中文摘要

本論文旨在改進直接能隙半導體折射係數計算公式。因為對於直接能隙化合物半導體的折射係數光譜（ ），目前只有很少的實驗數據且大部分都小於能隙（bandgap）以下。為了設計最佳化光電元件如光波導（waveguide），電制吸收調變器（electro-absorption modulator EAM）和Mach-Zendar 干涉儀（MZI）。我們必須利用有限的實驗數據，將折射係數光譜圖延伸到接近而且大於能隙的位置。
我們知道折射係數二次方為介電係數，所以我們將介電係數分解為能隙間（band-to-band）的吸收項和高能量振盪子（high energy single-oscillator）的吸收項。
對於在能隙間吸收項的吸收係數模型，我們再加入能帶延展效應的參數 （broadening parameter），再以Kramers-Kronig Transform理論求得能隙間吸收項的介電常數。而對於高能量振盪子的背景吸收項，我們先將有限的實驗數據扣掉能隙間吸收項，再以Sellemier’s Model擬合（fitting）出三個參數（A、B、C）的方程式。最後整合所有方程式，並將折射係數光譜延伸到接近而且大於能隙以上的位置。
我們成功地建立起完整的模型，並以砷化鎵來驗證我們的模型，其實驗數據與我們的模擬計算非常地吻合。我們進一步應用到直接能隙二元、三元和四元化合物半導體材料，將折射係數光譜延伸到接近而且大於能隙的位置以上，並得到很好的結果。

Abtract

In this thesis, our purpose is to improve the refractive index models for direct-gap semiconductors. For refractive index spectrum of direct-gap compound semiconductors, most experimental data is available only bellow the bandgap absorption edge. For used in the optimum design of eltro-optic devices, such as waveguide, electro-absorption modulator and Mach-Zehnder interferometer. We have to utilize a little experimental data to extend refractive index spectrum to near and just above the band-gap edge.
We have known that square of refractive index ( ) is dielectric constant ( ), so we decompose the dielectric constant ( ) into the part of band-to-band absorption and another part of single-oscillator high energy absorption.
For the part of band-to-band absorption, we added broadening parameter ( ) and used Kramers-Kronig relation to transform the absorption coefficient into dielectric constant. For another part of single-oscillator high energy background absorption, we first cut the absorption part form experimental data and then use Sellmeier’s equation to fit the residue data. Finally, recombine all equations and extend refractive index spectrum to near and just above the band-gap edge.
We successfully built whole model and confirm our model with GaAs. The calculation result on GaAs shows an excellent agreement with the reported experimental data. Furthermore, We apply our model to direct-gap binary、ternary and quaternary compounded materials and extend our model to near and just above the band-gap edge very well.

第一章 緒論………………………………………………………………..….1
1-1 前言….……………..........................................................………..1
1-2 動機……..….……………………………......….…………….…..2
1-3 論文架構………………………………………..…………….…..3

第二章 吸收係數之原理及推導過程…………………………………..4
2-1電子與光子交互作用之Hamiltonian函數…………………….....4
2-2單位光強度………………………….…………………..………...6
2-3躍遷機率………………………..……….………….……………..7
2-4應用Fermi-Golden Rule推導吸收係數……………………..……9

第三章 直接能隙半導體折射係數計算公式之改進……………...…14
3-1簡介………………….…….…..……..……………….………….14
3-2古典模型………………………………………………..……..…14
3-3改進能隙吸收邊的折射係數理論模型…………………..…..…16
3-3-1模型的來源………………………………………………...16
3-3-2理論吸收模型的建立與假設………...……………..……..18
3-3-3理論吸收模型之驗證…………………………….…….….21
3-4基本折射係數理論推導…………….…………………………...24
3-5 Kramers-Kronig Transform…………………………………..…..26


第四章 結果與分析……………………………………………..……………30
直接能隙半導體的二元化合物
…………………………………...………...………..………......32
………………………….…………….……….…………...……..34
………………………………………………………….………..36
直接能隙半導體的三元化合物
………………………………………………..…….…..38
直接能隙半導體的四元化合物 …………………...……………….......40
（X=0 , ）……………………………………...………..41
（X=0.3 , ）…………………………….………..42
（X=0.5 , ）…………………….……….…….…..43
（X=0.7 , ）…………………………….…….…..44
（X=1.0 , ）……………………………………………..45
（X=0.0∼1.0 比較圖）………………..………………….…..……..46
（X=0.0∼1.0 參數A, B, C）……………..…………………………48

第五章 結論…………………………………………………………….…….51

參考文獻………………………………………………………………..……..52

附錄A Kramer-Kronig轉換式……………………………………….……...54
A-1能隙間吸收係數光譜 ……………………..………….…...54
A-2 積分轉換式 …………………...…………..……….56
A-3 積分轉換式 …………………………...…….…….58
A-4留數定理………………………….…………..……………..…...60
A-5 Kramers-Kronig轉換式結果…………………………………....72

附錄B 方程式迴授及收斂求解…………………………….….…………….73

附錄C 方程式迴授及收斂求解…………………………….….………….…77
C-1 Matlab模擬理論吸收模型 .................................................77
C-2 Matlab模擬完整理論折射率模型 …….……………….....80

參考文獻

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