Abstract:The band structure of strongly correlated two-dimensional (2D) semimetal systems is found to be significantly affected by the spin-orbit coupling (SOC), resulting in SOC-induced Fermi surfaces. Dirac, Weyl and Majorana representations are used for the description of different semimetals, though the band structures of all these systems are very similar. We develop a theoretical approach to the band theory of two-dimensional semimetals within the Dirac-Hartree-Fock self-consistent field approximation. It reveals partially breaking symmetry of the Dirac cone affected by quasi-relativistic exchange interactions for 2D crystals with hexagonal symmetry. Fermi velocity becomes an operator within this approach, and elementary excitations have been calculated in the tight-binding approximation when taking into account the exchange interaction of π(p z )-electron with its three nearest π(p z )-electrons. These excitations are described by the massless Majorana equation instead of the Dirac one. The squared equation for this field is of the Klein-Gordon-Fock type. Such a feature of the band structure of 2D semimetals as the appearance of four pairs of nodes is shown to be described naturally within the developed formalism. Numerical simulation of band structure has been performed for the proposed 2D-model of graphene and a monolayer of Pb atoms.

No abstract

Problem of control over Klein-tunnelling states from electrostatically-confined graphene p - n junctions has been discussed. The lack of quasi-bound states, being the states with a finite life time, in a pseudo-Dirac-fermion model for the graphene quantum dot (GQD) is theoretically predicted as inapplicability of the so-called "resonance condition" leading to an inconsistent linear system corresponding to matching conditions. Within a pseudo-Dirac-Weyl fermion model GQD, the graphene charge carriers are topologically nontrivial and can be confined by a staircase-type potential due to competition between Zak curvature and centrifugal-force actions. The predicted topological effects elucidate experimentally observed resonances created by electron beam and laser pulse in crystalline arrays of single-walled carbon nanotubes as the Klein-tunnelling resonant states in the p - n graphene junctions. We present a robust approach to fabricate stable graphene p - n junctions by fine-tuning the topological effects.

The quantum field theory approach has been proposed for the description of graphene electronic properties. It generalizes massless Dirac fermion model and is based on the Dirac-Hartree-Fock self-consistent field approximation and assumption on antiferromagnetic ordering of graphene lattice. The developed approach allows asymmetric charged carriers in single layer graphene with partially degenerated Dirac cones.

In this work, a research has been carried out into the electronic properties of nanostructured graphene. We focus our attention on trapped states of the proposed systems such as spherical and toroidal graphene quantum dots. Using a continuum model, by solving the Dirac-Weyl equation, and applying periodic boundary conditions of two types, i.e. either with zigzag-edges only, or with both armchair-and zigzag-edges, we obtain analytical results for energy levels yielding self-similar energy bands located subsequently one after another on the energy scale. Only for toroidal quantum dots (owing to the lack of curvature) the distribution of electron density is similar to Bohr atomic orbitals. However, although the quasi-zero-energy band exists for both spherical and toroidal quantum dots, no electron density is present on this band for the toroidal quantum dot. This causes the formation of a pseudogap between the hole and electron bands, because of the absence of the electron density at the quantum dot center, like in the case of an ordinary atom. Conversely, the confinement of the charge-carrier density is observed for both geometries of graphene quantum dots.