Employing density-functional calculations we study single and double graphene layers on Si-and C-terminated 1 × 1 -6H-SiC surfaces. We show that, in contrast to earlier assumptions, the first carbon layer is covalently bonded to the substrate, and cannot be responsible for the graphenetype electronic spectrum observed experimentally. The characteristic spectrum of free-standing graphene appears with the second carbon layer, which exhibits a weak van der Waals bonding to the underlying structure. For Si-terminated substrate, the interface is metallic, whereas on C-face it is semiconducting or semimetallic for single or double graphene coverage, respectively.PACS numbers: 68.35. Ct, 68.47.Fg, The last years have witnessed an explosion of interest in the prospect of graphene-based nanometer-scale electronics [1,2,3,4]. Graphene, a single hexagonally ordered layer of carbon atoms, has a unique electronic band structure with the conic "Dirac points" at two inequivalent corners of the two-dimensional Brillouin zone. The electron mobility may be very high and lateral patterning with standard lithography methods allows device fabrication [1]. Two ways of obtaining graphene samples have been used up to now. In the first "mechanical" method, the carbon monolayers are mechanically split off the bulk graphite crystals and deposited onto a SiO 2 /Si substrate [4]. This way an almost "freestanding" graphene is produced, since the carbon monolayer is practically not coupled to the substrate. The second method uses epitaxial growth of graphite on singlecrystal silicon carbide (SiC). The ultrathin graphite layer is formed by vacuum graphitization due to Si depletion of the SiC surface [5]. This method has apparent technological advantages over the "mechanical" method, however it does not guarantee that an ultrathin graphite (or graphene) layer is electronically isolated from the substrate. Moreover, one expects a covalent coupling between both which may strongly modify the electronic properties of the graphene overlayer. Yet, experiments show that the transport properties of the interface are dominated by a single epitaxial graphene layer [1,2]. Most surprisingly, the electronic spectrum seems not to be affected much by the substrate. As in free-standing graphene one observes the "Dirac points" with the linear dispersion relation around them. The electron dynamics is governed by a Dirac-Weyl Hamiltonian with the Fermi velocity of graphene replacing the speed of light. This leads to an unusual sequence of Landau levels in a magnetic field and hence to peculiar features in the quantum Hall effect [1,4].

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene layers, many features of this system can be understood by interference conditions between the quantum states of the two layers, mathematically expressed as Diophantine problems. Based on this general analysis we demonstrate that while the Dirac cones from each layer are always effectively degenerate, the Fermi velocity $v_F$ of the Dirac cones decreases as $\theta\to 0^\circ$; the form we derive for $v_F(\theta)$ agrees with that found via a continuum approximation in Phys. Rev. Lett., 99:256802, 2007. From tight binding calculations for structures with $1.47^\circ \le \theta < 30^\circ$ we find agreement with this formula for $\theta \gtrsim 5^\circ$. In contrast, for $\theta \lesssim 5^\circ$ this formula breaks down and the Dirac bands become strongly warped as the limit $\theta \to 0$ is approached. For an ideal system of twisted layers the limit as $\theta\to0^\circ$ is singular as for $\theta > 0$ the Dirac point is fourfold degenerate, while at $\theta=0$ one has the twofold degeneracy of the $AB$ stacked bilayer. Interestingly, in this limit the electronic properties are in an essential way determined \emph{globally}, in contrast to the 'nearsightedness' [W. Kohn. Phys. Rev. Lett., 76:3168, 1996.] of electronic structure generally found in condensed matter.Comment: Article as to be published in Phys. Rev B. Main changes: K-point mapping tables fixed, several changes to presentation

We explore the consequences of a rotation between graphene layers for the electronic spectrum. We derive the commensuration condition in real space and show that the interlayer electronic coupling is governed by an equivalent commensuration in reciprocal space. The larger the commensuration cell, the weaker the interlayer coupling, with exact decoupling for incommensurate rotations and in the ! 0 limit. Furthermore, from first-principles calculations we determine that even for the smallest possible commensuration cell the decoupling is effectively perfect, and thus graphene layers will be seen to decouple for all rotation angles.

The diffusion of intrinsic defects in 3C-SiC is studied using an ab initio method based on density functional theory. The vacancies are shown to migrate on their own sublattice. The carbon splitinterstitials and the two relevant silicon interstitials, namely the tetrahedrally carbon-coordinated interstitial and the 110 -oriented split-interstitial, are found to be by far more mobile than the vacancies. The metastability of the silicon vacancy, which transforms into a vacancy-antisite complex in p-type and compensated material, kinetically suppresses its contribution to diffusion processes. The role of interstitials and vacancies in the self-diffusion is analyzed. Consequences for the dopant diffusion are qualitatively discussed. Our analysis emphasizes the relevance of mechanisms based on silicon and carbon interstitials.

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