Electronic properties of graphene with point defects (Review Article)

This review is an attempt to consistently examine the electronic spectrum of graphene containing defects (such as adsorbed atoms, substitutional atoms, vacancies) that can be adequately described using the Lifshitz model. Therefore, the known Hamiltonian of this model is chosen for the case of two-dimensional relativistic electrons, and the criteria for the appearance of an impurity resonance near the Dirac point are provided. The theory of concentration band structure transformation in graphene is outlined, from which it follows that a transport gap opens in the vicinity of the impurity resonance energy when a specific value of the impurity concentration is reached. Along the way, the question of whether or not it is possible (or impossible) for Dirac quasiparticles to become localized in such a spatially disordered system is analyzed. Based on this, it is feasible to explain and describe the recently observed in impure graphene phenomenon of metal-insulator transition, which turns out to be a direct consequence of the system’s Fermi energy falling into the domain of the transport gap. The concept of local spectrum rearrangement, which can also unfold as the defect concentration increases, is introduced and justified for graphene. We formulate the physical reasons why the minimum of graphene’s low-temperature conductivity dependence on the Fermi energy of electrons does correspond to the impurity resonance energy, and not the Dirac point, as it has been claimed in a number of theoretical and experimental studies. Furthermore, the mentioned minimum value proves to be not universal, but is dependent, instead, on the concentration of defects. The analytical considerations of the impurity effects are accompanied by numerical simulations of the examined system, and a complete correspondence between these two approaches is established as a result. In particular, the general scenarios of spectrum rearrangement, electron state localization, as well as of effects having a local nature, are confirmed.