多晶矽材料熱傳導係數之理論預測與蒙地卡羅法模擬驗證研究

Abstract

本研究提出一理論模型來預測具有任意晶粒尺寸分佈多晶材料的熱傳導係數。研究首先比對模型預測與3D全頻蒙地卡羅(Monte-Carlo)模擬結果，確定適當的等效直徑定義，以帶入晶界散射率理論模型中。再透過對不同尺寸的晶粒按體積百分比加權，計算系統等效晶界散射率。最終使用蘭道爾(Landauer)公式並考慮邊界散射率、晶界散射率和本質(intrinsic)散射率來預測具有任意晶粒尺寸分佈的多晶材料的等效熱傳導係數。 我們利用拉蓋爾(Laguerre)分割法構建了兩種雙尺寸多晶系統和兩種晶粒尺寸常數分布於76nm至145nm間和對數常態(log-normal)晶粒尺寸分佈的系統。並再次進行蒙地卡羅法模擬，以檢驗模型預測的準確性。結果顯示理論模型預測與模擬結果十分一致，在10K到300K的溫度範圍內最大偏差僅有7%左右。此外，在兩種雙尺寸多晶系統中：一種為晶粒在空間中完全混合，另一種則由兩個單一尺寸系統串聯組成，兩系統模擬結果和模型預測均完全一致，顯示晶粒的空間分佈對系統的等效熱導率影響甚微，合理化我們在理論中將不同晶粒尺寸的貢獻加總的處理流程。

This study presents a novel theoretical model to predict the thermal conductivity of polycrystalline materials with arbitrary grain size distributions. The model starts with determining appropriate characteristic diameters for calculating the grain boundary scattering rate associated with single-grain-size polycrystalline materials. These selected diameters are verified by an agreement between the model predictions and 3D full-spectrum Monte-Carlo (MC) simulation results. Next, we add up the contributions of grains of different sizes to the total grain boundary scattering rate weighted by their volume percentages. The effective thermal conductivity of polycrystalline materials with an arbitrary grain size distribution is finally calculated using Landauer formulation by taking into account the boundary scattering rate, grain boundary scattering rate, and intrinsic scattering rate. Two dual-size polycrystalline systems and two polycrystalline materials with a log-normal distribution and constant distribution ranging between diameters of 76nm and 145nm are constructed with Laguerre method. MC simulations are performed again to examine the accuracy of the proposed model. The agreement is excellent; the maximum deviation is only about 7% over the temperature range from 10K to 300K. In addition, the effective thermal conductivities of the two dual-size polycrystalline systems, one of which has grains completely mixed in space and the other of which consists of two single-grain-size systems connected in series, are in perfect agreement, implying that the spatial distribution of grains has little impact on the effective thermal conductivity of the system. This is consistent with the volume weighting method used to sum up the contributions of different grain sizes to the grain boundary scattering rate.