# 臺灣博碩士論文加值系統

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 熱在巨觀尺度下，遵從傅立葉熱傳方程式。當尺度縮小到微奈米等級的時候，利用傅立葉熱傳方程式分析模擬會跟實際上的物理量產生很大的出入。這代表了傅立葉熱傳方程式不再能適用於微尺度下探討分析真實熱傳導現象。一般在半導體或絕緣體材料中，熱是由熱載子(電子、聲子和光子)來傳遞的。其中最主要是由聲子來傳遞，本文主要利用聲子輻射熱傳方程式模擬在微尺度下薄膜內之暫態熱傳現象。 聲子輻射熱傳方程式為一多變數且非線性的積分微分方程式，在數學上極難求解，故其碰撞項常用一碰撞模式取代（採用鬆弛時間近似為一線性關係），可使得在數學處理上較容易。本文數值模擬方面，對於方向空間上應用分離座標法將方向餘弦離散化，對於位置空間上則應用高解析算則（WENO3和Min-Mod）於離散的位置格點上來分析。利用加權型基本不振盪算則(WENO3算則）配合分離座標法，以求解聲子輻射熱傳方程式，可以測試其準確性，並與其它高解析算則比較之。在分離座標法上，立體角方向數上的離散化會產生射線效應現象，使得模擬出來的暫態情形會出現如波浪分佈的現象，此時套用修正分離座標法來做分析，可以減少射線效應的影響而消除波浪狀的溫度分佈情形。本文利用聲子波茲曼方程式分析一維單層和雙層薄膜的問題，尺度介於奈米到微米之間，主要分析在暫態下之熱傳問題，並討論其穩態時的熱通量。
 The Phonon Boltzmann equation is a nonlinear, integral, differential equation with many variables. It is difficult to be solved mathematically, so the collision term is usually replaced with a collision model. This will make it easier to deal with. In this work, the direction space will be discretized by applying discrete ordinate method. After applying the discrete ordinate method, it becomes a set of differential equations. With this treatment, the difficulties of numerical calculation will be greatly reduced.Heat transport in semiconductor and dielectric materials is mainly by phonon. The transient heat conduction in microscale semiconductor thin films is investigated by using the equation of phonon radiative transfer (EPRT). The accuracy of using the simplified condition of equilibrium on the heat transfer characteristics is studied.Accurate numerical methods for transient heat conduction problems using EPRT are presented and compared with various known methods. The present method is a deterministic and direct solver in phase space where the discrete ordinate method is used for angular discretization and high resolution schemes are used for spatial discretization. Nanoscale heat conduction covering wide range of Knudsen numbers are computed.In this study, the Weighted Essentially Non-Oscillatory (WENO3) scheme in conjunction with discrete ordinate method was applied to solve the EPRT, and the explicit scheme for the EPRT was developed to solve the transient steady solutions.To mitigating ray effect, we introduce Modified Discrete Ordinate Method to solve EPRT. We can find that ray effect would be reduced and the solutions become more accurate.
 目 錄誌謝.......................................................Ⅰ中文摘要...................................................Ⅱ英文摘要..................................................Ⅲ目錄.......................................................Ⅳ附表目錄.................................................Ⅶ附圖目錄..................................................Ⅷ符號說明...................................................XI第一章 緒論................................................11.1 引言..................................................11.2 微觀熱傳導現象簡介...................................11.3 文獻回顧.............................................51.4 研究內容.............................................7第二章 聲子輻射熱傳理論.................................122.1 Liouville方程式.......................................122.2 Boltzmann方程式......................................142.3 鬆弛時間.............................................152.3.1 聲子間散射.....................................162.3.2 幾何散射.......................................172.4 聲子輻射熱傳方程式...................................182.5 邊界條件.............................................192.6 界面熱阻.............................................222.6.1 聲異理論模式(AMM).............................232.6.2 散異理論模式(DMM).............................242.6.3 散射聲異理論模式(SMAMM)......................26 2.7 射線效應(Ray Effect)...................................27 2.8 假散射(False Scattering).................................28第三章 數值方法..........................................333.1 離散座標法...........................................333.2 空間離散.............................................34 3.2.1 迎風算則.......................................34 3.2.2 雙曲線型守恆律算則.............................353.2.3 TVD算則(Min-Mod)..............................37 3.2.4 WENO3算則....................................383.3 時間離散.............................................41 3.3.1 Euler Method....................................41 3.3.2 TVD Runge-Kutta 時間積分算則...................413.4 修正分離座標法(MDOM)...............................423.5 無因次化.............................................44第四章 數值模擬結果與討論...............................514.1 一維鑽石板狀熱傳分析.................................52 4.1.1 數值方法之驗證.................................524.2 一維矽材料熱傳分析...................................54 4.2.1 等效熱傳導係數分析.............................54 4.2.2 高解析算則分析.................................56 4.2.3 修正分離座標法分析.............................564.3 一維雙層砷化鎵/砷化鋁材料熱傳分析...................57 4.3.1 界面熱阻與等效熱傳導係數分析..................57 4.3.2 高解析算則分析.................................59 4.3.3 修正分離座標法分析.............................59第五章 結論與建議........................................82參考文獻..................................................85附表目錄表1-1 熱載子的基本性質......................................9表1-2 各種理論模式適用範圍表................................9表3-1 鑽石物理材料性質.....................................47表3-2 不同 濃度的鑽石性質.................................47表3-3 三種薄膜材料的材料性質...............................48表4-1 單層矽材料等效熱傳導之比值關係.......................48表4-2 單層矽材料不同厚度下的等效熱傳導係數................48表4-3 雙層材料不同厚度下的界面熱阻與等效熱傳導係數........51附圖目錄圖1-1 薄膜中聲子的熱傳導...................................10圖1-2 聲子晶格振盪模型2D示意圖.............................10圖1-3 聲子晶格振盪模型3D示意圖.............................11圖2-1 相空間系綜演化示意圖.................................29圖2-2 空間立體角示意圖.....................................29圖2-3 三聲子過程示意圖.....................................29圖2-4 邊界鏡面反射.........................................30圖2-5 邊界擴散反射.........................................30圖2-6 AMM界面示意圖.......................................31 圖2-7 DMM界面示意圖.......................................31圖2-8 射線效應示意圖.......................................32圖2-9 假散射示意圖.........................................32圖3-1 迎風算則示意圖.......................................48圖3-2 Superposition示意圖.....................................49圖3-3 數值方法流程圖.......................................50圖4-1 鑽石薄膜平板示意圖...................................61圖4-2 雙層薄膜介質示意圖...................................61圖4-3 驗證鑽石膜厚在 下不同 濃度之穩態溫度分佈圖......62圖4-4 驗證鑽石膜厚在 下不同 濃度之穩態溫度分佈圖.......62圖4-5 驗證鑽石膜厚在 下不同 濃度之穩態溫度分佈圖......63圖4-6 鑽石膜厚 時灰體與 濃度0.07%之溫度比較分佈圖.....63圖4-7 鑽石膜厚 時灰體與 濃度0.07%之溫度比較分佈圖......64圖4-8 鑽石膜厚 時灰體與 濃度0.07%之溫度比較分佈圖.....64圖4-9 矽材料在膜厚 時之溫度分佈圖......................65圖4-10 矽材料在膜厚 時之熱通量分佈圖...................65圖4-11 矽材料在膜厚 時之溫度分佈圖.....................66圖4-12 矽材料在膜厚 時之熱通量分佈.....................66圖4-13 矽材料在膜厚 時之溫度分佈圖.......................67圖4-14 矽材料在膜厚 之熱通量分佈圖.......................67圖4-15 矽材料其等效熱傳導係數比值與特徵長度之關係圖.......68圖4-16 在厚度 時一階精度、MinMod及WENO3比較之溫度分佈圖..................................................69圖4-17 在厚度 時一階精度、MinMod及WENO3比較之熱通量分佈圖..................................................69圖4-18 方向數20像素200時MinMod與其MDOM比較之溫度分佈圖..................................................70圖4-19 方向數20像素200時MinMod與其MDOM比較之熱通量分佈圖.................................................70圖4-20 方向數20像素1000時MinMod與其MDOM比較之溫度分佈圖.................................................71圖4-21 方向數20像素1000時MinMod與其MDOM比較之熱通量分佈圖................................................71圖4-22 方向數20像素2000時MinMod與其MDOM比較之溫度分佈圖................................................72圖4-23 方向數20像素2000時MinMod與其MDOM比較之熱通量分佈圖................................................72圖4-24 方向數20像素200時WENO3與其MDOM比較之溫度分佈圖.................................................73圖4-25 方向數20像素200時WENO3與其MDOM比較之熱通量分佈圖. ................................................73圖4-26 方向數20像素1000時WENO3與其MDOM比較之溫度分佈圖. ................................................74圖4-27 方向數20像素1000時WENO3與其MDOM比較之熱通量分佈圖. ................................................74圖4-28 方向數20像素2000時WENO3與其MDOM比較之溫度分佈圖. ................................................75圖4-29 方向數20像素2000時WENO3與其MDOM比較之熱通量分佈圖. ................................................75圖4-30 GaAs/AlAs在厚度 不同時刻之溫度分佈圖..............76圖4-31 GaAs/AlAs在厚度 不同時刻之熱通量分佈圖.........76圖4-32 GaAs/AlAs在厚度 不同時刻之溫度分佈圖........77圖4-33 GaAs/AlAs在厚度 不同時刻之熱通量分佈圖..........77圖4-34 GaAs/AlAs在厚度 不同時刻之溫度分佈圖.........78圖4-35 GaAs/AlAs在厚度 不同時刻之熱通量分佈圖........78圖4-36 GaAs/AlAs在厚度 不同算則中之溫度分佈圖........79圖4-37 GaAs/AlAs在厚度 不同算則中之熱通量分佈圖.......79圖4-38 MinMod與其MDOM之溫度分佈圖.......................80圖4-39 MinMod與其MDOM之熱通量分佈圖.....................80圖4-40 WENO3與其MDOM之溫度分佈圖......................81圖4-41 WENO3與其MDOM之熱通量分佈圖....................81
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W., “Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws,” ICASE 25th Anniversary, November 1997.[22] Sakami, M. and Charette, A., “Application of A Modified Discrete Ordinates Method to Two-Dimensional Enclosures of Irregular Geometry,” Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 64, pp. 275-298, 2000.[23] Yang, R., Chen, G. and Dresselhaus, M. S., “Thermal Conductivity Modeling of Core-Shell and Tubular Nanowires,” Nano Letters, Vol. 5, pp. 1111-1115, 2005.[24] Yang, R., Chen, G., Laroche, M. and Taur, Y., “Simulation of Nanoscale Multidimensional Transient Heat Conduction Problems Using Ballistic-Diffusive Equations and Phonon Boltzmann Equation,” ASME Journal of Heat Transfer, Vol. 127, pp. 298-306, 2005.[25] Ju, Y. S., Hung, M. 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S., “Thermal Diffusivity Measurement of GaAs/AlGaAs Thin-Film Structures”, ASME Journal of Heat Transfer, Vol. 116,pp.325-331, 1994.[35] Yang, R. and Chen, G., “Thermal Conductivity Modeling of Periodic Two-Dimensional Nanocomposites”, Physical Review B, Vol. 69, pp.195316, 2004.[36] Chen, G., “Size and Interface Effects on Thermal Conductivity of Superlattices and Periodic Thin-Film Structures”, ASME Journal of Heat Transfer, Vol.119, pp.220-229, 1997.[37] 彭宇清,可壓縮流之高解析隱式算則的發展及其應用,國立台灣大學機械工程研究所博士論文,台北, 2000.[38] 劉靜,微米/奈米尺度熱傳學,北京,科學出版社,2001.[39] 王建評,界面熱阻對非平板薄膜系統熱傳導效應之分析,國立交通大學機械工程研究所碩士論文,新竹, 2002.[40] 施閔華,微奈米尺度薄膜熱傳現象之研究,國立交通大學機械工程研究所碩士論文,新竹, 2003.[41] 湯國樑,波茲曼模型方程式之高解析數值方法,國立台灣大學應用力學研究所博士論文,台北, 2005.
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 1 可壓縮流之高解析隱式算則的發展及其應用 2 微奈米尺度薄膜熱傳現象之研究 3 波茲曼模型方程式之高解析數值方法 4 界面熱阻對非平板薄膜系統熱傳導效應之分析 5 使用聲子波茲曼方程對緊密型奈米尺度複合物之熱傳模擬 6 使用聲子波茲曼方程對軸對稱奈米線之熱傳模擬 7 磷化鋁鎵銦發光二極體低溫之熱流特性 8 奈米尺度聲子熱傳之介面效應分析

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