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研究生:林猷穎
研究生(外文):Eu-Ying Lin
論文名稱:直接能隙半導體之精確吸收係數與折射係數光譜模型
論文名稱(外文):An accurate model for absorption and refractive-index spectra of direct band-gap semiconductors
指導教授:賴聰賢
指導教授(外文):Tsong-Sheng Lay
學位類別:博士
校院名稱:國立中山大學
系所名稱:光電工程學系研究所
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2009
畢業學年度:97
語文別:英文
論文頁數:140
中文關鍵詞:Kramers-Kronig轉換式激子折射係數吸收係數
外文關鍵詞:excitonKramers-Kronig transformrefractive indexabsorption
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我們發展一個新的吸收係數與折射係數模型,能夠在吸收邊上提供精確數值,並且能夠利用有限的直接能隙二元化合物半導體材料數據 (砷化鎵GaAs,磷化銦InP,砷化銦InAs),延伸到重要地三元化合物半導體材料 (砷化鋁鎵AlxGa1-XAs,砷化磷鎵In0.53Ga0.47As)。此新的吸收係數模型包含了關鍵的庫侖作用力與Urbach寬化的物理現象,我們透過線性模型來精確擬和吸收邊的吸收係數,並且建立一個新的寬化模型 (SCL-LSF) 來達到Urbach 寬化現象。此吸收係數模型的結果與實驗係數非常吻合。透過精確擬和能夠提供一組重要的半導體能隙參數,除此之外此吸收係數能夠達到完整Kramers-Kronig轉換式,能夠進一步得到精確的折射係數。
新的完整折射係數模型,能夠利用有限的折射係數實驗數據,將直接能隙三元化合物半導體材料 (砷化鋁鎵AlxGa1-XAs,砷化磷鎵In0.53Ga0.47As) 折射係數延伸到大於能隙以上的地方。透過一個單一震盪Sellmeier模型與吸收係數的Kramers-Kronig 轉換式能夠完整表現出折射係數在吸收邊上的激子躍遷物理現象。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。進一步,我們的模型只需要透過數個重要的能隙參數,就能組合出所有三元和四元直接能隙化合物半導體的精確吸收係數與折射係數。我們的模型與已發表的實驗數據非常符合,能夠提供未來半導體積體化光電元件 (電制吸收調變器EAM、電制折射係數調變器EOM) 之準確模擬與最佳化。
New improved model was developed to calculate the absorption and refractive-index spectra in the band-edge region for all the important compound semiconductors. An accurate absorption model including Coulomb interaction and Urbach-broadened band edge has been demonstrated for direct bandgap semiconductors. We have developed a accurate model in which a piecewise linear approximation is used for the shape of the absorption spectrum. We also propose a steep-edged compound Lorentzian line-shape function (SCL-LSF) for modeling the Urbach tail, and the line broadening of exciton absorptions. The results of applying this fitting procedure to the absorption spectra of GaAs, InP and InAs are presented, and a consistent set of band parameters are extracted. The analytical absorption model is suitable for a complete closed-form Kramers-Kronig transform of the absorption spectrum to obtain the refractive index spectrum.
A band-to-band Coulomb interaction model for the refractive index spectra is presented of AlxGa1-xAs for 0 < x < 0.412, and In0.53Ga0.47As ternary semiconductors at photon energies near and above the band gap. An accurate absorption model is used to calculate the contribution on the refractive index near band-edge region through a complete closed-form Kramers-Kronig transform. By including a single oscillator Sellmeier model for the high-energy absorption spectrum, closed-form expressions are obtained for the band-edge region refractive index. Both spectra are fully described in terms of a finite set of parameters that can be interpolated for all the important compound semiconductors. The refractive index spectra are extended beyond the band-gap energy and are in excellent agreement with the available experimental data. Our new model makes accurate modeling possible for devices such as electroabsorption and electrorefraction modulators.
Contents

Chapter 1 Introduction……………………………..………….…...……... 1
1-1 Background……………………..…………………….…….…........ 1
1-2 Progresses and challenges…………………………...………...…... 2
1-3 Overview of this dissertation …………………….………...…....... 5
References ……………………………………………….……………... 6

Chapter 2 Theoretical Fundamental……………………………...…..….. 7
2-1 Transition rate due to electron-photon interaction………………..... 7
2-2 The optical intensity……………………………..………………..... 9
2-3 Optical absorption coefficient according to Fermi-Golden rule….... 10
2-4 Absorption coefficient including the bound and continuum states... 14
2-5 Conventional line shape broadening……...………………….…….. 16
2-5-1 Single Lorentizian line-shape function……………….…..... 17
2-5-2 Gaussian line-shape function…………………..………....... 18
2-6 Momentum matrix element of a bulk semiconductor…………….... 19
References………………………………………………………....…….. 20

Chapter 3 Accurate model including Coulomb-enhanced and Urbach-broadened absorption spectrum of direct-gap semiconductors………………………………………….....….. 21
3-1 Introduction……………………………………………….……….. 21
3-2 Coulomb interaction………………………….…………….…..….. 23
3-3 The Urbach tail……………………..…………………....……........ 26

3-4 Conventional absorption model………………………….....….….. 29
3-5 The absorption model with a piecewise-linear approximation.......... 31
3-6 Line-shape function for band-edge broadening……….………….... 38
3-7 Fitting results for binary materials………………………....….….... 44
3-7-1 Fitting procedure (A):
When refractive-index data are available……………..….... 44
3-7-2 Fitting procedure (B):
When refractive-index data are not available…………....….49
References…………………………………………………...….....…….. 57

Chapter 4 A band-to-band Coulomb interaction model for
refractive index spectra of ternary semiconductors……….... 59
4-1 Introduction……………………………………...………......…….. 59
4-2 Theory of the model……………………………...………......…….. 60
4-3 Kramers-Kronig transform for refractive-index spectrum…...…….. 63
4-4 Fitting results for binary materials GaAs………………….....…...... 68
4-5 Fitting results for ternary materials………………………...…….... 70
4-5-1 AlxGa1-xAs for 0 < x < 0.412…………………..………..….. 70
4-5-2 InAs and In0.53Ga0.47As………………………...……..….…. 76
References……………………………………………….……......…….. 80

Chapter 5 Summary and future work………………..……......….....…... 83
5-1 Summary……………………………………...………...........……. 83
5-2 Suggestion...……………………………...………..................…….. 85
References……………………………………………….……......…….. 88

Appendices
A. The Kramers-Kronig relations………………………..…….….…..…89
B. Convolution of the delta function…………….…………….….......…92
C. Modulation spectroscopy on metamorphic InAs quantum dots.….......94
C-1. Electro-absorption (EA) setup……………...…………......…95
C-2. Photo-reflectance (PR) setup……………………….…......…97
D. Optical study of In0.38GaAsNx single-quantum-well structures…..…106
E. Confocal System………………………………...……………...…...113
F. Low temperature cryostat……………………………………........…115
G. Symbol lists……………………………...……...………………...…118

Publication list ……………………………..……….…................................ 120
Ch1 References

[1.1]M. J. Mondry, D. I. Babic, J. E. Bowers, and L. A. Coldren, IEEE Photon. Technol. Lett. 4, 627 (1992).
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Ch2 References

[2.1]L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits, John Wiley & Sons, New York (1995).
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[2.5]Jasprit Singh, Semiconductor optoelectronic physics and technology, McGraw-Hill, Inc. (1995).

Ch3 References

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Ch4 References

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[4.12]D. D. Sell, H. C. Casey Jr., and K. W. Wecht, J. Appl. Phys. 45, 2650 (1974).
[4.13]S. Adachi, J. Appl. Phys. 66, 6030 (1989).
[4.14]D. E. Aspnes, S. M. Kelso, R. A. Logan, and R. Bhat, J. Appl. Phys. 60, 754 (1986).
[4.15]D. E. Aspnes, and A. A. Studna, Phys. Rev. B. 27, 985 (1983).
[4.16]R. J. Elliott, Phys. Rev. 108, 1384 (1957).
[4.17]J. S. Blakemore, J. Appl. Phys. 53, R123 (1982).
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[4.23]H. C. Casey Jr., D. D. Sell, and K. W. Wecht, J. Appl. Phys. 46, 250 (1975).
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Ch5 References

[5.1]Shun Lien Chuang, Physics of Optoelectronic Devices, John Wiley & Sons, New York, (1995).
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[5.8]P. M. Young, P. M. Hui, and H. Ehrenreich, “Excitons and interband transitions in III-V semiconductor superlattices”, Phys. Rev. B. 44, 12969 (1991-I).
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