Ballistic-Diffusive Model for Heat Transport in Superlattices and the Minimum Effective Heat Conductivity

Abstract

There has been much interest in semiconductor superlattices because of showing very low thermal conductivities. This makes them especially suitable for applications in a variety of devices for thermoelectric generation of energy, heat control at the nanometric length scale, etc.Recent experiments have confirmed that the effective thermal conductivity of superlattices at room temperature have a minimum for very short periods (in the order of nanometers) as some kinetic calculations had anticipated previously. This work will show advances on a thermodynamic theory of heat transport in nanometric 1D multilayer systems by considering the separation of ballistic and diffusive heat fluxes, which are both described by Guyer-Krumhansl constitutive equations. The dispersion relations, as derived from the ballistic and diffusive heat transport equations, are used to derive an effective heat conductivity of the superlattice and to explain the minimum of the effective thermal conductivity.

Temperature distribution inside a porous silicon photonic mirror

“…Here it is relevant to mention that our model is minimal because the heat transport occurs in a 3D structure, and we are using only a 1D-transport description. Moreover, we are considering heat Fourier transport, while a detailed description may require a ballistic diffusive model [22,23]; but the qualitative behavior is consistent. In our model, we use the ratio of the multilayer effective thermal conductivity and the crystalline silicon thermal conductivity.…”

Internal structure and heat conduction in rigid solids: a two temperature approach

“…It is well known that in solid rigid conductors heat can be transported by one or more types of carriers [17,21,22,23,24]. Experiments in superlattices have made the wave properties of phonons evident and their effects have been interpreted within the coherent-incoherent phonon scheme [28,29,30,31].…”

“…For instance, this is done in transient ballistic-diffusive heat transport where the total internal energy per unit volume is the sum of longitudinal and transverse phononic branches [17,18]. Also, a similar separation is done in heat transport in common materials at the nanoscale, where the electrons and the phonons that transport energy are out of an equilibrium state during the heat transfer [19,20,21,22,23,24], or in nonlocal thermoelectric transport in thin layers [25]. In any case, however, this point will not deserve further consideration in this work.…”

Bragg Mirrors for Thermal Waves

“…Chen [19] used the CVe in a superlattice and calculated the dispersion relation using Bloch's theorem. Ballistic-diffusive models for heat transport in superlattice used the Guyer-Krumhansl equation since the thickness of the layers is comparable to the mean free path of the phonons [22].…”