Interplay between edge states and simple bulk defects in graphene nanoribbons

We study the interplay between the edge states and a single impurity in a zigzag graphene nanoribbon. We use tight-binding exact diagonalization techniques, as well as density functional theory calculations to obtain the eigenvalue spectrum, the eigenfunctions, as well the dependence of the local density of states (LDOS) on energy and position. We note that roughly half of the unperturbed eigenstates in the spectrum of the finite-size ribbon hybridize with the impurity state, and the corresponding eigenvalues are shifted with respect to their unperturbed values. The maximum shift and hybridization occur for a state whose energy is inverse proportional to the impurity potential; this energy is that of the impurity peak in the DOS spectrum. We find that the interference between the impurity and the edge gives rise to peculiar modifications of the LDOS of the nanoribbon, in particular to oscillations of the edge LDOS. These effects depend on the size of the system, and decay with the distance between the edge and the impurity.

Spin and Majorana polarization in topological superconducting wires

We study a one-dimensional wire with strong Rashba and Dresselhaus spin-orbit coupling (SOC), which supports Majorana fermions when subject to a Zeeman magnetic field and in proximity of a superconductor. Using both analytical and numerical techniques we calculate the electronic spin texture of the Majorana end states. We find that the spin polarization of these states depends on the relative magnitude of the Rashba and Dresselhaus SOC components. Moreover, we define and calculate a local Majorana polarization and Majorana density and argue that they can be used as order parameters to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes. We find that the local Majorana polarization is correlated to the transverse spin polarization, and we propose to test the presence of Majorana fermions in a 1D system by a spin-polarized density of states measurement.

Fourier Transform Scanning Tunneling Spectroscopy: the possibility to obtain constant energy maps and the band dispersion using a local measurement

We present here an overview of the Fourier Transform Scanning Tunneling spectroscopy technique (FT-STS). This technique allows one to probe the electronic properties of a two-dimensional system by analyzing the standing waves formed in the vicinity of defects. We review both the experimental and theoretical aspects of this approach, basing our analysis on some of our previous results, as well as on other results described in the literature. We explain how the topology of the constant energy maps can be deduced from the FT of dI/dV map images which exhibit standing waves patterns. We show that not only the position of the features observed in the FT maps, but also their shape can be explained using different theoretical models of different levels of approximation. Thus, starting with the classical and well known expression of the Lindhard susceptibility which describes the screening of electron in a free electron gas, we show that from the momentum dependence of the susceptibility we can deduce the topology of the constant energy maps in a joint density of states approximation (JDOS). We describe how some of the specific features predicted by the JDOS are (or are not) observed experimentally in the FT maps. The role of the phase factors which are neglected in the rough JDOS approximation is described using the stationary phase conditions. We present also the technique of the T-matrix approximation, which takes into account accurately these phase factors. This technique has been successfully applied to normal metals, as well as to systems with more complicated constant energy contours. We present results recently obtained on graphene systems which demonstrate the power of this technique, and the usefulness of local measurements for determining the band structure, the map of the Fermi energy and the constant-energy maps.

Metamorphosis and Taxonomy of Andreev Bound States

We analyze the spatial and energy dependence of the local density of states in a SNS junction. We model our system as a one-dimensional tight-binding chain which we solve exactly by numerical diagonalization. We calculate the dependence of the Andreev bound states on position, phase difference, gate voltage, and coupling with the superconducting leads. Our results confirm the physics predicted by certain analytical approximations, but reveal a much richer set of phenomena beyond the grasp of these approximations, such as the metamorphosis of the discrete states of the normal link (the normal bound states) into Andreev bound states as the leads become superconducting.

Dirac point metamorphosis from third-neighbor couplings in graphene

We study the band structure and the density of states of graphene in the presence of a next-to-nearest-neighbor coupling (N2) and a third-nearest-neighbor coupling (N3). We show that for values of N3 larger or equal to 1/3 of the value of the nearest-neighbor hopping (NN), extra Dirac points appear in the spectrum. If N3 is exactly equal to 1/3 NN, the new Dirac points are localized at the M points of the Brillouin zone and are hybrid: the electrons have a linear dispersion along the GammaM direction and a quadratic dispersion along the perpendicular direction MK. For larger values of N3 the new points have a linear dispersion, and are situated along the MK line. For a value of N3 equal to 1/2 NN, these points merge with the Dirac cones at the K points, yielding a gapless quadratic dispersion around K, while for larger values each quadratic point at K splits again into four Dirac points. The effects of changing the N2 coupling are not so dramatic. We calculate the density of states and we show that increasing the N3 coupling lowers the energy of the Van Hove singularities, and when N3 is larger than 1/3 NN the Van Hove singularities split in two, giving rise to extra singularities at low energies.

Superlattice of resonators on monolayer graphene created by intercalated gold nanoclusters

Here we report on a new type of ordering which allows to modify the electronic structure of a graphene monolayer (ML). We have intercalated small gold clusters between the top monolayer graphene and the buffer layer of epitaxial graphene. We show that these clusters perturb the quasiparticles on the ML graphene, and act as quantum dots creating a superlattice of resonators on the graphene ML, as revealed by a strong pattern of standing waves. A detailed analysis of the standing wave patterns using Fourier Transform Scanning Tunneling Spectroscopy strongly indicates that this phenomenon can arise from a strong modification of the band structure of graphene and (or) from Charge Density Waves (CDW)where a large extension of Van Hove singularities are involved.

The tunneling conductance between a superconducting STM tip and an out-of-equilibrium carbon nanotube

We calculate the current and differential conductance for the junction between a superconducting (SC) STM tip and a Luttinger liquid (LL). For an infinite single-channel LL, the SC coherence peaks are preserved in the tunneling conductance for interactions weaker than a critical value, while for strong interactions (g <0.38), they disappear and are replaced by cusp-like features. For a finite-size wire in contact with non-interacting leads, we find however that the peaks are restored even for extremely strong interactions. In the presence of a source-drain voltage the peaks/cusps split, and the split is equal to the voltage. At zero temperature, even very strong interactions do not smear the two peaks into a broader one; this implies that the recent experiments of Y.-F. Chen et. al. (Phys. Rev. Lett. 102, 036804 (2009)) do not rule out the existence of strong interactions in carbon nanotubes.

The local density of states in the presence of impurity scattering in graphene at high magnetic field

We study the Fourier transform of the local density of states (LDOS) in graphene in the presence of a single impurity at high magnetic field. We find that the most pronounced features occur for energies of the STM tip matching the Landau level energies. The Fourier transform of the LDOS shows regions of high intensity centered around the center and the corners of the Brillouin zone (BZ). The radial intensity dependence of these features is determined by the form of the wavefunctions of the electrons in the quantum Hall regime. Moreover, some of these regions break rotational symmetry, and their angular dependence is determined by the chirality of the graphene electrons. For the zeroth Landau level, the ratio between the features at the corners and center of the BZ depends on the nature of the disorder: it goes to zero for potential disorder, and is finite for hopping disorder. We believe that a comparison between our analysis and experiments will help understand the form of the quasiparticle wavefunction, as well as the nature of disorder in graphene.

Symmetry of standing waves generated by a point defect in epitaxial graphene

Using scanning tunneling microscopy (STM) and Fourier Transform STM (FT-STM), we have studied a point defect in an epitaxial graphene sample grown on silicon carbide substrate. This analysis allows us to extract the quasiparticle energy dispersion, and to give a first experimental proof of the validity of Fermi liquid theory in graphene for a wide range of energies from -800 meV to +800 meV. We also find evidence of a strong threefold anisotropy in the standing waves generated by the defect. We discuss possible relations between this anisotropy, the chirality of the electrons, and the asymmetry between graphenes two sublattices. All experimental measurements are compared and related to theoretical T-matrix calculations.