimal basis electron correlation energy in 2D semimetal, insulator, and semiconductors. We demonstrate that BE can predict lattice constants and bulk moduli for 2D systems with high precision. Furthermore, we highlight the capability of BE to treat electron correlation in twisted bilayer graphene superlattices with large unit cells containing hundreds of carbon atoms. We find that as the twist angle decreases toward the magic angle, the correlation energy initially decreases in magnitude, followed by a subsequent increase. We conclude that BE is a promising electronic structure method for future applications to 2D materials.

Strong electron-phonon coupling in magic-angle twisted bilayer graphene

The unusual properties of superconductivity in magic-angle twisted bilayer graphene (MATBG) have sparked considerable research interest1-13. However, despite the dedication of intensive experimental efforts and the proposal of several possible pairing mechanisms14-24, the origin of its superconductivity remains elusive. Here, by utilizing angle-resolved photoemission spectroscopy with micrometre spatial resolution, we reveal flat-band replicas in superconducting MATBG, where MATBG is unaligned with its hexagonal boron nitride substrate11. These replicas show uniform energy spacing, approximately 150 ± 15 meV apart, indicative of strong electron-boson coupling. Strikingly, these replicas are absent in non-superconducting twisted bilayer graphene (TBG) systems, either when MATBG is aligned to hexagonal boron nitride or when TBG deviates from the magic angle. Calculations suggest that the formation of these flat-band replicas in superconducting MATBG are attributed to the strong coupling between flat-band electrons and an optical phonon mode at the graphene K point, facilitated by intervalley scattering. These findings, although they do not necessarily put electron-phonon coupling as the main driving force for the superconductivity in MATBG, unravel the electronic structure inherent in superconducting MATBG, thereby providing crucial information for understanding the unusual electronic landscape from which its superconductivity is derived.

Low-Energy Optical Sum Rule in Moiré Graphene

Few layers of graphene at small twist angles have emerged as a fascinating platform for studying the problem of strong interactions in regimes with a nearly quenched single-particle kinetic energy and nontrivial band topology. Starting from the strong-coupling limit of twisted bilayer graphene with a vanishing single-electron bandwidth and interlayer tunneling between the same sublattice sites, we present an exact analytical theory of the Coulomb interaction-induced low-energy optical spectral weight at all integer fillings. In this limit, while the interaction-induced single-particle dispersion is finite, the optical spectral weight vanishes identically at integer fillings. We study corrections to the optical spectral weight by systematically including the effects of experimentally relevant strain-induced renormalization of the single-electron bandwidth and interlayer tunnelings between the same sublattice sites. Given the relationship between the optical spectral weight and the diamagnetic response that controls superconducting T_{c}, our results highlight the relative importance of specific parent insulating phases in enhancing the tendency towards superconductivity when doped away from integer fillings.

Direct magnetic imaging of fractional Chern insulators in twisted MoTe2

Orbital magnetization provides a sensitive probe of topology and interactions, with particularly rich phenomenology in Chern insulators in which the topological edge states carry large equilibrium currents. Here we use a nanoscale superconducting sensor1,2 to map the magnetic fringe fields in twisted bilayers of MoTe2, in which transport3,4 and optical sensing5,6 experiments have revealed the formation of fractional Chern insulator (FCI) states at zero magnetic field. We observe oscillations in the local magnetic field associated with fillings ν = -1, -2/3, -3/5, -4/7 and -5/9 of the first moiré hole band, consistent with the formation of FCIs at these fillings. We determine the local thermodynamic gaps of the most robust FCI state at ν = -2/3, finding -2/3Δ as large as 7 meV. We also characterize sample spatial disorder, which is dominated by both inhomogeneity in the effective unit cell area7 as well as inhomogeneity in the band edge offset and bound dipole moment. Our results highlight both the challenges posed by structural disorder in the study of twisted homobilayer moiré systems and the opportunities afforded by the robust nature of the underlying correlated topological states.

Local probe of bulk and edge states in a fractional Chern insulator

The fractional quantum Hall effect is a key example of topological quantum many-body phenomena, arising from the interplay between strong electron correlation, topological order and time-reversal symmetry breaking. Recently, a lattice analogue of the fractional quantum Hall effect at zero magnetic field has been observed, confirming the existence of a zero-field fractional Chern insulator (FCI). Despite this, the bulk-edge correspondence-a hallmark of a FCI featuring an insulating bulk with conductive edges-has not been directly observed. In fact, this correspondence has not been visualized in any system for fractional states owing to experimental challenges. Here we report the imaging of FCI edge states in twisted MoTe2 (t-MoTe2) using microwave impedance microscopy1. By tuning the carrier density, we observe the system evolving between metallic and FCI states, the latter of which exhibits insulating bulk and conductive edges, as expected from the bulk-boundary correspondence. Further analysis suggests the composite nature of the FCI edge states. We also observe the evolution of edge states across the topological phase transition as a function of interlayer electric field and reveal exciting prospects of neighbouring domains with different fractional orders. These findings pave the way for research into topologically protected one-dimensional interfaces between various anyonic states at zero magnetic field, such as gapped one-dimensional symmetry-protected phases with non-zero topological entanglement entropy, Halperin-Laughlin interfaces and the creation of non-abelian anyons.

Designing Moiré Patterns by Shearing

We analyze the elastic properties, structural effects, and low-energy physics of a sheared nanoribbon placed on top of graphene, which creates a gradually changing moiré pattern. By means of a classical elastic model we derive the strains in the ribbon and we obtain its electronic energy spectrum with a scaled tight-binding model. The size of the sheared region is determined by the balance between elastic and van der Waals energy, and different regimes are identified. Near the clamped edge, moderate strains and small twist angles lead to one-dimensional channels. Near the sheared edge, a long region behaves like magic angle twisted bilayer graphene (TBG), showing a sharp peak in the density of states, mostly isolated from the rest of the spectrum. We also calculate the band topology along the ribbon and we find that it is stable for large intervals of strains and twist angles. Together with the experimental observations, these results show that the sheared nanoribbon geometry is ideal for exploring superconductivity and correlated phases in TBG in the very sought-after regime of ultralow twist angle disorder.

Infrared single-photon detection with superconducting magic-angle twisted bilayer graphene

The moiré superconductor magic-angle twisted bilayer graphene (MATBG) shows exceptional properties, with an electron (hole) ensemble of only ~1011 carriers per square centimeter, which is five orders of magnitude lower than traditional superconductors (SCs). This results in an ultralow electronic heat capacity and a large kinetic inductance of this truly two-dimensional SC, providing record-breaking parameters for quantum sensing applications, specifically thermal sensing and single-photon detection. To fully exploit these unique superconducting properties for quantum sensing, here, we demonstrate a proof-of-principle experiment to detect single near-infrared photons by voltage biasing an MATBG device near its superconducting phase transition. We observe complete destruction of the SC state upon absorption of a single infrared photon even in a 16-square micrometer device, showcasing exceptional sensitivity. Our work offers insights into the MATBG-photon interaction and demonstrates pathways to use moiré superconductors as an exciting platform for revolutionary quantum devices and sensors.

Long-lived isospin excitations in magic-angle twisted bilayer graphene

Numerous correlated many-body phases, both conventional and exotic, have been reported in magic-angle twisted bilayer graphene (MATBG)1-24. However, the dynamics associated with these correlated states, crucial for understanding the underlying physics, remain unexplored. Here we combine exciton sensing and optical pump-probe spectroscopy to investigate the dynamics of isospin orders in MATBG with WSe2 substrate across the entire flat band, achieving sub-picosecond resolution. We observe remarkably slow isospin dynamics in a broad filling range around ν = 2 and between ν = -3 and -2, with lifetimes of up to 300 ps that decouple from the much faster cooling of electronic temperature (about 10 ps). This non-thermal behaviour demonstrates the presence of abnormally long-lived modes in the isospin degrees of freedom. This observation, not anticipated by theory, implies the existence of long-range propagating collective modes, strong isospin fluctuations and memory effects and is probably associated with an intervalley coherent or incommensurate Kekulé spiral ground state. We further demonstrate non-equilibrium control of the isospin orders previously found around integer fillings. Specifically, through ultrafast manipulation, it can be transiently shifted away from integer fillings. Our study demonstrates a unique probe of collective excitations in MATBG and paves the way for actively controlling non-equilibrium phenomena in moiré systems.

Cathodoluminescence from interlayer excitons in a 2D semiconductor heterobilayer

Photoluminescence has widely been used to study excitons in semiconducting transition metal dichalcogenide (MX2) monolayers, demonstrating strong light-matter interactions and locked spin and valley degrees of freedom. In heterobilayers composed of overlapping monolayers of two different MX2, an interlayer exciton can form, with the hole localised in one layer and the electron in the other. These interlayer excitons are long-lived, field-tunable, and can be trapped by moiré patterns formed at small twist angles between the layers. Here we demonstrate that emission from radiative recombination of interlayer excitons can be observed by cathodoluminescence from a WSe2/MoSe2heterobilayer encapsulated in hexagonal boron nitride. The higher spatial resolution of cathodoluminescence, compared to photoluminescence, allows detailed analysis of sample heterogeneity at the 100 s of nm lengthscales over which twist angles tend to vary in dry-transfer fabricated heterostructures.

Emergent phases in graphene flat bands

Electronic correlations in two-dimensional materials play a crucial role in stabilising emergent phases of matter. The realisation of correlation-driven phenomena in graphene has remained a longstanding goal, primarily due to the absence of strong electron-electron interactions within its low-energy bands. In this context, magic-angle twisted bilayer graphene has recently emerged as a novel platform featuring correlated phases favoured by the low-energy flat bands of the underlying moiré superlattice. Notably, the observation of correlated insulators and superconductivity, and the interplay between these phases have garnered significant attention. A wealth of correlated phases with unprecedented tunability was discovered subsequently, including orbital ferromagnetism, Chern insulators, strange metallicity, density waves, and nematicity. However, a comprehensive understanding of these closely competing phases remains elusive. The ability to controllably twist and stack multiple graphene layers has enabled the creation of a whole new family of moiré superlattices with myriad properties. Here, we review the progress and development achieved so far, encompassing the rich phase diagrams offered by these graphene-based moiré systems. Additionally, we discuss multiple phases recently observed in non-moiré multilayer graphene systems. Finally, we outline future opportunities and challenges for the exploration of hidden phases in this new generation of moiré materials.

Chern Insulator States with Tunable Chern Numbers in a Graphene Moiré Superlattice

Moiré superlattices, constituted by two-dimensional materials, demonstrate a variety of strongly correlated and topological phenomena including correlated insulators, superconductivity, and integer/fractional Chern insulators. In the realm of topological nontrivial Chern insulators within specific moiré superlattices, previous studies usually observe a single Chern number at a given filling factor in a device. Here we present the observation of gate-tunable Chern numbers within the Chern insulator state of an ABC-stacked trilayer graphene/hexagonal boron nitride moiré superlattice device. Near quarter filling, the moiré superlattice exhibits spontaneous valley polarization and distinct ferromagnetism associated with the Chern insulator states over a range of the displacement field. Surprisingly we find a transition of the Chern number from C = 3 to 4 as the displacement field is increased. Our observation of gate-tunable correlated Chern insulators suggests new ways to control and manipulate topological states in a moiré superlattice device.

Tunnel junctions based on interfacial two dimensional ferroelectrics

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

Emergent Correlated Phases in Rhombohedral Trilayer Graphene Induced by Proximity Spin-Orbit and Exchange Coupling

The impact of proximity-induced spin-orbit and exchange coupling on the correlated phase diagram of rhombohedral trilayer graphene (RTG) is investigated theoretically. By employing ab initio-fitted effective models of RTG encapsulated by transition metal dichalcogenides (spin-orbit proximity effect) and ferromagnetic Cr_{2}Ge_{2}Te_{6} (exchange proximity effect), we incorporate the Coulomb interactions within the random-phase approximation to explore potential correlated phases at different displacement fields and doping. We find a rich spectrum of spin-valley resolved Stoner and intervalley coherence instabilities induced by the spin-orbit proximity effects, such as the emergence of a spin-valley-coherent phase due to the presence of valley-Zeeman coupling. Similarly, proximity exchange removes the phase degeneracies by biasing the spin direction, enabling a magnetocorrelation effect-strong sensitivity of the correlated phases to the relative magnetization orientations (parallel or antiparallel) of the encapsulating ferromagnetic layers.

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.

Strain Engineering of Twisted Bilayer Graphene: The Rise of Strain-Twistronics

The layer-by-layer stacked van der Waals structures (termed vdW hetero/homostructures) offer a new paradigm for materials design-their physical properties can be tuned by the vertical stacking sequence as well as by adding a mechanical twist, stretch, and hydrostatic pressure to the atomic structure. In particular, simple twisting and stacking of two layers of graphene can form a uniform and ordered Moiré superlattice, which can effectively modulate the electrons of graphene layers and lead to the discovery of unconventional superconductivity and strong correlations. However, the twist angle of twisted bilayer graphene (tBLG) is almost unchangeable once the interlayer stacking is determined, while applying mechanical elastic strain provides an alternative way to deeply regulate the electronic structure by controlling the lattice spacing and symmetry. In this review, diverse experimental advances are introduced in straining tBLG by in-plane and out-of-plane modes, followed by the characterizations and calculations toward quantitatively tuning the strain-engineered electronic structures. It is further discussed that the structural relaxation in strained Moiré superlattice and its influence on electronic structures. Finally, the conclusion entails prospects for opportunities of strained twisted 2D materials, discussions on existing challenges, and an outlook on the intriguing emerging field, namely "strain-twistronics".

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.

Tunable quantum interferometer for correlated moiré electrons

Magic-angle twisted bilayer graphene can host a variety of gate-tunable correlated states - including superconducting and correlated insulator states. Recently, junction-based superconducting moiré devices have been introduced, enabling the study of the charge, spin and orbital nature of superconductivity, as well as the coherence of moiré electrons in magic-angle twisted bilayer graphene. Complementary fundamental coherence effects-in particular, the Little-Parks effect in a superconducting ring and the Aharonov-Bohm effect in a normally conducting ring - have not yet been reported in moiré devices. Here, we observe both phenomena in a single gate-defined ring device, where we can embed a superconducting or normally conducting ring in a correlated or band insulator. The Little-Parks effect is seen in the superconducting phase diagram as a function of density and magnetic field, confirming the effective charge of 2e. We also find that the coherence length of conducting moiré electrons exceeds several microns at 50 mK. In addition, we identify a regime characterized by h/e-periodic oscillations but with superconductor-like nonlinear transport.

De Haas-van Alphen spectroscopy and magnetic breakdown in moiré graphene

Quantum oscillations originating from the quantization of electron cyclotron orbits provide sensitive diagnostics of electron bands and interactions. We report on nanoscale imaging of the thermodynamic magnetization oscillations caused by the de Haas-van Alphen effect in moiré graphene. Scanning by means of superconducting quantum interference device (SQUID)-on-tip in Bernal bilayer graphene crystal axis-aligned to hexagonal boron nitride reveals large magnetization oscillations with amplitudes reaching 500 Bohr magneton per electron in weak magnetic fields, unexpectedly low frequencies, and high sensitivity to superlattice filling fraction. The oscillations allow us to reconstruct the complex band structure, revealing narrow moiré bands with multiple overlapping Fermi surfaces separated by unusually small momentum gaps. We identified sets of oscillations that violate the textbook Onsager Fermi surface sum rule, signaling formation of broad-band particle-hole superposition states induced by coherent magnetic breakdown.

Imaging quantum oscillations and millitesla pseudomagnetic fields in graphene

The exceptional control of the electronic energy bands in atomically thin quantum materials has led to the discovery of several emergent phenomena1. However, at present there is no versatile method for mapping the local band structure in advanced two-dimensional materials devices in which the active layer is commonly embedded in the insulating layers and metallic gates. Using a scanning superconducting quantum interference device, here we image the de Haas-van Alphen quantum oscillations in a model system, the Bernal-stacked trilayer graphene with dual gates, which shows several highly tunable bands2-4. By resolving thermodynamic quantum oscillations spanning more than 100 Landau levels in low magnetic fields, we reconstruct the band structure and its evolution with the displacement field with excellent precision and nanoscale spatial resolution. Moreover, by developing Landau-level interferometry, we show shear-strain-induced pseudomagnetic fields and map their spatial dependence. In contrast to artificially induced large strain, which leads to pseudomagnetic fields of hundreds of tesla5-7, we detect naturally occurring pseudomagnetic fields as low as 1 mT corresponding to graphene twisting by 1 millidegree, two orders of magnitude lower than the typical angle disorder in twisted bilayer graphene8-11. This ability to resolve the local band structure and strain at the nanoscale level enables the characterization and use of tunable band engineering in practical van der Waals devices.

Charge-4e superconductivity and chiral metal in 45°-twisted bilayer cuprates and related bilayers

The material realization of charge-4e/6e superconductivity (SC) is a big challenge. Here, we propose to realize charge-4e SC in maximally-twisted homobilayers, such as 45∘-twisted bilayer cuprates and 30∘-twisted bilayer graphene, referred to as twist-bilayer quasicrystals (TB-QC). When each monolayer hosts a pairing state with the largest pairing angular momentum, previous studies have found that the second-order interlayer Josephson coupling would drive chiral topological SC (TSC) in the TB-QC. Here we propose that, above the Tc of the chiral TSC, either charge-4e SC or chiral metal can arise as vestigial phases, depending on the ordering of the total- and relative-pairing-phase fields of the two layers. Based on a thorough symmetry analysis to get the low-energy effective Hamiltonian, we conduct a combined renormalization-group and Monte-Carlo study and obtain the phase diagram, which includes the charge-4e SC and chiral metal phases.

Energy dissipation on magic angle twisted bilayer graphene

Traditional Joule dissipation omnipresent in today's electronic devices is well understood while the energy loss of the strongly interacting electron systems remains largely unexplored. Twisted bilayer graphene (tBLG) is a host to interaction-driven correlated insulating phases, when the relative rotation is close to the magic angle (1.08∘). We report on low-temperature (5K) nanomechanical energy dissipation of tBLG measured by pendulum atomic force microscopy (p-AFM). The ultrasensitive cantilever tip acting as an oscillating gate over the quantum device shows dissipation peaks attributed to different fractional fillings of the flat energy bands. Local detection allows to determine the twist angle and spatially resolved dissipation images showed the existence of hundred-nanometer domains of different doping. Application of magnetic fields provoked strong oscillations of the dissipation signal at 3/4 band filling, identified in analogy to Aharonov-Bohm oscillations, a wavefunction interference present between domains of different doping and a signature of orbital ferromagnetism.

Gate-Defined Topological Josephson Junctions in Bernal Bilayer Graphene

Recent experiments on Bernal bilayer graphene (BLG) deposited on monolayer WSe_{2} revealed robust, ultraclean superconductivity coexisting with sizable induced spin-orbit coupling. Here, we propose BLG/WSe_{2} as a platform to engineer gate-defined planar topological Josephson junctions, where the normal and superconducting regions descend from a common material. More precisely, we show that if superconductivity in BLG/WSe_{2} is gapped and emerges from a parent state with intervalley coherence, then Majorana zero-energy modes can form in the barrier region upon applying weak in-plane magnetic fields. Our results spotlight a potential pathway for "internally engineered" topological superconductivity that minimizes detrimental disorder and orbital-magnetic-field effects.

Recent Advances in Moiré Superlattice Systems by Angle-Resolved Photoemission Spectroscopy

The last decade has witnessed a flourish in 2D materials including graphene and transition metal dichalcogenides (TMDs) as atomic-scale Legos. Artificial moiré superlattices via stacking 2D materials with a twist angle and/or a lattice mismatch have recently become a fertile playground exhibiting a plethora of emergent properties beyond their building blocks. These rich quantum phenomena stem from their nontrivial electronic structures that are effectively tuned by the moiré periodicity. Modern angle-resolved photoemission spectroscopy (ARPES) can directly visualize electronic structures with decent momentum, energy, and spatial resolution, thus can provide enlightening insights into fundamental physics in moiré superlattice systems and guides for designing novel devices. In this review, first, a brief introduction is given on advanced ARPES techniques and basic ideas of band structures in a moiré superlattice system. Then ARPES research results of various moiré superlattice systems are highlighted, including graphene on substrates with small lattice mismatches, twisted graphene/TMD moiré systems, and high-order moiré superlattice systems. Finally, it discusses important questions that remain open, challenges in current experimental investigations, and presents an outlook on this field of research.

Programming twist angle and strain profiles in 2D materials

Moiré superlattices in twisted two-dimensional materials have generated tremendous excitement as a platform for achieving quantum properties on demand. However, the moiré pattern is highly sensitive to the interlayer atomic registry, and current assembly techniques suffer from imprecise control of the average twist angle, spatial inhomogeneity in the local twist angle, and distortions caused by random strain. We manipulated the moiré patterns in hetero- and homobilayers through in-plane bending of monolayer ribbons, using the tip of an atomic force microscope. This technique achieves continuous variation of twist angles with improved twist-angle homogeneity and reduced random strain, resulting in moiré patterns with tunable wavelength and ultralow disorder. Our results may enable detailed studies of ultralow-disorder moiré systems and the realization of precise strain-engineered devices.

Magnetoplasmons in magic-angle twisted bilayer graphene

The magic-angle twisted bilayer graphene (MATBLG) has been demonstrated to exhibit exotic physical properties due to the special flat bands. However, exploiting the engineering of such properties by external fields is still in it infancy. Here we show that MATBLG under an external magnetic field presents a distinctive magnetoplasmon dispersion, which can be significantly modified by transferred momentum and charge doping. Along a wide range of transferred momentum, there exist special pronounced single magnetoplasmon and horizontal single-particle excitation modes near charge neutrality. We provide an insightful discussion of such unique features based on the electronic excitation of Landau levels quantized from the flat bands and Landau damping. Additionally, charge doping leads to peculiar multiple strong-weight magnetoplasmons. These characteristics make MATBLG a favorable candidate for plasmonic devices and technology applications.

Atomic Resolution Imaging of Highly Air-Sensitive Monolayer and Twisted-Bilayer WTe2

Bulk Td-WTe2 is a semimetal, while its monolayer counterpart is a two-dimensional (2D) topological insulator. Recently, electronic transport resembling a Luttinger liquid state was found in twisted-bilayer WTe2 (tWTe2) with a twist angle of ∼5°. Despite the strong interest in 2D WTe2 systems, little experimental information is available about their intrinsic microstructure, leaving obstacles in modeling their physical properties. The monolayer, and consequently tWTe2, are highly air-sensitive, and therefore, probing their atomic structures is difficult. In this study, we develop a robust method for atomic-resolution visualization of monolayers and tWTe2 obtained through mechanical exfoliation and fabrication. We confirm the high crystalline quality of mechanically exfoliated WTe2 samples and observe that tWTe2 with twist angles of ∼5 and ∼2° retains its pristine moiré structure without substantial deformations or reconstructions. The results provide a structural foundation for future electronic modeling of monolayer and tWTe2 moiré lattices.

Superconductivity and strong interactions in a tunable moiré quasicrystal

Electronic states in quasicrystals generally preclude a Bloch description1, rendering them fascinating and enigmatic. Owing to their complexity and scarcity, quasicrystals are underexplored relative to periodic and amorphous structures. Here we introduce a new type of highly tunable quasicrystal easily assembled from periodic components. By twisting three layers of graphene with two different twist angles, we form two mutually incommensurate moiré patterns. In contrast to many common atomic-scale quasicrystals2,3, the quasiperiodicity in our system is defined on moiré length scales of several nanometres. This 'moiré quasicrystal' allows us to tune the chemical potential and thus the electronic system between a periodic-like regime at low energies and a strongly quasiperiodic regime at higher energies, the latter hosting a large density of weakly dispersing states. Notably, in the quasiperiodic regime, we observe superconductivity near a flavour-symmetry-breaking phase transition4,5, the latter indicative of the important role that electronic interactions play in that regime. The prevalence of interacting phenomena in future systems with in situ tunability is not only useful for the study of quasiperiodic systems but may also provide insights into electronic ordering in related periodic moiré crystals6-12. We anticipate that extending this platform to engineer quasicrystals by varying the number of layers and twist angles, and by using different two-dimensional components, will lead to a new family of quantum materials to investigate the properties of strongly interacting quasicrystals.

Junctions and Superconducting Symmetry in Twisted Bilayer Graphene

Junctions provide a wealth of information on the symmetry of the order parameter of superconductors. We analyze junctions between a scanning tunneling microscope (STM) tip and superconducting twisted bilayer graphene (TBG) and TBG Josephson junctions (JJs). We compare superconducting phases that are even or odd under valley exchange (s- or f-wave). The critical current in mixed (s and f) JJs strongly depends on the angle between the junction and the lattice. In STM-TBG junctions, due to Andreev reflection, the f-wave leads to a prominent peak in subgap conductance, as seen in experiments.

Fast Twist Angle Mapping of Bilayer Graphene Using Spectroscopic Ellipsometric Contrast Microscopy

Twisted bilayer graphene provides an ideal solid-state model to explore correlated material properties and opportunities for a variety of optoelectronic applications, but reliable, fast characterization of the twist angle remains a challenge. Here we introduce spectroscopic ellipsometric contrast microscopy (SECM) as a tool for mapping twist angle disorder in optically resonant twisted bilayer graphene. We optimize the ellipsometric angles to enhance the image contrast based on measured and calculated reflection coefficients of incident light. The optical resonances associated with van Hove singularities correlate well to Raman and angle-resolved photoelectron emission spectroscopy, confirming the accuracy of SECM. The results highlight the advantages of SECM, which proves to be a fast, nondestructive method for characterization of twisted bilayer graphene over large areas, unlocking process, material, and device screening and cross-correlative measurement potential for bilayer and multilayer materials.

A Superconducting Micro-Magnetometer for Quantum Vortex in Superconducting Nanoflakes

Superconducting quantum interferometer device (SQUID) plays a key role in understanding electromagnetic properties and emergent phenomena in quantum materials. The technological appeal of SQUID is that its detection accuracy for the electromagnetic signal can precisely reach the quantum level of a single magnetic flux. However, conventional SQUID techniques normally can only be applied to a bulky sample and do not have the capability to probe the magnetic properties of micro-scale samples with small magnetic signals. Herein, it is demonstrated that, based on a specially designed superconducting nano-hole array, the contactless detection of magnetic properties and quantized vortices in micro-sized superconducting nanoflakes is realized. An anomalous hysteresis loop and a suppression of Little-Parks oscillation are observed in the detected magnetoresistance signal, which originates from the disordered distribution of the pinned vortices in Bi₂ Sr₂ CaCu₂ O_8+δ . Therefore, the density of pinning centers of the quantized vortices on such micro-sized superconducting samples can be quantitatively evaluated, which is technically inaccessible for conventional SQUID detection. The superconducting micro-magnetometer provides a new approach to exploring mesoscopic electromagnetic phenomena of quantum materials.

Tuning Multiple Landau Quantization in Transition-Metal Dichalcogenide with Strain

Landau quantization associated with the quantized cyclotron motion of electrons under magnetic field provides the effective way to investigate topologically protected quantum states with entangled degrees of freedom and multiple quantum numbers. Here we report the cascade of Landau quantization in a strained type-II Dirac semimetal NiTe₂ with spectroscopic-imaging scanning tunneling microscopy. The uniform-height surfaces exhibit single-sequence Landau levels (LLs) at a magnetic field originating from the quantization of topological surface state (TSS) across the Fermi level. Strikingly, we reveal the multiple sequence of LLs in the strained surface regions where the rotation symmetry is broken. First-principles calculations demonstrate that the multiple LLs attest to the remarkable lifting of the valley degeneracy of TSS by the in-plane uniaxial or shear strains. Our findings pave a pathway to tune multiple degrees of freedom and quantum numbers of TMDs via strain engineering for practical applications such as high-frequency rectifiers, Josephson diode and valleytronics.

Functional Renormalization Group Study of Superconductivity in Rhombohedral Trilayer Graphene

We employ a functional renormalization group approach to ascertain the pairing mechanism and symmetry of the superconducting phase observed in rhombohedral trilayer graphene. Superconductivity in this system occurs in a regime of carrier density and displacement field with a weakly distorted annular Fermi sea. We find that repulsive Coulomb interactions can induce electron pairing on the Fermi surface by taking advantage of momentum-space structure associated with the finite width of the Fermi sea annulus. The degeneracy between spin-singlet and spin-triplet pairing is lifted by valley-exchange interactions that strengthen under the RG flow and develop nontrivial momentum-space structure. We find that the leading pairing instability is d-wave-like and spin singlet, and that the theoretical phase diagram versus carrier density and displacement field agrees qualitatively with experiment.

Formation of Artificial Fermi Surfaces with a Triangular Superlattice on a Conventional Two-Dimensional Electron Gas

Imposing an external periodic electrostatic potential to the electrons confined in a quantum well makes it possible to engineer synthetic two-dimensional band structures, with electronic properties different from those in the host semiconductor. Here we report the fabrication and study of a tunable triangular artificial lattice on a GaAs/AlGaAs heterostructure where it is possible to transform from the original GaAs band structure and a circular Fermi surface to a new band structure with multiple artificial Fermi surfaces simply by altering a gate bias. For weak electrostatic modulation magnetotransport measurements reveal multiple quantum oscillations and commensurability oscillations due to the electron scattering from the artificial lattice. Increasing the strength of the modulation reveals new commensurability oscillations of the electrons from the artificial Fermi surface scattering from the triangular artificial lattice. These results show that low disorder gate-tunable lateral superlattices can be used to form artificial two-dimensional crystals with designer electronic properties.

Wrinkles, Ridges, Miura-Ori, and Moiré Patterns in MoSe2 Using Neural Networks

Effects of lateral compression on out-of-plane deformation of two-dimensional MoSe₂ layers are investigated. A MoSe₂ monolayer develops periodic wrinkles under uniaxial compression and Miura-Ori patterns under biaxial compression. When a flat MoSe₂ monolayer is placed on top of a wrinkled MoSe₂ layer, the van der Waals (vdW) interaction transforms wrinkles into ridges and generates mixed 2H and 1T phases and chain-like defects. Under a biaxial strain, the vdW interaction induces regions of Miura-Ori patterns in bilayers. Strained systems analyzed using a convolutional neural network show that the compressed system consists of semiconducting 2H and metallic 1T phases. The energetics, mechanical response, defect structure, and dynamics are analyzed as bilayers undergo wrinkle-ridge transformations under uniaxial compression and moiré transformations under biaxial compression. Our results indicate that in-plane compression can induce self-assembly of out-of-plane metasurfaces with controllable semiconducting and metallic phases and moiré patterns with unique optoelectronic properties.

Anderson's Theorem for Correlated Insulating States in Twisted Bilayer Graphene

The emergence of correlated insulating phases in magic-angle twisted bilayer graphene exhibits strong sample dependence. Here, we derive an Anderson theorem governing the robustness against disorder of the Kramers intervalley coherent (K-IVC) state, a prime candidate for describing the correlated insulators at even fillings of the moiré flat bands. We find that the K-IVC gap is robust against local perturbations, which are odd under PT, where P and T denote particle-hole conjugation and time reversal, respectively. In contrast, PT-even perturbations will in general induce subgap states and reduce or even eliminate the gap. We use this result to classify the stability of the K-IVC state against various experimentally relevant perturbations. The existence of an Anderson theorem singles out the K-IVC state from other possible insulating ground states.

A primer on twistronics: a massless Dirac fermion's journey to moiré patterns and flat bands in twisted bilayer graphene

The recent discovery of superconductivity in magic-angle twisted bilayer graphene (TBLG) has sparked a renewed interest in the strongly-correlated physics ofsp²carbons, in stark contrast to preliminary investigations which were dominated by the one-body physics of the massless Dirac fermions. We thus provide a self-contained, theoretical perspective of the journey of graphene from its single-particle physics-dominated regime to the strongly-correlated physics of the flat bands. Beginning from the origin of the Dirac points in condensed matter systems, we discuss the effect of the superlattice on the Fermi velocity and Van Hove singularities in graphene and how it leads naturally to investigations of the moiré pattern in van der Waals heterostructures exemplified by graphene-hexagonal boron-nitride and TBLG. Subsequently, we illuminate the origin of flat bands in TBLG at the magic angles by elaborating on a broad range of prominent theoretical works in a pedagogical way while linking them to available experimental support, where appropriate. We conclude by providing a list of topics in the study of the electronic properties of TBLG not covered by this review but may readily be approached with the help of this primer.