Horizontal shifts in the top layer of highly oriented pyrolytic graphite, induced by a scanning tunneling microscope (STM) tip, are presented. Excellent agreement is found between STM images and those simulated using density functional theory. First-principle calculations identify that the low-energy barrier direction of the top layer displacement is toward a structure where none of the carbon pz orbitals overlap, while the high-energy barrier direction is toward AA stacking. Each directional shift yields a real-space surface charge density similar to graphene; however the low-energy barrier direction requires only one bond length to convert ABA (Bernal) to ABC (rhombohedral).
Multi-layer graphene with rhombohedral stacking is a promising carbon phase possibly displaying correlated states like magnetism or superconductivity due to the occurrence of a flat surface band at the Fermi level. Recently, flakes of thickness up to 17 layers were tentatively attributed ABC sequences although the Raman fingerprint of rhombohedral multilayer graphene is currently unknown and the 2D resonant Raman spectrum of Bernal graphite not understood. We provide a first principles description of the 2D Raman peak in three and four layers graphene (all stackings) as well as in Bernal, rhom…
Tunneling atomic force microscopy (TUNA) was used at ambient conditions to measure the current-voltage ($I$-$V$) characteristics at clean surfaces of highly oriented graphite samples with Bernal and rhombohedral stacking orders. The characteristic curves measured on Bernal-stacked graphite surfaces can be understood with an ordinary self-consistent semiconductor modeling and quantum mechanical tunneling current derivations. We show that the absence of a voltage region without measurable current in the $I$-$V$ spectra is not a proof of the lack of an energy band gap. It can be induced by a surf…
We use an exact analytical technique [Phys. Rev. B \textbf{101}, 115405 (2020), Phys. Rev. B \textbf{102}, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the T-matrix formalism to study the quasiparticle interference patterns generated on the…
We study the electronic structure of multilayer graphenes with a mixture of Bernal and rhombohedral stacking and propose a general scheme to understand the electronic band structure of an arbitrary configuration. The system can be viewed as a series of finite Bernal graphite sections connected by stacking faults. We find that the low-energy eigenstates are mostly localized in each Bernal section, and, thus, the whole spectrum is well approximated by a collection of the spectra of independent sections. The energy spectrum is categorized into linear, quadratic and cubic bands corresponding to sp…
There has been a lot of excitement around the observation of superconductivity in twisted bilayer graphene, associated to flat bands close to the Fermi level. Such correlated electronic states also occur in multilayer rhombohedral stacked graphene (RG), which has been receiving increasing attention in the last years. In both natural and artificial samples however, multilayer stacked Bernal graphene (BG) occurs more frequently, making it desirable to determine what is their relative stability and under which conditions RG might be favored. Here, we study the energetics of BG and RG in bulk and …
We examine the quantum Hall effect in bilayer graphene grown on Cu substrates by chemical vapor deposition. Spatially resolved Raman spectroscopy suggests a mixture of Bernal (A-B) stacked and rotationally faulted (twisted) domains. Magnetotransport measurements performed on bilayer domains with a wide 2D band reveal quantum Hall states (QHSs) at filling factors $ν=4, 8, 12$ consistent with a Bernal stacked bilayer, while magnetotransport measurements in bilayer domains defined by a narrow 2D band show a superposition of QHSs of two independent monolayers. The analysis of the Shubnikov-de Haas…
Stackings in graphene have a pivotal role in properties to be discussed in the future, as seen in the recently found superconductivity of twisted bilayer graphene. Beyond bilayer graphene, the stacking order of multilayer graphene can be rhombohedral, which shows flat bands near the Fermi level that are associated with interesting phenomena, such as tunable conducting surface states expected to exhibit spontaneous quantum Hall effect, surface superconductivity, and even topological order. However, the difficulty in exploring rhombohedral graphenes is that in experiments, the alternative, hexag…
In rhombohedral-stacked few-layer graphene, the very flat energy bands near the charge neutrality point are unstable to electronic interactions, giving rise to states with spontaneous broken symmetries. Using transport measurements on suspended rhombohedral-stacked tetralayer graphene, we observe an insulating ground state with a large interaction-induced gap up to 80 meV. This gapped state can be enhanced by a perpendicular magnetic field, and suppressed by an interlayer potential, carrier density, or a critical temperature of ~ 40 K.
The most important bands for the evaluation of strain in graphene (the 2D and 2D prime modes) are investigated. It is shown that for Bernal-stacked bilayers, the two-phonon Raman features have three different components that can be assigned to processes originating solely from the top graphene layer, bottom graphene layer, and from a combination of processes originating both from the top and bottom layers. The individual components of the 2D and 2D prime modes are disentangled. The reported results enable addressing the properties of individual graphene layers in isotopically labelled turbostr…
Graphite crystals used to prepare graphene-based heterostructures are generally assumed to be defect free. We report here scanning tunneling microscopy results that show graphite commonly used to prepare graphene devices can contain a significant amount of native defects. Extensive scanning of the surface allows us to determine the concentration of native defects to be 6.6$\times$10$^8$ cm$^{-2}$. We further study the effects of these native defects on the electronic properties of Bernal-stacked bilayer graphene. We observe gate-dependent intravalley scattering and successfully compare our exp…
Controlling the size and shape of nanopores in two-dimensional materials is a key challenge in applications such as DNA sequencing, sieving, and quantum emission in artificial atoms. We here investigate experimentally and theoretically triangular vacancies in (unconventional) Bernal-stacked AB-h-BN formed using a high-energy electron beam. Due to the geometric configuration of AB-h-BN, triangular pores in different layers are aligned, and their sizes are controlled by the duration of the electron irradiation. Interlayer covalent bonding at the vacancy edge is not favored, as opposed to what oc…
The discovery of superconductivity and correlated electronic states in the flat bands of twisted bilayer graphene has raised a lot of excitement. Flat bands also occur in multilayer graphene flakes that present rhombohedral (ABC) stacking order on many consecutive layers. Although Bernal-stacked (AB) graphene is more stable, long-range ABC-ordered flakes involving up to 50 layers have been surprisingly observed in natural samples. Here we present a microscopic atomistic model, based on first-principles density functional theory calculations, that demonstrates how shear stress can produce long-…
The properties of semiconductors can be crucially impacted by midgap states induced by dopants, which can be native or intentionally incorporated in the crystal lattice. For Bernal-stacked bilayer graphene (BLG), which has a tunable bandgap, the existence of midgap states induced by dopants has been conjectured, but never confirmed experimentally. Here, we report scanning tunneling microscopy and spectroscopy results, supported by tight-binding calculations, that demonstrate the existence of midgap states in BLG. We show that the midgap state in BLG -- for which we demonstrate gate-tunability …
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 labelled bilayer graphene was synthesized where the two layers are composed of different carbon isotopes. Therefore, the frequency of the phonons in the Raman spectra is shifted in relation to each other. This technique was use…
Hexagonal boron nitride (hBN) is the supporting substrate of choice for two-dimensional material devices because it is atomically flat and chemically inert. However, due to the small size of mechanically exfoliated hBN flakes, electronic structure studies of 2D materials supported by hBN using angle-resolved photoemission spectroscopy (ARPES) are challenging. Here we investigate the electronic band structure of a Bernal-stacked bilayer graphene sheet on a hexagonal boron nitride (BLG/hBN) flake using nanospot ARPES (nanoARPES). By fitting high-resolution energy vs. momentum electronic band spe…
We analyze the ordered phases of Bernal stacked multilayer graphene in the presence of interaction induced band gaps due to sublattice symmetry breaking potentials, whose solutions can be analyzed in terms of light-mass and heavy-mass pseudospin doublets which have the same Chern numbers but opposite charge polarization directions. The application of a perpendicular external electric field reveals an effective Hund's rule for the ordering of the sublattice pseudospin doublets in a tetralayer, while a similar but more complex phase diagram develops with increasing layer number.
We present the first Raman spectroscopic study of Bernal bilayer graphene flakes under uniaxial tension. Apart from a purely mechanical behavior in flake regions where both layers are strained evenly, certain effects stem from inhomogeneous stress distribution across the layers. These phenomena such as the removal of inversion symmetry in bilayer graphene may have important implications in the band-gap engineering providing an alternative route to induce the formation of a band-gap.
Novel systems of layered graphene are attracting interest for theories and applications. The stability, band structures of few-layer graphite films, and their dependence on electric field applied along the c-axis are examined within the density functional theory. We predict that those of Bernal type and also rhombohedral type tri- and tetra-layer graphite films exhibit stability. The rhombohedral-type systems including AB-bilayer, show variable band gap induced by perpendicular electric field, whereas the other systems such as the Bernal-type films stay semi-metallic.
The existence of strong trigonal warping around the K point for the low energy electronic states in multilayer (N$\geq$2) graphene films and graphite is well established. It is responsible for phenomena such as Lifshitz transitions and anisotropic ballistic transport. The absolute orientation of the trigonal warping with respect to the center of the Brillouin zone is however not agreed upon. Here, we use quasiparticle scattering experiments on a gated bilayer graphene/hexagonal boron nitride heterostructure to settle this disagreement. We compare Fourier transforms of scattering interference m…
A quantum graph model for a single sheet of graphene is extended to bilayer and trilayer Bernal-stacked graphene; the spectra are characterized and the dispersion relations explicitly obtained; Dirac cones are then proven to be present only for trilayer graphene, although the bilayer has a gapless parabolic band component. Our model rigorously exhibits basic facts from tight-binding calculations, effective two-dimensional models and a $π$-orbital continuum model with nearest-neighbour tunneling that have been discussed in the physics literature.
The low-energy band structure of Bernal-stacked hexlayer graphene was investigated by the measuring transport properties of high-mobility graphene samples equipped with a top and a bottom gate electrode at low temperature and in a magnetic field. By analyzing the Landau fan diagram, it was found that the hexlayer graphene consisted of three different bilayer-like bands that overlap and form a semimetal. The electronic band structure was significantly influenced by the perpendicular electric field generated by the top and bottom gate voltages. The Not fan diagram shows splitting of zero-mode La…
Van der Waals (vdW) materials have greatly expanded our design space of heterostructures by allowing individual layers to be stacked at non-equilibrium configurations, for example via control of the twist angle. Such heterostructures not only combine characteristics of the individual building blocks, but can also exhibit emergent physical properties absent in the parent compounds through interlayer interactions. Here we report on a new family of emergent, nanometer-thick, semiconductor 2D ferroelectrics, where the individual constituents are well-studied non-ferroelectric monolayer transition …
Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin mo…
The complete lattice-layer entanglement structure of Bernal stacked bilayer graphene is obtained for the quantum system described by a tight-binding Hamiltonian which includes mass and bias voltage terms. Through a suitable correspondence with the parity-spin $SU(2)\otimes SU(2)$ structure of a Dirac Hamiltonian, when it brings up tensor and pseudovector external field interactions, the lattice-layer degrees of freedom can be mapped into such a parity-spin two-qubit basis which supports the interpretation of the bilayer graphene eigenstates as entangled ones in a lattice-layer basis. The Dirac…
Electronic properties of quantum dots (QDs) depend sensitively on their parent materials. Therefore, confined electronic states in graphene QDs (GQDs) of monolayer and Bernal-stacked bilayer graphene are quite different. Twisted bilayer graphene (TBG) is distinct from monolayer and Bernal-stacked bilayer graphene because of the new degree of freedom: twist angle. In the past few years, numerous efforts have been made to realize the GQDs of monolayer and Bernal-stacked bilayer graphene and achieved great success. Thus far, however, strategies for realizing GQDs of TBG have been elusive. Here, w…
We present the results of the calculations of longitudinal and Hall conductivities of AB-stacked bilayer graphene as a function of frequency, finite chemical potential, temperature both with and without magnetic fields on a base of 2- and 4-band effective models. The limited cases of the conductivities for direct current are derived. The relations being important for optoelectronic among Hall conductivities and Faraday, Kerr angles in the AB-bilayers samples in the electric and magnetic fields when the radiation passes across bilayer sheets on different kinds a substrate are obtained.
We investigate the competition between acoustic-phonon-mediated superconductivity and the long-range Coulomb interaction in moiréless graphene multilayers, specifically, Bernal bilayer graphene, rhombohedral trilayer graphene, and ABCA-stacked tetralayer graphene. In these graphene multilayers, the acoustic phonons can realize, through electron-phonon coupling, both spin-singlet and spin-triplet pairings, and the intra-sublattice pairings ($s$-wave spin-singlet and $f$-wave spin-triplet) are the dominant channels. Our theory naturally explains the distinct recent experimental findings in Berna…
Recent experiments indicate that the Bernal stacked graphene multilayer (BGM) have an interaction induced gapped (or pseudo gapped) ground state. Here, we propose that, due to the electron correlation, the BGM can be induced into a half metallic phase by applying a vertical electric field and doping. The half metallic states in even-layer and odd-layer BGMs have totally different behaviors, due to their different band structures. We systematically calculate the graphene tetralayer (4L-BGM) and trilayer (3L-BGM) as the typical examples of the even-layer and odd-layer BGMs, respectively. In 4L-B…
The band structure of ABC-stacked N-layer graphene comprises topologically corresponding flat surface and gapped bulk subbands, as a consequence of the unique stacking configuration. In this paper, the bulk subbands are for the first times ever obtained for arbitrary N. A non-perturbative effective Hamiltonian closed in the bulk subspace is derived and used. The gapped bulk subbands are shown to evolve towards the zero energy with increasing N and in the infinite limit, they touch linearly along a circle. This outcome is a manifestation of the dimensional crossover to a three-dimensional Dirac…
Trilayer graphene (TLG) exhibits rich novel electronic properties and extraordinary quantum Hall phenomena owning to enhanced electronic interactions and tunable chirality of its quasiparticles. Here, we report direct observation of chirality transition of quasiparticles at stacking solitons of TLG via spatial-resolved Landau level spectroscopy. The one-dimensional stacking solitons with width of the order of 10 nm separate adjacent Bernal-stacked TLG and rhombohedral-stacked TLG. By using high field tunneling spectra of scanning tunneling microscopy, we measured Landau quantization in both th…
Using a numerical tight-binding approach based on the Chebyshev-Bogoliubov-de Gennes method we describe Josephson junctions made of multilayer graphene contacted by top superconducting gates. Both Bernal (ABA) and rhombohedral (ABC) stacking are considered and we find that the type of stacking has a strong effect on the proximity effect and the supercurrent flow. For both cases the pair amplitude shows a polarization between dimer and non-dimer atoms, being more pronounced for rhombohedral stacking. Even though the proximity effect in non-dimer sites is enhanced when compared to single layer g…
Recent transport experiments have demonstrated that the rhombohedral stacking trilayer graphene is an insulator with an intrinsic gap of 6meV and the Bernal stacking trilayer one is a metal. We propose a Hubbard model with a moderate $U$ for layered graphene sheets, and show that the model well explains the experiments of the stacking dependent energy gap. The on-site Coulomb repulsion drives the metallic phase of the non-interacting system to a weak surface antiferromagnetic insulator for the rhombohedral stacking layers, but does not alter the metallic phase for the Bernal stacking layers.
Low-energy electronic properties of ABC-stacked graphite are studied by the tight-binding model. There are linear and parabolic bands with and without degeneracy. They show strongly anisotropic dispersions. ABC-stacked grahite is a semimetal due to slight overlapping near the Fermi level beween conduction and valence bands. The interlayer interactions could change the energy dispersion, state degeneracy, and positions of band-crossing and band-edge state. Density of states exhibit a shoulder structure, owing to band-edge states near or at high symmetric points. Low-frequency Coulomb excitation…
Multilayer graphene with rhombohedral and Bernal stacking are supposed to be metallic, as predicted by density functional theory calculations using semi-local functionals. However recent angular resolved photoemission and transport data have questioned this point of view. In particular, rhombohedral flakes are suggested to be magnetic insulators. Bernal flakes composed of an even number of layers are insulating, while those composed of an odd number of layers are pseudogapped. Here, by systematically benchmarking with plane waves codes, we develop very accurate all-electron Gaussian basis sets…
The interlayer coupling is emerging as a new parameter for tuning the physical properties of two-dimensional (2D) van der Waals materials. When two identical semiconductor monolayers are stacked with a twist angle, the periodic interlayer coupling modulation due to the moiré superlattice may endow exotic physical phenomena, such as moiré excitons and correlated electronic phases. To gain insight into these new phenomena, it is crucial to unveil the underlying coupling between atomic layers. Recently, the rhombohedral-stacked transition metal dichalcogenide (TMD) bilayer has attracted significa…
We study the effect of chiral-tunneling in Bernal and Rombhohedral stacked trilayer-graphene (3LG). Based on the chirality of the electronic bands, at the K-point, (Rombhohedral) Bernal-3LG exhibits 100% (50%) transparency across a heterojunction. Utilizing this property, we further investigate the effect of electron collimation in 3LG. Due to the difference in the Berry's phase, we show that, Rombhohedral-3LG is a better electron collimator, compared to monolayer and Bernal-bilayer graphene. Since, Bernal-3LG can be decomposed into two separate channels consisting of a monolayer and a modifie…
The electronic tunneling properties of the two stable forms of trilayer graphene (TLG), rhombohedral ABC and Bernal ABA, are examined for pn and pnp junctions as realized by using a single gate (SG) or a double gate (DG). For the rhombohedral form, due to the chirality of the electrons, the Klein paradox is found at normal incidence for SG devices while at high energy interband scattering between additional propagation modes can occur. The electrons in Bernal ABA TLG can have a monolayer- or bilayer-like character when incident on a SG device. Using a DG however both propagation modes will cou…
We demonstrate that surface relaxation, which is insignificant in trilayer graphene, starts to manifest in Bernal-stacked tetralayer graphene. Bernal-stacked few-layer graphene has been investigated by analyzing its Landau level spectra through quantum capacitance measurements. We find that in trilayer graphene, the interlayer interaction parameters were similar to that of graphite. However, in tetralayer graphene, the hopping parameters between the bulk and surface bilayers are quite different. This shows a direct evidence for the surface relaxation phenomena. In spite of the fact that the Va…
Correct defect quantification in graphene samples is crucial both for fundamental and applied re-search. Raman spectroscopy represents the most widely used tool to identify defects in graphene. However, despite its extreme importance the relation between the Raman features and the amount of defects in multilayered graphene samples has not been experimentally verified. In this study we intentionally created defects in single layer graphene, turbostratic bilayer graphene and Bernal stacked bilayer graphene by oxygen plasma. By employing isotopic labelling, our study reveals substantial differenc…
Displacement field stabilizes even-denominator and partonic fractional quantum Hall states in the $\mathcal{N}{=}2$ Landau levels of Bernal-stacked bilayer graphene
Institute of Mathematical Sciences（数学科学研究所）；Homi Bhabha National Institute（霍米·巴巴国立学院）；Department of Physics, Indian Institute of Technology Bombay（印度理工学院孟买分校物理系）；Theoretical Physics Division, Physical Research Laboratory（物理研究实验室理论物理部）
Few layer graphene (FLG) has been recently intensively investigated for its variable electronic properties defined by a local atomic arrangement. While the most natural layers arrangement in FLG is ABA (Bernal) stacking, a metastable ABC (rhombohedral) stacking characterized by a relatively high energy barrier can also occur. When both stacking occur in the same FLG device this results in in-plane heterostructure with a domain wall (DW). We show that ABC stacking in FLG can be controllably and locally turned into ABA stacking by two following approaches. In the first approach, Joule heating wa…
Stacking order in van der Waals materials determines the coupling between atomic layers and is therefore key to the materials' properties. By exploring different stacking orders, many novel physical phenomena have been realized in artificial vdW stacks. Recently, 2D ferroelectricity has been observed in zero-degree aligned hBN and graphene-hBN heterostructures, holding promise in a range of electronic applications. In those artificial stacks, however, the single domain size is limited by the stacking-angle misalignment to about 0.1 to 1 $μ$m, which is incompatible with most optical or optoelec…
The band structure and the optical conductivity of an ABA (Bernal-type) stacked graphene trilayer are calculated. It is shown that, under appropriate doping, a strong resonant peak develops in the optical conductivity, located at the frequency corresponding to approximately 1.4 times the interlayer hopping energy and caused by the 'nesting' of two nearly parabolic bands in the electronic spectrum. The intensity of this resonant absorption can be controlled by adjusting the gate voltage. The effect is robust with respect to increasing temperature.
