By means of an envelope function analysis, we perform a numerical investigation of the conductance behavior of a graphene structure consisting of two regions (dots) connected to the entrance and exit leads through constrictions and separated by a pot
ential barrier. We show that the conductance of the double dot depends on the symmetry of the structure and that this effect survives also in the presence of a low level of disorder, in analogy of what we had previously found for a double dot obtained in a semiconductor heterostructure. In graphene, this phenomenon is less dramatic and, in particular, conductance is not enhanced by the addition of symmetric constrictions with respect to that of the barrier alone.
The solution of differential problems, and in particular of quantum wave equations, can in general be performed both in the direct and in the reciprocal space. However, to achieve the same accuracy, direct-space finite-difference approaches usually i
nvolve handling larger algebraic problems with respect to the approaches based on the Fourier transform in reciprocal space. This is the result of the errors that direct-space discretization formulas introduce into the treatment of derivatives. Here, we propose an approach, relying on a set of sinc-based functions, that allows us to achieve an exact representation of the derivatives in the direct space and that is equivalent to the solution in the reciprocal space. We apply this method to the numerical solution of the Dirac equation in an armchair graphene nanoribbon with a potential varying only in the transverse direction.
We discuss the possibility of diffusive conduction and thus of suppression of shot noise by a factor 1/3 in mesoscopic semiconductor devices with two-dimensional and one-dimensional potential disorder, for which existing experimental results do not p
rovide a conclusive result. On the basis of our numerical analysis, we conclude that it is quite difficult to achieve diffusive transport over a reasonably wide parameter range, unless the device dimensions are increased up to the macroscopic scale. In addition, in the case of one-dimensional disorder, some mechanism capable of mode-mixing has to be present in order to reach or even approach the diffusive regime.
The k.p method is a semi-empirical approach which allows to extrapolate the band structure of materials from the knowledge of a restricted set of parameters evaluated in correspondence of a single point of the reciprocal space. In the first part of t
his review article we give a general description of this method, both in the case of homogeneous crystals (where we consider a formulation based on the standard perturbation theory, and Kanes approach) and in the case of non-periodic systems (where, following Luttinger and Kohn, we describe the single-band and multi-band envelope function method and its application to heterostructures). The following part of our review is completely devoted to the application of the k.p method to graphene and graphene-related materials. Following Andos approach, we show how the application of this method to graphene results in a description of its properties in terms of the Dirac equation. Then we find general expressions for the probability density and the probability current density in graphene and we compare this formulation with alternative existing representations. Finally, applying proper boundary conditions, we extend the treatment to carbon nanotubes and graphene nanoribbons, recovering their fundamental electronic properties.
