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.

Graphene sheets are a highly radiation-resistant material for prospective nuclear applications and nanoscale defect engineering. However, the precise mechanism of graphene radiation hardness has remained elusive. In this paper, we study the origin and nature of defects induced by gamma radiation in a graphene rolled-up plane. In order to reduce the environmental influence on graphene and reveal the small effects of gamma rays, we have synthesized a novel graphene-based nanocomposite material containing a bilayer of highly aligned carbon nanotube assemblies that have been decorated by organometallic compounds and suspended on nanoporous Al${}_2$O${}_3$ membranes. The bilayer samples were irradiated by gamma rays from a ${}^{137}$Cs source with a fluence rate of the order of ${10}^5$ m${}^{-2}$s${}^{-1}$. The interaction between the samples and gamma quanta results in the appearance of three characteristic photon escape peaks in the radiation spectra. We explain the mechanism of interaction between the graphene sheets and gamma radiation using a pseudo-Majorana fermion graphene model, which is a quasi-relativistic $N=3$-flavor graphene model with a Majorana-like mass term. This model admits the existence of giant charge carrier currents that are sufficient to neutralize the impact of ionizing radiation. Experimental evidence is provided for the prediction that the 661.7-keV gamma quanta transfer enough energy to the electron subsystem of graphene to bring about the deconfinement of the bound pseudo-Majorana modes and involve C atoms in a vortical motion of the electron density flows in the graphene plane. We explain the radiation hardness of graphene by the topological non-triviality of the pseudo-Majorana fermion configurations comprising the graphene charge carriers.

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.

A label-free high-performance electrochemical sensing of single nucleotide polymorphism in native genomic DNA samples from cancer tissues, in oncogene-specific complementary DNA and in...