Graphene bilayer with a twist: electronic structure

Tunable exciton funnel using Moiré superlattice in twisted van der Waals bilayer

A spatially varying bandgap drives exciton motion and can be used to funnel energy within a solid (Nat. Photonics 2012, 6, 866-872). This bandgap modulation can be created by composition variation (traditional heterojunction), elastic strain, or in the work shown next, by a small twist between two identical semiconducting atomic sheets, creating an internal stacking translation u(r) that varies gently with position r and controls the local bandgap Eg(u(r)). Recently synthesized carbon/boron nitride (Nat. Nanotechnol. 2013, 8, 119) and phosphorene (Nat. Nanotechnol. 2014, 9, 372) may be used to construct this twisted semiconductor bilayer that may be regarded as an in-plane crystal but an out-of-plane molecule, which could be useful in solar energy harvesting and electroluminescence. Here, by first-principles methods, we compute the bandgap map and delineate its material and geometric sensitivities. Eg(u(r)) is predicted to have multiple local minima ("funnel centers") due to secondary or even tertiary periodic structures in-plane, leading to a hitherto unreported pattern of multiple "exciton flow basins". A compressive strain or electric field will further enhance Eg-contrast in different regions of the pseudoheterostructure so as to absorb or emit even broader spectrum of light.

Four-fold Raman enhancement of 2D band in twisted bilayer graphene: evidence for a doubly degenerate Dirac band and quantum interference

We report the observation of a strong 2D band Raman in twisted bilayer graphene (tBLG) with large rotation angles under 638 nm and 532 nm visible laser excitations. The 2D band Raman intensity increased four-fold as opposed to the two-fold increase observed in single-layer graphene. The same tBLG samples also exhibited rotation-dependent G-line resonances and folded phonons under 364 nm UV laser excitation. We attribute this 2D band Raman enhancement to the constructive interference between two double-resonance Raman pathways, which were enabled by a nearly degenerate Dirac band in the tBLG Moiré superlattices.

Gating electron-hole asymmetry in twisted bilayer graphene

Electron-hole symmetry is one of the unique properties of graphene that is generally absent in most semiconductors because of the different conduction and valence band structures. Here we report on the manipulation of electron-hole symmetry in the low-energy band structure of twisted bilayer graphene, where symmetric saddle points form in the conduction and valence bands as a result of interlayer coupling. By applying a gate voltage to a twisted bilayer with a critical rotation angle, enhanced electron resonance between the two saddle points can be turned on or off, depending on the electron-hole symmetry near the saddle points. The appearance of a 2D(+) peak, a gate-tunable Raman feature found near the critical angle, indicates a reduction of Fermi velocity in the vicinity of the saddle point to/from which electrons are inelastically scattered by phonons in the round trip of the double-resonance process.

Raman identification of edge alignment of bilayer graphene down to the nanometer scale

The ideal edges of bilayer graphene (BLG) are that the edges of the top and bottom graphene layers (GLs) of BLG are well-aligned. Actually, the alignment distance between the edges of the top and bottom GLs of a real BLG can be as large as the submicrometer scale or as small as zero, which cannot be distinguished using an optical microscope. Here, we present a detailed Raman study on BLG at its edges. If the alignment distance of the top and bottom GLs of BLG is larger than the laser spot, the measured D mode at the edge of the top GL of BLG shows a similar spectral profile to that of disordered BLG. If the alignment distance is smaller than the laser spot, the D mode at a real BLG edge shows three typical spectral profiles similar to that at the edge of SLG, that of the well-aligned edge of BLG, or a combination of both. We show the sensitivity and ability of Raman spectroscopy to acquire the alignment distance between two edges of top and bottom GLs of BLG as small as several nanometers, which is far beyond the diffraction limit of a laser spot. This work opens the possibility to probe the edge alignment of multi-layer graphene.

Layer anti-ferromagnetism on bilayer honeycomb lattice

Bilayer honeycomb lattice, with inter-layer tunneling energy, has a parabolic dispersion relation, and the inter-layer hopping can cause the charge imbalance between two sublattices. Here, we investigate the metal-insulator and magnetic phase transitions on the strongly correlated bilayer honeycomb lattice by cellular dynamical mean-field theory combined with continuous time quantum Monte Carlo method. The procedures of magnetic spontaneous symmetry breaking on dimer and non-dimer sites are different, causing a novel phase transition between normal anti-ferromagnet and layer anti-ferromagnet. The whole phase diagrams about the magnetism, temperature, interaction and inter-layer hopping are obtained. Finally, we propose an experimental protocol to observe these phenomena in future optical lattice experiments.

Van Hove singularities and excitonic effects in the optical conductivity of twisted bilayer graphene

We report a systematic study of the optical conductivity of twisted bilayer graphene (tBLG) across a large energy range (1.2-5.6 eV) for various twist angles, combined with first-principles calculations. At previously unexplored high energies, our data show signatures of multiple van Hove singularities (vHSs) in the tBLG bands as well as the nonlinearity of the single layer graphene bands and their electron-hole asymmetry. Our data also suggest that excitonic effects play a vital role in the optical spectra of tBLG. Including electron-hole interactions in first-principles calculations is essential to reproduce the shape of the conductivity spectra, and we find evidence of coherent interactions between the states associated with the multiple vHSs in tBLG.

Evidence of van Hove singularities in ordered grain boundaries of graphene

It has long been under debate whether the electron transport performance of graphene could be enhanced by the possible occurrence of van Hove singularities in grain boundaries. Here, we provide direct experimental evidence to confirm the existence of van Hove singularity states close to the Fermi energy in certain ordered grain boundaries using scanning tunneling microscopy. The intrinsic atomic and electronic structures of two ordered grain boundaries, one with alternative pentagon and heptagon rings and the other with alternative pentagon pair and octagon rings, are determined. It is firmly verified that the carrier concentration and, thus, the conductance around ordered grain boundaries can be significantly enhanced by the van Hove singularity states. This finding strongly suggests that a graphene nanoribbon with a properly embedded ordered grain boundary can be a promising structure to improve the performance of graphene-based electronic devices.

Graphene–boron nitride superlattices: the role of point defects at the BN layer

We investigate, by means of first-principles calculations, the role of hBN point defects on the energetical stability and electronic structure of heterostructures composed of graphene atop hBN, rotated at angles of 13.17°, 9.43° and 7.34°. We consider, as possible point defects, boron and nitrogen vacancies and antisites, substitutional oxygen at the nitrogen site ON, substitutional carbon dimers, and nitrogen interstitials. The electronic and structural properties of all defects were analyzed. Among these, the most stable is ON, with negative formation energies at several possible rotation angles and chemical environments. Under such conditions, ON doping can raise the Fermi level of the neutral system by as much as 1 eV relative to graphene's Dirac point, reaching the band crossing between adjacent Dirac cones at the M point of the heterostructure Brillouin zone. This could lead to interesting electronic transport properties without the need for electrostatic doping.

Plasmonics in Dirac systems: from graphene to topological insulators

Recent developments in the emerging field of plasmonics in graphene and other Dirac systems are reviewed and a comprehensive introduction to the standard models and techniques is given. In particular, we discuss intrinsic plasmon excitation of single and bilayer graphene via hydrodynamic equations and the random phase approximation, but also comment on double and multilayer structures. Additionally, we address Dirac systems in the retardation limit and also with large spin–orbit coupling including topological insulators. Finally, we summarize basic properties of the charge, current and photon linear response functions in an appendix.

Proximity effect in graphene-topological-insulator heterostructures

We formulate a continuum model to study the low-energy electronic structure of heterostructures formed by graphene on a strong three-dimensional topological insulator (TI) for the cases of both commensurate and incommensurate stacking. The incommensurability can be due to a twist angle between graphene and the TI surface or a lattice mismatch between the two systems. We find that the proximity of the TI induces in graphene a strong enhancement of the spin-orbit coupling that can be tuned via the twist angle.

Lattice match and lattice mismatch models of graphene on hexagonal boron nitride from first principles

The interface between graphene and hexagonal boron nitride (h-BN) substrate plays an important role in device applications. Previously, theoretical studies have suggested that a small gap is opened at Dirac cones of graphene, but there is no detectable band gap in experiments. To explain the experimental result, we used two models from the views of lattice match and lattice mismatch between graphene and h-BN by first-principles calculations. We first studied the landscapes of the sliding energy surface (SES) and band gap of graphene on h-BN substrate within a lattice match approximation, which mimics continuously variable stacking sequences in a long-period graphene/BN Moiré superstructure arising from minor lattice mismatch. The plausibility of the long-period Moiré superstructure was evidenced by the smooth SES. The main features of the SES landscape can be captured by means of a simple registry index method. For most stacking patterns, the interactions between graphene and h-BN substrate open a band gap at the Dirac cones of graphene. However, there are special stacking modes in the landscape that preserve the Dirac cones of graphene. To further simulate the long-period graphene/BN Moiré superstructure observed in experiments, we also employed a rotation model within the lattice mismatch approximation. At the equilibrium interlayer spacing, the Dirac cones of graphene are preserved in all the rotational graphene/BN superstructures. The zero-band-gap feature is independent of the translation and rotation of graphene with respect to the h-BN substrate, which clearly agrees with the results of zero band gap in experiments.

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.

Multiple π-bands and Bernal stacking of multilayer graphene on C-face SiC, revealed by nano-Angle Resolved Photoemission

Only a single linearly dispersing π-band cone, characteristic of monolayer graphene, has so far been observed in Angle Resolved Photoemission (ARPES) experiments on multilayer graphene grown on C-face SiC. A rotational disorder that effectively decouples adjacent layers has been suggested to explain this. However, the coexistence of μm-sized grains of single and multilayer graphene with different azimuthal orientations and no rotational disorder within the grains was recently revealed for C-face graphene, but conventional ARPES still resolved only a single π-band. Here we report detailed nano-ARPES band mappings of individual graphene grains that unambiguously show that multilayer C-face graphene exhibits multiple π-bands. The band dispersions obtained close to the moreover clearly indicate, when compared to theoretical band dispersion calculated in the framework of the density functional method, Bernal (AB) stacking within the grains. Thus, contrary to earlier claims, our findings imply a similar interaction between graphene layers on C-face and Si-face SiC.

Heating Isotopically Labeled Bernal Stacked Graphene: A Raman Spectroscopy Study

One of the greatest issues of nanoelectronics today is how to control the heating of the components. Graphene is a promising material in this area, and it is essential to study its thermal properties. Here, the effect of heating a bilayer structure was investigated using in situ Raman spectroscopy. In order to observe the effects on each individual layer, an isotopically labeled bilayer graphene was synthesized where the two layers were composed of different carbon isotopes. Therefore, the frequency of the phonons in the Raman spectra was shifted in relation to each other. This technique was used to investigate the influence of different stacking order. It was found that in bilayer graphene grown by chemical vapor deposition (CVD), the two layers behave very similarly for both Bernal stacking and randomly oriented structures, while for transferred samples, the layers act more independently. This highlights a significant dependence on the sample preparation procedure.

Probing structural inhomogeneity of graphene layers via nonlinear optical scattering

Incoherent optical second harmonic generation (SHG) is studied from series of multilayer graphene samples of various thickness manufactured by chemical vapor deposition technique and deposited over 150 μm thick glass slides. Two different values of the correlation lengths are obtained from the linear and SHG indicatrices and reveal the existence of two types of optical scatterers. The first one is associated with homogeneous graphene areas, while the second one originates from wrinkles at the interdomain boundaries. Second harmonic imaging microscopy used to map the distribution of the second-order polarization at the nanoscale confirms the results of the nonlinear scattering data.

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.

Stacking order dependent second harmonic generation and topological defects in h-BN bilayers

The ability to control the stacking structure in layered materials could provide an exciting approach to tuning their optical and electronic properties. Because of the lower symmetry of each constituent monolayer, hexagonal boron nitride (h-BN) allows more structural variations in multiple layers than graphene; however, the structure-property relationships in this system remain largely unexplored. Here, we report a strong correlation between the interlayer stacking structures and optical and topological properties in chemically grown h-BN bilayers, measured mainly by using dark-field transmission electron microscopy (DF-TEM) and optical second harmonic generation (SHG) mapping. Our data show that there exist two distinct h-BN bilayer structures with different interlayer symmetries that give rise to a distinct difference in their SHG intensities. In particular, the SHG signal in h-BN bilayers is observed only for structures with broken inversion symmetry, with an intensity much larger than that of single layer h-BN. In addition, our DF-TEM data identify the formation of interlayer topological defects in h-BN bilayers, likely induced by local strain, whose properties are determined by the interlayer symmetry and the different interlayer potential landscapes.

Polycrystallinity and stacking in CVD graphene

Graphene, a truly two-dimensional hexagonal lattice of carbon atoms, possesses remarkable properties not seen in any other material, including ultrahigh electron mobility, high tensile strength, and uniform broadband optical absorption. While scientists initially studied its intrinsic properties with small, mechanically exfoliated graphene crystals found randomly, applying this knowledge would require growing large-area films with uniform structural and physical properties. The science of graphene has recently experienced revolutionary change, mainly due to the development of several large-scale growth methods. In particular, graphene synthesis by chemical vapor deposition (CVD) on copper is a reliable method to obtain films with mostly monolayer coverage. These films are also polycrystalline, consisting of multiple graphene crystals joined by grain boundaries. In addition, portions of these graphene films contain more than one layer, and each layer can possess a different crystal orientation and stacking order. In this Account, we review the structural and physical properties that originate from polycrystallinity and stacking in CVD graphene. To begin, we introduce dark-field transmission electron microscopy (DF-TEM), a technique which allows rapid and accurate imaging of key structural properties, including the orientation of individual domains and relative stacking configurations. Using DF-TEM, one can easily identify "lateral junctions," or grain boundaries between adjacent domains, as well as "vertical junctions" from the stacking of graphene multilayers. With this technique, we can distinguish between oriented (Bernal or rhombohedral) and misoriented (twisted) configurations. The structure of lateral junctions in CVD graphene is sensitive to growth conditions and is reflected in the material's electrical and mechanical properties. In particular, grain boundaries in graphene grown under faster reactant flow conditions have no gaps or overlaps, unlike more slowly grown films. These structural differences can affect the material's electrical properties: for example, better-connected grain boundaries are more electrically conductive. However, grain boundaries in general are mechanically weaker than pristine graphene, which is an order of magnitude stronger than CVD graphene based on indentation measurements performed with an atomic force microscope. Vertical junctions in multilayer CVD graphene have two key structural features. First, bilayer graphene (BLG) with Bernal stacking exists in two mirrored configurations (AB or AC) that also form isolated domains. Similarly, oriented trilayer graphene also has alternating ABA and ABC stacked layers. Second, in twisted multilayer graphene, stacked layers lack long-range atomic registry and can move freely relative to each other, which generates unique optical properties. In particular, an interlayer optical excitation produces strong Raman and absorption peaks, dependent on the twist angle. A better understanding of the structural and physical properties of grain boundaries and multilayers in CVD graphene is central to realizing the full potential of graphene in large-scale applications. In addition, these studies provide a model for characterizing other layered materials, such as hexagonal boron nitride and MoS2, where similar polycrystallinity and stacking are expected when grown in large areas.

Routes to rupture and folding of graphene on rough 6H-SiC(0001) and their identification

Twisted few layer graphene (FLG) is highly attractive from an application point of view, due to its extraordinary electronic properties. In order to study its properties, we demonstrate and discuss three different routes to in situ create and identify (twisted) FLG. Single layer graphene (SLG) sheets mechanically exfoliated under ambient conditions on 6H-SiC(0001) are modified by (i) swift heavy ion (SHI) irradiation, (ii) by a force microscope tip and (iii) by severe heating. The resulting surface topography and the surface potential are investigated with non-contact atomic force microscopy (NC-AFM) and Kelvin probe force microscopy (KPFM). SHI irradiation results in rupture of the SLG sheets, thereby creating foldings and bilayer graphene (BLG). Applying the other modification methods creates enlarged (twisted) graphene foldings that show rupture along preferential edges of zigzag and armchair type. Peeling at a folding over an edge different from a low index crystallographic direction can result in twisted BLG, showing a similar height as Bernal (or AA-stacked) BLG in NC-AFM images. The rotational stacking can be identified by a significant contrast in the local contact potential difference (LCPD) measured by KPFM.

Coexisting massive and massless Dirac fermions in symmetry-broken bilayer graphene

Charge carriers in bilayer graphene are widely believed to be massive Dirac fermions that have a bandgap tunable by a transverse electric field. However, a full transport gap, despite its importance for device applications, has not been clearly observed in gated bilayer graphene, a long-standing puzzle. Moreover, the low-energy electronic structure of bilayer graphene is widely held to be unstable towards symmetry breaking either by structural distortions, such as twist, strain, or electronic interactions that can lead to various ground states. Which effect dominates the physics at low energies is hotly debated. Here we show both by direct band-structure measurements and by calculations that a native imperfection of bilayer graphene, a distribution of twists whose size is as small as ~0.1°, is sufficient to generate a completely new electronic spectrum consisting of massive and massless Dirac fermions. The massless spectrum is robust against strong electric fields, and has a unusual topology in momentum space consisting of closed arcs having an exotic chiral pseudospin texture, which can be tuned by varying the charge density. The discovery of this unusual Dirac spectrum not only complements the framework of massive Dirac fermions, widely relevant to charge transport in bilayer graphene, but also supports the possibility of valley Hall transport.

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.

Hyperspectral imaging of structure and composition in atomically thin heterostructures

Precise vertical stacking and lateral stitching of two-dimensional (2D) materials, such as graphene and hexagonal boron nitride (h-BN), can be used to create ultrathin heterostructures with complex functionalities, but this diversity of behaviors also makes these new materials difficult to characterize. We report a DUV-vis-NIR hyperspectral microscope that provides imaging and spectroscopy at energies of up to 6.2 eV, allowing comprehensive, all-optical mapping of chemical composition in graphene/h-BN lateral heterojunctions and interlayer rotations in twisted bilayer graphene (tBLG). With the addition of transmission electron microscopy, we obtain quantitative structure-property relationships, confirming the formation of interfaces in graphene/h-BN lateral heterojunctions that are abrupt on a micrometer scale, and a one-to-one relationship between twist angle and interlayer optical resonances in tBLG. Furthermore, we perform similar hyperspectral imaging of samples that are supported on a nontransparent silicon/SiO2 substrate, enabling facile fabrication of atomically thin heterostructure devices with known composition and structure.

Chiral tunneling in a twisted graphene bilayer

The perfect transmission in a graphene monolayer and the perfect reflection in a Bernal graphene bilayer for electrons incident in the normal direction of a potential barrier are viewed as two incarnations of the Klein paradox. Here we show a new and unique incarnation of the Klein paradox. Owing to the different chiralities of the quasiparticles involved, the chiral fermions in a twisted graphene bilayer show an adjustable probability of chiral tunneling for normal incidence: they can be changed from perfect tunneling to partial or perfect reflection, or vice versa, by controlling either the height of the barrier or the incident energy. As well as addressing basic physics about how the chiral fermions with different chiralities tunnel through a barrier, our results provide a facile route to tune the electronic properties of the twisted graphene bilayer.

The electronic properties of bilayer graphene

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.

Free radical reactions in two dimensions: a case study on photochlorination of graphene

Graphene, a two-dimensional giant-molecule of sp(2) -bonded carbon atoms, provides a perfect platform for studying free radical reaction chemistry in two-dimensions, which holds promise to control the chemical functionality of graphene. Free-radical photochlorination of graphene is used as an example to investigate the thickness, stacking order, and single- and double-side dependent reactivity in graphene. Anomalously low reactivity is observed in the photochlorination of AB-stacked bilayer graphene in comparison with that of few-layer graphene. Double-sided chlorination of graphene shows higher reactivity and chlorine coverage than single-sided reaction. It is also experimentally and theoretically demonstrated that chlorine free radicals at low coverage prefer to form a stable charge-transfer complex with graphene, which highly enhances graphene's conductivity and simultaneously generates a pseudo-bandgap through noninvasive doping. Moreover, the initial accumulation of chlorine radicals is considered as the rate-determining step of photochlorination of graphene.

Effect of surface morphology on friction of graphene on various substrates

The friction of graphene on various substrates, such as SiO2, h-BN, bulk-like graphene, and mica, was investigated to characterize the adhesion level between graphene and the underlying surface. The friction of graphene on SiO2 decreased with increasing thickness and converged around the penta-layers due to incomplete contact between the two surfaces. However, the friction of graphene on an atomically flat substrate, such as h-BN or bulk-like graphene, was low and comparable to that of bulk-like graphene. In contrast, the friction of graphene folded onto bulk-like graphene was indistinguishable from that of mono-layer graphene on SiO2 despite the ultra-smoothness of bulk-like graphene. The characterization of the graphene's roughness before and after folding showed that the corrugation of graphene induced by SiO2 morphology was preserved even after it was folded onto an atomically flat substrate. In addition, graphene deposited on mica, when folded, preserved the same corrugation level as before the folding event. Our friction measurements revealed that graphene, once exfoliated from the bulk crystal, tends to maintain its corrugation level even after it is folded onto an atomically flat substrate and that ultra-flatness in both graphene and the substrate is required to achieve the intimate contact necessary for strong adhesion.

Twisting bilayer graphene superlattices

Bilayer graphene is an intriguing material in that its electronic structure can be altered by changing the stacking order or the relative twist angle, yielding a new class of low-dimensional carbon system. Twisted bilayer graphene can be obtained by (i) thermal decomposition of SiC; (ii) chemical vapor deposition (CVD) on metal catalysts; (iii) folding graphene; or (iv) stacking graphene layers one atop the other, the latter of which suffers from interlayer contamination. Existing synthesis protocols, however, usually result in graphene with polycrystalline structures. The present study investigates bilayer graphene grown by ambient pressure CVD on polycrystalline Cu. Controlling the nucleation in early stage growth allows the constituent layers to form single hexagonal crystals. New Raman active modes are shown to result from the twist, with the angle determined by transmission electron microscopy. The successful growth of single-crystal bilayer graphene provides an attractive jumping-off point for systematic studies of interlayer coupling in misoriented few-layer graphene systems with well-defined geometry.

Stacking stability, emergence of magnetization and electromechanical nanosensing in bilayer graphene nanoribbons

We study the electronic and magnetic structures of bilayer graphene nanoribbons (BGNRs) beyond the conventional AA and AB stackings, by using density functional theory within both local density and generalized gradient approximations (LDA and GGA). Our results show that, irrespective of the method chosen, stacking arrangements other than the conventional ones are most stable, and result in significant modification of BGNR characteristics. The most stable bilayer armchair and zigzag structures with a width of ~1 nm are semiconducting with band gaps of 0.04 and 0.05 eV, respectively. We show mechanical shift evolution of magnetic states and the emergence of magnetization upon mechanical deformation in bilayer zigzag GNRs. Band gap dependence on mechanical shift can be used to design accurate nanosensors.

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.

Optical probing of the electronic interaction between graphene and hexagonal boron nitride

Even weak van der Waals (vdW) adhesion between two-dimensional solids may perturb their various materials properties owing to their low dimensionality. Although the electronic structure of graphene has been predicted to be modified by the vdW interaction with other materials, its optical characterization has not been successful. In this report, we demonstrate that Raman spectroscopy can be utilized to detect a few percent decrease in the Fermi velocity (v(F)) of graphene caused by the vdW interaction with underlying hexagonal boron nitride (hBN). Our study also establishes Raman spectroscopic analysis which enables separation of the effects by the vdW interaction from those by mechanical strain or extra charge carriers. The analysis reveals that spectral features of graphene on hBN are mainly affected by change in v(F) and mechanical strain but not by charge doping, unlike graphene supported on SiO₂ substrates. Graphene on hBN was also found to be less susceptible to thermally induced hole doping.

Terahertz conductivity of twisted bilayer graphene

Using terahertz time-domain spectroscopy, the real part of optical conductivity [σ(1)(ω)] of twisted bilayer graphene was obtained at different temperatures (10-300 K) in the frequency range 0.3-3 THz. On top of a Drude-like response, we see a strong peak in σ(1)(ω) at ~2.7 THz. We analyze the overall Drude-like response using a disorder-dependent (unitary scattering) model, then attribute the peak at 2.7 THz to an enhanced density of states at that energy, which is caused by the presence of a van Hove singularity arising from a commensurate twisting of the two graphene layers.

Electronic hybridization of large-area stacked graphene films

Direct, tunable coupling between individually assembled graphene layers is a next step toward designer two-dimensional (2D) crystal systems, with relevance for fundamental studies and technological applications. Here we describe the fabrication and characterization of large-area (>cm(2)), coupled bilayer graphene on SiO(2)/Si substrates. Stacking two graphene films leads to direct electronic interactions between layers, where the resulting film properties are determined by the local twist angle. Polycrystalline bilayer films have a "stained-glass window" appearance explained by the emergence of a narrow absorption band in the visible spectrum that depends on twist angle. Direct measurement of layer orientation via electron diffraction, together with Raman and optical spectroscopy, confirms the persistence of clean interfaces over large areas. Finally, we demonstrate that interlayer coupling can be reversibly turned off through chemical modification, enabling optical-based chemical detection schemes. Together, these results suggest that 2D crystals can be individually assembled to form electronically coupled systems suitable for large-scale applications.

Origin of indirect optical transitions in few-layer MoS2, WS2, and WSe2

It has been well-established that single layer MX2 (M = Mo, W and X = S, Se) are direct gap semiconductors with band edges coinciding at the K point in contrast to their indirect gap multilayer counterparts. In few-layer MX2, there are two valleys along the Γ-K line with similar energy. There is little understanding on which of the two valleys forms the conduction band minimum (CBM) in this thickness regime. We investigate the conduction band valley structure in few-layer MX2 by examining the temperature-dependent shift of indirect exciton photoluminescence peak. Highly anisotropic thermal expansion of the lattice and the corresponding evolution of the band structure result in a distinct peak shift for indirect transitions involving the K and Λ (midpoint along Γ-K) valleys. We identify the origin of the indirect emission and concurrently determine the relative energy of these valleys.

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.

Correlation between (in)commensurate domains of multilayer epitaxial graphene grown on SiC(0001) and single layer electronic behavior

A systematic study of the evolution of the electronic behavior and atomic structure of multilayer epitaxial graphene (MEG) as a function of growth time was performed. MEG was obtained by sublimation of a 4H-SiC(0001(-)) substrate in an argon atmosphere. Raman spectroscopy and x-ray diffraction were carried out in samples grown for different times. For 30 min of growth the sample Raman signal is similar to that of graphite, while for 60 min the spectrum becomes equivalent to that of exfoliated graphene. Conventional x-ray diffraction reveals that all the samples have two different (0001) lattice spacings. Grazing incidence x-ray diffraction shows that thin films are composed of rotated (commensurate) structures formed by adjacent graphene layers. Thick films are almost completely disordered. This result can be directly correlated to the single layer electronic behavior of the films as observed by Raman spectroscopy. Finally, to understand the change in lattice spacings as a result of layer rotation, we have carried out first principles calculations (using density functional theory) of the observed commensurate structures.

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.

Slip corona surrounding bilayer graphene nanopore

The electronic and magnetic properties of bilayer graphene (BLG) depend on the stacking order between the two layers. We introduce a new conceptual structure of "slip corona" on BLG, which is a transition region between A-A stacking close to a nanopore composed of bilayer edges (BLEs) and A-B stacking far away. For an extremely small nanopore (diameter D(pore) < ~5 nm), both atomistic simulations and a continuum model reach consistent descriptions on the shape and size of this "corona" (diameter ~50 nm), which is much larger than the width of the typical dislocation core (~1 nm) in 3D metals or the nanopore itself, due to the weak van der Waals interactions and low interlayer shear resistance between two adjacent layers of graphene. The continuum model also suggests that the width of this "corona" from the BLE to the A-B stacking area would increase as D(pore) increases and converge to ~40 nm when D(pore) is more than ~80 nm. This large stacking transition region provides a new avenue for tailoring BLG properties.

Angle-dependent van Hove singularities in a slightly twisted graphene bilayer

Recent studies show that two low-energy van Hove singularities (VHSs) seen as two pronounced peaks in the density of states could be induced in a twisted graphene bilayer. Here, we report angle-dependent VHSs of a slightly twisted graphene bilayer studied by scanning tunneling microscopy and spectroscopy. We show that energy difference of the two VHSs follows ΔE(vhs)∼ℏν(F)ΔK between 1.0° and 3.0° [here ν(F)∼1.1 × 10(6) m/s is the Fermi velocity of monolayer graphene, and ΔK = 2Ksin(θ/2) is the shift between the corresponding Dirac points of the twisted graphene bilayer]. This result indicates that the rotation angle between graphene sheets does not result in a significant reduction of the Fermi velocity, which quite differs from that predicted by band structure calculations. However, around a twisted angle θ∼1.3°, the observed ΔE(vhs)∼0.11 eV is much smaller than the expected value ℏν(F)ΔK∼0.28 eV at 1.3°. The origin of the reduction of ΔE(vhs) at 1.3° is discussed.

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.

Electronic transport in two stacked graphene monolayers

We report on interlayer and lateral electronic transport measurements in two stacked graphene monolayers which have separate electrical contacts. The current-voltage characteristic across the two layers shows linear Ohmic behavior at zero magnetic field. At high magnetic fields, sequences of quantum Hall plateaus of the overlap region with filling factors 4, 8, and 12 are observed which can be explained by equilibration of the edge channel potentials of the individual graphene layers. An anomaly is observed at total filling factors ±2 in the overlap region. The I-V characteristic for interlayer transport turns nonlinear, and the Hall signal vanishes, indicating a magnetic field induced electrical decoupling of the two graphene layers.

Interfacial coupling in rotational monolayer and bilayer graphene on Ru(0001) from first principles

The interaction of graphene with metal is of critical importance for further optimization of the growth and transfer processes to achieve productive graphene. Here we report first-principles calculations with van der Waals corrections to address in-plane orientation effects on the geometric structure and electronic properties of monolayer and bilayer graphene on a Ru(0001) surface. We find that the recently measured slight rotation between monolayer graphene and Ru lattices minorly affects the characteristic geometric and electronic structures simulated to date for strict alignment. For epitaxial bilayer graphene, we unveil that a 25°-twisted bilayer graphene commensurate with Ru reproduces at best the hallmarks of free-standing electron-doped monolayer graphene as measured experimentally. At variance the classical Bernal stacking manifests the strongest interlayer coupling by destroying the Dirac point and exhibiting a graphite-like STM appearance. Our theoretical findings question the definite nature of the interfacial coupling of successive graphene layers grown on a strongly interacting metal substrate.

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.