Modeling Thermal Transport in Nano-Porous Semiconductors

Abstract

Thermal transport in nano-Porous material has drawn the attention of several research groups during the last decade due to the ability of such structures to tailor efficiently the thermal properties of materials and more specifically to lower drastically the thermal conductivity of semiconductors. The present chapter recalls the basics of thermal transport in porous media from different standpoints. After a short introduction and review of the literature, analytic models that characterize heat propagation in porous media are given. Their limitations, especially in what concerns heat carriers scattering with pores when characteristic sizes become very small is pointed out and alternatives are suggested. In a second time, Monte Carlo modeling techniques, which are well designed for mesoscopic length-scales, are introduced and their use for thermal conductivity appraisal of nano-porous media is discussed. Improvement of such technique to reduce computation time and to model thin films with high porosity is then exposed with Effective Monte Carlo model. Simulation results for silicon and germanium support this part. The last section of the chapter is devoted to Molecular Dynamic (MD) modeling of nano-porous structures. Again, practical details on Equilibrium MD are proposed with a specific attention paid to crystalline and amorphous phases modeling. Then simulation results for various kinds of a-Si and c-Si nano-porous structures are discussed before concluding on all these methods and models.

Appendix: Dispersion Properties and Relaxation Times

Parabolic fits $\omega = c_p K^2 + v_p K$ are used to describe the dispersion properties of Si and Ge along [100] axis, for LA and TA polarizations, as suggested by Pop et al. [62]. Parameters $c_p$ and $v_p$ are detailed in Table 9.3.

Table 9.3 Data used to fit Si and Ge dispersion curves

Table 9.4 Relaxation time parameters for Si and Ge

Relaxation-time constants are set to fit the experimental curve that gives the bulk material conductivity as a function of the temperature $k = f(T)$. Each scattering process has its own expression which depends of the frequency and the temperature [61]. Table 9.4 lists them for normal, umklapp and impurities scattering using the previous quadratic fits

$$\begin{aligned} \begin{array}{l} {\tau _{I}}^{-1}=B_I \omega ^4, {\tau _{N_{LA}}}^{-1}=B_L \omega ^2 T^3, {\tau _{U_{LA}}}^{-1}=B_L\omega ^2T^3 \\ {\tau _{N_{TA}}}^{-1}= \left\{ \begin{array}{l} B_{TN}\omega T^4\; \mathrm {if}\; \omega< \omega _{12},\\ 0\; \mathrm {else } \end{array} \right. \\ {\tau _{U_{TA}}}^{-1}= \left\{ \begin{array}{cc} 0\; \mathrm {if }\; \omega < \omega _{12},\\ B_{TU} \frac{\omega ^2}{\sinh {\frac{\hbar \omega }{k_BT}}}\; \mathrm {else} \end{array} \right. \end{array} \end{aligned}$$

(9.29)

$\omega _{12}$ is the frequency which corresponds to half of the Brillouin zone for the TA mode.