CALCULATION OF THERMAL CONDUCTIVITY FOR CRYSTALLINE NANOSTRUCTURES

We have introduced a theory for the calculation of thermodynamic characteristics for some characteristic nanostructures (ultrathin films and superlattices), using the adapted method of two-time temperature Green's functions. In this paper, he has determined the coefficient of thermal conductivity using the definition of free energy and then compared their temperature dependence to the thermal conductivity behavior of the bulk structures. For the observed nanostructures, the thermal conductivity coefficient values are almost equal at low temperatures, but at the same time, significantly lower than the bulk sample values. That result could be useful for the possible achievement of better superconducting conditions in the observed compound nanostructures.


INTRODUCTION
Elementary particles -mechanical oscillations -phonons are a subsystem that is always present when analyzing the conducting, semiconducting, or dielectric properties of the system.Accordingly, we will first analyze the kinetics of mechanical oscillations in nanoscopic-ultrathin films, which can be considered as a basis for investigating other properties of more complex nanostructures.In a way, this work represents a generalization of the previous research [1][2][3][4][5][6].We will start with the definition expression for the coefficient of thermal conductivity [7,8]: where D is the diffusion coefficient, Cspecific heat, and M is the mechanical density of the observed structure.The diffusion coefficient D (strictly, it is the diagonal matrix element of diffusion tensor Dij) will be found by the Kubo formula [7].The temperature dependence of nanostructure density will be determined by the two-time, temperature-dependent Green's function method [9].Using this method one can find the internal energy and the average value of the square of molecular displacements.All analyzes will be calculated regarding the presence of specific boundary conditions on its surfaces, which are responsible for the appearance of unusual effects and changes in the basic properties of these structures [6].

CALCULATION OF THE DIFFUSION
COEFFICIENT To determine the diffusion coefficient, we will start with the Kubo formula [7,8]: where i v ˆ and j v ˆ (i,j = x,y,z) are the velocity operators in Heisenberg representation,  is the perturbation parameter and the averages will be taken over the great canonical ensemble.We will find the correlation function , which represents the components of the molecular momentum.The Hamiltonian of the phonon subsystem of a superlattice with two motifs a and b (two ultrathin films) taken in harmonic and nearest neighbor's approximation [9], can be written as follows: n / are numbers of molecules in corresponding motive.Boundary conditions will be taken into account during the formation of a system of equations defining Green's function of the system.We have determined the following Green's function, which will be written in the next form: Because of valid relation: pi = M vi ; vi = dui / dt, in the expression determining this function, appears the second Green's function of a type [10].In this way, the correlation function of the Green's function is given by a general formula [11,12]: Green's function can be expressed as a sum of elementary fractions [10].In this way, we obtain the correlation function (2.3), i.e. corresponding velocity correlation function: (2.4) By the general formula (2.2) the diffusion coefficient is given by: (2.5) It is seen that the phonon diffusion coefficient of the superlattice, as well as that of the bulk, does not depend on temperature [13].

THERMODYNAMIC BEHAVIOUR OF SUPERLATTICE
The internal energy of the system is given by the standard formula [11][12][13]: where the phonon dispersion law is given by the basic formula: Phonon-reduced frequencies and intermolecular distance are expressed through the geometric mean of phonon frequencies, i.e. of intermolecular distances in separate motifs: Quantity θ is thermodynamic temperature θ= kB T.
After partial integration in (3.1) and introducing notations: we obtain the following expression: Since the specific heat is given by 2), we find that: Temperature dependence of the thermal capacity is determined by two specific terms.
The first term is "responsible" for the behavior of the system at low and high temperatures.The second term containing Z-functions characterizes temperature behavior in the middletemperature range.Based on the results of the previous research on the phonon contribution to the thermodynamic behavior of the ultra-thin film structures [1][2][3], and the well-known behavior of the bulk [7,[11][12][13], Fig. 1 shows the comparative display of the specific heat for the superlattice, ultra-thin film and the bulk structures in dependence of the reduced temperature: x=/m From Fig. 1, one can conclude that the behavior of the thermal capacity of a superlattice in a low-temperature range is similar to bulk ones.The temperature behavior of the thermal capacity of a superlattice in a middle-temperature region is similar to film ones.The difference in these capacities is most pronounced in the high-temperature area.Now we will approach the determination of the temperature behavior of the superlattice thermal conductivity.The expression for the dynamical density of superlattice: The primary text heading should be numbered by 1., 2., ... and should be in 12-pt., bold, capital letters, flush left with margin.The spacing from the text to the next heading is one line.

  
The averages of squares of displacements for molecular superlattices we can find as in Debye's representation.After integration, the expression for the density becomes: The diffusion coefficient is given by the relation (2.5), where: are the average value phonon frequencies.After elementary calculations, we obtain the diffusion coefficient: Introducing notations /m = x we reduce the expression (1.1) for thermal conductivity coefficient to the form: where Temperature dependence of the thermal conductivity coefficient is determined by two specific terms.Similar to the expression for thermal capacity, here is the first term which is "responsible" for the behavior of the system at low and high temperatures.Also, the second term containing Z-functions, characterizes temperature behavior in the middle-temperature range.A graphical presentation of the dependence of the relative thermal conductivity coefficient  From Fig. 2 it can be concluded that the behavior of the thermal conductivity coefficient of superlattice is similar to the ultrathin film one (it is higher than in bulk structure).The Difference is more expressive in the middle and high-temperature range.So, it can be concluded that the superlattices in the low-temperature range are somewhat better heat conductors than the bulk structures.At the same time, the heat conduction of the film is higher than in superlattices.On the other hand, in the hightemperature range superlattices are much better heat isolators than film structures and the corresponding infinite crystal structures.
As our results were obtained theoretically, we looked for experimental confirmation in the scientifically available literature.An extensive analysis of the available experimental results was carried out [14][15][16][17][18][19][20][21], of which we will in this paper present, for the sake of brevity, only a few of them.In the paper [14], the thermodynamic characteristics of micrometer SiN film structures were measured.Although these are not real nanostructures (because the thickness of these films is about 1.5 µm), film microstructures still show a similar behavior of specific heat and thermal conductivity as our theoretically obtained and predicted dependences for film structures, Fig. 3.

Journal of Sustainable Technologies and Materials
Figure 3. Thermal conductivity and heat capacity for 1.5 µm silicon nitride films [14] Since graphene is one of the most studied materials today, and it is a true member of nano-thin (film) structures, it was justified to compare our results with the thermodynamic behavior of graphene.In the paper [15], the specific heat was measured for graphene (nano-film structure of carbon atoms), AB-stacked bilayer graphene (AB-BLG), and graphite (which represents an essential bulk structure).It is visible that our calculations show a similar behavior of the dependence of the specific heat on the temperature for the film structure, compared to the bulk structures, in the domain of low temperatures (i.e. on the temperatures interesting for many phenomena -such as superconductivity).Figure 4, in the frame, shows the dependence of specific heat at high temperatures, which obeys the Dulong-Petit law.

CONCLUSION
The results obtained here show that the thermal conductivity coefficients of the film and the superlattice at low and high temperatures are significantly lower than the thermal conductivity coefficient of the corresponding bulk structures, where the thermal coefficient dependence from temperature is T 3 .This result is also practically applicable: a sandwich of several films would be a better thermal insulator than a bulk structure of the same thickness.Here presented theoretical results are compared with experimental data [14][15][16][17][18][19][20][21].Our theoretically predicted results are in satisfactory agreement with experimental data.According to the Viedeman-Frantz rule, electrical conductivity is proportional to thermal conductivity.This leads to the conclusion that films and superlattices are weaker electrical conductors than bulk structures of the same material at low and high temperatures.For future research would be interesting to evaluate the superconducting properties of observed structures.The general behavior for today's materials is that the worse conductors at the room temperature region, become the better superconductors at the low (cryogenic) temperatures region.In such a way, the ultrathin films and superlattices could be structures with high superconductivity potential, not only for the low-temperature but also for the hightemperature region.
b are the masses of molecules in first/second ultrathin films, VP/B are the potential energies of surface and bulk terms, u M p   ( u are molecular displacements), and b a on scaled temperature x=/ D is given the Fig.2.