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Proof assistants like Coq are increasingly popular to help mathematicians carry out proofs of the results they conjecture. However, formal proofs remain highly technical and are especially difficult to reuse. In this paper, we present a framework to carry out a posteriori script transformations. These transformations are meant to be applied as an automated post-processing step, once the proof has been completed. As an example, we present a transformation which takes an arbitrary large proof script and produces an equivalent single-line proof script, which can be executed by Coq in one single step. Other applications, such as fully expanding a proof script (for debugging purposes), removing all named hypotheses, etc. could be developed within this framework. We apply our tool to various Coq proof scripts, including some from the GeoCoq library., Comment: In Proceedings ADG 2023, arXiv:2401.10725

There are several ways to formally represent families of data, such as lambda terms, in a type theory such as the dependent type theory of Coq. Mathematical representations are very compact ones and usually rely on the use of dependent types, but they tend to be difficult to handle in practice. On the contrary, implementations based on a larger (and simpler) data structure combined with a restriction property are much easier to deal with. In this work, we study several families related to lambda terms, among which Motzkin trees, seen as lambda term skeletons, closable Motzkin trees, corresponding to closed lambda terms, and a parameterized family of open lambda terms. For each of these families, we define two different representations, show that they are isomorphic and provide tools to switch from one representation to another. All these datatypes and their associated transformations are implemented in the Coq proof assistant. Furthermore we implement random generators for each representation, using the QuickChick plugin., Comment: Under review

We study how to formalize in the Coq proof assistant the smallest projective space PG(3,2). We then describe formally the spreads and packings of PG(3,2), as well as some of their properties. The formalization is rather straightforward, however as the number of objects at stake increases rapidly, we need to exploit some symmetry arguments as well as smart proof techniques to make proof search and verification faster and thus tractable using the Coq proof assistant. This work can be viewed as a first step towards formalizing projective spaces of higher dimension, e.g. PG(4,2), or larger order, e.g. PG(3,3)., Comment: In Proceedings ADG 2021, arXiv:2112.14770

Several tools have been developed to enhance automation of theorem proving in the 2D plane. However, in 3D, only a few approaches have been studied, and to our knowledge, nothing has been done in higher dimensions. In this paper, we present a few examples of incidence geometry theorems in dimensions 3, 4, and 5. We then prove them with the help of a combinatorial prover based on matroid theory applied to geometry., Comment: In Proceedings ADG 2021, arXiv:2112.14770

Recently, we developed an automated theorem prover for projective incidence geometry. This prover, based on a combinatorial approach using matroids, proceeds by saturation using the matroid rules. It is designed as an independent tool, implemented in C, which takes a geometric configuration as input and produces as output some Coq proof scripts: the statement of the expected theorem, a proof script proving the theorem and possibly some auxiliary lemmas. In this document, we show how to embed such an external tool as a plugin in Coq so that it can be used as a simple tactic., Comment: In Proceedings PxTP 2021, arXiv:2107.01544

Graphene holds promises for exploring exotic superconductivity with Dirac-like fermions. Making graphene a superconductor at large scales is however a long-lasting challenge. A possible solution relies on epitaxially-grown graphene, using a superconducting substrate. Such substrates are scarce, and usually destroy the Dirac character of the electronic band structure. Using electron diffraction (reflection high-energy, and low-energy), scanning tunneling microscopy and spectroscopy, atomic force microscopy, angle-resolved photoemission spectroscopy, Raman spectroscopy, and density functional theory calculations, we introduce a strategy to induce superconductivity in epitaxial graphene $via$ a remote proximity effect, from the rhenium substrate through an intercalated gold layer. Weak graphene-Au interaction, contrasting with the strong undesired graphene-Re interaction, is demonstrated by a reduced graphene corrugation, an increased distance between graphene and the underlying metal, a linear electronic dispersion and a characteristic vibrational signature, both latter features revealing also a slight $p$ doping of graphene. We also reveal that the main shortcoming of the intercalation approach to proximity superconductivity is the creation of a high density of point defects in graphene (10$^{14}$~cm$^{-2}$). Finally, we demonstrate remote proximity superconductivity in graphene/Au/Re(0001), at low temperature., Comment: 13 pages, 6 figures, to appear in 2D Materials

Intercalation of alkali atoms within the lamellar transition metal dichalcogenides is a possible route toward a new generation of batteries. It is also a way to induce structural phase transitions authorizing the realization of optical and electrical switches in this class of materials. The process of intercalation has been mostly studied in three-dimensional dichalcogenide films. Here, we address the case of a single-layer of molybdenum disulfide (MoS$_2$), deposited on a gold substrate, and intercalated with cesium (Cs) in ultra-clean conditions (ultrahigh vacuum). We show that intercalation decouples MoS$_2$ from its substrate. We reveal electron transfer from Cs to MoS$_2$, relative changes in the energy of the valence band maxima, and electronic disorder induced by structural disorder in the intercalated Cs layer. Besides, we find an abnormal lattice expansion of MoS$_2$, which we relate to immediate vicinity of Cs. Intercalation is thermally activated, and so is the reverse process of de-intercalation. Our work opens the route to a microscopic understanding of a process of relevance in several possible future technologies, and shows a way to manipulate the properties of two-dimensional dichalcogenides by "under-cover" functionalization., Comment: 6 figures

A broad variety of defects has been observed in two-dimensional materials. Many of these defects can be created by top-down methods such as electron irradiation or chemical etching, while a few of them are created along bottom-up processes, in particular during the growth of the material, in which case avoiding their formation can be challenging. This occurs e.g. with dislocations, Stone-Wales defects, or atomic vacancies in graphene. Here we address a defect that has been observed repeatedly since 2007 in epitaxial graphene on metal surfaces like Ru(0001) and Re(0001), but whose nature has remained elusive thus far. This defect has the appearance of a vacant hill in the periodically nanorippled topography of graphene, which comes together with a moir{\'e} pattern. Based on atomistic simulations and scanning tunneling microscopy/spectroscopy measurements, we argue that such defects are topological in nature and that their core is a stacking fault patch, either in graphene, surrounded by loops of non-hexagonal carbon rings, or in the underlying metal. We discuss the possible origin of these defects in relation with recent reports of metastable polycyclic carbon molecules forming upon graphene growth. Like other defects, the vacant hills may be considered as deleterious in the perspective of producing high quality graphene. However, provided they can be organized in graphene, they might allow novel optical, spin, or electronic properties to be engineered.

Background: Chronic myeloid leukemia (CML) results from hematopoietic stem cell transformation by the bcr-abl chimeric oncogene, encoding a 210 kDa protein with constitutive tyrosine kinase activity. In spite of the efficiency of tyrosine kinase inhibitors (TKI; Imatinib), other strategies are explored to eliminate CML leukemia stem cells, such as calcium pathways. Results: In this work, we showed that Store-Operated Calcium Entry (SOCE) and thrombin induced calcium influx were decreased in Bcr-Abl expressing 32d cells (32d-p210). The 32d-p210 cells showed modified Orai1/STIM1 ratio and reduced TRPC1 expression that could explain SOCE reduction. Decrease in SOCE and thrombin induced calcium entry was associated to reduced Nuclear Factor of Activated T cells (NFAT) nucleus translocation in 32d-p210 cells. We demonstrated that SOCE blockers enhanced cell mobility of 32d-p210 cells and reduced the proliferation rate in both 32d cell lines. TKI treatment slightly reduced the thrombin-induced response, but imatinib restored SOCE to the wild type level. Bcr-Abl is also known to deregulate Protein Kinase C (PKC), which was described to modulate calcium entries. We showed that PKC enhances SOCE and thrombin induced calcium entries in control cells while this effect is lost in Bcr-Abl-expressing cells. Conclusion: The tyrosine kinase activity seems to regulate calcium entries probably not directly but through a global cellular reorganization involving a PKC pathway. Altogether, calcium entries are deregulated in Bcr-Abl-expressing cells and could represent an interesting therapeutic target in combination with TKI.

Up to date, inertial migration of particles in microflows has demonstrated a great potential for a wide range of applications. In particular, this phenomenon is used to achieve particle separation or sorting in a suspension. Recent works reported that the focusing mode of particles can be modified in a polydisperse suspension. Nevertheless, the impact of the particle sizes in a mixture on their inertial migration has been rarely studied up to now. Thus, we have investigated in this work the influence of bidispersity on the lateral migration of the particles towards equilibrium positions and on their longitudinal ordering into trains. Different changes in the particles behavior were observed when the ratio between the particle sizes (dp1/dp2) varied from 1.64 to 4.58.

KEY WORDS suspensions, inertial focusing, particle-laden flows, high Reynolds numbers SHORT SUMMARY The inertial migration of particles in square channel flows at the micro-scale has been deeply investigated in the last two decades. The well-known four equilibrium positions are located near the center of each channel face at moderate Reynolds numbers [1]. More recently, Miura et al. [2] revealed experimentally the presence of eight equilibrium positions in millimetric square channels for Reynolds numbers higher than 250. The aim of the present work is to extend these results obtained at millimeter scale to the micrometer scale. To this end, in situ visualization of particles flowing in square micro-channels at Reynolds numbers ranging from 5 to 300 have been conducted and analyzed.

Miniaturization of continuous fluorescence detection is a challenging task due to the multiple and sensitive parameters intervening in the process. By analyzing fluorescence sensing architectures proposed during the last two years, this work has the goal to identify some trends in the process of fluorescence miniaturization for in-time liquid detection. A lack of postulated strategies regarding the miniaturization process was observed and this review tries to answer partially to this need. The identified integration strategies excel in fulfilling partially the desired functions of a fully autonomous miniaturized detector and further research is needed in order to develop sensing micro-system being capable to step outside of the lab world.

Finite-size neutrally buoyant particles in a channel flow are known to accumulate at specific equilibrium positions or spots in the channel cross-section if the flow inertia is finite at the particle scale. Experiments in different conduit geometries have shown that while reaching equilibrium locations, particles tend also to align regularly in the streamwise direction. In this paper, the Force Coupling Method was used to numerically investigate the inertia-induced particle alignment, using square channel geometry. The method was first shown to be suitable to capture the quasi-steady lift force that leads to particle cross-streamline migration in channel flow. Then the particle alignment in the flow direction was investigated by calculating the particle relative trajectories as a function of flow inertia and of the ratio between the particle size and channel hydraulic diameter. The flow streamlines were examined around the freely rotating particles at equilibrium, revealing stable small-scale vortices between aligned particles. The streamwise inter-particle spacing between aligned particles at equilibrium was calculated and compared to available experimental data in square channel flow (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017)). The new result highlighted by our numerical simulations is that the inter-particle spacing is unconditionally stable only for a limited number of aligned particles in a single train, the threshold number being dependent on the confinement (particle-to-channel size ratio) and on the Reynolds number. For instance, when the particle Reynolds number is $\approx1$ and the particle-to-channel height size ratio is $\approx0.1$, the maximum number of stable aligned particles per train is equal to 3. This agrees with statistics realized on the experiments of (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017))., Comment: 13 pages, 13 figures

Few- and single-layer MoS2 host substantial densities of defects. They are thought to influence the doping level, the crystal structure, and the binding of electron-hole pairs. We disentangle the concomitant spectroscopic expression of all three effects and identify to what extent they are intrinsic to the material or extrinsic to it, i.e., related to its local environment. We do so by using different sources of MoS2 -- a natural one and one prepared at high pressure and high temperature -- and different substrates bringing varying amounts of charged impurities and by separating the contributions of internal strain and doping in Raman spectra. Photoluminescence unveils various optically active excitonic complexes. We discover a defect-bound state having a low binding energy of 20 meV that does not appear sensitive to strain and doping, unlike charged excitons. Conversely, the defect does not significantly dope or strain MoS2. Scanning tunneling microscopy and density functional theory simulations point to substitutional atoms, presumably individual nitrogen atoms at the sulfur site. Our work shows the way to a systematic understanding of the effect of external and internal fields on the optical properties of two-dimensional materials.

In crystal growth, surfactants are additive molecules used in dilute amount or as dense, permeable layers to control surface morphologies. Here, we investigate the properties of a strikingly different surfactant: a two-dimensional and covalent layer with close atomic packing, graphene. Using in situ, real time electron microscopy, scanning tunneling microscopy, kinetic Monte Carlo simulations, and continuum mechanics calculations, we reveal why metallic atomic layers can grow in a two-dimensional manner below an impermeable graphene membrane. Upon metal growth, graphene dynamically opens nanochannels called wrinkles, facilitating mass transport, while at the same time storing and releasing elastic energy via lattice distortions. Graphene thus behaves as a mechanically active, deformable surfactant. The wrinkle-driven mass transport of the metallic layer intercalated between graphene and the substrate is observed for two graphene-based systems, characterized by different physico-chemical interactions, between graphene and the substrate, and between the intercalated material and graphene. The deformable surfactant character of graphene that we unveil should then apply to a broad variety of species, opening new avenues for using graphene as a two-dimensional surfactant forcing the growth of flat films, nanostructures and unconventional crystalline phases.

Moir\'e superlattices in graphene supported on various substrates have opened a new avenue to engineer graphene's electronic properties. Yet, the exact crystallographic structure on which their band structure depends remains highly debated. In this scanning tunneling microscopy and density functional theory study, we have analysed graphene samples grown on multilayer graphene prepared onto SiC and on the close-packed surfaces of Re and Ir with ultra-high precision. We resolve small-angle twists and shears in graphene, and identify large unit cells comprising more than 1,000 carbon atoms and exhibiting non-trivial nanopatterns for moir\'e superlattices, which are commensurate to the graphene lattice. Finally, a general formalism applicable to any hexagonal moir\'e is presented to classify all reported structures., Comment: 14 pages, 6 figures

Rotated graphene bilayers form an exotic class of nanomaterials with fascinating electronic properties governed by the rotation angle theta. For large rotation angles, the electron eigenstates are restricted to one layer and the bilayer behaves like two decoupled graphene layer. At intermediate angles, Dirac cones are preserved but with a lower velocity and van Hove singularities are induced at energies where the two Dirac cones intersect. At very small angles, eigenstates become localized in peculiar moire zones. We analyse here the effect of an asymmetric doping for a series of commensurate rotated bilayers on the basis of tight binding calculations of their band dispersions, density of states, participation ratio and diffusive properties. While a small doping level preserves the theta dependence of the rotated bilayer electronic structure, larger doping induces a further reduction of the band velocity in the same way of to a further reduction of the rotation angle., Comment: 12 pages, 12 figures

Graphene outstanding properties created a huge interest in the condensed matter community and unprecedented fundings at the international scale in the hope of application developments. Recently, there have been several reports of incomplete removal of the polymer resists used to transfer as-grown graphene from one substrate to another, resulting in altered graphene transport properties. Finding a large-scale solution to clean graphene from adsorbed residues is highly desirable and one promising possibility would be to use hydrogen plasmas. In this spirit, we couple here quantum and classical molecular dynamics simulations to explore the kinetic energy ranges required by atomic hydrogen to selectively etch a simple residue, a CH3 group, without irreversibly damaging the graphene. For incident energies in the 2-15 eV range, the CH3 radical can be etched by forming a volatile CH4 compound which leaves the surface, either in the CH4 form or breaking into CH3+H fragments, without further defect formation. At this energy, adsorption of H atoms on graphene is possible and further annealing will be required to recover pristine graphene., Comment: 9 figures, 27 pages

We report a convergent surface polymerization reaction scheme on Au(111), based on a triple aldol condensation, yielding a carbon-rich, covalent nanoporous two-dimensional network. The reaction is not self-poisoning and proceeds up to a full surface coverage. The deposited precursor molecules 1,3,5-tri(4'-acetylphenyl) first form supramolecular assemblies that are converted to the porous covalent network upon heating. The formation and structure of the network and of the intermediate steps are studied with scanning tunneling microscopy, Raman spectroscopy and density functional theory., Comment: 1 Scheme, 5 Figures

Staking layered materials revealed to be a very powerful method to tailor their electronic properties. It has indeed been theoretically and experimentally shown that twisted bilayers of graphene (tBLG) with a rotation angle $\theta$, forming Moir\'e pattern, confine electrons in a tunable way as a function of $\theta$. Here, we study electronic structure and transport in tBLG using tight-binding numerical calculations in commensurate twisted bilayer structures and a pertubative continuous theory, which is valid for not too small angles ($\theta > \sim 2^\circ $). This two approaches allow to understand the effect of $\theta$ on the local density of states, the electron lifetime due to disorder, the dc-conducitivity and the quantum correction of the conductivity due to multiple scattering effects. We distinguished the cases where disorder is equaly distributed in the two layer or only in one layer. When only one layer is disordered, diffusion properties depends strongly on $\theta$, showing thus the effect of Moir\'e electronic localisation at intermediate angles, $\sim 2^\circ < \theta < \sim 20^\circ$.

We report a new way to strongly couple graphene to a superconductor. The graphene monolayer has been grown directly on top of a superconducting Re(0001) thin film and characterized by scanning tunneling microscopy and spectroscopy. We observed a moir\'e pattern due to the mismatch between Re and graphene lattice parameters that we have simulated with ab initio calculations. The density of states around the Fermi energy appears to be position dependent on this moir\'e pattern. Tunneling spectroscopy performed at 50 mK shows that the superconducting behavior of graphene on Re is well described by the Bardeen-Cooper-Schrieffer theory and stands for a very good interface between the graphene and its metallic substrate., Comment: 5 pages, 5 figures

We investigate the graphene-impurity interaction problem by combining experimental - scanning tunneling microscopy (STM) and spectroscopy (STS) - and theoretical - Anderson impurity model and density functional theory (DFT) calculations - techniques. We use graphene on the SiC(000-1)(2x2)_C reconstruction as a model system. The SiC substrate reconstruction is based on silicon adatoms. Graphene mainly interacts with the dangling bonds of these adatoms which act as impurities. Graphene grown on SiC(000-1)(2x2)_C shows domains with various orientations relative to the substrate so that very different local graphene/Si adatom stacking configurations can be probed on a given grain. The position and width of the adatom (impurity) state can be analyzed by STM/STS and related to its local environment owing to the high bias electronic transparency of graphene. The experimental results are compared to Anderson's model predictions and complemented by DFT calculations for some specific local environments. We conclude that the adatom resonance shows a smaller width and a larger shift toward the Dirac point for an adatom at the center of a graphene hexagon than for an adatom just on top of a C graphene atom., Comment: 13 pages, 6 figures, Accepted for publication in Phys. Rev. B

Extensive scanning tunnelling microscopy and spectroscopy experiments complemented by first principles and parameterized 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{\deg} to 10{\deg}) of rotation angles in our graphene on 6H-SiC(000-1) samples. From the variation of the energy separation of the vHs with 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 calculations, which test the role of the periodic modulation and absolute value of the interlayer distance. Finally, we clarify the origin of the related moir\'e corrugation detected in the STM images.

Rotated graphene multilayers form a new class of graphene related systems with electronic properties that drastically depend on the rotation angles. It has been shown that bilayers behave like two isolated graphene planes for large rotation angles. For smaller angles, states in the Dirac cones belonging to the two layers interact resulting in the appearance of two van Hove singularities. States become localised as the rotation angle decreases and the two van Hove singularities merge into one peak at the Dirac energy. Here we go further and consider bilayers with very small rotation angles. In this case, well defined regions of AA stacking exist in the bilayer supercell and we show that states are confined in these regions for energies in the [-\gamma_t, +\gamma_t] range with \gamma_t the interplane mean interaction. As a consequence, the local densities of states show discrete peaks for energies different from the Dirac energy., Comment: 8 pages

It has been shown that the first C layer on the SiC(0001)(2{\times}2)C surface already exhibits graphene-like electronic structure, with linear pi bands near the Dirac point. Indeed, the (2{\times}2)C reconstruction, with a Si adatom and C restatom structure, efficiently passivates the SiC(0001) surface thanks to an adatom/restatom charge transfer mechanism. Here, we study the effects of interface modifications on the graphene layer using density functional theory calculations. The modifications we consider are inspired from native interface defects observed by scanning tunneling microscopy. One H atom per 4 {\times} 4 SiC cell (5 {\times} 5 graphene cell) is introduced in order to saturate a restatom dangling bond and hinder the adatom/restatom charge transfer. As a consequence, the graphene layer is doped with electrons from the substrate and the interaction with the adatom states slightly increases. Native interface defects are therefore likely to play an important role in the doping mechanism on the C terminated SiC substrates. We also conclude that an efficient passivation of the C face of SiC by H requires a complete removal of the reconstruction. Otherwise, at variance with the Si terminated SiC substrates, the presence of H at the interface would increase the graphene/substrate interaction., Comment: 8 pages, 4 figures, accepted for publication in Phys. Rev. B

The atomistic structure of the graphene buffer layer on Si-terminated SiC is studied using a modified environment-dependent interatomic potential (EDIP). The investigation of equilibrium state by conjuguate gradients suffers from a complex multi-minima energy surface. A dedicated procedure is therefore presented to provide a suitable initial configuration on the way to the minimum. The result forms an hexagonal pattern with unsticked rods to release the misfit with the surface. The structure presents an agreement with the global pattern obtained by experiments and even with the details of an ab initio calculation., Comment: 8 pages, 12 figures

We present an investigation of the atomic and electronic structure of graphene monolayer islands on the 6H-SiC(000-1)(3x3) (SiC(3x3)) surface reconstruction using scanning tunneling microscopy (STM) and spectroscopy (STS). The orientation of the graphene lattice changes from one island to the other. In the STM images, this rotational disorder gives rise to various superlattices with periods in the nm range. We show that those superlattices are moir\'e patterns (MPs) and we correlate their apparent height with the stacking at the graphene/SiC(3x3) interface. The contrast of the MP in STM images corresponds to a small topographic modulation of the graphene layer. From STS measurements we find that the substrate surface presents a 1,5 eV wide bandgap encompassing the Fermi level. This substrate surface bandgap subsists below the graphene plane. The tunneling spectra are spatially homogeneous on the islands within the substrate surface gap, which shows that the MPs do not impact the low energy electronic structure of graphene. We conclude that the SiC(3 x 3) reconstruction efficiently passivates the substrate surface and that the properties of the graphene layer which grows on top of it should be similar to those of the ideal material., Comment: 24 pages, 6 figures, To be published in Phys. Rev. B

Graphene, the two-dimensional form of carbon presents outstanding electronic and transport properties. This gives hope for the development of applications in nanoelectronics. However, for industrial purpose, graphene has to be supported by a substrate. We focus here on the graphene-on-SiC system to discuss how the SiC substrate interacts with the graphene layer and to show the effect of the interface on graphene atomic and electronic structures., Comment: 3 pages, 3 figures

We study the electronic structure of two Dirac electron gazes coupled by a periodic Hamiltonian such as it appears in rotated graphene bilayers. Ab initio and tight-binding approaches are combined and show that the spatially periodic coupling between the two Dirac electron gazes can renormalize strongly their velocity. We investigate in particular small angles of rotation and show that the velocity tends to zero in this limit. The localization is confirmed by an analysis of the eigenstates which are localized essentially in the AA zones of the Moire patterns., Comment: 4 pages, 5 figures

The atomic and electronic structures of a graphene layer on top of the $(2\times2)$ reconstruction of the SiC (000$\bar{1}$) surface are studied from ab initio calculations. At variance with the (0001) face, no C bufferlayer is found here. Si adatoms passivate the substrate surface so that the very first C layer presents a linear dispersion characteristic of graphene. A small graphene-substrate interaction remains in agreement with scanning tunneling experiments (F.Hiebel et al. {\it Phys. Rev. B} {\bf 78} 153412 (2008)). The stacking geometry has little influence on the interaction which explains the rotational disorder observed on this face., Comment: 4 pages, 3 figures, additional material

An investigation of the early stage formation of graphene on the C-face of 6H-SiC is presented. We show that the sublimation of few atomic layers of Si out of the SiC substrate is not homogeneous. In good agreement with the results of theoretical calculations it starts from defective sites, mainly dislocations that define nearly circular flakes, which have a pyramidal, volcano-like, shape with a center chimney where the original defect was located. At higher temperatures, complete conversion occurs but, again, it is not homogeneous. Within the sample surface the intensity of the Raman G and 2D bands, evidences non-homogeneous thickness., Comment: 12 pages, 3 figures

Graphene exhibits unconventional two-dimensional electronic properties resulting from the symmetry of its quasiparticles, which leads to the concepts of pseudospin and electronic chirality. Here we report that scanning tunneling microscopy can be used to probe these unique symmetry properties at the nanometer scale. They are reflected in the quantum interference pattern resulting from elastic scattering off impurities, and they can be directly read from its fast Fourier transform. Our data, complemented by theoretical calculations, demonstrate that the pseudospin and the electronic chirality in epitaxial graphene on SiC(0001) correspond to the ones predicted for ideal graphene., Comment: 4 pages, 3 figures, minor changes

We have analyzed by Scanning Tunnelling Microscopy (STM) thin films made of few (3-5) graphene layers grown on the C terminated face of 6H-SiC in order to identify the nature of the azimuthal disorder reported in this material. We observe superstructures which are interpreted as Moir\'e patterns due to a misorientation angle between consecutive layers. The presence of stacking faults is expected to lead to electronic properties reminiscent of single layer graphene even for multilayer samples. Our results indicate that this apparent electronic decoupling of the layers can show up in STM data., Comment: 20 pages

Graphene outstanding properties directly come from its pecular electronic structure and thus from the honeycomb lattice symmetry. The way interaction with the substrate impact this lattice is of primary importance. This is peculiarly true for epitaxial graphene because of the SiC substrate. The advantage of this system that produces macroscopic samples in registry with a substrate could turn to a major drawback if the graphene lattice reveals to be strongly distorted. Extensive ab initio calculations supported by Scanning Tunneling Microscopy experiments, demonstrate here that the substrate indeed induces a strong nanostructuration of the interface carbon layer. It propagates to the above C layer where it generates incommensurate ripples in the honeycomb lattice., Comment: submitted to Phys. Rev. B, figures here are in a low resolution

We examine the stacking order of multilayer graphene grown on the SiC$(000\bar{1})$ surface using low-energy electron diffraction and surface X-ray diffraction. We show that the films contain a high density of rotational stacking faults caused by three types of rotated graphene: sheets rotated $30^\circ$ and $\pm 2.20^\circ$ relative to the SiC substrate. These angles are unique because they correspond to commensurate phases of layered graphene, both with itself and with the SiC substrate. {\it Ab intio} calculations show that these rotational phases electronically decouple adjacent graphene layers. The band structure from graphene at fault boundaries displays linear energy dispersion at the $K$-point (Dirac cones), nearly identical to that of a single graphene sheet., Comment: 5 pages 4 figures

We present a scanning tunneling microscopy (STM) study of a gently-graphitized 6H-SiC(0001) surface in ultra high vacuum. From an analysis of atomic scale images, we identify two different kinds of terraces, which we unambiguously attribute to mono- and bilayer graphene capping a C-rich interface. At low temperature, both terraces show $(\sqrt{3}\times \sqrt{3})$ quantum interferences generated by static impurities. Such interferences are a fingerprint of $\pi$-like states close to the Fermi level. We conclude that the metallic states of the first graphene layer are almost unperturbed by the underlying interface, in agreement with recent photoemission experiments (A. Bostwick et al., Nature Physics 3, 36 (2007)), Comment: 4 pages, 3 figures submitted

Recent transport measurements on thin graphite films grown on SiC show large coherence lengths and anomalous integer quantum Hall effects expected for isolated graphene sheets. This is the case eventhough the layer-substrate epitaxy of these films implies a strong interface bond that should induce perturbations in the graphene electronic structure. Our DFT calculations confirm this strong substrate-graphite bond in the first adsorbed carbon layer that prevents any graphitic electronic properties for this layer. However, the graphitic nature of the film is recovered by the second and third absorbed layers. This effect is seen in both the (0001)and $(000\bar{1})$ 4H SiC surfaces. We also present evidence of a charge transfer that depends on the interface geometry. It causes the graphene to be doped and gives rise to a gap opening at the Dirac point after 3 carbon layers are deposited in agreement with recent ARPES experiments (T.Ohta et al, Science {\bf 313} (2006) 951).