臺灣博碩士論文加值系統

本論文內容主要的工作是介紹K‧P理論之應用，進而推導出Luttinger-Kohn模型，進而將此模型應用在求解應變之下的帶結構圖以及求解其等效質量上。第一章為引言。第二章我們介紹了應變和應力的關係，以及在不同方向上之雙軸應變。第三章介紹K‧P理論以及單一能帶之K‧P理論的推導，並且利用Lowdin Perturbation Theory推導出Luttinger-Kohn模型，進而推導出在受應變下之的Pikus-Bir Hamiltonian，進而利用Luttinger-Kohn模型和Pikus-Bir Hamiltonian推導出在雙軸應變下之磊晶層成長在基底方向(001)的色散關係式以及其所對應之等效質量。第四章則為結論。

The major work of this thesis is the introduction to the k‧p theory of application. And then derive the Luttinger
-Kohn model. Thus this model is applied to solve the strain under the band structure as well as solving its equivalent mass. Chapter I is an introduction. Chapter II describes
the relationship between the strain and stress, and the biaxial strain in different directions. The chapter III describes the k‧p theory and the derivation of the single band theory, and the use of the Lowdin Perturbation Theory to derive out-of Luttinger-Kohn model. In turn is derived in the Pikus-Bir Hamiltonian by the strain. Luttinger-Kohn model and the Pikus-Bir Hamiltonian derive the dispersion relation of epitaxial layer growth under biaxial strain on the substrate direction (001) and their corresponding equivalent mass. Chapter IV is the conclusion.

目錄
Chapter1 前言 …………………………………………………………1
Chapter2 應變與應力 …………………………………………………2
2-1 應變 …………………………………………………………………2
2-2 應力 …………………………………………………………………5
2-3 應變和應力的關係 …………………………………………………7
2-3.1 米勒指標 …………………………………………………………8
2-3.2 磊晶層成長在不同基板方向上之雙軸應變之結果 ……………8
Chapter3 能帶理論之微擾理論及其應用 ……………………………11
3-1 k⋅ p method ………………………………………………………11
3-2 單一能帶下之k ⋅ p method …………………………………… 13
3-3 Lowdin perturbation theory ………………………………… 18
3-4 不考慮Spin-orbit interaction 之P-like 價帶結構 ………25
3-5 考慮Spin-orbit interaction 下之Γ 點導帶、價帶結構……29
3-6 考慮Spin-orbit interaction 下之Γ 點附近之價帶結構 … 36
3-7 Pikus − Bir Hamiltonian…………………………………… 42
3-8 在忽略自旋軌道分裂帶之耦合下的帶結 ……………………… 46
Chapter4 結論………………………………………………………… 54
參考資料…………………………………………………………………55

圖目錄
Fig.1 非應變座標系和應變座標系 ……………………………………2
Fig.2 可變形連續物質內部各種可能應力 ……………………………5
Fig.3 立方體的米勒指標 ………………………………………………8
Fig.4 磊晶層受基板應變的影響(a) 圖左:磊晶層受壓應力(b) 圖右:磊晶層受張應力 …………………………………………………………9
Fig.5 帶模型:導帶為class A，其它帶為class B …………………22
Fig.6 帶模型:價帶為class A，其它帶為class B …………………23
Fig.7 帶模型:價帶為class A，其它帶為class B …………………25
Fig.8 在Γ點下，有Spin-orbit interaction和無Spin-orbit interaction 之結果……………………………………………………34
Fig.9 考慮自旋與軌道偶合下之帶結構圖……………………………37
Fig.10 雙軸應力下之壓縮情況( l s a > a )………………………49
Fig.11 雙軸應力下之伸張情況( l s a < a )………………………50
Fig.12 磊晶層1 x x In Ga As − 材料在InP 基板上之壓縮和伸張應變下之能帶圖 ……………………………………………………… 51

表目錄
表1:為不同鑽石結構及閃鋅礦結構之半導體的Luttinger
parameters ………………………………………………………41
表2:GaAs、InAs 和InP 之參數表……………………………………52

﹝1﹞E.O. Kane. “Energy band structure in p-type germanium and silicon. ”,
J.Phys.Chem.Solids, vol. 1, no. 1-2, pages 82-99, 1956.
﹝2﹞J.M. Hinckley, and J. Singh. “Influence of substrate composition and crystallographic orientation on the band structure of pseudomorphic Si-Ge alloy films.” ,Physical Review B, vol. 42, pages 3546-3566, 1990.
﹝3﹞P. Lodin, “A note on the quantum-mechanical perturbation theory.” ,J. Chern. Phys. 19,pages 1396-1401,1951.
﹝4﹞Baldereschi, A., Lipari, N.O. “ Spherical model of shallow acceptor states in semiconductors. ”
﹝5﹞Mayer, H., R‥ossler, U. “ Nonparabolicity in the conduction band of II-VI semiconductors. ”,Solid State Commun. 87(2), pages 81–84,1993.
﹝6﹞G. E. Pikus and G. L. Bir, “Effects of deformation on the hole energy spectrum of germanium and silicon.” ,Sov. Phys i-Solid State 1, pages 1502-1517,1960.
﹝7﹞G. L. Bir and G. E. Pikus, “Symmetry and Strain-Induced Effects in Semicon
-ductors,Wiley.”, New York, 1974.
﹝8﹞Jasprit Singh, “Physics of semiconductors and their heterostructures.” ,MaGraw
-Hill, Inc. pages 229-238,1993.
﹝9﹞S. L. Chuang, “Efficient band-structure calculations of strained quantum wells using a two-by-two Hamiltonian.” ,Phys. Rev.B,vol.43, pages 9649-9661,1991.
﹝10﹞ Y. P. Chao and S. L. Chuang, “Spin-orbit-coupling effects on the valence band structure of strained semiconductor quantum wells. ” , Phys. Rev. B,vol.46, pages 4110-4122,1992.
﹝11﹞ Jasprit Singh, “Physics of semiconductors and their heterostructures”, MaGraw-Hill, Inc.,1993, chapter 4, 5,7.
﹝12﹞Shun Lien Chuang, “Physics of optoelectronics devices”, John Wiley & Sons, Inc., 1995,chapter 4.
﹝13﹞Supriyo Datta, “Quantum Phenomena”,Addison-Wesley Publishing
Company, Inc.1989,chapter 6.
﹝14﹞倪澤恩, “基礎固態物理學”,第六章,五南出版社,台北市,2011。
﹝15﹞C. Kittel. “Introduction to solid state physics”, John Wiley & Sons, 8 edition, 2005.chapter 5.
﹝16﹞ Stephan-Enzo Ungersbock, “Advanced Modeling of Strained CMOS Technology” ,2007.chapter 3.
﹝17﹞Viktor Sverdlov, “Strain-Induced Effects in Advanced MOSFETs”, springer, 2011.chapter 3,6,8,9.
﹝18﹞ Lok C. Lew Yan Voon,Morten Willatzen ,“The k‧p method”,springer
,2009.chapter 2,3,4,5,6,7