Temperature Distribution through a Nanofilm by Means of a Ballistic-Diffusive Approach

As microelectronic devices are important in many applications, their heat management needs to be improved, in order to prolong their lifetime, and to reduce the risk of damage. In nanomaterials, heat transport shows different behaviors than what can be observed at macroscopic sizes. Studying heat transport through nanofilms is a necessary tool for nanodevice thermal management. This work proposes a thermodynamic model incorporating both ballistic, introduced by non-local effects, and diffusive phonon transport. Extended thermodynamics principles are used in order to develop a constitutive equation for the ballistic behavior of heat conduction at small-length scales. Being an irreversible process, the present two-temperature model contains a one-way transition of ballistic to diffusive phonons as time proceeds. The model is compared to the classical Fourier and Cattaneo laws. These laws were not able to present the non-locality that our model shows, which is present in cases when the length scale of the material is of the same order of magnitude or smaller than the phonon mean free path, i.e., when the Knudsen number Kn <= O(1). Moreover, for small Kn numbers, our model predicted behaviors close to that of the classical laws, with a weak temperature jump at both sides of the nanofilm. However, as Kn increases, the behavior changes completely, the ballistic component becomes more important, and the temperature jump at both sides of the nanofilms becomes more pronounced. For comparison, a model using Fourier's and Cattaneo's laws with an effective thermal conductivity has shown, with reasonable qualitative comparison for small Knudsen numbers and large times.