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Allotropes of carbon, including one-dimensional carbon nanotubes and two-dimensional graphene sheets, continue to draw attention as promising platforms for probing the physics of electrons in lower dimensions. Recent research has shown that the electronic properties of graphene multilayers are exquisitely sensitive to the relative orientation between sheets, and in the bilayer case exhibit strong electronic correlations when close to a magic twist angle. Here, we investigate the electronic properties of a carbon nanotube deposited on a graphene sheet by deriving a low-energy theory that accounts both for rotations and rigid displacements of the nanotube with respect to the underlying graphene layer. We show that this heterostructure is described by a translationally invariant, a periodic or a quasi-periodic Hamiltonian, depending on the orientation and the chirality of the nanotube. Furthermore, we find that, even for a vanishing twist angle, rigid displacements of a nanotube with respect to a graphene substrate can alter its electronic structure qualitatively. Our results identify a promising new direction for strong correlation physics in low dimensions.
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The electronic structure of graphene on Cu(111) and Cu(100) single crystals is investigated using low energy electron microscopy, low energy electron diffraction and angle resolved photoemission spectroscopy. On both substrates the graphene is rotationally disordered and interactions between the graphene and substrate lead to a shift in the Dirac crossing of $sim$ -0.3 eV and the opening of a $sim$ 250 meV gap. Exposure of the samples to air resulted in intercalation of oxygen under the graphene on Cu(100), which formed a ($sqrt{2} times 2sqrt{2}$)R45$^{rm o}$ superstructure. The effect of this intercalation on the graphene $pi$ bands is to increase the offset of the Dirac crossing ($sim$ -0.6 eV) and enlarge the gap ($sim$ 350 meV). No such effect is observed for the graphene on Cu(111) sample, with the surface state at $Gamma$ not showing the gap associated with a surface superstructure. The graphene film is found to protect the surface state from air exposure, with no change in the effective mass observed.
We investigate the adsorption of graphene sheets on h-BN substrates by means of first-principles calculations in the framework of adiabatic connection fluctuation-dissipation theory in the random phase approximation. We obtain adhesion energies for different crystallographic stacking configurations and show that the interlayer bonding is due to long-range van der Waals forces. The interplay of elastic and adhesion energies is shown to lead to stacking disorder and moire structures. Band structure calculations reveal substrate induced mass terms in graphene which change their sign with the stacking configuration. The dispersion, absolute band gaps and the real space shape of the low energy electronic states in the moire structures are discussed. We find that the absolute band gaps in the moire structures are at least an order of magnitude smaller than the maximum local values of the mass term. Our results are in agreement with recent STM experiments.
We have studied the discrete electronic spectrum of closed metallic nanotube quantum dots. At low temperatures, the stability diagrams show a very regular four-fold pattern that allows for the determination of the electron addition and excitation energies. The measured nanotube spectra are in excellent agreement with theoretical predictions based on the nanotube band structure. Our results permit the complete identification of the electron quantum states in nanotube quantum dots.
We report experimental measurements of electronic Raman scattering under resonant conditions by electrons in individual single-walled carbon nanotubes (SWNTs). The inelastic Raman scattering at low frequency range reveals a single particle excitation feature and the dispersion of electronic structure around the center of Brillouin zone of a semiconducting SWNT (14, 13) is extracted.
The localization length and density of states of carbon nanotubes are evaluated within the tight-binding approximation. By comparison with the corresponding results for the square lattice tubes, it is found that the hexagonal structure affects strongly the behaviors of the density of states and localization lengths of carbon nanotubes.
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