The character of propagation in graphene nanostructures of quasi-flexural phonons, whose dispersion law differs from that of sound, is analyzed. Based on the calculation of the frequency dependences of the group velocities and the values of the path of quasiparticles for one period of oscillation, the frequency intervals are established at which: i) phonons propagate freely along all directions of reciprocal space—the propagon zone; ii) phonon propagation along some directions is impossible—diffuse zone; iii) phonons are localized at the nodes of the honeycomb lattice—the locon zone. A comparison is made with a similar classification of phonons in a three-dimensional crystal of cubic symmetry.

The nature of flexural rigidity of graphene monolayers and the formation of the quasi-flexural modes in graphite and graphene nanofilms is explained. It is shown, that in the major part of the band of quasi-continuous spectrum, quasi-flexural modes do not practically couple with the modes polarized in the plane of the layer. The frequencies of the propagone zone are determined for the phonons with various polarizations. The existence of a high-frequency propagation zone is demonstrated for quasi-flexural phonons. The conditions for the formation of quasi-local vibrations and boson peaks are analyzed for modes with different polarization. The conditions for the formation of the quasi-local vibrations and boson peaks in the modes of different polarization are analyzed.

The nature of flexural rigidity of graphene monolayers and a formation of the quasiflexural modes in a graphite and the graphene nanofilms is explained. It is shown, that in the major part of the band of quasicontinuous spectrum, quasiflexural modes do not practically couple with the modes polarized in the plane of the layer. The frequencies of propagone zone are determined for the phonons with various polarizations. The existence of high-frequency propagation zone is demonstrated for quasi-flexural phonons. The conditions for a formation of quasi-local vibrations and boson peaks are analyzed for modes with different polarization. The conditions for the formation of quasi-local vibrations and boson peaks in modes of different polarization are analyzed.

The densities of phonon states and their related vibrational thermodynamic characteristics, such as heat capacity, root-mean-square displacement of atoms, and thermal expansion, are calculated and analyzed at the microscopic level for graphite and graphene nanoformations, such as nanofilms and nanotubes. The simulation model is based on experimental data without a priori assumptions as to the nature and potentials of interatomic interactions, and is compared to them only after, which yields a good agreement. The quasi-flexural and torsional modes, which are inherent to graphene nanotubes, are considered and their contribution to the low temperature vibrational characteristics are analyzed. The impact that extended defects, such as graphite intercalation with transition metals, step-edges on graphene nanofilms, and edges of a graphene single layer on a substrate, have on the phonon spectrum and vibrational characteristics is analyzed. The calculated results are compared to the experimental data.

Among the large number of both fundamental and ordinary studies of various physical parts of nanomaterials, the study of graphene structures, such as nanofilms (bigraphene, trigraphene) and nanotubes is undoubtedly a priority. At first sight, from a practical point of view, the vibrational characteristics of these structures do not seem to be as relevant as the spectra of some of the other quasiparticle excitations and their properties (e.g., electronic and magnetic), this is clearly not the case. First of all, the phonon spectrum determines the stability of the structure, which is especially important for the building of nanostructures. Secondly, some properties that are manifested by investigated nanoobjects, or those that are already being applied and are currently being put into effect, or those that are still in demand (such as increased conductivity of graphene structures), manifest themselves with a significant and even decisive participation of phonons. One more thought is the theoretical background of the dynamic stability of carbon nanostructures, which is very important for “fine tuning” the process of their synthesis and determining the operating conditions of devices based on this structures. In addition, an analysis of the rms amplitudes of atomic displacements makes it real to define the limits of applicability of the harmonic approximation when considering vibrations of graphite and bigraphene, trigraphene, i.e., the use of phonons and phonons modes to explain heat capacity, root-mean-square (rms) atomic displacements, thermal expansion of the concepts themselves, which is not a priori obvious.

The total electron densities of states for graphene nanoribbons with edges of different chirality, as well as the electron local densities of states for individual atoms in these nanoribbons, are calculated and analyzed. There are sharp resonance peaks near the Fermi level in the total electron densities of states of graphene nanoribbons with zigzag edges, which emerge only in the local densities of atoms from the sublattice that goes directly to the nearest edge (i.e., whose atoms have dangling bonds). Semiconducting gaps appear in the spectra of graphene nanobands with armchair chirality edges having a number of constituent atomic lines that is either a multiple of three, or gives a remainder of one when divided by three. The width of this gap only depends on the width of the nanoribbon, and is the same for all its atoms. The electron spectra of graphene nanoribbons with armchair-chirality edges have a metallic behavior if the number of atomic lines gives a remainder of two when divided by three. However, semiconducting gaps still manifest on the local densities of the atoms belonging to some lines of such nanoribbons. Published under license by AIP Publishing.

The conditions of the existence and the main characteristics (frequencies, intensities, and attenuation parameters) of discrete vibrational levels caused by structural defects of linear chains of inert gas atoms adsorbed on the surface of nanotubes aggregated in nanobundles, are calculated and analyzed. Discrete vibrational levels lying both above and below the band of the quasi-continuous spectrum of the chain are considered. Analytical expressions obtained for the frequencies of these discrete levels make it possible to determine with high precision the defect parameters (difference in the interaction with the substrate and with the atoms of the chain) basing on optical measurements.

This chapter contains sections titled: Introduction Electron and Phonon Spectra of Ultrathin Graphene Nanofilms Effect of Defects to Electron and Phonon Spectra Conclusion

The features of phonon spectra and their effect on the vibrational heat capacity of linear chains of inert gas atoms adsorbed onto a substrate, which is the surface of nanotubes bound to a nanobundle. The influence of the substrate results both in a shift of the lower limit of the chain spectrum from zero, and in mechanical stress in the chain (its extension or compression) also. It is shown that in the case of a compressed chain, the non-central interaction between atoms is negative (repulsive), it results in a shift of the lower boundary of the spectrum of transverse vibrations to low frequencies and to a shortening of the part of the specific heat temperature dependence in which this dependence is close to exponential. Heterogeneity of the nanobundle structure can cause a change in the distances between atoms of the chain. It is shown both and analytically and numerically, that as a result of it, discrete levels with frequencies both above and below the quasi-continuous spectrum band can appear in the phonon spectrum of the chain. The discrete levels with frequencies below the quasi-continuous spectrum band lead to a further shortening of the temperature interval at which the temperature dependence of the specific heat is close to the exponential one.

We show theoretically a possibility to increase the superconducting transition temperature T-c of graphene by introducing defects. Eventually, the peak of local electron densities shifts to the Fermi level with T-c above ambient temperature in line with recent observations.

We perform analytical and numerical analysis of the electronic and phonon spectrum evolution of graphene during formation of a boundary with a “zigzag” chirality. It is determined, that the excited gap wave has a relativistic dispersion near the Fermi level that propagates along the boundary and decays with distance from it. Both properties and formation of the wave is considered. It is shown that the wave propagation occurs only along the atoms of the sub-lattice, which contains atoms with bonds broken during the boundary formation. The gap wave forms narrow resonance peaks in the local density of states of the sublattice atoms. It is shown, that the boundary formation on a graphene layer with this chirality similarly affects the phonon modes polarized normal to the layer, forming narrow maxima with frequencies nearing that of the quasiflexural phonons with the quasiwave vector at the K-point of the first Brillouin zone. This way, the formation of the “zigzag”-boundary increases both the number of charge carriers as well as the number of phonons with a large group velocity that can cause a large contribution to the electron-phonon interactions.

Based on calculations conducted on a microscopic level, the phonon heat capacity of ultrathin graphene nanofilms such as bigraphene and trigraphene, and single-wall graphene nanotubes, is quantitatively described. The nature of the flexural stiffness of graphene monolayers is analyzed, and the temperature intervals at which the shape of the temperature dependence of heat capacity is determined by contributions made by flexural vibrations are identified. The contribution to the phonon heat capacity derived from graphene nanotube flexural waves that propagate along the surface thereof is analyzed, as are the bending vibrations of the tube as a whole one-dimensional object, and the contribution from torsional vibrations.

The electron local density of states (LDOS) are calculated for graphene with isolated vacancies, divacancies and vacancy group of four nearest-neighbor vacancies. A strong anisotropy of behavior of LDOS near Fermi level is demonstrated for atoms near defect. Effect of next-to-nearest neighbor interaction on the properties of graphene with vacancies is established.

Phonon and electron spectra of metallic bigraphene are analyzed in the presence of step-edge crystal imperfection. Different geometries of step-edge are considered. The dynamic planar stability of the considered structure is proved for temperatures above the ambient. The number of phonon states is shown to grow near the K-point of the first Brillouin zone, compared to pristine graphene. It is found, that this type of defects causes substantially nonuniform distribution of electron states and the pronounced increase in the number of states with energies close to Fermi energy can be expected in electron spectrum of the graphene-based compounds. The performed calculations are in good agreement with inelastic neutron, x-ray and Raman measurements.

Calculations on a microscopic level are used to explain the experimentally observed negative linear thermal expansion along some directions in a number of crystalline compounds with complicated lattices and anisotropic interactions between atoms. Anomalies in the temperature dependence of the coefficient of linear thermal expansion are analyzed in layered crystals made up of monatomic layers (graphite and graphene nanofilms) and multilayer “sandwiches” (transition metal dichalcogenides), in multilayered crystal structures such as high-temperature superconductors where the anisotropy of the interatomic interactions is not conserved in the long-range order, and in graphene nanotubes. The theoretical calculations are compared with data from x-ray, neutron diffraction, and dilatometric measurements.

Abstract. The negative expansion observed at low temperatures in specific crystallographic directions of complex structures with anisotropic interactions is explained by microscopic analysis within the tight-binding, quasi-harmonic approximation of lattice dynamics. The results agree with measurements on multilayer sandwiches of transition-metal dichalcogenides, high-temperature superconductors and relevant nanostructures including the carbon-based nanotubes.

The atomic dynamics of linear chains embedded in a crystalline matrix or adsorbed on its surface is studied. A linear chain formed by substitutional impurities in a surface layer and at the same time offsetting from this layer was analyzed particularly. This system models the actively studied experimentally structures in which gas molecules are adsorbed on the walls of the bundles of carbon nanotubes located in certain medium. It is shown that the quasi-1D features are typical for the chains in which the interatomic interaction is higher than the interaction between the atoms of the chain and the atoms of the crystal matrix. On the local phonon density of atoms of the chain, the transition to quasi-one-dimensional behavior has the form of the kink. In other words, it is the first (lowest-frequency) van Hove singularity, which in 3D structures (the system under consideration is generally three-dimensional) corresponds to the transition from closed to open constant-frequency (quasi-plane) surfaces. The local phonon densities of atoms in the chain have one-dimensional character at frequencies higher than the frequency of the van Hove singularity. The rms-amplitude of embedded chains atoms vibrations is calculated and the behavior of the atomic vibrations contribution in the low-temperature heat capacity of the system is analyzed.

This is an analysis of the properties of quasi-local vibrations, and the conditions of the formation thereof, in a realistic model of the crystal lattice on a microscopic scale. The evolution of quasi-local vibrations with an increase in the concentration of impurity atoms, is examined. It is shown that the formation of boson peaks occurs mainly due to the additional dispersion of high-velocity acoustic phonons (connected to the atomic vibrations of the main lattice), caused by the scattering of these phonons by the quasi-local vibrations localized at the impurities. We demonstrate a connection between the boson peaks in disordered systems, and the first van Hove singularity, in regular crystal structures. We analyze the manifestation of quasi-local vibrations and boson peaks, as it relates to the behavior of low-temperature heat capacity, and how it changes with an increasing impurity concentration.

We calculate and analyze, within a model that allows to obtain an analytical approximation using the method of Jacobi matrices, the electronic spectra of graphene with impurity atoms, especially with boron substitutional impurity. We present analytical expressions for the conditions of the existence of electronic local discrete levels due to the presence of a substitutional impurity in graphene. (C) 2012 Elsevier Ltd. All rights reserved.

It is well known that graphene monolayers cannot exist as planar objects in the free state, because in flat 2D-crystals the mean-square amplitudes of the atoms in the direction normal to the layer plane diverge even at T =0 (see, e.g., [3]). So we can study and practically apply only such graphene, which is deposited on a certain substrate providing the stability of the plane carbon nanofilms (see, e.g., [4-6]). Only small flakes can be detached from the sub‐ strate and these flakes immediately acquire a corrugated shape [7]. When studying the elec‐ tronic properties of graphene a dielectric substrate is often used. The presence of the substrate greatly increases the occurrence of various defects in graphene and carbon nano‐ films. Our investigations make it possible to predict the general properties of phonon and electron spectra for graphene and bigraphene containing different defects.