Graphene bilayer with a twist: electronic structure

Geometric and Conventional Contribution to the Superfluid Weight in Twisted Bilayer Graphene

By tuning the angle between graphene layers to specific "magic angles" the lowest energy bands of twisted bilayer graphene (TBLG) can be made flat. The flat nature of the bands favors the formation of collective ground states and, in particular, TBLG has been shown to support superconductivity. When the energy bands participating in the superconductivity are well isolated, the superfluid weight scales inversely with the effective mass of such bands. For flat band systems one would therefore conclude that even if superconducting pairing is present, most of the signatures of the superconducting state should be absent. This conclusion is at odds with the experimental observations for TBLG. We calculate the superfluid weight for TBLG taking into account both the conventional contribution and the contribution arising from the quantum geometry of the bands. We find that both contributions are larger than one would expect treating the bands as well isolated, that at the magic angle the geometric contribution is larger than the conventional one, and that for small deviations away from the magic angle the conventional contribution is larger than the geometric one. Our results show that, despite the flatness of the bands the superfluid weight in TBLG is finite and consistent with experimental observations. We also show how the superfluid weight can be tuned by varying the chemical potential and the twist angle opening the possibility to tune the nature of the superconducting transition between the standard BCS transition and the Berezinskii-Kosterlitz-Thouless transition.

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.

In-plane anisotropy in twisted bilayer graphene probed by Raman spectroscopy

Monolayer graphene has high symmetrical crystal structure and exhibits in-plane isotropic physical properties. However, twisted bilayer graphene (tBLG) is expected to differ physically, due to the broken symmetry introduced by the interlayer coupling between adjacent graphene layers. This symmetry breaking is usually accompanied by in-plane anisotropy in their electrical, optical and thermal properties. However, the existence of in-plane anisotropy in tBLG has remained evasive until now. Here, an unambiguous identification of the in-plane anisotropy in tBLG is established by angle-resolved polarized Raman spectroscopy. It was found that the double-resonant two-dimensional band is anisotropic. The degree of in-plane anisotropy is found to be dependent on the misorientation angles, which is two- and four-fold for tBLG with misorientation angles of 15° and 20°, respectively. This finding adds a new dimension to the properties of graphene, which opens a possibility to the development of graphene-based angle-resolved photonics and electronics.

Intrinsically undamped plasmon modes in narrow electron bands

Surface plasmons in 2-dimensional electron systems with narrow Bloch bands feature an interesting regime in which Landau damping (dissipation via electron-hole pair excitation) is completely quenched. This surprising behavior is made possible by strong coupling in narrow-band systems characterized by large values of the "fine structure" constant [Formula: see text] Dissipation quenching occurs when dispersing plasmon modes rise above the particle-hole continuum, extending into the forbidden energy gap that is free from particle-hole excitations. The effect is predicted to be prominent in moiré graphene, where at magic twist-angle values, flat bands feature [Formula: see text] The extinction of Landau damping enhances spatial optical coherence. Speckle-like interference, arising in the presence of disorder scattering, can serve as a telltale signature of undamped plasmons directly accessible in near-field imaging experiments.

Correlated insulating and superconducting states in twisted bilayer graphene below the magic angle

The emergence of flat bands and correlated behaviors in "magic angle" twisted bilayer graphene (tBLG) has sparked tremendous interest, though its many aspects are under intense debate. Here we report observation of both superconductivity and the Mott-like insulating state in a tBLG device with a twist angle of ~0.93°, which is smaller than the magic angle by 15%. At an electron concentration of ±5 electrons/moiré unit cell, we observe a narrow resistance peak with an activation energy gap ~0.1 meV. This indicates additional correlated insulating state, and is consistent with theory predicting a high-energy flat band. At doping of ±12 electrons/moiré unit cell we observe resistance peaks arising from the Dirac points in the spectrum. Our results reveal that the "magic" range of tBLG is in fact larger than what is previously expected, and provide a wealth of new information to help decipher the strongly correlated phenomena observed in tBLG.

Electrically Tunable Flat Bands and Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers provide a versatile platform to engineer metamaterials with novel emergent properties by exploiting the resulting geometric moiré superlattice. Such superlattices are known to host bulk valley currents at tiny angles (α≈0.3°) and flat bands at magic angles (α≈1°). We show that tuning the twist angle to α^{*}≈0.8° generates flat bands away from charge neutrality with a triangular superlattice periodicity. When doped with ±6 electrons per moiré cell, these bands are half-filled and electronic interactions produce a symmetry-broken ground state (Stoner instability) with spin-polarized regions that order ferromagnetically. Application of an interlayer electric field breaks inversion symmetry and introduces valley-dependent dispersion that quenches the magnetic order. With these results, we propose a solid-state platform that realizes electrically tunable strong correlations.

Maximized electron interactions at the magic angle in twisted bilayer graphene

The electronic properties of heterostructures of atomically thin van der Waals crystals can be modified substantially by moiré superlattice potentials from an interlayer twist between crystals^1,2. Moiré tuning of the band structure has led to the recent discovery of superconductivity^3,4 and correlated insulating phases⁵ in twisted bilayer graphene (TBG) near the 'magic angle' of twist of about 1.1 degrees, with a phase diagram reminiscent of high-transition-temperature superconductors. Here we directly map the atomic-scale structural and electronic properties of TBG near the magic angle using scanning tunnelling microscopy and spectroscopy. We observe two distinct van Hove singularities (VHSs) in the local density of states around the magic angle, with an energy separation of 57 millielectronvolts that drops to 40 millielectronvolts with high electron/hole doping. Unexpectedly, the VHS energy separation continues to decrease with decreasing twist angle, with a lowest value of 7 to 13 millielectronvolts at a magic angle of 0.79 degrees. More crucial to the correlated behaviour of this material, we find that at the magic angle, the ratio of the Coulomb interaction to the bandwidth of each individual VHS (U/t) is maximized, which is optimal for electronic Cooper pairing mechanisms. When doped near the half-moiré-band filling, a correlation-induced gap splits the conduction VHS with a maximum size of 6.5 millielectronvolts at 1.15 degrees, dropping to 4 millielectronvolts at 0.79 degrees. We capture the doping-dependent and angle-dependent spectroscopy results using a Hartree-Fock model, which allows us to extract the on-site and nearest-neighbour Coulomb interactions. This analysis yields a U/t of order unity indicating that magic-angle TBG is moderately correlated. In addition, scanning tunnelling spectroscopy maps reveal an energy- and doping-dependent three-fold rotational-symmetry breaking of the local density of states in TBG, with the strongest symmetry breaking near the Fermi level and further enhanced when doped to the correlated gap regime. This indicates the presence of a strong electronic nematic susceptibility or even nematic order in TBG in regions of the phase diagram where superconductivity is observed.

All Magic Angles in Twisted Bilayer Graphene are Topological

We show that the electronic structure of the low-energy bands in the small angle-twisted bilayer graphene consists of a series of semimetallic and topological phases. In particular, we are able to prove, using an approximate low-energy particle-hole symmetry, that the gapped set of bands that exist around all magic angles have a nontrivial topology stabilized by a magnetic symmetry, provided band gaps appear at fillings of ±4 electrons per moiré unit cell. The topological index is given as the winding number (a Z number) of the Wilson loop in the moiré Brillouin zone. Furthermore, we also claim that, when the gapped bands are allowed to couple with higher-energy bands, the Z index collapses to a stable Z_{2} index. The approximate, emergent particle-hole symmetry is essential to the topology of graphene: When strongly broken, nontopological phases can appear. Our Letter underpins topology as the crucial ingredient to the description of low-energy graphene. We provide a four-band short-range tight-binding model whose two lower bands have the same topology, symmetry, and flatness as those of the twisted bilayer graphene and which can be used as an effective low-energy model. We then perform large-scale (11000 atoms per unit cell, 40 days per k-point computing time) ab initio calculations of a series of small angles, from 3° to 1°, which show a more complex and somewhat different evolution of the symmetry of the low-energy bands than that of the theoretical moiré model but which confirm the topological nature of the system.

Flatbands and Perfect Metal in Trilayer Moiré Graphene

We investigate the electronic structure of a twisted multilayer graphene system forming a moiré pattern. We consider small twist angles separating the graphene sheets and develop a low-energy theory to describe the coupling of Dirac Bloch states close to the K point in each individual plane. Extending beyond the bilayer case, we show that, when the ratio of the consecutive twist angles is rational, a periodicity emerges in quasimomentum space with moiré Bloch bands even when the system does not exhibit a crystalline lattice structure in real space. For a trilayer geometry, we find flatbands in the spectrum at certain rotation angles. Performing a symmetry analysis of the band model for the trilayer, we prove that the system is a perfect metal in the sense that it is gapless at all energies. This striking result originates from the three Dirac cones which can only gap in pairs and produce bands with an infinite connectivity. It also holds quite generally for multilayer graphene with an odd number of planes under the condition of C_{2z}T symmetry.

Twisted Bilayer Graphene: A Phonon-Driven Superconductor

We study the electron-phonon coupling in twisted bilayer graphene (TBG), which was recently experimentally observed to exhibit superconductivity around the magic twist angle θ≈1.05°. We show that phonon-mediated electron attraction at the magic angle is strong enough to induce a conventional intervalley pairing between graphene valleys K and K^{'} with a superconducting critical temperature T_{c}∼1  K, in agreement with the experiment. We predict that superconductivity can also be observed in TBG at many other angles θ and higher electron densities in higher moiré bands, which may also explain the possible granular superconductivity of highly oriented pyrolytic graphite. We support our conclusions by ab initio calculations.

Ferromagnetic Mott state in Twisted Graphene Bilayers at the Magic Angle

We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with fourfold degeneracy. After calculating the maximally localized superlattice Wannier wave functions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures.

Atomic and electronic reconstruction at the van der Waals interface in twisted bilayer graphene

Control of the interlayer twist angle in two-dimensional van der Waals (vdW) heterostructures enables one to engineer a quasiperiodic moiré superlattice of tunable length scale^1-8. In twisted bilayer graphene, the simple moiré superlattice band description suggests that the electronic bandwidth can be tuned to be comparable to the vdW interlayer interaction at a 'magic angle'⁹, exhibiting strongly correlated behaviour. However, the vdW interlayer interaction can also cause significant structural reconstruction at the interface by favouring interlayer commensurability, which competes with the intralayer lattice distortion^10-16. Here we report atomic-scale reconstruction in twisted bilayer graphene and its effect on the electronic structure. We find a gradual transition from an incommensurate moiré structure to an array of commensurate domains with soliton boundaries as we decrease the twist angle across the characteristic crossover angle, θ_c ≈ 1°. In the solitonic regime (θ < θ_c) where the atomic and electronic reconstruction become significant, a simple moiré band description breaks down and the secondary Dirac bands appear. On applying a transverse electric field, we observe electronic transport along the network of one-dimensional topological channels that surround the alternating triangular gapped domains. Atomic and electronic reconstruction at the vdW interface provide a new pathway to engineer the system with continuous tunability.

A hidden Markov model for describing turbostratic disorder applied to carbon blacks and graphene

A mathematical framework is presented to represent turbostratic disorder in materials like carbon blacks, smectites and twistedn-layer graphene. In particular, the set of all possible disordered layers, including rotated, shifted and curved layers, forms a stochastic sequence governed by a hidden Markov model. The probability distribution over the set of layer types is treated as an element of a Hilbert space and, using the tools of Fourier analysis and functional analysis, expressions are developed for the scattering cross sections of a broad class of disordered materials.

Origin of Magic Angles in Twisted Bilayer Graphene

Twisted bilayer graphene (TBG) was recently shown to host superconductivity when tuned to special "magic angles" at which isolated and relatively flat bands appear. However, until now the origin of the magic angles and their irregular pattern have remained a mystery. Here we report on a fundamental continuum model for TBG which features not just the vanishing of the Fermi velocity, but also the perfect flattening of the entire lowest band. When parametrized in terms of α∼1/θ, the magic angles recur with a remarkable periodicity of Δα≃3/2. We show analytically that the exactly flat band wave functions can be constructed from the doubly periodic functions composed of ratios of theta functions-reminiscent of quantum Hall wave functions on the torus. We further report on the unusual robustness of the experimentally relevant first magic angle, address its properties analytically, and discuss how lattice relaxation effects help justify our model parameters.

Magnetic impurity resonance states for different pairing symmetries in twisted bilayer graphene

In this work, we study the magnetic impurity resonance states in the superconducting phase of 'magic' angle twisted bilayer graphene for different pairing symmetries. Using a two-orbital model on the emergent honeycomb lattice, we find that the resonance states are dramatically different for [Formula: see text]-wave pairing and topological nontrivial pairings. When the magnetic impurity is located at one site of the emergent honeycomb lattice, i.e. the center of the AB spot of the moiré pattern, the spacial distributions of the resonance states will break both the threefold and twofold rotation symmetries of [Formula: see text] point group for pairing symmetries which belong to the irreducible representations of this point group. When the magnetic impurity is located at the center of the emergent honeycomb lattice i.e. the center of the AA spot of the moiré pattern, the appearance of resonance peak at the position close to the impurity can be considered as a strong evidence of non-[Formula: see text]-wave pairing.

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.

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

We investigate the electronic structure of the flat bands induced by moiré superlattices and electric fields in nearly aligned ABC trilayer graphene (TLG) boron-nitride (BN) interfaces where Coulomb effects can lead to correlated gapped phases. Our calculations indicate that valley-spin resolved isolated superlattice flat bands that carry a finite Chern number C=3 proportional to the layer number can appear near charge neutrality for appropriate perpendicular electric fields and twist angles. When the degeneracy of the bands is lifted by Coulomb interactions, these topological bands can lead to anomalous quantum Hall phases that embody orbital and spin magnetism. Narrow bandwidths of ∼10  meV achievable for a continuous range of twist angles θ≲0.6° with moderate interlayer potential differences of ∼50  meV make the TLG-BN systems a promising platform for the study of electric-field tunable Coulomb-interaction-driven spontaneous Hall phases.

Ultraflatbands and Shear Solitons in Moiré Patterns of Twisted Bilayer Transition Metal Dichalcogenides

Ultraflatbands in twisted bilayers of two-dimensional materials have the potential to host strong correlations, including the Mott-insulating phase at half-filling of the band. Using first-principles density functional theory calculations, we show the emergence of ultraflatbands at the valence band edge in twisted bilayer MoS_{2}, a prototypical transition metal dichalcogenide. The computed band widths, 5 and 23 meV for 56.5° and 3.5° twist angles, respectively, are comparable to that of twisted bilayer graphene near "magic" angles. Large structural transformations in the moiré patterns lead to formation of shear solitons at stacking boundaries and strongly influence the electronic structure. We extend our analysis for twisted bilayer MoS_{2} to show that flatbands can occur at the valence band edge of twisted bilayer WS_{2}, MoSe_{2}, and WSe_{2} as well.

Atomic Structure and Mechanical Properties of Twisted Bilayer Graphene

We studied the atomic structure and mechanical properties of twisted bilayer graphene with a different twist angle using molecular dynamic simulations. The two layers are corrugated after energy minimization. We found two different modes of corrugation. The mechanical properties are tested both in-plane and perpendicular to the plane. The in-plane properties are dominated by the orientation of graphene. The perpendicular properties depend on the twist angle, as the larger the twist angle, the higher the intrinsic strength.

Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene

We present a theory of phonon-mediated superconductivity in near magic angle twisted bilayer graphene. Using a microscopic model for phonon coupling to moiré band electrons, we find that phonons generate attractive interactions in both s- and d-wave pairing channels and that the attraction is strong enough to explain the experimental superconducting transition temperatures. Before including Coulomb repulsion, the s-wave channel is more favorable; however, on-site Coulomb repulsion can suppress s-wave pairing relative to d wave. The pair amplitude varies spatially with the moiré period, and is identical in the two layers in the s-wave channel but phase shifted by π in the d-wave channel. We discuss experiments that can distinguish the two pairing states.

Evaluation of Exfoliated Graphite to Graphene in Polyamide 66 Using Novel High Shear Elongational Flow

Graphene has been publicized as the game changing material of this millennium. To this day, scalable production leading to exceptional material properties has been difficult to attain. Most methods require harsh chemicals, which result in destroying the graphene surface. A method was developed, exploiting high speed elongational flow in a novel designed batch mixer; creating a distribution of pristine few to many layer graphene flakes. The method focuses on exfoliating in a molten polyamide 66 (PA66) matrix, creating a graphene reinforced polymer matrix composite (G-PMC). The process revealed that high speed elongational flow was able to create few layer graphene. Graphite exfoliation was found driven in part by diffusion, leading to intercalation of PA66 in graphite. The intercalated structure lead to increases in the hydrogen bonding domain, creating anisotropic crystal domains. The thermal stability of the G-PMC was found to be dependent to the degree of exfoliation, PA66 crystal structure and composite morphology. The aim of this research is to characterize uniquely produced graphene containing polymer matrix composites using a newly created elongational flow field. Using elongational flow, graphite will be directly exfoliated into graphene within a molten polymer.

Electrostatic effects, band distortions, and superconductivity in twisted graphene bilayers

Bilayer graphene twisted by a small angle shows a significant charge modulation away from neutrality, as the charge in the narrow bands near the Dirac point is mostly localized in a fraction of the Moiré unit cell. The resulting electrostatic potential leads to a filling-dependent change in the low-energy bands, of a magnitude comparable to or larger than the bandwidth. These modifications can be expressed in terms of new electron-electron interactions, which, when expressed in a local basis, describe electron-assisted hopping terms. These interactions favor superconductivity at certain fillings.

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.

Electron quantum metamaterials in van der Waals heterostructures

In recent decades, scientists have developed the means to engineer synthetic periodic arrays with feature sizes below the wavelength of light. When such features are appropriately structured, electromagnetic radiation can be manipulated in unusual ways, resulting in optical metamaterials whose function is directly controlled through nanoscale structure. Nature, too, has adopted such techniques-for example in the unique colouring of butterfly wings-to manipulate photons as they propagate through nanoscale periodic assemblies. In this Perspective, we highlight the intriguing potential of designer structuring of electronic matter at scales at and below the electron wavelength, which affords a new range of synthetic quantum metamaterials with unconventional responses. Driven by experimental developments in stacking atomically layered heterostructures-such as mechanical pick-up/transfer assembly-atomic-scale registrations and structures can be readily tuned over distances smaller than characteristic electronic length scales (such as the electron wavelength, screening length and electron mean free path). Yet electronic metamaterials promise far richer categories of behaviour than those found in conventional optical metamaterial technologies. This is because, unlike photons, which scarcely interact with each other, electrons in subwavelength-structured metamaterials are charged and strongly interact. As a result, an enormous variety of emergent phenomena can be expected and radically new classes of interacting quantum metamaterials designed.

Confined states in graphene quantum blisters

Bilayer graphene samples may exhibit regions where the two layers are locally delaminated forming a so-called quantum blister in the graphene sheet. Electron and hole states can be confined in this graphene quantum blisters (GQB) by applying a global electrostatic bias. We scrutinize the electronic properties of these confined states under the variation of interlayer bias, coupling, and blister's size. The spectra display strong anti-crossings due to the coupling of the confined states on upper and lower layers inside the blister. These spectra are layer localized where the respective confined states reside on either layer or equally distributed. For finite angular momentum, this layer localization can be at the edge of the blister and corresponds to degenerate modes of opposite momenta. Furthermore, the energy levels in GQB exhibit electron-hole symmetry that is sensitive to the electrostatic bias. Finally, we demonstrate that confinement in GQB persists even in the presence of a variation in the inter-layer coupling.

Critical Annealing Temperature for Stacking Orientation of Bilayer Graphene

It is rarely reported that stacking orientations of bilayer graphene (BLG) can be manipulated by the annealing process. Most investigators have painstakingly fabricated this BLG by chemical vapor deposition growth or mechanical means. Here, it is discovered that, at ≈600 °C, called the critical annealing temperature (CAT), most stacking orientations collapse into strongly coupled or AB-stacked states. This phenomenon is governed (i) macroscopically by the stress generation and release in top graphene domains, evolving from mild ripples to sharp billows in certain local areas, and (ii) microscopically by the principle of minimal potential obeyed by carbon atoms that have acquired sufficient thermal energy at CAT. Conspicuously, evolutions of stacking orientations in Raman mappings under various annealing temperatures are observed. Furthermore, MoS₂ synthesized on BLG is used to directly observe crystal orientations of top and bottom graphene layers. The finding of CAT provides a guide for the fabrication of strongly coupled or AB-stacked BLG, and can be applied to aligning other 2D heterostructures.

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.

Dirac electrons in a dodecagonal graphene quasicrystal

Quantum states of quasiparticles in solids are dictated by symmetry. We have experimentally demonstrated quantum states of Dirac electrons in a two-dimensional quasicrystal without translational symmetry. A dodecagonal quasicrystalline order was realized by epitaxial growth of twisted bilayer graphene rotated exactly 30°. We grew the graphene quasicrystal up to a millimeter scale on a silicon carbide surface while maintaining the single rotation angle over an entire sample and successfully isolated the quasicrystal from a substrate, demonstrating its structural and chemical stability under ambient conditions. Multiple Dirac cones replicated with the 12-fold rotational symmetry were observed in angle-resolved photoemission spectra, which revealed anomalous strong interlayer coupling with quasi-periodicity. Our study provides a way to explore physical properties of relativistic fermions with controllable quasicrystalline orders.

Interfacial engineering in graphene bandgap

Graphene exhibits superior mechanical strength, high thermal conductivity, strong light-matter interactions, and, in particular, exceptional electronic properties. These merits make graphene an outstanding material for numerous potential applications. However, a graphene-based high-performance transistor, which is the most appealing application, has not yet been produced, which is mainly due to the absence of an intrinsic electronic bandgap in this material. Therefore, bandgap opening in graphene is urgently needed, and great efforts have been made regarding this topic over the past decade. In this review article, we summarise recent theoretical and experimental advances in interfacial engineering to achieve bandgap opening. These developments are divided into two categories: chemical engineering and physical engineering. Chemical engineering is usually destructive to the pristine graphene lattice via chemical functionalization, the introduction of defects, doping, chemical bonds with substrates, and quantum confinement; the latter largely maintains the atomic structure of graphene intact and includes the application of an external field, interactions with substrates, physical adsorption, strain, electron many-body effects and spin-orbit coupling. Although these pioneering works have not met all the requirements for electronic applications of graphene at once, they hold great promise in this direction and may eventually lead to future applications of graphene in semiconductor electronics and beyond.

Electronic Spectrum of Twisted Graphene Layers under Heterostrain

We demonstrate that stacking layered materials allows a strain engineering where each layer is strained independently, which we call heterostrain. We combine detailed structural and spectroscopic measurements with tight-binding calculations to show that small uniaxial heterostrain suppresses Dirac cones and leads to the emergence of flat bands in twisted graphene layers (TGLs). Moreover, we demonstrate that heterostrain reconstructs, much more severely, the energy spectrum of TGLs than homostrain for which both layers are strained identically, a result which should apply to virtually all van der Waals structures opening exciting possibilities for straintronics with 2D materials.

Intralayer and interlayer electron-phonon interactions in twisted graphene heterostructures

The understanding of interactions between electrons and phonons in atomically thin heterostructures is crucial for the engineering of novel two-dimensional devices. Electron–phonon (el–ph) interactions in layered materials can occur involving electrons in the same layer or in different layers. Here we report on the possibility of distinguishing intralayer and interlayer el–ph interactions in samples of twisted bilayer graphene and of probing the intralayer process in graphene/h-BN by using Raman spectroscopy. In the intralayer process, the el–ph scattering occurs in a single graphene layer and the other layer (graphene or h-BN) imposes a periodic potential that backscatters the excited electron, whereas for the interlayer process the el–ph scattering occurs between states in the Dirac cones of adjacent graphene layers. Our methodology of using Raman spectroscopy to probe different types of el–ph interactions can be extended to study any kind of graphene-based heterostructure.

Electron–phonon interactions in van der Waals layered materials can occur either within the same layer (intralayer) or in different layers (interlayer). Here, the authors use multi-wavelength Raman spectroscopy to probe intra- and inter-layer electron–phonon interactions in twisted graphene heterostructures.

Synthesis of a Helical Bilayer Nanographene

A rigid, inherently chiral bilayer nanographene has been synthesized as both the racemate and enantioenriched M isomer (with 93 % ee) in three steps from established helicenes. This folded nanographene is composed of two hexa-peri-hexabenzocoronene layers fused to a [10]helicene, with an interlayer distance of 3.6 Å as determined by X-ray crystallography. The rigidity of the helicene linker forces the layers to adopt a nearly aligned AA-stacked conformation, rarely observed in few-layer graphene. By combining the advantages of nanographenes and helicenes, we have constructed a bilayer system of 30 fused benzene rings that is also chiral, rigid, and remains soluble in common organic solvents. We present this as a molecular model system of bilayer graphene, with properties of interest in a variety of potential applications.

Raman spectroscopy of graphene-based materials and its applications in related devices

Graphene-based materials exhibit remarkable electronic, optical, and mechanical properties, which has resulted in both high scientific interest and huge potential for a variety of applications. Furthermore, the family of graphene-based materials is growing because of developments in preparation methods. Raman spectroscopy is a versatile tool to identify and characterize the chemical and physical properties of these materials, both at the laboratory and mass-production scale. This technique is so important that most of the papers published concerning these materials contain at least one Raman spectrum. Thus, here, we systematically review the developments in Raman spectroscopy of graphene-based materials from both fundamental research and practical (i.e., device applications) perspectives. We describe the essential Raman scattering processes of the entire first- and second-order modes in intrinsic graphene. Furthermore, the shear, layer-breathing, G and 2D modes of multilayer graphene with different stacking orders are discussed. Techniques to determine the number of graphene layers, to probe resonance Raman spectra of monolayer and multilayer graphenes and to obtain Raman images of graphene-based materials are also presented. The extensive capabilities of Raman spectroscopy for the investigation of the fundamental properties of graphene under external perturbations are described, which have also been extended to other graphene-based materials, such as graphene quantum dots, carbon dots, graphene oxide, nanoribbons, chemical vapor deposition-grown and SiC epitaxially grown graphene flakes, composites, and graphene-based van der Waals heterostructures. These fundamental properties have been used to probe the states, effects, and mechanisms of graphene materials present in the related heterostructures and devices. We hope that this review will be beneficial in all the aspects of graphene investigations, from basic research to material synthesis and device applications.

Chiral Response of Twisted Bilayer Graphene

We present an effective (minimal) theory for chiral two-dimensional materials. These materials possess an electromagnetic coupling without exhibiting a topological gap. As an example, we study the response of doped twisted bilayers, unveiling unusual phenomena in the zero frequency limit. An in-plane magnetic field induces a huge paramagnetic response at the neutrality point and, upon doping, also gives rise to a substantial longitudinal Hall response. The system also accommodates nontrivial longitudinal plasmonic modes that are associated with a longitudinal magnetic moment, thus endowing them with a chiral character. Finally, we note that the optical activity can be considerably enhanced upon doping and our general approach would enable systematic exploration of 2D material heterostructures with optical activity.

Real-Space Imaging of the Tailored Plasmons in Twisted Bilayer Graphene

We report a systematic plasmonic study of twisted bilayer graphene (TBLG)-two graphene layers stacked with a twist angle. Through real-space nanoimaging of TBLG single crystals with a wide distribution of twist angles, we find that TBLG supports confined infrared plasmons that are sensitively dependent on the twist angle. At small twist angles, TBLG has a plasmon wavelength comparable to that of single-layer graphene. At larger twist angles, the plasmon wavelength of TBLG increases significantly with apparently lower damping. Further analysis and modeling indicate that the observed twist-angle dependence of TBLG plasmons in the Dirac linear regime is mainly due to the Fermi-velocity renormalization, a direct consequence of interlayer electronic coupling. Our work unveils the tailored plasmonic characteristics of TBLG and deepens our understanding of the intriguing nano-optical physics in novel van der Waals coupled two-dimensional materials.

Seebeck Coefficient of a Single van der Waals Junction in Twisted Bilayer Graphene

When two planar atomic membranes are placed within the van der Waals distance, the charge and heat transport across the interface are coupled by the rules of momentum conservation and structural commensurability, leading to outstanding thermoelectric properties. Here we show that an effective "interlayer phonon drag" determines the Seebeck coecient (S) across the van der Waals gap formed in twisted bilayer graphene (tBLG). The cross-plane thermovoltage, which is nonmonotonic in both temperature and density, is generated through scattering of electrons by the out-of-plane layer breathing (ZO'/ZA₂) phonon modes and differs dramatically from the expected Landauer-Buttiker formalism in conventional tunnel junctions. The tunability of the cross-plane Seebeck effect in van der Waals junctions may be valuable in creating a new genre of versatile thermoelectric systems with layered solids.

Quantum transport across van der Waals domain walls in bilayer graphene

Bilayer graphene can exhibit deformations such that the two graphene sheets are locally detached from each other resulting in a structure consisting of domains with different van der Waals inter-layer coupling. Here we investigate how the presence of these domains affects the transport properties of bilayer graphene. We derive analytical expressions for the transmission probability, and the corresponding conductance, across walls separating different inter-layer coupling domains. We find that the transmission can exhibit a valley-dependent layer asymmetry and that the domain walls have a considerable effect on the chiral tunnelling properties of the charge carriers. We show that transport measurements allow one to obtain the strength with which the two layers are coupled. We perform numerical calculations for systems with two domain walls and find that the availability of multiple transport channels in bilayer graphene significantly modifies the conductance dependence on inter-layer potential asymmetry.

Electrically Controllable Magnetism in Twisted Bilayer Graphene

Twisted graphene bilayers develop highly localized states around AA-stacked regions for small twist angles. We show that interaction effects may induce either an antiferromagnetic or a ferromagnetic (FM) polarization of said regions, depending on the electrical bias between layers. Remarkably, FM-polarized AA regions under bias develop spiral magnetic ordering, with a relative 120° misalignment between neighboring regions due to a frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges naturally without the need of spin-orbit coupling, and competes with the more conventional lattice-antiferromagnetic instability, which interestingly develops at smaller bias under weaker interactions than in monolayer graphene, due to Fermi velocity suppression. This rich and electrically controllable magnetism could turn twisted bilayer graphene into an ideal system to study frustrated magnetism in two dimensions.

Continuous Films of Self-Assembled Graphene Quantum Dots for n-Type Doping of Graphene by UV-Triggered Charge Transfer

The demands to examine components serving as one of the active layers in heterostructures of 2D materials have been recently increasing. Nanomaterials synthesized from a solution process and their self-assembly can provide a promising route to build a new type of mixed dimensional heterostructures, and several methodologies have been reported previously to construct 2D assemblies from colloidal nanostructures in solution. Graphene quantum dots (GQDs), receiving much interest due to the tunable optical band gap and the capability of chemical functionalization, are considered as emerging nanomaterials for various optoelectronic and biological applications. This study fabricates a closely packed GQDs film (GQDF) from colloidal solutions using a solvent-assisted Langmuir Blodgett method, and investigates the optical and electrical characteristics of the heterostacked graphene/GQD film (G/GQDF) structures. It is observed that the GQDF plays a role not only as a buffer layer that isolates Chemical Vapor Deposited graphene (CVD graphene) from undesired p-doping but also as a photoactive layer that triggers n-doping of the heterostacked CVD graphene film. The n-doping density of the G/GQDF device is proportional to UV irradiation time, but its carrier mobility remains constant regardless of doping densities, which are unique characteristics that have not been observed in other doping methods.

Evolution of the electronic band structure of twisted bilayer graphene upon doping

The electronic band structure of twisted bilayer graphene develops van Hove singularities whose energy depends on the twist angle between the two layers. Using Raman spectroscopy, we monitor the evolution of the electronic band structure upon doping using the G peak area which is enhanced when the laser photon energy is resonant with the energy separation of the van Hove singularities. Upon charge doping, the Raman G peak area initially increases for twist angles larger than a critical angle and decreases for smaller angles. To explain this behavior with twist angle, the energy separation of the van Hove singularities must decrease with increasing charge density demonstrating the ability to modify the electronic and optical properties of twisted bilayer graphene with doping.

Superlubricity of a graphene/MoS2 heterostructure: a combined experimental and DFT study

Graphene and other two-dimensional materials have been proved to be able to offer low friction. Here we assembled van der Waals heterostructures with graphene and molybdenum disulphide monolayers. The Raman spectrum together with a modified linear chain model indicate a two-orders-of-magnitude decrease in the interlayer lateral force constant, as compared with their homogeneous bilayers, indicating a possible routine to achieve superlubricity. The decrease in the interlayer lateral force constant is consistent with the ultrasmall potential energy corrugation during sliding, which is derived from density functional theory calculations. The potential energy corrugation is found to be determined by the sliding-induced interfacial charge density fluctuation, suggesting a new perspective to understand the physical origin of the atomic scale friction of two-dimensional materials.

Substrate Doping Effect and Unusually Large Angle van Hove Singularity Evolution in Twisted Bi- and Multilayer Graphene

Graphene has demonstrated great potential in new-generation electronic applications due to its unique electronic properties such as large carrier Fermi velocity, ultrahigh carrier mobility, and high material stability. Interestingly, the electronic structures can be further engineered in multilayer graphene by the introduction of a twist angle between different layers to create van Hove singularities (vHSs) at adjustable binding energy. In this work, using angle-resolved photoemission spectroscopy with sub-micrometer spatial resolution, the band structures and their evolution are systematically studied with twist angle in bilayer and trilayer graphene sheets. A doping effect is directly observed in graphene multilayer system as well as vHSs in bilayer graphene over a wide range of twist angles (from 5° to 31°) with wide tunable energy range over 2 eV. In addition, the formation of multiple vHSs (at different binding energies) is also observed in trilayer graphene. The large tuning range of vHS binding energy in twisted multilayer graphene provides a promising material base for optoelectrical applications with broadband wavelength selectivity from the infrared to the ultraviolet regime, as demonstrated by an example application of wavelength selective photodetector.

Tunable moiré bands and strong correlations in small-twist-angle bilayer graphene

According to electronic structure theory, bilayer graphene is expected to have anomalous electronic properties when it has long-period moiré patterns produced by small misalignments between its individual layer honeycomb lattices. We have realized bilayer graphene moiré crystals with accurately controlled twist angles smaller than 1° and studied their properties using scanning probe microscopy and electron transport. We observe conductivity minima at charge neutrality, satellite gaps that appear at anomalous carrier densities for twist angles smaller than 1°, and tunneling densities-of-states that are strongly dependent on carrier density. These features are robust up to large transverse electric fields. In perpendicular magnetic fields, we observe the emergence of a Hofstadter butterfly in the energy spectrum, with fourfold degenerate Landau levels, and broken symmetry quantum Hall states at filling factors ±1, 2, 3. These observations demonstrate that at small twist angles, the electronic properties of bilayer graphene moiré crystals are strongly altered by electron-electron interactions.