適應性相場模式在多晶高非均向性晶體生長之研究

Abstract

在自然界的物理現象中，像是冰晶生長和一般工業長晶程序中，模擬高非均向性樹枝狀晶體生長及多晶矽生長是很重要的一個課題。相場模式在模擬固化問題一直是很強大的工具，但是在三維高非均向性物質的模擬上仍然很難克服；此外，二維多晶生長模擬之前在實驗室已經有很顯著的發展，故發展三維多晶模擬的模式也是個重要的課題。因此本論文中，吾人利用三維多晶模式進行模擬，並且發展出高非均向性物質的模擬方法。 首先，在我們適應性相場模擬中，發展了一個簡單解決三維高非均向性物質的方法。二維和三維中，晶體平衡形狀和解析解幾乎一致。純物質樹枝狀晶體在不同過冷度和非均向性強度下比較可以看出，非均向性愈強的樹枝狀尖端愈尖，過冷度愈高其形狀則愈接近平衡形狀。接著則是利用了Pusztai等人[1]的三維多晶模擬的模式進行模擬。第一步先利用晶界溝來和解析接觸角及解析形狀比較，得到一致性的結果並且接觸角和解析解的計算也是接近的。再來則是利用Karma等人[2]的模式，經過參數選取的不同，可以重現之前實驗室在晶粒競爭上的定性研究行為，最後則是一個簡單三維多晶晶粒碰撞後的晶態觀察。而第三部分，定量模擬上也有顯著的突破，和實驗觀察到的矽晶{111}面及界面不穩定中的波長，都能得到相同的數量級，而其中長速仍然和實驗上有著一倍的差距，歸因於實驗上在散熱項的熱擴散係數及降溫溫梯的估計和模擬上的理論值有些微的差距。

The modeling of dendritic crystal growth of highly anisotropic materials is important in understanding the physics for nature phenomena, such as snow growth, and industrial processes, such as polycrystalline silicon growth. Phase field modeling has emerged as a powerful tool in such a simulation, but the description of the anisotropic function for three-dimensional (3D) interfacial energy without miss-orientation remains quite difficult. Besides, polycrystalline two-dimensional (2D) simulation has been successfully present in previous work in our laboratory. Another model, which can be solved for three-dimensional polycrystalline simulation, is applied. In this thesis, three main parts are separated for our results. First, a simple method is proposed in our adaptive phase field model. In 2D, and 3D, the simulated equilibrium shapes with different anisotropic strengths are first compared with the exact solutions and they are in good agreement. The dendritic crystal growth of a pure material with different anisotropic strengths at various overcoolings is further simulated. For the second part, polycrystalline model proposed by Pusztai et al. [1] is applied for solving the problems, such as the correctness of the model by grain boundary groove, the grain competition including the kinetic effect, and the two seeds impingement in 3D. Finally, a quantitative analysis for the contract between the simulation and the experiment is further discussed. The tendency of the silicon facet formation and the order of magnitude are well- comparable, although not extremely accurate. Some estimation for the physical and experimental properties are believed for this slightly divergence between simulation and experiment. The long and short of it, a method of highly anisotropic function and a application of polycrystalline in 3D are workable.