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研究生： 呂中偉
Chung-wei Lu
論文名稱： 以熱交換器法生長氧化鋁單晶之模擬分析
Numerical Simulation of Sapphire Crystal Growth using HEM
指導教授： 陳志臣
Jyh-Chen Chen
口試委員:
學位類別： 博士
Doctor
系所名稱： 工學院 - 機械工程學系
Department of Mechanical Engineering
畢業學年度： 90
語文別： 中文
論文頁數： 118
中文關鍵詞： 熱交換器法 、氧化鋁單晶 、數值模擬 、單晶生長
外文關鍵詞： sapphire, HEM, numerical simluation, crystal growth
相關次數： 點閱：14  下載：0 
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    • 長晶實驗是一項昂貴的實驗程序，尤其是類似熱交換器長晶法(HEM)這種生長大型晶體的方法。對於這種問題的解決之道，是使用數值模擬的方式先期獲得一些基本的資料，再反覆推知所需的生長條件以減少實驗所費的時間。由於熱交換器長晶法(HEM)主要是控制溫度梯度來控制整個晶體生長的過程，因此加熱爐所提供的整體熱場設計、熱交換器取熱方式的設計等熱流控制問題對於生長品質優良之氧化鋁單晶來說相當重要。
    為了確保生長sapphire單晶的品質我們須掌握坩堝內之生長情況；因此使用模擬的方法對坩堝內之情形加以分析，以獲得各項熱流場情形資訊。本研究我們選用以有限元素法（FEM）為基礎的套裝軟體FIDAP，進行模擬分析工作。在模擬研究的初期，我們使用準穩態的方式針對不同生長變數如爐體內環境溫度及熱傳係數、熱交換器內之溫度、熱傳係數及取熱區大小，對生長氧化鋁單晶過程之熱流場及固液界面形狀的影響作深入的研究，各種模擬的結果與過去的實驗比較是吻合的。之後我們亦分析有關溫度梯度大小、不同坩堝形狀等因素，對生長氧化鋁單晶過程之熱流場及固液界面形狀的影響，結果發現愈高的正向溫度梯度對HEM長晶愈有利。另由於高溫狀況下生長材料之熱傳導係數值一般均無法確知，所以我們也對此進行分析比較，發現較小的熱傳導係數值會有較大的固液界面凸出率；最後我們也針對非等向性材料性質之影響，使用ANSYS套裝軟體進行三維模擬相關之研究，發現不同軸向的熱傳導係數值會影響生長的凸出率。
    這些分析結果將可作為HEM系統單晶生長進行時之重要參考指標，並可為將來深入研究HEM單晶生長機制的基礎。未來的研究方向則可依據真實的生長情形推定相關參數值後，再針對合於實際狀態的長晶情形加以分析，同時藉由模擬結果與實驗比對的過程，希望未來能對整個HEM長晶系統作出細部的改善，以期能得到更佳之系統穩定性，而使長晶過程更順利，而能生長出高純度、大尺寸氧化鋁單晶。

    Sapphire single crystals are widely used in variety of modern high-tech applications. Among crystal growth methods, the heat exchanger method (HEM) is a good commercial method for growing the larger, high-optical-quality sapphire.
    The finite element software FIDAP is employed to study the temperature and velocity distribution and the interface shape during sapphire crystal growth process using HEM. In the present study, the energy input to the crucible by the radiation and convection inside the furnace and the energy output through the heat exchanger is modeled by the convection boundary conditions. The associations of the various parameters are studied. It is found that the contact angle is obtuse before the solid-melt interface touches the sidewall of the crucible. Therefore, hot spots always appear in this process. The maximum convexity decreases significantly when the cooling-zone radius (RC) increases. The maximum convexity also decreases significantly as the combined convection coefficient inside the furnace (hI) decreases.
    In the second part, the effects of the thermal distribution in a HEM crystal growth system on the convexity of the melt-crystal interface and the rate of crystal size increase have been performed to investigate. A higher environmental temperature generates higher maximum convexity. During the environmental temperature reduction process, the crystal quickly enlarges in size as the environmental temperature approaches the melting point of the growing crystal. Therefore, decreases in the environmental temperature must be slow to obtain a constant rate of crystal size increase. An upward temperature gradient decreases the convexity of the melt-crystal interface and the hot spot area. An upward temperature gradient case is more adapted to the maintenance of a constant increase in the crystal size during a decreased in the furnace temperature to near the melting point of the growing crystal.
    The influences of crucible wall and the conductivity of the sapphire on the growth process have been also investigated in the present study,. We found that a lower conductivity of sapphire generates higher maximum convexity. The finite element software ANSYS is also employed to study the effects of sapphire’s anistropic conductivity.
    These results will be the important index of the HEM crystal growth process and for further study.

    目 錄 摘 要 I 英文摘要 II 致 謝 IV 符 號 表 V 圖 表 目 錄 X 第一章 緒 論 1 1.1熱交換器法生長氧化鋁單晶之簡介 3 1.2 文獻回顧 6 1.3 研究動機 12 1.4 對於結果與討論的說明 14 第二章HEM長晶模擬的分析模式 16 2.1影響長晶熱流場的因素 16 2.2二維熱對流模擬之模式 (MODELLING) 20 2.3三維模擬之方程式 23 第三章HEM長晶模擬的數值方法 24 3.1 FIDAP的主要特色 24 3.2模擬的項目 28 第四章 參數之間的關係 30 4.1取熱參數（HC及TREF）的影響 30 4.2 取熱區大小（R/RC）的影響 33 4.3 加熱參數（HI及TIREF）的影響 34 第五章 各種參數對長晶的影響 37 5.1溫度梯度的影響 37 5.2不同坩鍋形狀的影響 48 5.3不同坩鍋壁的影響 51 第六章 單晶材料熱傳導係數之影響 55 6.1固液區熱傳導係數同時改變之影響 57 6.2單獨改變固液區熱傳導係數之影響 59 6.3單晶材料非等向性熱傳道係數的影響 60 第七章 三維單晶生長熱流場的分析 63 7.1三維解與二維解之比較 63 7.2非等向性熱傳導係數 66 第八章 結論 70 參考文獻 115 附錄一：HEM長晶設備與長晶模擬之關係 1 A.1供熱與絕熱系統 2 A.2真空系統 3 A.3控制系統 4 A.4熱交換器系統 5 A.5量測觀察系統 6 A.6長晶過程及現象之探討 7 附錄二：加熱棒與坩堝間視角因子(VIEW FACTOR)的計算 8 圖 表 目 錄 表1.1 氧化鋁(Sapphire)特性表 74 表3.1 影響HEM長晶之要項表 75 表4.1 HEM長晶製程參數表 76 表5.1 接近真實長晶狀況時HEM長晶製程參數表 77 表5.2計算熱傳導係數值影響時所使用之物理性質表 77 圖1-1： HEM系統示意圖 78 圖1-2：熱交換器法長晶爐爐體的基本結構 79 圖2-1：(a) HEM長晶示意圖；(b)模擬區域示意圖 81 圖4-1：鳥籠式加熱元件圖 82 圖4-2： hC(W/m2K)改變對速度場及溫度場的影響情形 83 圖4-3： hC 增加與熱交換器取走熱量的對應圖 83 圖4-4： Tref 對速度場及溫度場的影響 84 圖4-5：當生長的單晶體積相同時Tref與hc值的對應圖 84 圖4-6：固液界面的凸出率對應hC且為Tref 函數時之關係圖 85 圖4-7：固液界面的凸出率對應hC且為R/RC函數時之關係圖 85 圖4-8：不同 R/RC狀況下全區之等溫線圖與熔液區之流線圖 86 圖4-9：固液界面的凸出率對應hI之關係圖 87 圖4-10：固液界面的凸出率對應Tiref之關係圖 87 圖5-1：二種溫度梯度分佈情形 88 圖5-2：當hc = 300 W/m2K的狀況下，長晶坩堝內之溫度場、流場及固液界面隨著Tiref 變化的情形 89 圖5-3：單晶生長固液界面形狀相對於爐內環境溫度Tiref的關係圖 90 圖5-4：當不同hc時，單晶生長體積(Vc)對應Tb的關係圖 90 圖5-5：當不同hc時，凸出率(D)對應爐內環境溫度(Tiref)的變化情形 91 圖5-6：當不同Tb時，凸出率(D)對應hc的變化情形 91 圖5-7：當不同ΔT時，凸出率(D)對應hc的變化情形 92 圖5-8：當不同ΔT時，坩堝內之溫度場、流場及固液界面分佈情形 92 圖5-9：Tb=7k時，不同ΔT條件下，凸出率D對應hc變化的情形 93 圖5-10：單晶最大高度(Dc)對應hc之關係圖 93 圖5-11：當Tb=7k，hc=300w/m2K狀況下最大凸率(Dmax)對應 ΔT的關係圖 94 圖5-12：當ΔT=10K，不同Tb情形下凸出率對應hc的變化圖 94 圖5-13：當ΔT=10K不同hc時，凸出率(D)對應Tb的變化圖 95 圖5-14：在ΔT=0 K及ΔT=10 K下，凸出率(D)對應Tb的變化情形 95 圖5-15：在ΔT=10K及hc=200W/m2K時單晶生長固液界面外形對應Tb下降之變化情形 96 圖5-16：在ΔT=10K時hc=200 W/m2K及hc=300 W/m2K之情形下，Vc對應Tb之變化情形 96 圖5-17：在hc=200 W/m2K時ΔT=0K及ΔT=10K之情形下，Vc對應Tb之變化情形 97 圖5-18：使用不同形狀之坩堝模擬所得之結果 98 圖5-19：圓柱形與非圓柱形坩堝其凸出率(D)隨Tiref降低之變化情形 99 圖5-20：坩堝具有不同曲率半徑時，其凸出率對應Tiref下降之關係圖 99 圖5-21：坩堝底部之曲率半徑大小對生長單晶之最大凸出率之影響 100 圖5-22：坩堝壁厚度對長晶熱流場與固液界面形狀的影響 101 圖5-23：坩堝壁厚度對應最大固液界面凸出率的變化圖 102 圖5-25：坩堝壁材料熱傳導係數對應最大固液界面凸出率的變化圖 104 圖5-26：坩堝壁上製作溝槽對長晶熱流場與固液界面形狀的影響 105 圖6-2：km=ks以降溫方式生長，不同k值對應Dmax之分佈圖 106 圖6-3：km=3.5 W/mK時不同ks值對應Dmax之分佈圖，並將圖6-2的結果併入比較 107 圖6-4：ks=3.5 W/mK時不同km值對應Dmax之分佈圖，並將圖6-2的結果併入比較 107 圖6-5：ksz=3.5 W/mK時不同ksr值對應Dmax之分佈圖 108 圖6-6：ksr=3.5W/mK時不同ksz值對應Dmax之分佈圖 108 圖7-1：三維網格點配置的影響 109 圖7-2：三維縱切面的固液界面形狀圖 109 圖7-3：平行Z軸切面上的速度分佈圖 110 圖7-4：固液界面之高度不同時，在Z軸相同高度橫切面之UZ速度分佈圖 110 圖7-5：利用CCD所觀察到之坩堝頂部之影像示意圖 111 圖7-6：更密的車輪狀(spoke type)網格點配置上視圖 111 圖7-7：km=ksx=ksy=3.5 W/m K，ksz=4.5 W/m K的情形下，晶體外形隨著Tiref下降而變動的情形 112 圖7-8：三軸向之熱傳導係數均不相同，km=3.5 W/m K, ksx=4.0 W/m K, ksy=3.0 W/m K, ksz=4.5 W/m K的情形下，由不同視角方向觀看晶體外形 113 圖7-9：以a軸方向生長，軸向之熱傳導係數以參考文件[50]推算出km=3.9 W/m K,ksx=4.5 W/m K, ksy=3.9 W/m K，ksz=4.5 W/m K的情形下，由不同視角觀看晶體外形 114

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