Gate-Tunable Topological Flat Bands in Trilayer Graphene Boron-Nitride Moiré Superlattices

Orbital Hall conductivity in a Graphene Haldane and Haldane Haldane bilayers

We investigate the orbital Hall conductivity in bilayer graphene (G/G) by modifying one or both the layers as Haldane type ([Formula: see text] : Graphene/Haldane and [Formula: see text] : Haldane/Haldane) with the inclusion of next nearest neighbour (NNN) hopping strength ([Formula: see text]) and flux (ϕ). It is observed that the low energy bands of [Formula: see text] and [Formula: see text] are isolated with a gap at charge neutrality with the next nearest neighbour (NNN) hopping term [Formula: see text]. The time reversal (TR) symmetry breaking with [Formula: see text] induces a large orbital magnetic moment ([Formula: see text]) for the [Formula: see text] band in [Formula: see text] and [Formula: see text] bilayers. This TR symmetry breaking, modulated by the [Formula: see text] strength, leads to the emergence of Orbital Ferromagnetism and Valley Orbital Magnetism within the BZ for the Haldane single layer as well for both [Formula: see text] and [Formula: see text]. We show that for the applied longitudinal electric fields, the intrinsic angular momentum ([Formula: see text]) gives the orbital current ([Formula: see text]) along a transverse direction and generates the orbital Hall conductivity (OHC). We further show that the orbital magnetic polarity leads the Haldane single layer to Orbital Chern Insulator. Interestingly, the orbital Hall conductivities are finite and exhibit a large plateau in the gap over the occupied bands. Moreover, the accumulation of orbital magnetic moment of the bands in Haldane graphene bilayer shows Orbital Hall Insulator and Orbital Chern Insulators with large plateaus. Similarly, we show that in the hetero-bilayers, one of the layers of the Haldane type generates the orbital magnetism and induces the OHC. We conclude that the isolated bands in Haldane graphene bilayers with external stimuli are of an orbital nature and have various orbital Hall phases.

Electric Field-Defined Superlattices in Bilayer Graphene: Formation of Topological Bands in Two Dimensions

An electric field applied to the Bernal-stacked bilayer graphene opens an energy gap; its reversal in some regions creates domain walls and leads to the appearance of one-dimensional chiral gapless states localized at the walls. Here, we investigate the energy structure of bilayer graphene with superlattice potential defined by an external electric field. The calculations are performed within an atomistic π-electron tight-binding approximation. We study one-dimensional and two-dimensional superlattices formed by arrays of electric-field walls in the zigzag and armchair directions and investigate different field polarizations. Chiral gapless states discretize due to the superlattice potential and transform into minibands in the energy gap. As the main result, we show that the minibands can cross at the Fermi level for some field polarizations. This leads to a new kind of two-dimensional gapless states of topological character that form Dirac-like cones at the crossing points. This also has application potential: changing the field polarization can close the energy gap and change the character of the superlattice from semiconducting to metallic.

Emergent Symmetry and Valley Chern Insulator in Twisted Double-Bilayer Graphene

Theoretical calculations show that twisted double bilayer graphene (TDBG) under a transverse electric field develops a valley Chern number 2 at charge neutrality. Using thermodynamic and thermal activation measurements we report the experimental observation of a universal closing of the charge neutrality gap in the Hofstadter spectrum of TDBG at 1/2 magnetic flux per unit cell, in agreement with theoretical predictions for a valley Chern number 2 gap. Our theoretical analysis of the experimental data shows that the interaction energy, while larger than the flat-band bandwidth in TDBG near 1° does not alter the emergent valley symmetry or the single-particle band topology.

Spontaneous emergence of straintronics effects and striped stacking domains in untwisted three-layer epitaxial graphene

Emergent electronic phenomena, from superconductivity to ferroelectricity, magnetism, and correlated many-body band gaps, have been observed in domains created by stacking and twisting atomic layers of Van der Waals materials. In graphene, emergent properties have been observed in ABC stacking domains obtained by exfoliation followed by expert mechanical twisting and alignment with the desired orientation, a process very challenging and nonscalable. Here, conductive atomic force microscopy shows in untwisted epitaxial graphene grown on SiC the surprising presence of striped domains with dissimilar conductance, a contrast that demonstrates the presence of ABA and ABC domains since it matches exactly the conductivity difference observed in ABA/ABC domains in twisted exfoliated graphene and calculated by density functional theory. The size and geometry of the stacking domains depend on the interplay between strain, solitons crossing, and shape of the three-layer regions. Interestingly, we demonstrate the growth of three-layer regions in which the ABA/ABC stacking domains self-organize in stable stripes of a few tens of nanometers. The growth-controlled production of isolated and stripe-shaped ABA/ABC domains open the path to fabricate quantum devices on these domains. These findings on self-assembly formation of ABA/ABC epitaxial graphene stripes on SiC without the need of time-consuming and nonscalable graphene exfoliation, alignment, and twisting provide different potential applications of graphene in electronic devices.

Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene

In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to an intimate connection between valley and layer degrees of freedom. Adding a moiré superlattice potential enriches the BBG physics with the formation of topological minibands - potentially leading to tunable exotic quantum transport. Here, we present magnetotransport measurements of a high-quality bilayer graphene-hexagonal boron nitride (hBN) heterostructure. The zero-degree alignment generates a strong moiré superlattice potential for the electrons in BBG and the resulting Landau fan diagram of longitudinal and Hall resistance displays a Hofstadter butterfly pattern with a high level of detail. We demonstrate that the intricate relationship between valley and layer degrees of freedom controls the topology of moiré-induced bands, significantly influencing the energetics of interacting quantum phases in the BBG superlattice. We further observe signatures of field-induced correlated insulators, helical edge states and clear quantizations of interaction-driven topological quantum phases, such as symmetry broken Chern insulators.

Twist Proximity-Endowed Large Figure of Merit in a Band-Modulated CrI3/1T-MoS2 Moiré Superlattice

Moiré superlattices with a robust twist proximity effect in the low-dimensional regime can facilitate nanoscale thermoelectric devices. In pristine systems, the low efficiency and lack of proficient control of thermoelectric properties impede desirable advancements in the field of energy conversion. In the present study, we demonstrate remarkable macroscopic thermoelectric response as a consequence of microscopic band structure modulation via the twist proximity in an engineered CrI3/1T-MoS2 moiré superlattice. The local twist effect, which leads to the microscopic phenomena of electron localization, results in a comprehensive electronic structure modulation. Consequently, these local effects convolute the macroscopic thermoelectric effect. Additionally, flat bands and angle-dependent metallic to semiconducting transitions are observed at 10.89, 23.41, and 30° twist angles. We correlate the observed phenomenon with the augmented spin-charge transport and interconversion via the twist proximity effect in its semiconducting phase. The estimated ultralow electronic and lattice thermal conductivities further corroborate with the observed large figure of merit and Seebeck coefficient. The maximum values of the Seebeck coefficient and figure of merit are estimated to be ∼413 μV/K and ∼4.3 at 200 K for 30° under the constant time relaxation approach. The twist-endowed outstanding thermoelectric effect in moiré superlattices with band modulation unveils a distinctive approach to establish efficient thermoelectric devices.

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.

Engineering the Band Topology in a Rhombohedral Trilayer Graphene Moiré Superlattice

Moiré superlattices have become a fertile playground for topological Chern insulators, where the displacement field can tune the quantum geometry and Chern number of the topological band. However, in experiments, displacement field engineering of spontaneous symmetry-breaking Chern bands has not been demonstrated. Here in a rhombohedral trilayer graphene moiré superlattice, we use a thermodynamic probe and transport measurement to monitor the Chern number evolution as a function of the displacement field. At a quarter filling of the moiré band, a novel Chern number of three is unveiled to compete with the well-established number of two upon turning on the electric field and survives when the displacement field is sufficiently strong. The transition can be reconciled by a nematic instability on the Fermi surface due to the pseudomagnetic vector field potentials associated with moiré strain patterns. Our work opens more opportunities to active control of Chern numbers in van der Waals moiré systems.

Ultra-flat bands at large twist angles in group-V twisted bilayer materials

Flat bands in 2D twisted materials are key to the realization of correlation-related exotic phenomena. However, a flat band often was achieved in the large system with a very small twist angle, which enormously increases the computational and experimental complexity. In this work, we proposed group-V twisted bilayer materials, including P, As, and Sb in the β phase with large twist angles. The band structure of twisted bilayer materials up to 2524 atoms has been investigated by a deep learning method DeepH, which significantly reduces the computational time. Our results show that the bandgap and the flat bandwidth of twisted bilayer β-P, β-As, and β-Sb reduce gradually with the decreasing of twist angle, and the ultra-flat band with bandwidth approaching 0 eV is achieved. Interestingly, we found that a twist angle of 9.43° is sufficient to achieve the band flatness for β-As comparable to that of twist bilayer graphene at the magic angle of 1.08°. Moreover, we also find that the bandgap reduces with decreasing interlayer distance while the flat band is still preserved, which suggests interlayer distance as an effective routine to tune the bandgap of flat band systems. Our research provides a feasible platform for exploring physical phenomena related to flat bands in twisted layered 2D materials.

Metal-Semiconductor Behavior along the Line of Stacking Order Change in Gated Multilayer Graphene

We investigated gated multilayer graphene with stacking order changes along the armchair direction. We consider that some layers cracked to release shear strain at the stacking domain wall. The energy cones of graphene overlap along the corresponding direction in the k-space, so the topological gapless states from different valleys also overlap. However, these states strongly interact and split due to atomic-scale defects caused by the broken layers, yielding an effective energy gap. We find that for some gate voltages, the gap states cross and the metallic behavior along the stacking domain wall can be restored. In particular cases, a flat band appears at the Fermi energy. We show that for small variations in the gate voltage, the charge occupying this band oscillates between the outer layers.

Stacking Order Engineering of Two-Dimensional Materials and Device Applications

Stacking orders in 2D van der Waals (vdW) materials dictate the relative sliding (lateral displacement) and twisting (rotation) between atomically thin layers. By altering the stacking order, many new ferroic, strongly correlated and topological orderings emerge with exotic electrical, optical and magnetic properties. Thanks to the weak vdW interlayer bonding, such highly flexible and energy-efficient stacking order engineering has transformed the design of quantum properties in 2D vdW materials, unleashing the potential for miniaturized high-performance device applications in electronics, spintronics, photonics, and surface chemistry. This Review provides a comprehensive overview of stacking order engineering in 2D vdW materials and their device applications, ranging from the typical fabrication and characterization methods to the novel physical properties and the emergent slidetronics and twistronics device prototyping. The main emphasis is on the critical role of stacking orders affecting the interlayer charge transfer, orbital coupling and flat band formation for the design of innovative materials with on-demand quantum properties and surface potentials. By demonstrating a correlation between the stacking configurations and device functionality, we highlight their implications for next-generation electronic, photonic and chemical energy conversion devices. We conclude with our perspective of this exciting field including challenges and opportunities for future stacking order engineering research.

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.

Emerging Characteristics and Properties of Moiré Materials

In recent years, scientists have conducted extensive research on Moiré materials and have discovered some compelling properties. The Moiré superlattice allows superconductivity through flat-band and strong correlation effects. The presence of flat bands causes the Moiré material to exhibit topological properties as well. Modulating electronic interactions with magnetic fields in Moiré materials enables the fractional quantum Hall effect. In addition, Moiré materials have ferromagnetic and antiferromagnetic properties. By tuning the interlayer coupling and spin interactions of the Moiré superlattice, different magnetic properties can be achieved. Finally, this review also discusses the applications of Moiré materials in the fields of photocurrent, superconductivity, and thermoelectricity. Overall, Moiré superlattices provide a new dimension in the development of two-dimensional materials.

Chiral Decomposition of Twisted Graphene Multilayers with Arbitrary Stacking

We formulate the chiral decomposition rules that govern the electronic structure of a broad family of twisted N + M multilayer graphene configurations that combine arbitrary stacking order and a mutual twist. We show that at the magic angle in the chiral limit the low-energy bands of such systems are composed of chiral pseudospin doublets that are energetically entangled with two flat bands per valley induced by the moiré superlattice potential. The analytic construction is supported by explicit numerical calculations based on realistic parametrization. We further show that vertical displacement fields can open energy gaps between the pseudospin doublets and the two flat bands, such that the flat bands may carry nonzero valley Chern numbers. These results provide guidelines for the rational design of topological and correlated states in generic twisted graphene multilayers.

Strong gate-tunability of flat bands in bilayer graphene due to moiré encapsulation between hBN monolayers

When using hexagonal boron-nitride (hBN) as a substrate for graphene, the resulting moiré pattern creates secondary Dirac points. By encapsulating a multilayer graphene within aligned hBN sheets the controlled moiré stacking may offer even richer benefits. Using advanced tight-binding simulations on atomistically-relaxed heterostructures, here we show that the gap at the secondary Dirac point can be opened in selected moiré-stacking configurations, and is independent of any additional vertical gating of the heterostructure. On the other hand, gating can broadly tune the gap at the principal Dirac point, and may thereby strongly compress the first moiré mini-band in width against the moiré-induced gap at the secondary Dirac point. We reveal that in hBN-encapsulated bilayer graphene this novel mechanism can lead to isolated bands flatter than 10 meV under moderate gating, hence presenting a convenient pathway towards electronically-controlled strongly-correlated states on demand.

Two Dimensional Heterostructures for Optoelectronics: Current Status and Future Perspective

Researchers have found various families of two-dimensional (2D) materials and associated heterostructures through detailed theoretical work and experimental efforts. Such primitive studies provide a framework to investigate novel physical/chemical characteristics and technological aspects from micro to nano and pico scale. Two-dimensional van der Waals (vdW) materials and their heterostructures can be obtained to enable high-frequency broadband through a sophisticated combination of stacking order, orientation, and interlayer interactions. These heterostructures have been the focus of much recent research due to their potential applications in optoelectronics. Growing the layers of one kind of 2D material over the other, controlling absorption spectra via external bias, and external doping proposes an additional degree of freedom to modulate the properties of such materials. This mini review focuses on current state-of-the-art material design, manufacturing techniques, and strategies to design novel heterostructures. In addition to a discussion of fabrication techniques, it includes a comprehensive analysis of the electrical and optical properties of vdW heterostructures (vdWHs), particularly emphasizing the energy-band alignment. In the following sections, we discuss specific optoelectronic devices, such as light-emitting diodes (LEDs), photovoltaics, acoustic cavities, and biomedical photodetectors. Furthermore, this also includes a discussion of four different 2D-based photodetector configurations according to their stacking order. Moreover, we discuss the challenges that remain to be addressed in order to realize the full potential of these materials for optoelectronics applications. Finally, as future perspectives, we present some key directions and express our subjective assessment of upcoming trends in the field.

Spontaneous time-reversal symmetry breaking in twisted double bilayer graphene

Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayer graphene sheets with a twist between them. Gate voltages applied to top and back gates of a tDBG device tune both the flatness and topology of the electronic bands, enabling an unusual level of experimental control. Metallic states with broken spin and valley symmetries have been observed in tDBG devices with twist angles in the range 1.2–1.3°, but the topologies and order parameters of these states have remained unclear. We report the observation of an anomalous Hall effect in the correlated metal state of tDBG, with hysteresis loops spanning hundreds of mT in out-of-plane magnetic field (B_⊥) that demonstrate spontaneously broken time-reversal symmetry. The B_⊥ hysteresis persists for in-plane fields up to several Tesla, suggesting valley (orbital) ferromagnetism. At the same time, the resistivity is strongly affected by even mT-scale values of in-plane magnetic field, pointing to spin-valley coupling or to a direct orbital coupling between in-plane field and the valley degree of freedom.

Twisted double bilayer graphene (tDBG) comprises two Bernal-stacked bilayer graphene sheets with a twist between them. Here, the authors report a strong anomalous Hall effect in the correlated-metal regime of tDBG, indicating time reversal symmetry breaking from orbital ferromagnetism, likely associated with valley polarization.

Efficient Optical Modulation of Exciton State Population in Monolayer MoS2 at Room Temperature

The modulation of exciton energy and state density of layer-structured transition metal dichalcogenides (TMDs) is required for diverse optoelectronic device applications. Here, the spontaneous inversion of exciton state population in monolayer MoS₂ is observed by turning the pump light power. The excitons prefer to exist in low energy state under low pump power, but reverse under high pump power. To discuss the mechanism in depth, we propose a semiclassical model by combining the rate equation and photo-exciton interaction. Considering the modifying of exciton-exciton annihilation, the spontaneous inversion of exciton state population is phenomenologically described.

Ultra-strong spin-orbit coupling and topological moiré engineering in twisted ZrS2 bilayers

We predict that twisted bilayers of 1T-ZrS2 realize a novel and tunable platform to engineer two-dimensional topological quantum phases dominated by strong spin-orbit interactions. At small twist angles, ZrS2 heterostructures give rise to an emergent and twist-controlled moiré Kagome lattice, combining geometric frustration and strong spin-orbit coupling to give rise to a moiré quantum spin Hall insulator with highly controllable and nearly-dispersionless bands. We devise a generic pseudo-spin theory for group-IV transition metal dichalcogenides that relies on the two-component character of the valence band maximum of the 1T structure at Γ, and study the emergence of a robust quantum anomalous Hall phase as well as possible fractional Chern insulating states from strong Coulomb repulsion at fractional fillings of the topological moiré Kagome bands. Our results establish group-IV transition metal dichalcogenide bilayers as a novel moiré platform to realize strongly-correlated topological phases in a twist-tunable setting.

Influence of Hexagonal Boron Nitride on Electronic Structure of Graphene

By performing first-principles calculations, we studied hexagonal-boron-nitride (hBN)-supported graphene, in which moiré structures are formed due to lattice mismatch or interlayer rotation. A series of graphene/hBN systems has been studied to reveal the evolution of properties with respect to different twisting angles (21.78°, 13.1°, 9.43°, 7.34°, 5.1°, and 3.48°). Although AA- and AB-stacked graphene/hBN are gapped at the Dirac point by about 50 meV, the energy gap of the moiré graphene/hBN, which is much more asymmetric, is only about several meV. Although the Dirac cone of graphene residing in the wide gap of hBN is not much affected, the calculated Fermi velocity is found to decrease with the increase in the moiré super lattice constant due to charge transfer. The periodic potential imposed by hBN modulated charge distributions in graphene, leading to the shift of graphene bands. In agreement with experiments, there are dips in the calculated density of states, which get closer and closer to the Fermi energy as the moiré lattice grows larger.

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.

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.

Electronic Properties of Oxidized Graphene: Effects of Strain and an Electric Field on Flat Bands and the Energy Gap

A multiscale modeling and simulation approach, including first-principles calculations, ab initio molecular dynamics simulations, and a tight binding approach, is employed to study band flattening of the electronic band structure of oxidized monolayer graphene. The width of flat bands can be tuned by strain, the external electric field, and the density of functional groups and their distribution. A transition to a conducting state is found for monolayer graphene with impurities when it is subjected to an electric field of ∼1.0 V/Å. Several parallel impurity-induced flat bands appear in the low-energy spectrum of monolayer graphene when the number of epoxy groups is changed. The width of the flat band decreases with an increase in tensile strain but is independent of the electric field strength. Here an alternative and easy route for obtaining band flattening in thermodynamically stable functionalized monolayer graphene is introduced. Our work discloses a new avenue for research on band flattening in monolayer graphene.

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.

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.

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

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

Tunable conductance and spin filtering in twisted bilayer copper phthalocyanine molecular devices

We investigate theoretically the quantum transport properties of a twisted bilayer copper phthalocyanine (CuPc) molecular device, in which the bottom-layer CuPc molecule is connected to V-shaped zigzag-edged graphene nanoribbon electrodes. Based on a non-equilibrium Green's function approach in combination with density-functional theory, we find that the twist angle effectively modulates the electron interaction between the bilayer CuPc molecules. HOMO (highest occupied molecular orbital)-LUMO (lowest unoccupied molecular orbital) gap, spin filtering efficiency (SFE) and spin-dependent conductance of the bilayer CuPc molecular device could be modulated by changing the twist angle. The conductance reaches its maximum when the twist angle θ is 0° while the largest SFE is achieved when θ = 60°. The twist angle-induced exotic transport phenomena can be well explained by analyzing the transmission spectra, molecular energy level spectra and scattering states of the twisted bilayer CuPc molecular device. The tunable conductance, HOMO-LUMO gap and spin filtering versus twist angle are helpful for predicting how a two-molecule system may behave with twist angle.

Moiré Patterns in 2D Materials: A Review

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

Tunable Optical Properties of Thin Films Controlled by the Interface Twist Angle

Control of materials properties has been the driving force of modern technologies. So far, materials properties have been modulated by their composition, structure, and size. Here, by using cathodoluminescence in a scanning transmission electron microscope, we show that the optical properties of stacked, >100 nm thick hexagonal boron nitride (hBN) films can be continuously tuned by their relative twist angles. Due to the formation of a moiré superlattice between the two interface layers of the twisted films, a new moiré sub-band gap is formed with continuously decreasing magnitude as a function of the twist angle, resulting in tunable luminescence wavelength and intensity increase of >40×. Our results demonstrate that moiré phenomena extend beyond monolayer-based systems and can be preserved in a technologically relevant, bulklike material at room temperature, dominating optical properties of hBN films for applications in medicine, environmental, or information technologies.

Topological flat bands in twisted trilayer graphene

Twisted trilayer graphene (TLG) may be the simplest realistic system so far, which has flat bands with nontrivial topology. Here, we give a comprehensive calculation about its band structures and the band topology, i.e., valley Chern number of the nearly flat bands, with the continuum model. With realistic parameters, the magic angle of twisted TLG is about 1.12°, at which two nearly flat bands appears. Unlike the twisted bilayer graphene, a small twist angle can induce a tiny gap at all the Dirac points, which can be enlarged further by a perpendicular electric field. The valley Chern numbers of the two nearly flat bands in the twisted TLG depends on the twist angle θ and the perpendicular electric field E_⊥. Considering its topological flat bands, the twisted TLG should be an ideal experimental platform to study the strongly correlated physics in topologically nontrivial flat band systems. And, due to its reduced symmetry, the correlated states in twisted TLG should be quite different from that in twisted bilayer graphene and twisted double bilayer graphene.

Current-Induced Reversal of Anomalous Hall Conductance in Twisted Bilayer Graphene

It is observed experimentally that the sign of the Hall resistance can be flipped by a dc electric current in the twisted bilayer graphene (TBG) at 3/4 filling of the fourfold degenerate conduction flat bands. The experiment implies a switching of the valley polarization (VP) and topology in TBG. Here we present a theory on the current-induced switching of VP and topology. The presence of current in the bulk causes the redistribution of electron occupation in bands near the Fermi energy, which then deforms and shifts the band dispersion due to the Coulomb interaction. Above a critical current, the original occupied and empty bands can be swapped, resulting in the switching of VP and topology.

Structural and Electronic Properties of Heterostructures Composed of Antimonene and Monolayer MoS2

Antimonene is found to be a promising material for two-dimensional optoelectronic equipment due to its broad band gap and high carrier mobility. The van der Waals heterostructure, as a unique structural unit for the study of photoelectric properties, has attracted great attention. By using ab initio density functional theory with van der Waals corrections, we theoretically investigated the structural and electronic properties of the heterostructures composed of antimonene and monolayer MoS₂. Our results revealed that the Sb/MoS₂ hetero-bilayer is an indirect semiconductor with type-II band alignment, which implies the spatial separation of photogenerated electron–hole pairs. Due to the weak van der Waals interlayer interactions between the adjacent sheets of the hetero-bilayer systems, the band structures of isolated antimonene and monolayer MoS₂ are preserved. In addition, a tunable band gap in Sb/MoS₂ hetero-bilayer can be realized by applying in-plane biaxial compressing/stretching. When antimonene and monolayer MoS₂ are stacked into superlattices, the indirect semiconductors turn into direct semiconductors with the decreased band gaps. Our results show that the antimonene-based hybrid structures are good candidate structures for photovoltaic devices.

Emergence of Highly Linearly Polarized Interlayer Exciton Emission in MoSe2/WSe2 Heterobilayers with Transfer-Induced Layer Corrugation

The availability of accessible fabrication methods based on deterministic transfer of atomically thin crystals has been essential for the rapid expansion of research into van der Waals heterostructures. An inherent issue of these techniques is the deformation of the polymer carrier film during the transfer, which can lead to highly nonuniform strain induced in the transferred two-dimensional material. Here, using a combination of optical spectroscopy, atomic force, and Kelvin probe force microscopy, we show that the presence of nanometer scale wrinkles formed due to transfer-induced stress relaxation can lead to strong changes in the optical properties of MoSe₂/WSe₂ heterostructures and the emergence of linearly polarized interlayer exciton photoluminescence. We attribute these changes to local breaking of crystal symmetry in the nanowrinkles, which act as efficient accumulation centers for interlayer excitons due to the strain-induced interlayer band gap reduction. Surface potential images of the rippled heterobilayer samples acquired using Kelvin probe force microscopy reveal variations of the local work function consistent with strain-induced band gap modulation, while the potential offset observed at the ridges of the wrinkles shows a clear correlation with the value of the tensile strain estimated from the wrinkle geometry. Our findings highlight the important role of the residual strain in defining optical properties of van der Waals heterostructures and suggest effective approaches for interlayer exciton manipulation by local strain engineering.

Twistronics in Graphene, from Transfer Assembly to Epitaxy

The twistronics, which is arising from the moiré superlattice of the small angle between twisted bilayers of 2D materials like graphene, has attracted much attention in the field of 2D materials and condensed matter physics. The novel physical properties in such systems, like unconventional superconductivity, come from the dispersionless flat band that appears when the twist reaches some magic angles. By tuning the filling of the fourfold degeneracy flat bands, the desired effects are induced due to the strong correlation of the degenerated Bloch electrons. In this article, we review the twistronics in twisted bi- and multi-layer graphene (TBG and TMG), which is formed both by transfer assembly of exfoliated monolayer graphene and epitaxial growth of multilayer graphene on SiC substrates. Starting from a brief history, we then introduce the theory of flat band in TBG. In the following, we focus on the major achievements in this field: (a) van Hove singularities and charge order; (b) superconductivity and Mott insulator in TBG and (c) transport properties in TBG. In the end, we give the perspective of the rising materials system of twistronics, epitaxial multilayer graphene on the SiC.

Flat Bands and Mechanical Deformation Effects in the Moiré Superlattice of MoS2-WSe2 Heterobilayers

It has recently been shown that quantum-confined states can appear in epitaxially grown van der Waals material heterobilayers without a rotational misalignment (θ = 0°), associated with flat bands in the Brillouin zone of the moiré pattern formed due to the lattice mismatch of the two layers. Peaks in the local density of states and confinement in a MoS₂/WSe₂ system was qualitatively described only considering local stacking arrangements, which cause band edge energies to vary spatially. In this work, we report the presence of large in-plane strain variation across the moiré unit cell of a θ = 0° MoS₂/WSe₂ heterobilayer and show that inclusion of strain variation and out-of-plane displacement in density functional theory calculations greatly improves their agreement with the experimental data. We further explore the role of a twist angle by showing experimental data for a twisted MoS₂/WSe₂ heterobilayer structure with a twist angle of θ = 15°, which exhibits a moiré pattern but no confinement.

Flatband in a three-dimensional tungsten nitride compound

Herein, the flatband of a W₁N₂ crystal is theoretically investigated. It is revealed that the flatband can be well-described by a tight-binding model of the N₁₂ skeleton, where the dispersion of the flatband is governed by both the ppσ bonding strength (V_ppσ) and the ppπ bonding strength (V_ppπ) between the nearest-neighbor N atoms. It is also found that the proper strength of the ppπ bonding between neighboring N atoms plays a prime role in the formation of the flatband. In addition, when the compound is doped with holes, the electrons at the flatband are fully polarized, showing a ferromagnetic character. This behavior has a weak correlation with the on-site Coulomb interaction U. Moreover, several three-dimensional compounds possessing flatbands in the whole k space are predicted.

Ferromagnetism in Narrow Bands of Moiré Superlattices

Many graphene moiré superlattices host narrow bands with nonzero valley Chern numbers. We provide analytical and numerical evidence for a robust spin and/or valley polarized insulator at total integer band filling in nearly flat bands of several different moiré materials. In the limit of a perfectly flat band, we present analytical arguments in favor of the ferromagnetic state substantiated by numerical calculations. Further, we numerically evaluate its stability for a finite bandwidth. We provide exact diagonalization results for models appropriate for ABC trilayer graphene aligned with hBN, twisted double bilayer graphene, and twisted bilayer graphene aligned with hBN. We also provide DMRG results for a honeycomb lattice with a quasiflat band and nonzero Chern number, which extend our results to larger system sizes. We find a maximally spin and valley polarized insulator at all integer fillings when the band is sufficiently flat. We also show that interactions may induce effective dispersive terms strong enough to destabilize this state. These results still hold in the case of zero valley Chern number (for example, trivial side of TLG/hBN). We give an intuitive picture based on extended Wannier orbitals, and emphasize the role of the quantum geometry of the band, whose microscopic details may enhance or weaken ferromagnetism in moiré materials.

Topology-Bounded Superfluid Weight in Twisted Bilayer Graphene

While regular flat bands are good for enhancing the density of states and hence the gap, they are detrimental to the superfluid weight. We show that the predicted nontrivial topology of the two lowest flat bands of twisted bilayer graphene (TBLG) plays an important role in the enhancement of the superfluid weight and hence of superconductivity. We derive the superfluid weight (phase stiffness) of the TBLG superconducting flat bands with a uniform pairing, and show that it can be expressed as an integral of the Fubini-Study metric of the flat bands. This mirrors results already obtained for nonzero Chern number bands even though the TBLG flat bands have zero Chern number. We further show that the metric integral is lower bounded by the topological C_{2z}T Wilson loop winding number of TBLG flat bands, which renders that the superfluid weight is also bounded by this topological index. In contrast, trivial flat bands have a zero superfluid weight. The superfluid weight is crucial in determining the Berezinskii-Kosterlitz-Thouless transition temperature of the superconductor. Based on the transition temperature measured in TBLG experiments, we estimate the topological contribution of the superfluid weight in TBLG.

Particle-Hole Duality, Emergent Fermi Liquids, and Fractional Chern Insulators in Moiré Flatbands

Moiré flatbands, occurring, e.g., in twisted bilayer graphene at magic angles, have attracted ample interest due to their high degree of experimental tunability and the intriguing possibility of generating novel strongly interacting phases. Here we consider the core problem of Coulomb interactions within fractionally filled spin and valley polarized Moiré flatbands and demonstrate that the dual description in terms of holes, which acquire a nontrivial hole dispersion, provides key physical intuition and enables the use of standard perturbative techniques for this strongly correlated problem. In experimentally relevant examples such as ABC stacked trilayer and twisted bilayer graphene aligned with boron nitride, it leads to emergent interaction-driven Fermi liquid states at electronic filling fractions down to around 1/3 and 2/3, respectively. At even lower filling fractions, the electron density still faithfully tracks the single-hole dispersion while exhibiting distinct non-Fermi liquid behavior. Most saliently, we provide microscopic evidence that high temperature fractional Chern insulators can form in twisted bilayer graphene aligned with hexagonal boron nitride.

Collective Excitations of Quantum Anomalous Hall Ferromagnets in Twisted Bilayer Graphene

We present a microscopic theory for collective excitations of quantum anomalous Hall ferromagnets (QAHF) in twisted bilayer graphene. We calculate the spin magnon and valley magnon spectra by solving Bethe-Salpeter equations and verify the stability of QAHF. We extract the spin stiffness from the gapless spin wave dispersion and estimate the energy cost of a skyrmion-antiskyrmion pair, which is found to be comparable in energy with the Hartree-Fock gap. The valley wave mode is gapped, implying that the valley polarized state is more favorable compared to the valley coherent state. Using a nonlinear sigma model, we estimate the valley ordering temperature, which is considerably reduced from the mean-field transition temperature due to thermal excitations of valley waves.

Signatures of tunable superconductivity in a trilayer graphene moiré superlattice

Understanding the mechanism of high-transition-temperature (high-T_c) superconductivity is a central problem in condensed matter physics. It is often speculated that high-T_c superconductivity arises in a doped Mott insulator¹ as described by the Hubbard model^2-4. An exact solution of the Hubbard model, however, is extremely challenging owing to the strong electron-electron correlation in Mott insulators. Therefore, it is highly desirable to study a tunable Hubbard system, in which systematic investigations of the unconventional superconductivity and its evolution with the Hubbard parameters can deepen our understanding of the Hubbard model. Here we report signatures of tunable superconductivity in an ABC-trilayer graphene (TLG) and hexagonal boron nitride (hBN) moiré superlattice. Unlike in 'magic angle' twisted bilayer graphene, theoretical calculations show that under a vertical displacement field, the ABC-TLG/hBN heterostructure features an isolated flat valence miniband associated with a Hubbard model on a triangular superlattice^5,6 where the bandwidth can be tuned continuously with the vertical displacement field. Upon applying such a displacement field we find experimentally that the ABC-TLG/hBN superlattice displays Mott insulating states below 20 kelvin at one-quarter and one-half fillings of the states, corresponding to one and two holes per unit cell, respectively. Upon further cooling, signatures of superconductivity ('domes') emerge below 1 kelvin for the electron- and hole-doped sides of the one-quarter-filling Mott state. The electronic behaviour in the ABC-TLG/hBN superlattice is expected to depend sensitively on the interplay between the electron-electron interaction and the miniband bandwidth. By varying the vertical displacement field, we demonstrate transitions from the candidate superconductor to Mott insulator and metallic phases. Our study shows that ABC-TLG/hBN heterostructures offer attractive model systems in which to explore rich correlated behaviour emerging in the tunable triangular Hubbard model.

Signatures of tunable superconductivity in a trilayer graphene moiré superlattice

Understanding the mechanism of high-transition-temperature (high-T_c) superconductivity is a central problem in condensed matter physics. It is often speculated that high-T_c superconductivity arises in a doped Mott insulator¹ as described by the Hubbard model^2-4. An exact solution of the Hubbard model, however, is extremely challenging owing to the strong electron-electron correlation in Mott insulators. Therefore, it is highly desirable to study a tunable Hubbard system, in which systematic investigations of the unconventional superconductivity and its evolution with the Hubbard parameters can deepen our understanding of the Hubbard model. Here we report signatures of tunable superconductivity in an ABC-trilayer graphene (TLG) and hexagonal boron nitride (hBN) moiré superlattice. Unlike in 'magic angle' twisted bilayer graphene, theoretical calculations show that under a vertical displacement field, the ABC-TLG/hBN heterostructure features an isolated flat valence miniband associated with a Hubbard model on a triangular superlattice^5,6 where the bandwidth can be tuned continuously with the vertical displacement field. Upon applying such a displacement field we find experimentally that the ABC-TLG/hBN superlattice displays Mott insulating states below 20 kelvin at one-quarter and one-half fillings of the states, corresponding to one and two holes per unit cell, respectively. Upon further cooling, signatures of superconductivity ('domes') emerge below 1 kelvin for the electron- and hole-doped sides of the one-quarter-filling Mott state. The electronic behaviour in the ABC-TLG/hBN superlattice is expected to depend sensitively on the interplay between the electron-electron interaction and the miniband bandwidth. By varying the vertical displacement field, we demonstrate transitions from the candidate superconductor to Mott insulator and metallic phases. Our study shows that ABC-TLG/hBN heterostructures offer attractive model systems in which to explore rich correlated behaviour emerging in the tunable triangular Hubbard model.

Topological Insulators in Twisted Transition Metal Dichalcogenide Homobilayers

We show that moiré bands of twisted homobilayers can be topologically nontrivial, and illustrate the tendency by studying valence band states in ±K valleys of twisted bilayer transition metal dichalcogenides, in particular, bilayer MoTe_{2}. Because of the large spin-orbit splitting at the monolayer valence band maxima, the low energy valence states of the twisted bilayer MoTe_{2} at the +K (-K) valley can be described using a two-band model with a layer-pseudospin magnetic field Δ(r) that has the moiré period. We show that Δ(r) has a topologically nontrivial skyrmion lattice texture in real space, and that the topmost moiré valence bands provide a realization of the Kane-Mele quantum spin-Hall model, i.e., the two-dimensional time-reversal-invariant topological insulator. Because the bands narrow at small twist angles, a rich set of broken symmetry insulating states can occur at integer numbers of electrons per moiré cell.