V. A. Osipov

The continuum gauge field-theory model for low-energy electronic states of icosahedral fullerenes

Abstract. The low-energy electronic structure of icosahedral fullerenes is studied within the field-theory model. In the field model, the pentagonal rings in the fullerene are simulated by two kinds of gauge fields. The first one, non-abelian field, follows from so-called K spin rotation invariance for the spinor field while the second one describes the elastic flow due to pentagonal apical disclinations. For fullerene molecule, these fluxes are taken into account by introducing an effective field due to magnetic monopole placed at the center of a sphere. Additionally, the spherical geometry of the fullerene is incorporated via the spin connection term. The exact analytical solution of the problem (both for the eigenfunctions and the energy spectrum) is found.

Small-angle scattering from the Cantor surface fractal on the plane and the Koch snowflake

The small-angle scattering (SAS) from the Cantor surface fractal on the plane and Koch snowflake is considered. We develop the construction algorithm for the Koch snowflake, which makes possible the recurrence relation for the scattering amplitude. The surface fractals can be decomposed into a sum of surface mass fractals for arbitrary fractal iteration, which enables various approximations for the scattering intensity. It is shown that for the Cantor fractal, one can neglect with good accuracy the correlations between the mass fractal amplitudes, while for the Koch snowflake, these correlations are important. It is shown that nevertheless, correlations can be built in the mass fractal amplitudes, which explains the decay of the scattering intensity I(q) ∼ qDs-4, with 1 < Ds < 2 being the fractal dimension of the perimeter. The curve I(q)q4-Ds is found to be log-periodic in the fractal region with a period equal to the scaling factor of the fractal. The log-periodicity arises from the self-similarity of the sizes of basic structural units rather than from correlations between their distances. A recurrence relation is obtained for the radius of gyration of the Koch snowflake, which is solved in the limit of infinite iterations. The present analysis allows us to obtain additional information from SAS data, such as the edges of the fractal regions, the fractal iteration number and the scaling factor.

Electronic properties of disclinated flexible membrane beyond the inextensional limit: application to graphene

A gauge-theory approach to describe Dirac fermions on a disclinated flexible membrane beyond the inextensional limit is formulated. The elastic membrane is considered as an embedding of a 2D surface into R(3). The disclination is incorporated through an SO(2) gauge vortex located at the origin, which results in a metric with a conical singularity. A smoothing of the conical singularity is accounted for by replacing a disclinated rigid plane membrane with a hyperboloid of near-zero curvature pierced at the tip by the SO(2) vortex. The embedding parameters are chosen to match the solution to the von Karman equations. A homogeneous part of that solution is shown to stabilize the theory. The modification of the Landau states and density of electronic states of the graphene membrane due to elasticity is discussed.

Scattering from generalized Cantor fractals

A fractal with a variable fractal dimension, which is a generalization of the well known triadic Cantor set, is considered. In contrast with the usual Cantor set, the fractal dimension is controlled using a scaling factor, and can vary from zero to one in one dimension and from zero to three in three dimensions. The intensity profile of small-angle scattering from the generalized Cantor fractal in three dimensions is calculated. The system is generated by a set of iterative rules, each iteration corresponding to a certain fractal generation. Small-angle scattering is considered from monodispersive sets, which are randomly oriented and placed. The scattering intensities represent minima and maxima superimposed on a power law decay, with the exponent equal to the fractal dimension of the scatterer, but the minima and maxima are damped with increasing polydispersity of the fractal sets. It is shown that, for a finite generation of the fractal, the exponent changes at sufficiently large wave vectors from the fractal dimension to four, the value given by the usual Porod law. It is shown that the number of particles of which the fractal is composed can be estimated from the value of the boundary between the fractal and Porod regions. The radius of gyration of the fractal is calculated analytically.

Scattering from surface fractals in terms of composing mass fractals

We argue that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of surface fractal is shown to be a sum of the amplitudes of composing mass fractals. Various approximations for the scattering intensity of surface fractal are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensity

Electronic states of zigzag graphene nanoribbons with edges reconstructed with topological defects

I(q) \propto q^{D_{\mathrm{s}}-6}

Review Article: Tunneling-based graphene electronics: Methods and examples

, where

Small-angle scattering from the deterministic fractal systems1

2 < D_{\mathrm{s}} < 3

Small-angle scattering from three-phase systems: Investigation of the crossover between mass fractal regimes

is the surface fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution

Prospects for investigating deterministic fractals: Extracting additional information from small angle scattering data

d N(r) \propto r^{-\tau} dr