基於晶格波茲曼法之三維微尺度聲子能傳建模與計算

Abstract

隨著科技日新月異的進步，近年來製程技術的迅速發展，半導體產業、光電產業及微機電系統的需求，使微型化成為趨勢，當流體之特徵長度與分子之平均自由徑 (Mean free path) 逐漸接近，尺寸效應已不能再忽略，用於研究巨觀熱傳現象之主導方程式 Fourier Law 將不適用。其中原因為在巨觀下，分子可視為連續體，然而在微尺度下描述聲子分布狀態之主導方程為Boltzmann 傳輸方程（Boltzmann transport equation, BTE）。 在本文的研究中，應用三維聲子晶格Boltzmann-BGK模擬微尺度熱傳問題，藉由Hermite多項式的展開，得到Bose-Einstein統計平衡態的分布函數，並採用3D模型來分析，利用週期性邊界條件取出週期性單元作為模擬區域。現今聲子晶格Boltzmann法已證實可應用在二維的微尺度熱傳問題。而在三維情況進行分析，藉由建模以模擬的方式，隨著Knudsen數的改變，流場將呈現不同的特性，經由碰撞和遷移過程，得到新的分布函數，計算溫度分布。由研究結果可發現，導體材料不僅有尺寸效應，於微尺度下材料界面產生的界面散射將使熱傳導係數下降及溫度分布產生滑移。

The feature size of electronic devices in current integrated circuits has become comparable to or even smaller than the mean free path (MFP) of the energy carrier. It has been well known that the continuum-based Fourier heat conduction law may lead to erroneous results when the phonon mean free path becomes comparable or larger than the characteristic size of the material studied. The prevailing approach to calculate the thermal conductivity of semiconductors and dielectric materials is based on phonon Boltzmann transport equation as the energy carriers at temperatures of interest are phonons. In this thesis, an efficient three-dimensional lattice Boltzmann method based on the phonon Boltzmann-BGK transport equation is developed for solving energy transfer in the subcontinuum regime. The lattice Boltzmann method is derived by expanding the phonon Bose-Einstein distribution in the tensor Hermite polynomials space and standard D3Q19 lattice model is employed. Depending on the order of expansion adopted, different level of approximation of the transport phenomenon can be modeled. The implementations of periodic boundary conditions and constant temperature wall conditions are described. The present phonon lattice Boltzmann method is validated against existing direct discrete ordinate method by modeling and simulation of thermal conductivities for two-dimensional problems. Modeling and simulations of the micro-/nano-scale phonon energy transfer covering broad range of Knudsen numbers range are presented. It is found that the present lattice Boltzmann method solutions are in good agreement with those from the discrete ordinate phonon Boltzmann solver. Results suggest that reducing feature size will decrease the thermal conductivity, and temperature will become non-continuum distributions in the interface. And the effective thermal conductivity changes not only with the length of the thin film, but also with the boundary thermal resistance.