Kinetic-Collective Model (KCM) allows to calculate the thermal conductivity of semiconductor taking in consideration the effect of Normal scattering.

Thermal conductivity in the Kinetic-Collective Model has two contributions

The kinetic contribution

and the collective contribution

The constant 1/3 is inculded in the expressions as de total velocity is used. This assumption is only valid in isotropic matrials like the ones of the present work. $C_{0} $ corresponds to the specific heat.

The kinetic mean free path is calculated with the Mathiessen rule as:

using only the resistive scattering mechanisms, that are umklapp $\tau_{U}$, impurity $\tau_{I}$ and boundary $\tau_{B}$ mean free times. The collective mean free time is obtained considering only intrinsic scattering rates, thus boundary scatteing is not considered:

where the intrinsic proceses are:

The wheighting function for averaging, that includes the especific heat is

The switching factor is defined as:

that determines the kinetic and collective contributions in terms of the resistive and normal integrated scattering rates:

A derivation of these terms from Boltzmann Transport Equation can be obtained elsewhere \cite{Guyer1966a}

[1] X. Alvarez and D. Jou, Applied Physics Letters 90, (2007).

[2] Z. M. Zhang, Nano/Microscale Heat Transfer (Graw-Hill, Mc, 2007).

[3] R. A. Guyer and J. A. Krumhansl, Physical Review 148, 766 (1966).