使用聲子波茲曼方程對軸對稱奈米線之熱傳模擬

Abstract

在半導體內，能量是透過原子或晶格振動來傳遞的，這些振動是靠著物質波來做傳遞，這些能量被量子化後稱之為聲子。在巨觀尺度下，半導體的熱傳是符合傅立葉定律的，但在微觀尺度下，傅立葉定律就無法用來描述微觀下的熱傳行為，這時就必須使用聲子輻射傳輸方程式來探討微觀尺度下的熱傳行為。 聲子輻射傳輸方程式為非線性且包含了微分項與積分項的方程式，要直接求解並不容易。透過BGK方程式(Bhantnagar-Gross-Krook Equation)將碰撞項簡化後，會較好處理。 本文主要探討的是圓柱座標下軸對稱的奈米線在不同材料排列方式下的熱傳特性，分別為:單層圓柱、空心圓柱、同心圓柱、多層圓柱、緊密型圓柱。在數值方法上，方向使用離散座標法(Discrete Ordinate Method)將方向餘弦離散化，在空間上則使用一階迎風算則(Upwind Scheme)來分析問題。由研究結果可以發現，奈米線的等效熱傳導係數不但受到徑向與軸向尺度的影響，也受到界面散射影響甚鉅。

Energy transport in semiconductor is basicly by atomic or crystal vibrations, These vibrations travel within material waves. The energy is quantized and each quantam is called a phonon. On a macroscopic scale, the heat transfer in semiconductor mainly obey the Fourier law. However, on a microscopic scale the heat transfer will no longer follow the Fourier law. Instead, the equation of phonon radiative transfer (EPRT) is developed to study the heat transfer under microscopic scale. EPRT is a nonlinear equation with intergral and differential terms, which is difficult to solve directly. If we simplify the collision term by Bhatnagar-Gross-Krook Equation, the equation will be easy to solve. This article is mainly about the heat transfer in the symmetric semiconductor nanowires with different material arrangement under cylindrical coordinates. Several geometries are studied including : single layer nanowires, tubular nanowires, core-shell nanowires, multi-layer nanowires, and composite nanowires. The discrete ordinate method is used for angular discretization; and upwind scheme is used for spatial discretization. The results show that the effective thermal conductivity changes not only with the radius and the length of the nanowires, but also with the boundary thermal resistance.