Intrinsic quantized anomalous Hall effect in a moiré heterostructure

Mesoscopic Transport of Quantum Anomalous Hall Effect in the Submicron Size Regime

The quantum anomalous Hall (QAH) effect has been demonstrated in two-dimensional topological insulator systems incorporated with ferromagnetism. However, a comprehensive understanding of mesoscopic transport in submicron QAH devices has not yet been established. Here we fabricated miniaturized QAH devices with channel widths down to 600 nm, where the QAH features are still preserved. A backscattering channel is formed in narrow QAH devices through percolative hopping between 2D compressible puddles. Large resistance fluctuations are observed in narrow devices near the coercive field, which is associated with collective interference between intersecting paths along domain walls when the device geometry is smaller than the phase coherence length L_{ϕ}. Through measurement of size-dependent breakdown current, we confirmed that the chiral edge states are confined at the physical boundary with its width on the order of Fermi wavelength.

Floquet topological phases with high Chern numbers in a periodically driven extended Su-Schrieffer-Heeger model

The high Chern number phases with the Chern number |C| > 1 are observed in this study of a periodically driven extended Su-Schrieffer-Heeger (E-SSH) model with a cyclic parameter. Besides the standard intra-dimer and the nearest-neighbor inter-dimer hopping of the SSH model, an additional next-nearest-neighbor hopping is considered in the E-SSH model. The cyclic parameter, which plays the role of a synthetic dimension, is invoked as a modulation of the hopping strengths. A rigorous analysis of different phase diagrams has shown multiple Floquet topological phase transitions among the high Chern number phases. These phase transitions can be controlled by the strength and frequency of the periodic driving. Instead of applying perturbation theory, the whole analysis is done by Floquet replica technique. This gives a freedom to study high as well as low-frequency effects on the system by considering less or more number of photon sectors. This system can be experimentally realized through a pulse sequence scheme in the optical lattice setup.

The Magnetic Genome of Two-Dimensional van der Waals Materials

Magnetism in two-dimensional (2D) van der Waals (vdW) materials has recently emerged as one of the most promising areas in condensed matter research, with many exciting emerging properties and significant potential for applications ranging from topological magnonics to low-power spintronics, quantum computing, and optical communications. In the brief time after their discovery, 2D magnets have blossomed into a rich area for investigation, where fundamental concepts in magnetism are challenged by the behavior of spins that can develop at the single layer limit. However, much effort is still needed in multiple fronts before 2D magnets can be routinely used for practical implementations. In this comprehensive review, prominent authors with expertise in complementary fields of 2D magnetism (i.e., synthesis, device engineering, magneto-optics, imaging, transport, mechanics, spin excitations, and theory and simulations) have joined together to provide a genome of current knowledge and a guideline for future developments in 2D magnetic materials research.

Observation of chiral and slow plasmons in twisted bilayer graphene

Moiré superlattices have led to observations of exotic emergent electronic properties such as superconductivity and strong correlated states in small-rotation-angle twisted bilayer graphene (tBLG)^1,2. Recently, these findings have inspired the search for new properties in moiré plasmons. Although plasmon propagation in the tBLG basal plane has been studied by near-field nano-imaging techniques^3-7, the general electromagnetic character and properties of these plasmons remain elusive. Here we report the direct observation of two new plasmon modes in macroscopic tBLG with a highly ordered moiré superlattice. Using spiral structured nanoribbons of tBLG, we identify signatures of chiral plasmons that arise owing to the uncompensated Berry flux of the electron gas under optical pumping. The salient features of these chiral plasmons are shown through their dependence on optical pumping intensity and electron fillings, in conjunction with distinct resonance splitting and Faraday rotation coinciding with the spectral window of maximal Berry flux. Moreover, we also identify a slow plasmonic mode around 0.4 electronvolts, which stems from the interband transitions between the nested subbands in lattice-relaxed AB-stacked domains. This mode may open up opportunities for strong light-matter interactions within the highly sought after mid-wave infrared spectral window⁸. Our results unveil the new electromagnetic dynamics of small-angle tBLG and exemplify it as a unique quantum optical platform.

Family of Ideal Chern Flatbands with Arbitrary Chern Number in Chiral Twisted Graphene Multilayers

We consider a family of twisted graphene multilayers consisting of n-untwisted chirally stacked layers, e.g., AB, ABC, etc, with a single twist on top of m-untwisted chirally stacked layers. Upon neglecting both trigonal warping terms for the untwisted layers and the same sublattice hopping between all layers, the resulting models generalize several remarkable features of the chiral model of twisted bilayer graphene (CTBG). In particular, they exhibit a set of magic angles which are identical to those of CTBG at which a pair of bands (i) are perfectly flat, (ii) have Chern numbers in the sublattice basis given by ±(n,-m) or ±(n+m-1,-1) depending on the stacking chirality, and (iii) satisfy the trace condition, saturating an inequality between the quantum metric and the Berry curvature, and thus realizing ideal quantum geometry. These are the first higher Chern bands that satisfy (iii) beyond fine-tuned models or combinations of Landau levels. We show that ideal quantum geometry is directly related to the construction of fractional quantum Hall model wave functions. We provide explicit analytic expressions for the flatband wave functions at the magic angle in terms of the CTBG wave functions. We also show that the Berry curvature distribution in these models can be continuously tuned while maintaining perfect quantum geometry. Similar to the study of fractional Chern insulators in ideal C=1 bands, these models pave the way for investigating exotic topological phases in higher Chern bands for which no Landau level analog is available.

Correlated States in Strained Twisted Bilayer Graphenes Away from the Magic Angle

Graphene moiré superlattice formed by rotating two graphene sheets can host strongly correlated and topological states when flat bands form at so-called magic angles. Here, we report that, for a twisting angle far away from the magic angle, the heterostrain induced during stacking heterostructures can also create flat bands. Combining a direct visualization of strain effect in twisted bilayer graphene moiré superlattices and transport measurements, features of correlated states appear at "non-magic" angles in twisted bilayer graphene under the heterostrain. Observing correlated states in these "non-standard" conditions can enrich the understanding of the possible origins of the correlated states and widen the freedom in tuning the moiré heterostructures and the scope of exploring the correlated physics in moiré superlattices.

Unusual magnetotransport in twisted bilayer graphene

SignificanceWhen two sheets of graphene are twisted to the magic angle of 1.1∘, the resulting flat moiré bands can host exotic correlated electronic states such as superconductivity and ferromagnetism. Here, we show transport properties of a twisted bilayer graphene device at 1.38∘, far enough above the magic angle that we do not expect exotic correlated states. Instead, we see several unusual behaviors in the device's resistivity upon tuning both charge carrier density and perpendicular magnetic field. We can reproduce these behaviors with a surprisingly simple model based on Hofstadter's butterfly. These results shed light on the underlying properties of twisted bilayer graphene.

Exciton Proliferation and Fate of the Topological Mott Insulator in a Twisted Bilayer Graphene Lattice Model

A topological Mott insulator (TMI) with spontaneous time-reversal symmetry breaking and nonzero Chern number has been discovered in a real-space effective model for twisted bilayer graphene (TBG) at 3/4 filling in the strong coupling limit [1]. However, the finite temperature properties of such a TMI state remain illusive. In this work, employing the state-of-the-art thermal tensor network and the perturbative field-theoretical approaches, we obtain the finite-T phase diagram and the dynamical properties of the TBG model. The phase diagram includes the quantum anomalous Hall and charge density wave phases at low T, and an Ising transition separating them from the high-T symmetric phases. Because of the proliferation of excitons-particle-hole bound states-the transitions take place at a significantly reduced temperature than the mean-field estimation. The exciton phase is accompanied with distinctive experimental signatures in such as in charge compressibilities and optical conductivities close to the transition. Our work explains the smearing of the many-electron state topology by proliferating excitons and opens an avenue for controlled many-body investigations on finite-temperature states in the TBG and other quantum moiré systems.

Domain Formation Driven by the Entropy of Topological Edge Modes

In this Letter we study interacting systems with spontaneous discrete symmetry breaking, where the degenerate symmetry-broken states are topologically distinct gapped phases. Edge modes appear at domain walls between the two topological phases. In the presence of a weak disorder field conjugate to the order parameter, we find that the entropy of the edge modes drives a thermal transition between a gapped uniform phase and a phase with a disorder-induced domain structure. We characterize this transition using a phenomenological Landau functional, and corroborate our conclusions with a concrete microscopic model. Finally, we discuss the possibilities of experimental signatures of this phase transition, and propose graphene-based moiré heterostructures as candidate materials in which such a phase transition can be detected.

Global Phase Diagram of the Normal State of Twisted Bilayer Graphene

We investigate the full doping and strain-dependent phase diagram of the normal state of magic-angle twisted bilayer graphene (TBG). Using comprehensive Hartree-Fock calculations, we show that at temperatures where superconductivity is absent the global phase structure can be understood based on the competition and coexistence between three types of intertwined orders: a fully symmetric phase, spatially uniform flavor-symmetry-breaking states, and an incommensurate Kekulé spiral (IKS) order. For small strain, the IKS phase, recently proposed as a candidate order at all nonzero integer fillings of the moiré unit cell, is found to be ubiquitous for noninteger doping as well. We demonstrate that the corresponding electronic compressibility and Fermi surface structure are consistent with the "cascade" physics and Landau fans observed experimentally. This suggests a unified picture of the phase diagram of TBG in terms of IKS order.

Dative Epitaxy of Commensurate Monocrystalline Covalent van der Waals Moiré Supercrystal

Realizing van der Waals (vdW) epitaxy in the 1980s represents a breakthrough that circumvents the stringent lattice matching and processing compatibility requirements in conventional covalent heteroepitaxy. However, due to the weak vdW interactions, there is little control over film qualities by the substrate. Typically, discrete domains with a spread of misorientation angles are formed, limiting the applicability of vdW epitaxy. Here, the epitaxial growth of monocrystalline, covalent Cr₅ Te₈ 2D crystals on monolayer vdW WSe₂ by chemical vapor deposition is reported, driven by interfacial dative bond formation. The lattice of Cr₅ Te₈ , with a lateral dimension of a few tens of micrometers, is fully commensurate with that of WSe₂ via 3 × 3 (Cr₅ Te₈ )/7 × 7 (WSe₂ ) supercell matching, forming a single-crystalline moiré superlattice. This work establishes a conceptually distinct paradigm of thin-film epitaxy, termed "dative epitaxy", which takes full advantage of covalent epitaxy with chemical bonding for fixing the atomic registry and crystal orientation, while circumventing its stringent lattice matching and processing compatibility requirements; conversely, it ensures the full flexibility of vdW epitaxy, while avoiding its poor orientation control. Cr₅ Te₈ 2D crystals grown by dative epitaxy exhibit square magnetic hysteresis, suggesting minimized interfacial defects that can serve as pinning sites.

Spectroscopy signatures of electron correlations in a trilayer graphene/hBN moiré superlattice

ABC-stacked trilayer graphene/hexagonal boron nitride moiré superlattice (TLG/hBN) has emerged as a playground for correlated electron physics. We report spectroscopy measurements of dual-gated TLG/hBN using Fourier transform infrared photocurrent spectroscopy. We observed a strong optical transition between moiré minibands that narrows continuously as a bandgap is opened by gating, indicating a reduction of the single-particle bandwidth. At half-filling of the valence flat band, a broad absorption peak emerges at ~18 milli-electron volts, indicating direct optical excitation across an emerging Mott gap. Similar photocurrent spectra are observed in two other correlated insulating states at quarter- and half-filling of the first conduction band. Our findings provide key parameters of the Hubbard model for the understanding of electron correlation in TLG/hBN.

Engineering Proximity Exchange by Twisting: Reversal of Ferromagnetic and Emergence of Antiferromagnetic Dirac Bands in Graphene/Cr_{2}Ge_{2}Te_{6}

We investigate the twist-angle and gate dependence of the proximity exchange coupling in twisted graphene on monolayer Cr_{2}Ge_{2}Te_{6} from first principles. The proximitized Dirac band dispersions of graphene are fitted to a model Hamiltonian, yielding effective sublattice-resolved proximity-induced exchange parameters (λ_{ex}^{A} and λ_{ex}^{B}) for a series of twist angles between 0° and 30°. For aligned layers (0° twist angle), the exchange coupling of graphene is the same on both sublattices, λ_{ex}^{A}≈λ_{ex}^{B}≈4  meV, while the coupling is reversed at 30° (with λ_{ex}^{A}≈λ_{ex}^{B}≈-4  meV). Remarkably, at 19.1° the induced exchange coupling becomes antiferromagnetic: λ_{ex}^{A}<0, λ_{ex}^{B}>0. Further tuning is provided by a transverse electric field and the interlayer distance. The predicted proximity magnetization reversal and emergence of an antiferromagnetic Dirac dispersion make twisted graphene/Cr_{2}Ge_{2}Te_{6} bilayers a versatile platform for realizing topological phases and for spintronics applications.

Manipulating Exchange Bias in a Van der Waals Ferromagnet

Spintronics applications of thin-film magnets require control and design of specific magnetic properties. Exchange bias, originating from the pinning of spins in a ferromagnet by these of an antiferromagnet, is a part of the highly important elements for spintronics applications. Here, an exchange bias of ≈90 mT in a van der Waals ferromagnet encapsulated by two antiferromagnets at 5 K, the value of which is highly tunable by the field coolings, is reported. The non-antisymmetric dependence of exchange bias on field cooling is explained through considering an uncompensated interfacial magnetic layer of an antiferromagnet with a noncollinear spin texture, and a weak antiferromagnetic order in the oxidized layer, at two ferromagnet/antiferromagnet interfaces. This work opens up new routes toward designing and controlling 2D spintronic devices made of atomically thin van der Waals magnets.

Degradation of Phonons in Disordered Moiré Superlattices

The elastic collective modes of a moiré superlattice arise not from vibrations of a rigid crystal but from the relative displacement between the constituent layers. Despite their similarity to acoustic phonons, these modes, called phasons, are not protected by any conservation law. Here, we show that disorder in the relative orientation between the layers and thermal fluctuations associated with their sliding motion degrade the propagation of sound in the moiré superlattice. Specifically, the phason modes become overdamped at low energies and acquire a finite gap, which displays a universal dependence on the twist-angle variance. Thus, twist-angle inhomogeneity is manifested not only in the noninteracting electronic structure of moiré systems, but also in their phononlike modes. More broadly, our results have important implications for the electronic properties of twisted moiré systems that are sensitive to the electron-phonon coupling.

Nanoscale Control of One-Dimensional Confined States in Strongly Correlated Homojunctions

Construction of lateral junctions is essential to generate one-dimensional (1D) confined potentials that can effectively trap quasiparticles. A series of remarkable electronic phases in one dimension, such as Wigner crystallization, are expected to be realized in such junctions. Here, we demonstrate that we can precisely tune the 1D-confined potential with an in situ manipulation technique, thus providing a dynamic way to modify the correlated electronic states at the junctions. We confirm the existence of 1D-confined potential at the homojunction of two single-layer 1T-NbSe₂ islands. Such potential is structurally sensitive and shows a nonmonotonic function of their interspacing. Moreover, there is a change of electronic properties from the correlated insulator to the generalized 1D Wigner crystallization while the confinement becomes strong. Our findings not only establish the capability to fabricate structures with dynamically tunable properties, but also pave the way toward more exotic correlated systems in low dimensions.

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.

Spin-orbit-driven ferromagnetism at half moiré filling in magic-angle twisted bilayer graphene

Strong electron correlation and spin-orbit coupling (SOC) can have a profound influence on the electronic properties of materials. We examined their combined influence on a two-dimensional electronic system at the atomic interface between magic-angle twisted bilayer graphene and a tungsten diselenide crystal. We found that strong electron correlation within the moiré flatband stabilizes correlated insulating states at both quarter and half filling, and that SOC transforms these Mott-like insulators into ferromagnets, as evidenced by a robust anomalous Hall effect with hysteretic switching behavior. The coupling between spin and valley degrees of freedom could be demonstrated through control of the magnetic order with an in-plane magnetic field or a perpendicular electric field. Our findings establish an experimental knob to engineer topological properties of moiré bands in twisted bilayer graphene and related systems.

Visualization of Charge-Density-Wave Reconstruction and Electronic Superstructure at the Edge of Correlated Insulator 1T-NbSe2

Edges of low-dimensional quantum systems have profound effects on both fundamental research and device functionality. Real-space investigation of the microscopic edge structures and understanding the edge-modulated electronic properties are of great essence. Here we report the nanoscale structural reconstruction at the atomically sharp edge of a charge-density-wave (CDW) correlated insulator 1T-NbSe₂ and the induced electronic properties. We find the CDW unit cells at the edge of single layer (SL) 1T-NbSe₂ evolve from the well-defined CDW order in bulk and spontaneously reconstruct into the quartet along the edge. Moreover, we capture an anomalous electronic superstructure along the edge, the periodicity of which is four times that of ordinary CDW lattice. Our findings provide a way to design the one-dimensional electronic superstructure in 2D quantum materials.

Charge-Density-Wave-Induced Peak-Dip-Hump Structure and the Multiband Superconductivity in a Kagome Superconductor CsV_{3}Sb_{5}

The entanglement of charge density wave (CDW), superconductivity, and topologically nontrivial electronic structure has recently been discovered in the kagome metal AV_{3}Sb_{5} (A=K, Rb, Cs) family. With high-resolution angle-resolved photoemission spectroscopy, we study the electronic properties of CDW and superconductivity in CsV_{3}Sb_{5}. The spectra around K[over ¯] is found to exhibit a peak-dip-hump structure associated with two separate branches of dispersion, demonstrating the isotropic CDW gap opening below E_{F}. The peak-dip-hump line shape is contributed by linearly dispersive Dirac bands in the lower branch and a dispersionless flat band close to E_{F} in the upper branch. The electronic instability via Fermi surface nesting could play a role in determining these CDW-related features. The superconducting gap of ∼0.4  meV is observed on both the electron band around Γ[over ¯] and the flat band around K[over ¯], implying the multiband superconductivity. The finite density of states at E_{F} in the CDW phase is most likely in favor of the emergence of multiband superconductivity, particularly the enhanced density of states associated with the flat band. Our results not only shed light on the controversial origin of the CDW, but also offer insights into the relationship between CDW and superconductivity.

Spin-Polarized Nematic Order, Quantum Valley Hall States, and Field-Tunable Topological Transitions in Twisted Multilayer Graphene Systems

We theoretically study the correlated insulator states, quantum anomalous Hall (QAH) states, and field-induced topological transitions between different correlated states in twisted multilayer graphene systems. Taking twisted bilayer-monolayer graphene and twisted double-bilayer graphene as examples, we show that both systems stay in spin-polarized, C_{3z}-broken insulator states with zero Chern number at 1/2 filling of the flat bands under finite displacement fields. In some cases these spin-polarized, nematic insulator states are in the quantum valley Hall (QVH) phase by virtue of the nontrivial band topology of the systems. The spin-polarized insulator state is quasidegenerate with the valley polarized state if only the dominant intravalley Coulomb interaction is included. Such quasidegeneracy can be split by atomic on-site interactions such that the spin-polarized, nematic state become the unique ground state. Such a scenario applies to various twisted multilayer graphene systems at 1/2 filling, thus can be considered as a universal mechanism. Moreover, under vertical magnetic fields, the orbital Zeeman splittings and the field-induced change of charge density in twisted multilayer graphene systems would compete with the atomic Hubbard interactions, which can drive transitions from spin-polarized zero-Chern-number states to valley-polarized QAH states with small onset magnetic fields.

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.

Interlayer Interactions in 1D Van der Waals Moiré Superlattices

Studying two-dimensional (2D) van der Waals (vdW) moiré superlattices and their interlayer interactions have received surging attention after recent discoveries of many new phases of matter that are highly tunable. Different atomistic registry between layers forming the inner and outer nanotubes can also form one-dimensional (1D) vdW moiré superlattices. In this review, experimental observations and theoretical perspectives related to interlayer interactions in 1D vdW moiré superlattices are summarized. The discussion focuses on double-walled carbon nanotubes (DWNTs), a model 1D vdW moiré system, and the authors highlight the new optical features emerging from the non-trivial strong interlayer coupling effect and the unique physics in 1D DWNTs. Future directions and questions in probing the intriguing physical phenomena in 1D vdW moiré superlattices such as, correlated physics in different 1D moiré systems beyond DWNTs are proposed and discussed.

Science of 2.5 dimensional materials: paradigm shift of materials science toward future social innovation

The past decades of materials science discoveries are the basis of our present society - from the foundation of semiconductor devices to the recent development of internet of things (IoT) technologies. These materials science developments have depended mainly on control of rigid chemical bonds, such as covalent and ionic bonds, in organic molecules and polymers, inorganic crystals and thin films. The recent discovery of graphene and other two-dimensional (2D) materials offers a novel approach to synthesizing materials by controlling their weak out-of-plane van der Waals (vdW) interactions. Artificial stacks of different types of 2D materials are a novel concept in materials synthesis, with the stacks not limited by rigid chemical bonds nor by lattice constants. This offers plenty of opportunities to explore new physics, chemistry, and engineering. An often-overlooked characteristic of vdW stacks is the well-defined 2D nanospace between the layers, which provides unique physical phenomena and a rich field for synthesis of novel materials. Applying the science of intercalation compounds to 2D materials provides new insights and expectations about the use of the vdW nanospace. We call this nascent field of science '2.5 dimensional (2.5D) materials,' to acknowledge the important extra degree of freedom beyond 2D materials. 2.5D materials not only offer a new field of scientific research, but also contribute to the development of practical applications, and will lead to future social innovation. In this paper, we introduce the new scientific concept of this science of '2.5D materials' and review recent research developments based on this new scientific concept.

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.

Cascades between Light and Heavy Fermions in the Normal State of Magic-Angle Twisted Bilayer Graphene

We present a framework for understanding the cascade transitions and the Landau level degeneracies of twisted bilayer graphene. The Coulomb interaction projected onto narrow bands causes the charged excitations at an integer filling to disperse, forming new bands. If the excitation moves the filling away from the charge neutrality point, then it has a band minimum at the moiré Brillouin zone center with a small mass that compares well with the experiment; if towards the charge neutrality point, then it has a much larger mass and a higher degeneracy. At a nonzero density away from an integer filling the excitations interact. The system on the small mass side has a large bandwidth and forms a Fermi liquid. On the large mass side the bandwidth is narrow, the compressibility is negative and the Fermi liquid is likely unstable. This explains the observed sawtooth features in compressibility, the Landau fans pointing away from charge neutrality and their degeneracies. The framework sets the stage for superconductivity at lower temperatures.

Oscillations of the Spacing between van Hove Singularities Induced by sub-Ångstrom Fluctuations of Interlayer Spacing in Graphene Superlattices

Physical properties of two-dimensional van der Waals (vdWs) structures depend sensitively on both stacking orders and interlayer interactions. Yet, in most cases studied to date, the interlayer interaction is considered to be a "static" property of the vdWs structures. Here we demonstrate that applying a scanning tunneling microscopy (STM) tip pulse on twisted bilayer graphene (TBG) can induce sub-Ångstrom fluctuations of the interlayer separation in the TBG, which are equivalent to dynamic vertical external pressure of about 10 GPa on the TBG. The sub-Ångstrom fluctuations of the interlayer separation result in large oscillations of the energy separations between two van Hove singularities (VHSs) in the TBG. The period of the oscillations of the VHSs spacing is extremely long, about 500-1000 sec, attributing to tip-induced local stress in the atomic-thick TBG. Our result provides an efficient method to tune and measure the physical properties of the vdWs structures dynamically.

Theory of Correlated Insulators and Superconductivity in Twisted Bilayer Graphene

We introduce and analyze a model that sheds light on the interplay between correlated insulating states, superconductivity, and flavor-symmetry breaking in magic angle twisted bilayer graphene. Using a variational mean-field theory, we determine the normal-state phase diagram of our model as a function of the band filling. The model features robust insulators at even integer fillings, occasional weaker insulators at odd integer fillings, and a pattern of flavor-symmetry breaking at noninteger fillings. Adding a phonon-mediated intervalley retarded attractive interaction, we obtain strong-coupling superconducting domes, whose structure is in qualitative agreement with experiments. Our model elucidates how the intricate form of the interactions and the particle-hole asymmetry of the electronic structure determine the phase diagram. It also explains how subtle differences between devices may lead to the different behaviors observed experimentally. A similar model can be applied with minor modifications to other moiré systems, such as twisted trilayer graphene.

Correlated states in doubly-aligned hBN/graphene/hBN heterostructures

Interfacial moiré superlattices in van der Waals vertical assemblies effectively reconstruct the crystal symmetry, leading to opportunities for investigating exotic quantum states. Notably, a two-dimensional nanosheet has top and bottom open surfaces, allowing the specific case of doubly aligned super-moiré lattice to serve as a toy model for studying the tunable lattice symmetry and the complexity of related electronic structures. Here, we show that by doubly aligning a graphene monolayer to both top and bottom encapsulating hexagonal boron nitride (h-BN), multiple conductivity minima are observed away from the main Dirac point, which are sensitively tunable with respect to the small twist angles. Moreover, our experimental evidences together with theoretical calculations suggest correlated insulating states at integer fillings of -5, -6, -7 electrons per moiré unit cell, possibly due to inter-valley coherence. Our results provide a way to construct intriguing correlations in 2D electronic systems in the weak interaction regime.

Detection of graphene's divergent orbital diamagnetism at the Dirac point

[Figure: see text].

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.

Exact Landau Level Description of Geometry and Interaction in a Flatband

Flatbands appear in many condensed matter systems, including the two-dimensional electron gas in a high magnetic field, correlated materials, and moiré heterostructures. They are characterized by intrinsic geometric properties such as the Berry curvature and Fubini-Study metric. The influence of the band geometry on electron-electron interaction is difficult to understand analytically because the geometry is in general nonuniform in momentum space. In this work, we study the topological flatband of Chern number C=1 with a momentum-dependent but positive definite Berry curvature that fluctuates in sync with Fubini-Study metric. We derive an exact correspondence between such ideal flatbands and Landau levels and show that the band geometry fluctuation gives rise to a new type of interaction in the corresponding Landau levels that depends on the center of mass of two particles. We characterize such interactions by generalizing the usual Haldane pseudopotentials. This mapping gives exact zero-energy ground states for short-ranged repulsive generalized pseudopotentials in flatbands, in analogy to fractional quantum Hall systems. Driving the center-of-mass interactions beyond the repulsive regime leads to a dramatic reconstruction of the ground states towards gapless phases. The generalized pseudopotential could be a useful basis for future numerical studies.

Evidence for unconventional superconductivity in twisted bilayer graphene

The emergence of superconductivity and correlated insulators in magic-angle twisted bilayer graphene (MATBG) has raised the intriguing possibility that its pairing mechanism is distinct from that of conventional superconductors^1-4, as described by the Bardeen-Cooper-Schrieffer (BCS) theory. However, recent studies have shown that superconductivity persists even when Coulomb interactions are partially screened^5,6. This suggests that pairing in MATBG might be conventional in nature and a consequence of the large density of states of its flat bands. Here we combine tunnelling and Andreev reflection spectroscopy with a scanning tunnelling microscope to observe several key experimental signatures of unconventional superconductivity in MATBG. We show that the tunnelling spectra below the transition temperature T_c are inconsistent with those of a conventional s-wave superconductor, but rather resemble those of a nodal superconductor with an anisotropic pairing mechanism. We observe a large discrepancy between the tunnelling gap Δ_T, which far exceeds the mean-field BCS ratio (with 2Δ_T/k_BT_c ~ 25), and the gap Δ_AR extracted from Andreev reflection spectroscopy (2Δ_AR/k_BT_c ~ 6). The tunnelling gap persists even when superconductivity is suppressed, indicating its emergence from a pseudogap phase. Moreover, the pseudogap and superconductivity are both absent when MATBG is aligned with hexagonal boron nitride. These findings and other observations reported here provide a preponderance of evidence for a non-BCS mechanism for superconductivity in MATBG.

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.

Fractional Chern insulators in magic-angle twisted bilayer graphene

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states that may provide a new avenue towards manipulating non-Abelian excitations. Early theoretical studies1-7 have predicted their existence in systems with flat Chern bands and highlighted the critical role of a particular quantum geometry. However, FCI states have been observed only in Bernal-stacked bilayer graphene (BLG) aligned with hexagonal boron nitride (hBN)8, in which a very large magnetic field is responsible for the existence of the Chern bands, precluding the realization of FCIs at zero field. By contrast, magic-angle twisted BLG9-12 supports flat Chern bands at zero magnetic field13-17, and therefore offers a promising route towards stabilizing zero-field FCIs. Here we report the observation of eight FCI states at low magnetic field in magic-angle twisted BLG enabled by high-resolution local compressibility measurements. The first of these states emerge at 5 T, and their appearance is accompanied by the simultaneous disappearance of nearby topologically trivial charge density wave states. We demonstrate that, unlike the case of the BLG/hBN platform, the principal role of the weak magnetic field is merely to redistribute the Berry curvature of the native Chern bands and thereby realize a quantum geometry favourable for the emergence of FCIs. Our findings strongly suggest that FCIs may be realized at zero magnetic field and pave the way for the exploration and manipulation of anyonic excitations in flat moiré Chern bands.

Unraveling Strain Gradient Induced Electromechanical Coupling in Twisted Double Bilayer Graphene Moiré Superlattices

Moiré superlattices of 2D materials with a small twist angle are thought to exhibit appreciable flexoelectric effect, though unambiguous confirmation of their flexoelectricity is challenging due to artifacts associated with commonly used piezoresponse force microscopy (PFM). For example, unexpectedly small phase contrast (≈8°) between opposite flexoelectric polarizations is reported in twisted bilayer graphene (tBG), though theoretically predicted value is 180°. Here a methodology is developed to extract intrinsic moiré flexoelectricity using twisted double bilayer graphene (tDBG) as a model system, probed by lateral PFM. For small twist angle samples, it is found that a vectorial decomposition is essential to recover the small intrinsic flexoelectric response at domain walls from a large background signal. The obtained threefold symmetry of commensurate domains with significant flexoelectric response at domain walls is fully consistent with the theoretical calculations. Incommensurate domains in tDBG with relatively large twist angles can also be observed by this technique. A general strategy is provided here for unraveling intrinsic flexoelectricity in van der Waals moiré superlattices while providing insights into engineered symmetry breaking in centrosymmetric materials.

Putting a twist on spintronics

[Figure: see text].

Correlation-Induced Triplet Pairing Superconductivity in Graphene-Based Moiré Systems

Motivated by the possible non-spin-singlet superconductivity in the magic-angle twisted trilayer graphene experiment, we investigate the triplet-pairing superconductivity arising from a correlation-induced spin-fermion model of Dirac fermions with spin, valley, and sublattice degrees of freedom. We find that the f-wave pairing is favored due to the valley-sublattice structure, and the superconducting state is time-reversal symmetric, fully gapped, and nontopological. With a small in-plane magnetic field, the superconducting state becomes partially polarized, and the transition temperature can be slightly enhanced. Our results apply qualitatively to Dirac fermions for the triplet-pairing superconductivity in graphene-based moiré systems, which is fundamentally distinct from triplet superconductivity in ^{3}He and ferromagnetic superconductors.

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.

Competing Zero-Field Chern Insulators in Superconducting Twisted Bilayer Graphene

The discovery of magic angle twisted bilayer graphene has unveiled a rich variety of superconducting, magnetic, and topologically nontrivial phases. Here, we show that the zero-field states at odd integer filling factors in h-BN nonaligned devices are consistent with symmetry broken Chern insulators, as is evidenced by the observation of the anomalous Hall effect near moiré cell filling factor ν=+1. The corresponding Chern insulator has a Chern number C=±1 and a relatively high Curie temperature of T_{c}≈4.5  K. In a perpendicular magnetic field above B>0.5  T we observe a transition of the ν=+1 Chern insulator from Chern number C=±1 to C=3, characterized by a quantized Hall plateau with R_{yx}=h/3e^{2}. These observations demonstrate that interaction-induced symmetry breaking leads to zero-field ground states that include almost degenerate and closely competing Chern insulators, and that states with larger Chern numbers couple most strongly to the B field. In addition, the device reveals strong superconducting phases with critical temperatures of up to T_{c}≈3.5  K. By providing the first demonstration of a system that allows gate-induced transitions between magnetic and superconducting phases, our observations mark a major milestone in the creation of a new generation of quantum electronics.

Weak-Field Hall Resistivity and Spin-Valley Flavor Symmetry Breaking in Magic-Angle Twisted Bilayer Graphene

Near a magic twist angle, the lowest energy conduction and valence bands of bilayer graphene moiré superlattices become extremely narrow. The band dispersion that remains is sensitive to the moiré's strain pattern, nonlocal tunneling between layers, and filling-factor-dependent Hartree and exchange band renormalizations. In this Letter, we analyze the influence of these band-structure details on the pattern of flavor symmetry breaking observed in this narrow band system and on the associated pattern of Fermi surface reconstructions revealed by weak-field Hall and Shubnikov-de Haas magnetotransport measurements.

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.

Acoustic-Phonon-Mediated Superconductivity in Rhombohedral Trilayer Graphene

Motivated by the observation of two distinct superconducting phases in the moiréless ABC-stacked rhombohedral trilayer graphene, we investigate the electron-acoustic-phonon coupling as a possible pairing mechanism. We predict the existence of superconductivity with the highest T_{c}∼3  K near the Van Hove singularity. Away from the Van Hove singularity, T_{c} remains finite in a wide range of doping. In our model, the s-wave spin-singlet and f-wave spin-triplet pairings yield the same T_{c}, while other pairing states have negligible T_{c}. Our theory provides a simple explanation for the two distinct superconducting phases in the experiment and suggests that superconductivity and other interaction-driven phases (e.g., ferromagnetism) can have different origins.

Tailoring the Band Structure of Twisted Double Bilayer Graphene with Pressure

Twisted two-dimensional structures open new possibilities in band structure engineering. At magic twist angles, flat bands emerge, which gave a new drive to the field of strongly correlated physics. In twisted double bilayer graphene dual gating allows changing of the Fermi level and hence the electron density and also allows tuning of the interlayer potential, giving further control over band gaps. Here, we demonstrate that by application of hydrostatic pressure, an additional control of the band structure becomes possible due to the change of tunnel couplings between the layers. We find that the flat bands and the gaps separating them can be drastically changed by pressures up to 2 GPa, in good agreement with our theoretical simulations. Furthermore, our measurements suggest that in finite magnetic field due to pressure a topologically nontrivial band gap opens at the charge neutrality point at zero displacement field.

Magnetic Topological Insulator Heterostructures: A Review

Topological insulators (TIs) provide intriguing prospects for the future of spintronics due to their large spin-orbit coupling and dissipationless, counter-propagating conduction channels in the surface state. The combination of topological properties and magnetic order can lead to new quantum states including the quantum anomalous Hall effect that was first experimentally realized in Cr-doped (Bi,Sb)₂ Te₃ films. Since magnetic doping can introduce detrimental effects, requiring very low operational temperatures, alternative approaches are explored. Proximity coupling to magnetically ordered systems is an obvious option, with the prospect to raise the temperature for observing the various quantum effects. Here, an overview of proximity coupling and interfacial effects in TI heterostructures is presented, which provides a versatile materials platform for tuning the magnetic and topological properties of these exciting materials. An introduction is first given to the heterostructure growth by molecular beam epitaxy and suitable structural, electronic, and magnetic characterization techniques. Going beyond transition-metal-doped and undoped TI heterostructures, examples of heterostructures are discussed, including rare-earth-doped TIs, magnetic insulators, and antiferromagnets, which lead to exotic phenomena such as skyrmions and exchange bias. Finally, an outlook on novel heterostructures such as intrinsic magnetic TIs and systems including 2D materials is given.

Dichotomy of Electron-Phonon Coupling in Graphene Moiré Flat Bands

Graphene moiré superlattices are outstanding platforms to study correlated electron physics and superconductivity with exceptional tunability. However, robust superconductivity has been measured only in magic-angle twisted bilayer graphene (MA-TBG) and magic-angle twisted trilayer graphene (MA-TTG). The absence of a superconducting phase in certain moiré flat bands raises a question on the superconducting mechanism. In this work, we investigate electronic structure and electron-phonon coupling in graphene moiré superlattices based on atomistic calculations. We show that electron-phonon coupling strength λ is dramatically different among graphene moiré flat bands. The total strength λ is very large (λ>1) for MA-TBG and MA-TTG, both of which display robust superconductivity in experiments. However, λ is an order of magnitude smaller in twisted double bilayer graphene (TDBG) and twisted monolayer-bilayer graphene (TMBG) where superconductivity is reportedly rather weak or absent. We find that the Bernal-stacked layers in TDBG and TMBG induce sublattice polarization in the flat-band states, suppressing intersublattice electron-phonon matrix elements. We also obtain the nonadiabatic superconducting transition temperature T_{c} that matches well with the experimental results. Our results clearly show a correlation between strong electron-phonon coupling and experimental observations of robust superconductivity.

Quantum anomalous Hall octet driven by orbital magnetism in bilayer graphene

The quantum anomalous Hall (QAH) effect-a macroscopic manifestation of chiral band topology at zero magnetic field-has been experimentally realized only by the magnetic doping of topological insulators^1-3 and the delicate design of moiré heterostructures^4-8. However, the seemingly simple bilayer graphene without magnetic doping or moiré engineering has long been predicted to host competing ordered states with QAH effects^9-11. Here we explore states in bilayer graphene with a conductance of 2 e² h^-1 (where e is the electronic charge and h is Planck's constant) that not only survive down to anomalously small magnetic fields and up to temperatures of five kelvin but also exhibit magnetic hysteresis. Together, the experimental signatures provide compelling evidence for orbital-magnetism-driven QAH behaviour that is tunable via electric and magnetic fields as well as carrier sign. The observed octet of QAH phases is distinct from previous observations owing to its peculiar ferrimagnetic and ferrielectric order that is characterized by quantized anomalous charge, spin, valley and spin-valley Hall behaviour⁹.

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.