Dirac electrons in graphene-based quantum wires and quantum dots

Abstract

In this paper we analyse the electronic properties of Dirac electrons in finite-size ribbons and in circular and hexagonal quantum dots. We show that due to the formation of sub-bands in the ribbons it is possible to spatially localize some of the electronic modes using a p-n-p junction. We also show that scattering of confined Dirac electrons in a narrow channel by an infinitely massive wall induces mode mixing, giving a qualitative reason for the fact that an analytical solution to the spectrum of Dirac electrons confined in a square box has not yet been found. A first attempt to solve this problem is presented. We find that only the trivial case k = 0 has a solution that does not require the existence of evanescent modes. We also study the spectrum of quantum dots of graphene in a perpendicular magnetic field. This problem is studied in the Dirac approximation, and its solution requires a numerical method whose details are given. The formation of Landau levels in the dot is discussed. The inclusion of the Coulomb interaction among the electrons is considered at the self-consistent Hartree level, taking into account the interaction with an image charge density necessary to keep the back-gate electrode at zero potential. The effect of a radial confining potential is discussed. The density of states of circular and hexagonal quantum dots, described by the full tight-binding model, is studied using the Lanczos algorithm. This is necessary to access the detailed shape of the density of states close to the Dirac point when one studies large systems. Our study reveals that zero-energy edge states are also present in graphene quantum dots. Our results are relevant for experimental research in graphene nanostructures. The style of writing is pedagogical, in the hope that newcomers to the subject will find this paper a good starting point for their research.

Boundary Conditions in the Dirac Approach to Graphene Devices

“…Note that Ref. 22 had already suggested that this could be the case. In that reference, MIT bag boundary conditions are rather called Berry-Mondragón boundary conditions, because they were also studied in Ref.…”

“…Solving this boundary value problem is a simple exercise (see, for instance, Ref. 22). The first outcome is that zigzag boundary conditions (α = ± π 2 ) allow for an infinite amount of zero modes, as expected from the facts that they do not satisfy the Lopatinski-Shapiro condition and that we are now treating a compact region with a smooth boundary.…”

Electron-electron interactions in graphene field-induced quantum dots in a high magnetic field

“…These compressible strips are known to affect and alter the electronic properties of quantum dots and antidots [25][26][27]. Even though certain aspects of electron-electron interactions have been considered previously [28,29], the effect of magnetic-field-induced modification of the confining potential on the eigenspectrum of graphene quantum dots has not been studied before, and the present paper represents a step in this direction.…”

“…We calculate V mod in the same way as V (r), where, however, we set B = 0, and therefore instead of the DOS for graphene in a magnetic field, Eq. (4), we use a linear DOS describing graphene for the case of B = 0 [28],…”

Quantum Mechanics on Surfaces

“…The tentative framework presented here might furthermore also help shed light on a fundamental problem connected with the understanding of quantum mechanics on surfaces following Dirac's quantization prescription (2). It is well known that the Dirac quantization scheme does not produce an unique expression for the induced quantum geometric potential V S (15)(16)(17). It is claimed that this result is not related to improper choice of coordinates, but emerges solely due to operator ordering issues (15).…”