It is proved that the acoustic-type dispersion of bending mode in graphene is generated by the fluctuation interaction between in-plane and out-of-plane terms in the free energy arising with account of non-linear components in the graphene strain tensor. In doing so we use an original adiabatic approximation based on the alleged (confirmed a posteriori) significant difference of sound speeds for in-plane and bending modes. The explicit expression for the bending sound speed depending only on the graphene mass density, in-plane elastic constants and temperature is deduced as well as the characteristics of the microscopic corrugations of graphene. The obtained results are in good quantitative agreement with the data of real experiments and computer simulations.
Proceeding from the model of a two-dimensional elastic continuum, we describe the characteristic features of thermal expansion of graphene using an approach that goes beyond the quasi-harmonic approximation. The negative value of the thermal expansion coefficient of graphene at low temperatures and its sign reversal at T ≈ 1000 K are established. It is shown that the bending vibrational mode plays a decisive role in peculiarities of the thermal contraction/expansion of graphene and that the contribution of this mode to the thermal expansion coefficient does not depend on the sample size, due to the "sound" nature of the bending mode long-wave dispersion. The obtained results allow giving a quantitative description of the known data of numerical experiments on the thermal expansion of graphene in the entire interval of the MD simulations.
The phonon density of states (DOS) of graphene with different types of point defects (carbon isotopes, substitution atoms, vacancies) is considered. Using a solvable model which is based on the harmonic approximation and the assumption that the elastic forces act only between nearest neighboring ions we calculate corrections to the graphene DOS dependent on the type and concentration of defects. In particular the correction due to isotopic dimers is determined. It is shown that a relatively small concentration of defects may lead to significant and specific changes in the DOS, especially at low frequencies, near the Van Hove points and in the vicinity of the K points of the Brillouin zone. In some cases defects generate one or several narrow gaps near the critical points of the phonon DOS as well as resonance states in the Brillouin zone regular points. All types of defects are characterized by the appearance of one or more additional Van Hove peaks near the (Dirac) K points and their singular contribution may be comparable with the effect of electron-phonon interaction. Besides, for low frequencies and near the critical points the relative change in density of states may be many times higher than the concentration of defects.
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