Diffusion of phonons through (along and across) the ultrathin crystalline films

Abstract

The momentum-momentum Green's functions were used to calculate the diffusion tensor.  These functions enter into Kubo's formula defining diffusion properties of the system.  Boundary conditions are included through the Hamiltonian of the film-structure.  The elements of the diffusion tensor express discrete behavior of the dispersion law  The diffusion tensor elements are temperature independent.

The influence of the surface parameter changes onto the phonon states in ultrathin crystalline films

“…we get the system of (N z + 1) nonhomogeneous algebraic-difference equations with the same number of undetermined Green's functions [13][14][15][16][17][18][19][20][21][22][23][24][25]:…”

“…In this respect, particular emphasis should be given to the research carried out in order to examine the influence of the phonon subsystem on the properties of the graphene, epitaxially grown on metals and transition carbides, which is mandatory in order to tailor graphenes mechanical properties [9][10][11][12][13]. We have recently published the results of the research of phonon diffusion through ultrathin crystal films [14], where we had to use the phonon dispersion law. Due to complications with analytic, we decided to consider only "ideal" film structures.…”

Influence of the phonon subsystem onto the mechanical and thermodynamic properties of low-dimensional materials

“…It is considered that there is an interaction between atoms in boundary layers of the film and external areas (substrate and e.g. air), disregarding that along  z direction there are no atoms belonging to the film; however, boundary atoms are coupled through changed Hooke's forces with the atoms of the external environment [12][13][14][15][16][17][18][19]. In accordance with these conditions, elastic constants which describe the interaction between atoms of boundary surfaces and external environment are modified with appropriate coefficients  and  .…”

“…Allowed phonon energies will be determined by following the same procedure described for the massive crystalline samples and by applying the partial spatial Fourier-transformation along z  direction, where the translational symmetry has been interrupted. In this way, we get the system of   z N1  nonhomogeneous algebraic-difference equations with the same number of undetermined Green's functions [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]:…”

Process of phonon diffusion through crystalline structures

“…Using the presented procedure for calculation of Green's function, which is in detail explained in our paper [21], one can write the following equation:…”