In this paper, research has been carried out on 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 (GQDs). 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 the toroidal quantum dot (owing to the lack of curvature) the distribution of electron density is like 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 GQDs.

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 $\pi ({\mathrm{p}}_z)$ -electron with its three nearest $\pi ({\mathrm{p}}_z)$ -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.

Within the earlier developed high-energy- $\overrightarrow{k}·\overrightarrow{p}$ -Hamiltonian approach to describe graphene-like materials, the simulations of non-Abelian Zak phases and band structure of the quasi-relativistic graphene model with a number of flavors $N=3$ have been performed in approximations with and without gauge fields (flavors). It has been shown that a Zak-phases set for non-Abelian Majorana-like excitations (modes) in Dirac valleys of the quasi-relativistic graphene model is the cyclic group ${\mathbf{Z}}_{12}$ . This group is deformed into ${\mathbf{Z}}_8$ at sufficiently high momenta due to deconfinement of the modes. Since the deconfinement removes the degeneracy of the eightfolding valleys, Weyl nodes and antinodes emerge. We offer that a Majorana-like mass term of the quasi-relativistic model affects the graphene band structure in the following way. Firstly, the inverse symmetry emerges in the graphene model with Majorana-like mass term, and secondly the mass term shifts the location of Weyl nodes and antinodes into the region of higher energies.

Within the earlier developed high-energy-$\overrightarrow{k}·\overrightarrow{p}$-Hamiltonian approach to describe graphene-like materials, the simulations of band structure, non-Abelian Zak phases and the complex conductivity of graphene have been performed. The quasi-relativistic graphene model with a number of flavors (gauge fields) $N_F=3$ in two approximations (with and without a pseudo-Majorana mass term) has been utilized as a ground for the simulations. It has been shown that Zak-phases set for the non-Abelian Majorana-like excitations (modes) in graphene represent the cyclic ${\mathbb{Z}}_{12}$ and this group is deformed into a smaller one ${\mathbb{Z}}_8$ at sufficiently high momenta due to a deconfinement of the modes. Simulations of complex longitudinal low-frequency conductivity have been performed with a focus on effects of spatial dispersion. A spatial periodic polarization in the graphene models with the pseudo Majorana charge carriers is offered.

Graphene model with a high-energy k · p-Hamiltonian has been proposed. A feature of the model is the increasing spin-orbital coupling at relativistic exchange interactions of electron-hole...