Quantum interference at the twist boundary in graphene

Quantum Valley Hall Effect without Berry Curvature

The quantum valley Hall effect (QVHE) is characterized by the valley Chern number (VCN) in a way that one-dimensional (1D) chiral metallic states are guaranteed to appear at the domain walls (DW) between two domains with opposite VCN for a given valley. Although in the case of QVHE, the total Berry curvature (BC) of the system is zero, the BC distributed locally around each valley makes the VCN well defined as long as intervalley scattering is negligible. Here, we propose a new type of valley-dependent topological phenomenon that occurs when the BC is strictly zero at each momentum. Such zero Berry curvature (ZBC) QVHE is characterized by the valley Euler number (VEN) which is computed by integrating the Euler curvature around a given valley in two-dimensional (2D) systems with space-time inversion symmetry. 1D helical metallic states can be topologically protected at the DW between two domains with the opposite VENs when the DW configuration preserves either the mirror symmetry with respect to the DW or the combination of the DW space-time inversion and chiral symmetries. We establish the fundamental origin of ZBC QVHE. Also, by combining tight-binding model study and first-principles calculations, we propose stacked hexagonal bilayer lattices including h-BX (X=As, P) and large-angle twisted bilayer graphenes as candidate systems with robust helical DW states protected by VEN.

Emergent phases in graphene flat bands

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

Band separation and electric field prediction in square Bravais-Moiré photonic crystals

In this study, we address three key challenges in photonic crystals: modeling of isolated flat bands, electric field prediction, and band separation in dispersion relations. Using twisted square Bravais lattices at specific angles, we create Bravais-Moiré photonic crystals exhibiting unique characteristics. These include band pairing and parallelism in certain Brillouin zones, enabling predictable electric field behavior and identification of isolated, flat band pairs within extensive band gaps. We apply advanced Shape theory-based classification methods for precise band separation, offering significant contributions to photonics research and light manipulation applications.

Torsional force microscopy of van der Waals moirés and atomic lattices

In a stack of atomically thin van der Waals layers, introducing interlayer twist creates a moiré superlattice whose period is a function of twist angle. Changes in that twist angle of even hundredths of a degree can dramatically transform the system's electronic properties. Setting a precise and uniform twist angle for a stack remains difficult; hence, determining that twist angle and mapping its spatial variation is very important. Techniques have emerged to do this by imaging the moiré, but most of these require sophisticated infrastructure, time-consuming sample preparation beyond stack synthesis, or both. In this work, we show that torsional force microscopy (TFM), a scanning probe technique sensitive to dynamic friction, can reveal surface and shallow subsurface structure of van der Waals stacks on multiple length scales: the moirés formed between bi-layers of graphene and between graphene and hexagonal boron nitride (hBN) and also the atomic crystal lattices of graphene and hBN. In TFM, torsional motion of an Atomic Force Microscope (AFM) cantilever is monitored as it is actively driven at a torsional resonance while a feedback loop maintains contact at a set force with the sample surface. TFM works at room temperature in air, with no need for an electrical bias between the tip and the sample, making it applicable to a wide array of samples. It should enable determination of precise structural information including twist angles and strain in moiré superlattices and crystallographic orientation of van der Waals flakes to support predictable moiré heterostructure fabrication.

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.

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

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

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

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

Local atomic-morphology-resolved edge states in twisted bilayer graphene nanoribbons

We study the properties of edge states for a selected (10,1)[(4,3)] twisted bilayer graphene (TBG) nanoribbon with minimal edges but a majority of zigzag edges. By using the tight-binding and Green's function methods, we find a remarkable rule of a local electronic transfer for the edge states. As the energy away from the Fermi level, the transfer is in the order of convex AB-, concave AB-, concave AA- and convex AA-stacked regions of the ribbon curve edges. We illustrate that this rule comes from the difference in interlayer couplings among the four types of local geometries at edges. Further, an in-plane transverse electric field can rearrange the edge bands and enlarge the energy regimes, leading to the lowest energy states modified from AB-stacked edge states to AA-stacked ones. The realignment of the edge bands results from the interplay between the interlayer coupling and the potential difference induced by the transverse electric field, which results in different bonding and antibonding edge states, i.e. the edge bands. In contrast, the total energy regime of the edge bands remain nearly unchanged under a relative strong off-plane perpendicular electric field, and the typical AA-stacked edge states are still maintained even the rotational symmetry of two layers is broken. Until a sufficiently strong value, the TBG nanoribbon tends to behave as two noninteracting monolayer ribbons except for a band distortion in low-energy regime. The conductance spectra reflects the edge bands well. We also discussed the influence of edge defects in the TBG nanoribbon on transport properties. It is found that the contributed conductance of each type of edge states shows different degrees of suppression for a monatomic vacancy in the corresponding region of edges.

Moiré metasurfaces for dynamic beamforming

Recent advances in digitally programmable metamaterials have accelerated the development of reconfigurable intelligent surfaces (RIS). However, the excessive use of active components (e.g., pin diodes and varactor diodes) leads to high costs, especially for those operating at millimeter-wave frequencies, impeding their large-scale deployments in RIS. Here, we introduce an entirely different approach-moiré metasurfaces-to implement dynamic beamforming through mutual twists of two closely stacked metasurfaces. The superposition of two high-spatial-frequency patterns produces a low-spatial-frequency moiré pattern through the moiré effect, which provides the surface impedance profiles to generate desired radiation patterns. We demonstrate experimentally that the direction of the radiated beams can continuously sweep over the entire reflection space along predesigned trajectories by simply adjusting the twist angle and the overall orientation. Our work opens previously unexplored directions for synthesizing far-field scattering through the direct contact of mutually twisted metallic patterns with different plane symmetry groups.

Various defects in graphene: a review

Pristine graphene has been considered one of the most promising materials because of its excellent physical and chemical properties. However, various defects in graphene produced during synthesis or fabrication hinder its performance for applications such as electronic devices, transparent electrodes, and spintronic devices. Due to its intrinsic bandgap and nonmagnetic nature, it cannot be used in nanoelectronics or spintronics. Intrinsic and extrinsic defects are ultimately introduced to tailor electronic and magnetic properties and take advantage of their hidden potential. This article emphasizes the current advancement of intrinsic and extrinsic defects in graphene for potential applications. We also discuss the limitations and outlook for such defects in graphene.

Dirac Magic and Lifshitz Transitions in AA-Stacked Twisted Multilayer Graphene

We uncover a new type of magic-angle phenomena when an AA-stacked graphene bilayer is twisted relative to another graphene system with band touching. In the simplest case this constitutes a trilayer system formed by an AA-stacked bilayer twisted relative to a single layer of graphene. We find multiple anisotropic Dirac cones coexisting in such twisted multilayer structures at certain angles, which we call "Dirac magic." We trace the origin of Dirac magic angles to the geometric structure of the twisted AA-bilayer Dirac cones relative to the other band-touching spectrum in the moiré reciprocal lattice. The anisotropy of the Dirac cones and a concomitant cascade of saddle points induce a series of topological Lifshitz transitions that can be tuned by the twist angle and perpendicular electric field. We discuss the possibility of direct observation of Dirac magic as well as its consequences for the correlated states of electrons in this moiré system.

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.

Moiré, Euler and self-similarity - the lattice parameters of twisted hexagonal crystals

A real-space approach for the calculation of the moiré lattice parameters for superstructures formed by a set of rotated hexagonal 2D crystals such as graphene or transition-metal dichalcogenides is presented. Apparent moiré lattices continuously form for all rotation angles, and their lattice parameter to a good approximation follows a hyperbolical angle dependence. Moiré crystals, i.e. moiré lattices decorated with a basis, require more crucial assessment of the commensurabilities and lead to discrete solutions and a non-continuous angle dependence of the moiré-crystal lattice parameter. In particular, this lattice parameter critically depends on the rotation angle, and continuous variation of the angle can lead to apparently erratic changes of the lattice parameter. The solutions form a highly complex pattern, which reflects number-theoretical relations between formation parameters of the moiré crystal. The analysis also provides insight into the special case of a 30° rotation of the constituting lattices, for which a dodecagonal quasicrystalline structure forms.

Twist dependent magneto-optical response in twisted bilayer graphene

By employing a linearised Boltzmann equation, we calculate the magneto-optical properties of twisted bilayer graphene using non-magnetic wave functions. Both transverse and longitudinal responses are calculated up to the second order in applied magnetic field with their twist angle and Fermi level dependence examined. We find that increasing the twist angle increases the transverse metallic response so long as the Fermi level remains below the upper conduction band. Interlayer transitions provide an appreciable enhancement when the Fermi level traverses the gap between the two conduction bands. Interlayer transitions are also responsible for a nonzero anomalous Hall conductivity in this model. As the Fermi level moves towards zero, the longitudinal response begins to dominate and a highly anisotropic negative magneto-resistance is observed.

Trigonal quasicrystalline states in [Formula: see text] rotated double moiré superlattices

We study the lattice configuration and electronic structure of a double moiré superlattice, which is composed of a graphene layer encapsulated by two other layers in a way such that the two hexagonal moiré patterns are arranged in a dodecagonal quasicrystalline configuration. We show that there are between 0 and 4 such configurations depending on the lattice mismatch between graphene and the encapsulating layer. We then reveal the resonant interaction, which is distinct from the conventional 2-, 3-, 4-wave mixing of moiré superlattices, that brings together and hybridizes twelve degenerate Bloch states of monolayer graphene. These states do not fully satisfy the dodecagonal quasicrystalline rotational symmetry due to the symmetry of the wave vectors involved. Instead, their wave functions exhibit trigonal quasicrystalline order, which lacks inversion symmetry, at the energies much closer to the charge neutrality point of graphene.

Plasmon-Enhanced Near-Field Chirality in Twisted van der Waals Heterostructures

It is shown that chiral plasmons, characterized by a longitudinal magnetic moment accompanying the longitudinal charge plasmon, lead to electromagnetic near-fields that are also chiral. For twisted bilayer graphene, we estimate that the near-field chirality of screened plasmons can be several orders of magnitude larger than that of the related circularly polarized light. The chirality also manifests itself in a deflection angle that is formed between the direction of the plasmon propagation and its Poynting vector. Twisted van der Waals heterostructures might thus provide a novel platform to promote enantiomer-selective physio-chemical processes in chiral molecules without the application of a magnetic field or external nanopatterning that break time-reversal, mirror plane, or inversion symmetry, respectively.

Superconductivity in graphene induced by the rotated layer

Recent discoveries in graphene bilayers have revealed that when one of the layers is rotated by a specific angle, superconductivity emerges. We provide an explanation for this phenomenon. We find that due to the layer rotation the spinors and the nodal Dirac points also depend on the rotation angle. The combination of rotation and projection of the spinors in the vicinity of the two Dirac nodes is modified in such a way that a repulsive interaction becomes attractive in certain directions. The change from a repulsive interaction to an attractive one gives rise to a one-dimensional system described by the double sine-Gordon model for charge-density-waves and spin gap superconductivity. The competition between the two models is controlled by the chemical potential which acts as a bias field.

Opportunities and Challenges in Twisted Bilayer Graphene: A Review

Two-dimensional (2D) materials exhibit enhanced physical, chemical, electronic, and optical properties when compared to those of bulk materials. Graphene demands significant attention due to its superior physical and electronic characteristics among different types of 2D materials. The bilayer graphene is fabricated by the stacking of the two monolayers of graphene. The twisted bilayer graphene (tBLG) superlattice is formed when these layers are twisted at a small angle. The presence of disorders and interlayer interactions in tBLG enhances several characteristics, including the optical and electrical properties. The studies on twisted bilayer graphene have been exciting and challenging thus far, especially after superconductivity was reported in tBLG at the magic angle. This article reviews the current progress in the fabrication techniques of twisted bilayer graphene and its twisting angle-dependent properties.

Moiré Flat Bands in Twisted Double Bilayer Graphene

We investigate twisted double bilayer graphene (TDBG), a four-layer system composed of two AB-stacked graphene bilayers rotated with respect to each other by a small angle. Our ab initio band structure calculations reveal a considerable energy gap at the charge-neutrality point that we assign to the intrinsic symmetric polarization (ISP). We then introduce the ISP effect into the tight-binding parametrization and perform calculations on TDBG models that include lattice relaxation effects down to very small twist angles. We identify a narrow region around the magic angle characterized by a manifold of remarkably flat bands gapped out from other states even without external electric fields. To understand the fundamental origin of the magic angle in TDBG, we construct a continuum model that points to a hidden mathematical link to the twisted bilayer graphene model, thus indicating that the band flattening is a fundamental feature of TDBG and is not a result of external fields.

Higher-Order Topological Insulator in Twisted Bilayer Graphene

Higher-order topological insulators are newly proposed topological phases of matter, whose bulk topology manifests as localized modes at two- or higher-dimensional lower boundaries. In this Letter, we propose the twisted bilayer graphenes with large angles as higher-order topological insulators, hosting topological corner charges. At large commensurate angles, the intervalley scattering opens up the bulk gap and the corner states occur at half filling. Based on both first-principles calculations and analytic analysis, we show the striking results that the emergence of the corner states do not depend on the choice of the specific angles as long as the underlying symmetries are intact. Our results show that the twisted bilayer graphene can serve as a robust candidate material of a two-dimensional higher-order topological insulator.

Emergent Geometric Frustration and Flat Band in Moiré Bilayer Graphene

So far the physics of moiré graphene bilayers at large, incommensurate rotation angles has been considered uninteresting. It has been held that the interlayer coupling in such structures is weak and the system can be thought of as a pair of decoupled single graphene sheets to a good approximation. Here, we demonstrate that for large rotation angles near commensurate ones, the interlayer coupling, far from being weak, is able to completely localize electrons to within a large scale, geometrically frustrated network of topologically protected modes. The emergent geometric frustration of the system gives rise to completely flat bands, with strong correlation physics as a result. All of this arises although in the lattice structure no large scale pattern appears to the unguided eye. Sufficiently close to commensuration the low-energy physics of this remarkable system has an exact solution.

Kohn-Luttinger Superconductivity in Twisted Bilayer Graphene

We show that the recently observed superconductivity in twisted bilayer graphene (TBG) can be explained as a consequence of the Kohn-Luttinger (KL) instability which leads to an effective attraction between electrons with originally repulsive interaction. Usually, the KL instability takes place at extremely low energy scales, but in TBG, a doubling and subsequent strong coupling of the van Hove singularities (vHS) in the electronic spectrum occurs as the magic angle is approached, leading to extended saddle points in the highest valence band with almost perfect nesting between states belonging to different valleys. The highly anisotropic screening induces an effective attraction in a p-wave channel with odd parity under the exchange of the two disjoined patches of the Fermi line. We also predict the appearance of a spin-density wave instability, adjacent to the superconducting phase, and the opening of a gap in the electronic spectrum from the condensation of spins with wave vector corresponding to the nesting vector close to the vHS.

Chiral Spin Density Wave and d+id Superconductivity in the Magic-Angle-Twisted Bilayer Graphene

We model the newly synthesized magic-angle-twisted bilayer graphene superconductor with two p_{x,y}-like Wannier orbitals on the superstructure honeycomb lattice, where the hopping integrals are constructed via the Slater-Koster formulism by symmetry analysis. The characteristics exhibited in this simple model are well consistent with both the rigorous calculations and experiment observations. A van Hove singularity and Fermi-surface (FS) nesting are found in the doping levels relevant to the correlated insulator and unconventional superconductivity revealed experimentally, based on which we identify the two phases as weak-coupling FS instabilities. Then, with repulsive Hubbard interactions turned on, we performed random-phase-approximation based calculations to identify the electron instabilities. As a result, we find chiral d+id topological superconductivity bordering the correlated insulating state near half-filling, identified as noncoplanar chiral spin-density wave ordered state, featuring the quantum anomalous Hall effect. The phase diagram obtained in our approach is qualitatively consistent with experiments.

Formation and Mechanical Behavior of Nanocomposite Superstructures from Interlayer Bonding in Twisted Bilayer Graphene

We report a comprehensive study on the design of two-dimensional graphene-diamond nanocomposite superstructures formed through interlayer covalent bonding of twisted bilayer graphene with commensurate bilayers. The interlayer bonding is induced by patterned hydrogenation that leads to the formation of superlattices of two-dimensional nanodiamond domains embedded between the two graphene layers. We generalize a rigorous algorithm for the formation of all possible classes of these superstructures: the structural parameters employed to design such carbon nanocomposites include the commensurate bilayer's twist angle, the stacking type of the nanodomains where the interlayer bonds are formed, the interlayer bond pattern, and the interlayer C-C bond density that is proportional to the concentration of sp³-hybridized interlayer-bonded C atoms. We also analyze systematically the mechanical behavior of these nanocomposite superstructures on the basis of molecular-dynamics simulations of uniaxial tensile straining tests according to a reliable interatomic bond-order potential. We identify a range of structural parameters over which the fracture of these superstructures is ductile, mediated by void formation, growth, and coalescence, contrary to the typical brittle fracture of graphene. We introduce a ductility metric as an order parameter for the brittle-to-ductile transition, demonstrate its direct dependence on the fraction of sp³-hybridized interlayer-bonded C atoms, and show that increasing the fraction of interlayer-bonded C atoms beyond a critical level in certain classes of these superstructures induces their ductile mechanical response.

Externally Controlled Magnetism and Band Gap in Twisted Bilayer Graphene

We theoretically study the effects of electron-electron interaction in twisted bilayer graphene in a transverse dc electric field. When the twist angle is not very small, the electronic spectrum of the bilayer consists of four Dirac cones inherited from each graphene layer. An applied bias voltage leads to the appearance of two holelike and two electronlike Fermi surface sheets with perfect nesting among electron and hole components. Such a band structure is unstable with respect to the exciton band-gap opening due to the screened Coulomb interaction. The exciton order parameter is accompanied by spin-density-wave order. The gap depends on the twist angle and can be varied by a bias voltage. This result correlates well with recent transport measurements [J.-B. Liu et al., Sci. Rep. 5, 15285 (2015)SRCEC32045-232210.1038/srep15285]. Our proposal allows the coexistence of (i) an externally controlled semiconducting gap and (ii) a nontrivial multicomponent magnetic order. This is interesting for both fundamental research and applications.

Coherent commensurate electronic states at the interface between misoriented graphene layers

Graphene and layered materials in general exhibit rich physics and application potential owing to their exceptional electronic properties, which arise from the intricate π-orbital coupling and the symmetry breaking in twisted bilayer systems. Here, we report room-temperature experiments to study electrical transport across a bilayer graphene interface with a well-defined rotation angle between the layers that is controllable in situ. This twisted interface is artificially created in mesoscopic pillars made of highly oriented pyrolytic graphite by mechanical actuation. The overall measured angular dependence of the conductivity is consistent with a phonon-assisted transport mechanism that preserves the electron momentum of conduction electrons passing the interface. The most intriguing observations are sharp conductivity peaks at interlayer rotation angles of 21.8° and 38.2°. These angles correspond to a commensurate crystalline superstructure leading to a coherent two-dimensional (2D) electronic interface state. Such states, predicted by theory, form the basis for a new class of 2D weakly coupled bilayer systems with hitherto unexplored properties and applications.

Observation of van Hove Singularities in Twisted Silicene Multilayers

Interlayer interactions perturb the electronic structure of two-dimensional materials and lead to new physical phenomena, such as van Hove singularities and Hofstadter's butterfly pattern. Silicene, the recently discovered two-dimensional form of silicon, is quite unique, in that silicon atoms adopt competing sp(2) and sp(3) hybridization states leading to a low-buckled structure promising relatively strong interlayer interaction. In multilayer silicene, the stacking order provides an important yet rarely explored degree of freedom for tuning its electronic structures through manipulating interlayer coupling. Here, we report the emergence of van Hove singularities in the multilayer silicene created by an interlayer rotation. We demonstrate that even a large-angle rotation (>20°) between stacked silicene layers can generate a Moiré pattern and van Hove singularities due to the strong interlayer coupling in multilayer silicene. Our study suggests an intriguing method for expanding the tunability of the electronic structure for electronic applications in this two-dimensional material.

Selectively enhanced photocurrent generation in twisted bilayer graphene with van Hove singularity

Graphene with ultra-high carrier mobility and ultra-short photoresponse time has shown remarkable potential in ultrafast photodetection. However, the broad and weak optical absorption (∼2.3%) of monolayer graphene hinders its practical application in photodetectors with high responsivity and selectivity. Here we demonstrate that twisted bilayer graphene, a stack of two graphene monolayers with an interlayer twist angle, exhibits a strong light–matter interaction and selectively enhanced photocurrent generation. Such enhancement is attributed to the emergence of unique twist-angle-dependent van Hove singularities, which are directly revealed by spatially resolved angle-resolved photoemission spectroscopy. When the energy interval between the van Hove singularities of the conduction and valance bands matches the energy of incident photons, the photocurrent generated can be significantly enhanced (up to ∼80 times with the integration of plasmonic structures in our devices). These results provide valuable insight for designing graphene photodetectors with enhanced sensitivity for variable wavelength.

Graphene has the high carrier mobility and short photoresponse time required for efficient photodetection, but broad and weak optical absorption are severe drawbacks. Here, the authors show that twisted bilayer graphene with van Hove singularities exhibits a strong light-matter interaction and selectively enhanced photocurrent generation.

Friedel oscillation near a van Hove singularity in two-dimensional Dirac materials

We consider Friedel oscillation in the two-dimensional Dirac materials when the Fermi level is near the van Hove singularity. Twisted graphene bilayer and the surface state of topological crystalline insulator are the representative materials which show low-energy saddle points that are feasible to probe by gating. We approximate the Fermi surface near saddle point with a hyperbola and calculate the static Lindhard response function. Employing a theorem of Lighthill, the induced charge density [Formula: see text] due to an impurity is obtained and the algebraic decay of [Formula: see text] is determined by the singularity of the static response function. Although a hyperbolic Fermi surface is rather different from a circular one, the static Lindhard response function in the present case shows a singularity similar with the response function associated with circular Fermi surface, which leads to the [Formula: see text] at large distance R. The dependences of charge density on the Fermi energy are different. Consequently, it is possible to observe in twisted graphene bilayer the evolution that [Formula: see text] near Dirac point changes to [Formula: see text] above the saddle point. Measurements using scanning tunnelling microscopy around the impurity sites could verify the prediction.

Spectroscopic characterization of charge carrier anisotropic motion in twisted few-layer graphene

Graphene, a layer of carbon atoms in a honeycomb lattice, captures enormous interest as probably the most promising component of future electronics thanks to its mechanical robustness, flexibility, and unique charge carrier quasiparticles propagating like massless high energy Dirac fermions. If several graphene layers form a stack, the interaction between them is, on the one hand, weak, allowing realization of various registries between the layers and, on the other hand, strong enough for a wide range tuning of the electronic properties. Here we grow few layer graphene with various number of layers and twist configurations and address the electronic properties of individual atomic layers in single microscopic domains using angle-resolved photoelectron spectromicroscopy. The dependence of the interlayer coupling on the twist angle is analyzed and, in the domains with tri-layers and more, if different rotations are present, the electrons in weaker coupled adjacent layers are shown to have different properties manifested by coexisting van Hove singularities, moiré superlattices with corresponding superlattice Dirac points, and charge carrier group velocity renormalizations. Moreover, pronounced anisotropy in the charge carrier motion, opening a possibility to transform strongly coupled graphene bilayers into quasi one-dimensional conductors, is observed.

Electronic Properties of Bilayer Graphene Strongly Coupled to Interlayer Stacking and an External Electric Field

Bilayer graphene (BLG) with a tunable band gap appears interesting as an alternative to graphene for practical applications; thus, its transport properties are being actively pursued. Using density functional theory and perturbation analysis, we investigated, under an external electric field, the electronic properties of BLG in various stackings relevant to recently observed complex structures. We established the first phase diagram summarizing the stacking-dependent gap openings of BLG for a given field. We further identified high-density midgap states, localized on grain boundaries, even under a strong field, which can considerably reduce the overall transport gap.

Origin of band gaps in graphene on hexagonal boron nitride

Recent progress in preparing well-controlled two-dimensional van der Waals heterojunctions has opened up a new frontier in materials physics. Here we address the intriguing energy gaps that are sometimes observed when a graphene sheet is placed on a hexagonal boron nitride substrate, demonstrating that they are produced by an interesting interplay between structural and electronic properties, including electronic many-body exchange interactions. Our theory is able to explain the observed gap behaviour by accounting first for the structural relaxation of graphene’s carbon atoms when placed on a boron nitride substrate, and then for the influence of the substrate on low-energy π-electrons located at relaxed carbon atom sites. The methods we employ can be applied to many other van der Waals heterojunctions.

Graphene doesn’t usually have a bandgap but one can appear when the two-dimensional material is placed on a hexagonal boron nitride substrate. Jung et al. now develop a theory that indicates that this occurs because the graphene’s carbon atoms structurally relax when placed on boron nitride.

Sharp variations in the electronic properties of graphene deposited on the h-BN layer

Investigation of the complex structure based on the graphene monolayer and the twisted BN monolayer was carried out.

Thermal conductivity of twisted bilayer graphene

We have investigated experimentally the thermal conductivity of suspended twisted bilayer graphene. The measurements were performed using an optothermal Raman technique. It was found that the thermal conductivity of twisted bilayer graphene is lower than that of monolayer graphene and the reference, Bernal stacked bilayer graphene in the entire temperature range examined (∼300-700 K). This finding indicates that the heat carriers - phonons - in twisted bilayer graphene do not behave in the same manner as that observed in individual graphene layers. The decrease in the thermal conductivity found in twisted bilayer graphene was explained by the modification of the Brillouin zone due to plane rotation and the emergence of numerous folded phonon branches that enhance the phonon Umklapp and normal scattering. The results obtained are important for understanding thermal transport in two-dimensional systems.

Tailoring the electronic structure in bilayer molybdenum disulfide via interlayer twist

Molybdenum disulfide bilayers with well-defined interlayer twist angle were constructed by stacking single-crystal monolayers. Varying interlayer twist angle results in strong tuning of the indirect optical transition energy and second-harmonic generation and weak tuning of direct optical transition energies and Raman mode frequencies. Electronic structure calculations show the interlayer separation changes with twist due to repulsion between sulfur atoms, resulting in shifts of the indirect optical transition energies. These results show that interlayer alignment is a crucial variable in tailoring the properties of two-dimensional heterostructures.

Rotational disorder in twisted bilayer graphene

Conventional means of stacking two-dimensional (2D) crystals inevitably leads to imperfections. To examine the ramifications of these imperfections, rotational disorder and strain are quantified in twisted bilayer graphene (TBG) using a combination of Raman spectroscopic and low-energy electron diffraction imaging. The twist angle between TBG layers varies on the order of 2° within large (50-100 μm) single-crystalline grains, resulting in changes of the emergent Raman response by over an order of magnitude. Rotational disorder does not evolve continuously across the large grains but rather comes about by variations in the local twist angles between differing contiguous subgrains, ∼ 1 μm in size, that themselves exhibit virtually no twist angle variation (ΔΘ ∼ 0.1°). Owing to weak out-of-plane van der Waals bonding between azimuthally rotated graphene layers, these subgrains evolve in conjunction with the 0.3% strain variation observed both within and between the atomic layers. Importantly, the emergent Raman response is altered, but not removed, by these extrinsic perturbations. Interlayer interactions are therefore resilient to strain and rotational disorder, a fact that gives promise to the prospect of designer 2D solid heterostructures created via transfer processes.

Coupled Dirac fermions and neutrino-like oscillations in twisted bilayer graphene

The low-energy quasiparticles in graphene can be described by a Dirac-Weyl Hamiltonian for massless fermions, hence graphene has been proposed to be an effective medium to study exotic phenomena originally predicted for relativistic particle physics, such as Klein tunneling and Zitterbewegung. In this work, we show that another important particle-physics phenomenon, the neutrino oscillation, can be studied and observed in a particular graphene system, namely, twisted bilayer graphene. It has been found that graphene layers grown epitaxially on SiC or by the chemical vapor deposition method on metal substrates display a stacking pattern with adjacent layers rotated by an angle with respect to each other. The quasiparticle states in two distinct graphene layers act as neutrinos with two flavors, and the interlayer interaction between them induces an appreciable coupling between these two "flavors" of massless fermions, leading to neutrino-like oscillations. In addition, our calculation shows that anisotropic transport properties manifest in a specific energy window, which is accessible experimentally in twisted bilayer graphene. Combining two graphene layers enables us to probe the rich physics involving multiple interacting Dirac fermions.

Anomalous optical phonon splittings in sliding bilayer graphene

We study the variations of electron-phonon coupling and their spectroscopic consequences in response to the sliding of two layers in bilayer graphene using first-principles calculations and a model Hamiltonian. Our study shows that the long wavelength optical phonon modes change in a sensitive and unusual way depending on the symmetry as well as the parity of sliding atomic structures and that, accordingly, Raman- and infrared-active optical phonon modes behave differently upon the direction and size of the sliding. The renormalization of phonon modes by the interlayer electronic coupling is shown to be crucial to explain their anomalous behavior upon the sliding. Also, we show that the crystal symmetry change due to the sliding affects the polarized Stokes Raman scattering intensity, which can be utilized to detect tiny misalignment of graphene layers using spectroscopic tools.

Interaction between graphene layers and the mechanisms of graphite's superlubricity and self-retraction

Graphene layer-layer interaction is explored as a function of the misorientation angle. A stepwise potential energy surface (PES), where the optimized commensurate configuration (AB stacking) corresponds to the global minimum and all incommensurate configurations correspond to nearly equal energies, is shown. The stepwise behavior is attributed to the alternating appearance of AB and AA stacking-like areas and the transition areas between them. Further, the PES of most incommensurate configurations is found to be ultra-smooth. Based on this, the puzzling experimental observation of graphite flake self-retraction is successfully explained.

Breakdown of the interlayer coherence in twisted bilayer graphene

Coherent motion of electrons in Bloch states is one of the fundamental concepts of charge conduction in solid-state physics. In layered materials, however, such a condition often breaks down for the interlayer conduction, when the interlayer coupling is significantly reduced by, e.g., a large interlayer separation. We report that complete suppression of coherent conduction is realized even in an atomic length scale of layer separation in twisted bilayer graphene. The interlayer resistivity of twisted bilayer graphene is much higher than the c-axis resistivity of Bernal-stacked graphite and exhibits strong dependence on temperature as well as on external electric fields. These results suggest that the graphene layers are significantly decoupled by rotation and incoherent conduction is a main transport channel between the layers of twisted bilayer graphene.

Phonon-mediated interlayer conductance in twisted graphene bilayers

Conduction between graphene layers is suppressed by momentum conservation whenever the layer stacking has a rotation. Here we show that phonon scattering plays a crucial role in facilitating interlayer conduction. The resulting dependence on orientation is radically different than previously expected, and far more favorable for device applications. At low temperatures, we predict diode-like current-voltage characteristics due to a phonon bottleneck. Simple scaling relationships give a good description of the conductance as a function of temperature, doping, rotation angle, and bias voltage, reflecting the dominant role of the interlayer beating phonon mode.

Unraveling the intrinsic and robust nature of van Hove singularities in twisted bilayer graphene by scanning tunneling microscopy and theoretical analysis

Extensive scanning tunneling microscopy and spectroscopy experiments complemented by first-principles and parametrized tight binding calculations provide a clear answer to the existence, origin, and robustness of van Hove singularities (vHs) in twisted graphene layers. Our results are conclusive: vHs due to interlayer coupling are ubiquitously present in a broad range (from 1° to 10°) of rotation angles in our graphene on 6H-SiC(000-1) samples. From the variation of the energy separation of the vHs with the rotation angle we are able to recover the Fermi velocity of a graphene monolayer as well as the strength of the interlayer interaction. The robustness of the vHs is assessed both by experiments, which show that they survive in the presence of a third graphene layer, and by calculations, which test the role of the periodic modulation and absolute value of the interlayer distance. Finally, we clarify the role of the layer topographic corrugation and of electronic effects in the apparent moiré contrast measured on the STM images.

Evidence for interlayer coupling and moiré periodic potentials in twisted bilayer graphene

We report a study of the valence band dispersion of twisted bilayer graphene using angle-resolved photoemission spectroscopy and ab initio calculations. We observe two noninteracting cones near the Dirac crossing energy and the emergence of van Hove singularities where the cones overlap for large twist angles (>5°). Besides the expected interaction between the Dirac cones, minigaps appeared at the Brillouin zone boundaries of the moiré superlattice formed by the misorientation of the two graphene layers. We attribute the emergence of these minigaps to a periodic potential induced by the moiré. These anticrossing features point to coupling between the two graphene sheets, mediated by moiré periodic potentials.

Engineering the electronic structure of graphene

Graphene exhibits many unique electronic properties owing to its linear dispersive electronic band structure around the Dirac point, making it one of the most studied materials in the last 5-6 years. However, for many applications of graphene, further tuning its electronic band structure is necessary and has been extensively studied ever since graphene was first isolated experimentally. Here we review the major progresses made in electronic structure engineering of graphene, namely by electric and magnetic fields, chemical intercalation and adsorption, stacking geometry, edge-chirality, defects, as well as strain.

Fractal Landau-level spectra in twisted bilayer graphene

The Hofstadter butterfly spectrum for Landau levels in a two-dimensional periodic lattice is a rare example exhibiting fractal properties in a truly quantum system. However, the observation of this physical phenomenon in a conventional material will require a magnetic field strength several orders of magnitude larger than what can be produced in a modern laboratory. It turns out that for a specific range of rotational angles twisted bilayer graphene serves as a special system with a fractal energy spectrum under laboratory accessible magnetic field strengths. This unique feature arises from an intriguing electronic structure induced by the interlayer coupling. Using a recursive tight-binding method, we systematically map out the spectra of these Landau levels as a function of the rotational angle. Our results give a complete description of LLs in twisted bilayer graphene for both commensurate and incommensurate rotational angles and provide quantitative predictions of magnetic field strengths for observing the fractal spectra in these graphene systems.