在我們的工作中以氫原子鈍化球狀矽奈米晶體表面並使用sp d s3 5 *緊束縛模型計算 直徑範圍由 0.5nm 至 7.6nm 的矽奈米晶體電子結構與能隙，再由矽奈米晶體的波函數求得 矽奈米晶體的光吸收振子強度。在考慮大範圍的直徑變化下，我們能獲得與其他理論方法 相當接近能隙結果。在量子侷限效應作用下矽奈米晶體電子結構中導電帶的電子基態能量 與電子激發態能量隨著奈米晶體直徑變小而上升，價電帶的電洞基態能量與電洞激發態 能量隨著奈米晶體直徑減少而下降。因為塊材矽導電帶底部在倒空間中六個 X 點上且電 子共有兩種自旋態，所以直徑較大的矽奈米晶體基態能量形成十二重簡併。而塊材矽價
電帶頂部位在倒空間點，其基態能量有重電洞與輕電洞且電子共有兩種自旋態，所以
矽奈米晶體基態能量形成四重簡併。
在光吸收收振子強度結果中，我們發現量子侷限效應使得光吸收能量峰值隨奈米 晶體直徑變大而紅移(red shift)，不同的光吸收能量峰值間的能量差距也隨直徑變大而減 小。但矽奈米晶體光吸收振子強度卻隨著直徑變大而快速下降並由偶極矩陣元素中原子 位置偶極矩所主導。
參考文獻
[1] H. Takagi, H. Ogawa, Y. Yamazaki, A. Ishizaki, and T. Nakagiri, “Quantum size effects on photoluminescence in ultrafine Si particles”, Appl. Phys. Lett. 56, 2379 (1990) [2] L. T. Canham, “Silicon quantum wire array fabrication by electrochemical and chemical
dissolution of wafers”, Appl. Phys. Lett. 57, 1046 (1990)
[3] V. Lehmann and U. Gosele, “Porous silicon formation: A quantum wire effect”, Appl.
Phys. Lett. 58, 856 (1991)
[4] L. Pavesi, L. Dal Negro, C. Mazzoleni, G. Franzó, and F. Priolo, “Optical gain in silicon nanocrystals”, Nature 408, 440 (2000)
[5] H. Rong, A. Liu, R. Jones, O. Cohen, D. Hak, A. Fang, and M. Paniccia, “An all-silicon Raman laser”, Nature 433, 292 (2005)
[6] S. K. Kim, C. H. Cho, B. H. Kim, S. J. Park, and J. W. Lee, “Electrical and optical characteristics of silicon nanocrystal solar cells”, Appl. Phys.Lett. 95, 143120 (2009) [7] I. P. Wurfl, X. Hao, A. Gentle, D. H. Kim, G. Conibeer, and M. A. Green, “Si
nanocrystal p-i-n diodes fabricated on quartz substrates for third generation solar cell applications”, Appl. Phys.Lett. 95, 153506 (2009)
[8] M. V. Wolkin, J. Jorne, P. M. Fauchet, G. Allan, and C. Delerue, “Electronic States and Luminescence in Porous Silicon Quantum Dots: The Role of Oxygen”, Phys. Rev. Lett.
82, 197 (1999)
[9] Z. Zhou, L. Brus, and R. Friesner, “Electronic Structure and Luminescence of 1.1- and 1.4-nm Silicon Nanocrystals: Oxide Shell versus Hydrogen Passivation”, Nano Lett. 3, 163 (2003)
[10] M. Luppi and S. Ossicini, “Ab initio study on oxidized silicon clusters and silicon
nanocrystals embedded in SiO2: Beyond the quantum confinement effect”, Phys. Rev. B 71, 035340 (2005)
[11] I. Sagnes, H. Halimaoui, G. Vincent, and P. A. Badoz, “Optical absorption evidence of a quantum size effect in porous silicon”, Appl. Phys. Lett. 62, 1155 (1993)
[12] K. Y. Kuo, S. W. Hsu, W. L. Chuang, and P. T. Lee, “Formation of nano-crystalline Si quantum dots in ZnO thin-films using a ZnO/Si multilayer structure”, Materials Letters 68 463 (2012)
[13] S. Y. Ren, and J. D. Dow, “Hydrogenated Si clusters: Band formation with increasing size”, Phys. Rev. B 45, 6492 (1992)
[14] Y. M. Niquet, C. Delerue, G. Allan, and M. Lannoo, “Method for tight-binding parametrization: Application to silicon nanostructures”, Phys. Rev. B 62, 5109 (2000) [15] G. Allan, Y. M. Niquet, and C. Delerue, “Quantum confinement energies in zinc-blende
III–V and group IV semiconductors”, Appl. Phys. Lett. 77, 639 (2000)
[16] J. See, P. Dollfus, and S. Galdin, “Comparison between a sp3d5 tight-binding and an effective-mass description of silicon quantum dots”, Phys. Rev. B 66, 193307 (2002)
[17] M. Nishida, “Electronic state calculations of Si quantum dots: Oxidation effects”, Phys.
Rev. B 69, 165324 (2004)
[18] L. E. Ramos, J. Furthmuller, and F. Bechstedt, “Effect of backbond oxidation on silicon nanocrystallites”, Phys. Rev. B 70, 033311 (2004)
[19] D. Konig, J. Rudd, M. A. Green, and G. Conibeer, “Role of the interface for the electronic structure of Si quantum dots”, Phys. Rev. B 78, 035339 (2008)
[20] H. Lu, Y. J. Zhao, X. B. Yang, and H. Xu, “Theoretical investigation of structural stability and electronic properties of hydrogenated silicon nanocrystals: Size, shape, and surface reconstruction”, Phys. Rev. B 86, 085440 (2012)
[21] P. Hapala, K. Kusova, I. Pelant, and P. Jelnek, “Theoretical analysis of electronic band structure of 2- to 3-nm Si nanocrystals”, Phys. Rev. B 87, 195420 (2013)
[22] A. Zunger, and L. W. Wang, “Theory of Silicon nanostructures”, Applied Surface Science 102 350 (1996)
[23] D. V. Melnikov and J. R. Chelikowsky, “Electron affinities and ionization energies in Si and Ge nanocrystals”, Phys. Rev. B 69, 113305 (2004)
[24] T. Takagahara and K. Takeda, “Theory of the quantum confinement effect on excitons in quantum dots of indirect-gap materials”, Phys. Rev. B 46, 15578 (1992)
[25] C. R. S. da Silva and A. Fazzio, “Formation and structural properties of the
amorphous-crystal interface in a nanocrystalline system”, Phys. Rev. B 64, 075301 (2001)
[26] L. Kong and L. J. Lewis, “Structure and energetics of c-SiO/a-SiO2 systems: Planar interfaces and embedded Si nanocrystals”, Phys. Rev. B 77, 085204 (2008)
[27] C. Delerue, G. Allan, and M. Lannoo, “Theoretical aspects of the luminescence of porous silicon”, Phys. Rev. B 48, 11024 (1993)
[28] N. A. Hill and K. B. Whaley, “Size Dependence of Excitons in Silicon Nanocrystals”, Phys. Rev. Lett. 75, 1130 (1995)
[29] G. Allan, C. Delerue, and Y. M. Niquet, “Luminescence polarization of silicon nanocrystals”, Phys. Rev. B 63, 205301 (2001)
[30] K. Nishio and J. Koga, “Theoretical study of light-emission properties of amorphous silicon quantum dots”, Phys. Rev. B 67, 195304 (2003)
[31] F. Trani, G. Cantele, D. Ninno, and G. Iadonisi, “Tight-binding calculation of the optical absorption cross section of spherical and ellipsoidal silicon nanocrystals”, Phys. Rev. B 72, 075423 (2005)
[32] F. Trani, D. Ninno and G. Iadonisi, “Role of local fields in the optical properties of silicon nanocrystals using the tight binding approach”, Phys. Rev. B 75, 033312 (2007) [33] M. Nishida, “Electronic state calculations of ultrasmall Si quantum boxes: Quasialloying
and surface effects on the electronic and optical properties”, Phys. Rev. B 81, 235306 (2010)
[34] H. C. Weissker, J. Furthmuller, and F. Bechstedt, “Optical properties of Ge and Si nanocrystallites from ab initio calculations. II. Hydrogenated nanocrystallites”, Phys.
Rev. B 65, 155328 (2001)
[35] V. Kocevski, O. Eriksson, and J. Rusz, “Transition between direct and indirect band gap in silicon nanocrystals”, Phys. Rev. B 87, 245401 (2013)
[36] S. Wippermann, M. Voros, D. Rocca, A. Gali, G. Zimanyi and G. Galli, “High-Pressure Core Structures of Si Nanoparticles for Solar Energy Conversion”, Phys. Rev. Lett. 110, 046804 (2013)
[37] L. E. Ramos, J. Paier, G. Kresse and F. Bechstedt, “Optical spectra of Si nanocrystallites:
Bethe-Salpeter approach versus time-dependent ensity-functional theory”, Phys. Rev. B 78, 195423 (2008)
[38] A. D. Zdetsis, “One-nanometer luminous silicon nanoparticles: Possibility of a fullerene interpretation”, Phys. Rev. B 79, 195437 (2009)
[39] L. Koponen, L. O. Tunturivuori, M. J. Puska, and R. M. Nieminen, “Effect of the surrounding oxide on the photoabsorption spectra of Si nanocrystals”, Phys. Rev. B 79, 235332 (2009)
[40] A. J.Williamson, J. C. Grossman, R. Q. Hood, A. Puzder, and G. Galli, “Quantum Monte Carlo Calculations of Nanostructure Optical Gaps: Application to Silicon Quantum Dots”, Phys. Rev. Lett. 89, 196803 (2002)
[41] J. C. SLATER AND G. F. KOSTER, “Simplified LCAO Method for the Periodic Potential Problem”, Phys. Rev. 94, 1498 (1954)
[42] David J. Griffiths. "Introduction to quantum mechanics", 2 nd Ed. , Pearson Prentice Hall, pp. 188 (2005)
[43] S. Lee, F. Oyafuso, P. v. Allmen, and G. Klimeck, “Boundary conditions for the
electronic structure of finite-extent embedded semiconductor nanostructures”, Phys. Rev.
B 69, 045316 (2004)
[44] P. E. Batson and J. R. Heath, “Electron Energy Loss Spectroscopy of Single Silicon Nanocrystals: The Conduction Band”, Phys. Rev. Lett. 71, 911 (1993)
[45] R. N. Pereira,1, D. J. Rowe, R. J. Anthony, and U. Kortshagen, “Oxidation of freestanding silicon nanocrystals probed with electron spin resonance of interfacial dangling bonds”, Phys. Rev. B 83, 155327 (2011)
[46] W. A. Harrison. "Electronic Structure and the Properties of Solids", Dover Publications , p48,100,505(1989)
[47] J. Perez-Condea, A.K. Bhattacharjee, “Electronic structure of CdTe nanocrystals: a tight-binding study”, Solid State Communications 110 259 (1999)
[48] John H. Davies. "The physics of low-dimensional semiconductors", Cambridge University Press , chapter 8 (1998)
[49] J. J. Sakurai, "Modern Quantum Mechanics Revised Edition", Addison-Wesley Publishing Company , p338. (1994)
[50] K. Leung and K. B. Whaley, “Electron-hole interactions in silicon nanocrystals”, Phys.
Rev. B 56, 7455 (1997)
[51] K. Leung, S. Pokrant and K. B. Whaley, “Exciton fine structure in CdSe nanoclusters”, Phys. Rev. B 57, 12291 (1998)
[52] J. P. Conde, “Exciton states and optical properties of CdSe nanocrystals”, Phys. Rev. B 63, 245318 (2001)
[53] S. Lee, J. Kim, L. Jonsson, and J. W. Wilkins, “Many-body levels of optically excited and multiply charged InAs nanocrystals modeled by semiempirical tight binding”, Phys.
Rev. B 66, 235307 (2002)
[54] G. W. Bryant, and W. Jaskolski, “Tight-binding theory of quantum-dot quantum wells:
Single-particle effects and near-band-edge structure”, Phys. Rev. B 67, 205320 (2003) [55] S. Schulz, S. Schumacher, and G. Czycholl, “Tight-binding model for semiconductor
quantum dots with a wurtzite crystal structure: From one-particle properties to Coulomb correlations and optical spectra”, Phys. Rev. B 73, 245327 (2006)
[56] J. G. Diaz and G. W. Bryant, “Electronic and optical fine structure of GaAs nanocrystals:
The role of d orbitals in a tight-binding approach”, Phys. Rev. B 73, 075329 (2006)
[57] A. F. van Driel, G. Allan, C. Delerue, P. Lodahl, W. L. Vos, and D. Vanmaekelbergh,
“Frequency-Dependent Spontaneous Emission Rate from CdSe and CdTe Nanocrystals:
Influence of Dark States”, Phys. Rev. Lett. 95, 236804 (2005)
[58] Shun Lien Chuang, "Physics of Photonic Devices Second Edition", Wiley , p350-351.
(1994)
[59] T. B. Boykin, G. Klimeck, R. C. Bowen, and F. Oyafuso, “Diagonal parameter shifts due to nearest-neighbor displacements in empirical tight-binding theory”, Phys. Rev. B 66,
125207 (2002)
[60] P.Vogl, H. P. Hjalmarson, and J. D. Dow, “A SEMI-EMPIRICAL TIGHT-BINDING THEORY OF THE ELECTRONIC STRUCTURE OF SEMICONDUCTORS”, J. Phys.
Chem. Solids 44, 365 (1983)
[61] O. Madelung, Semiconducters Physics of Group IV Elements and III-V Compounds, edited by Landolt-Bornstein, New Series, Group III, Vol. 17, Pt. a (Springer-Verlag, Berlin, 1992)
[62] T. B. Boykin, “Valence band effective-mass expressions in the sp3d5s* empirical tight-binding model applied to a Si and Ge parametrization”, Phys. Rev. B 69, 115201 (2004)
[63] Y. Zheng, C. Rivas, R. Lake, K. Alam, T. B. Boykin, and G. Klimeck, “Electronic
Properties of Silicon Nanowires”, IEEE TRANSACTIONS ON ELECTRON DEVICES, 52, 1097 (2005)
[64] 黃威智,“為發展太陽能電池所進行之矽奈米粒子電子結構的研究”，國立交通大 學，碩士論文，民國102 年
[65] C. S. Garoufalis and A. D. Zdetsis, “High Level Ab Initio Calculations of the Optical Gap of Small Silicon Quantum Dots”, Phys. Rev. Lett. 87, 276402 (2001)
[66] F. A. Reboredo, A. Franceschetti, and A. Zunger, “Dark excitons due to direct Coulomb interactions in silicon quantum dots”, Phys. Rev. B 61, 13073 (2000)
[67] G. Belomoin, J. Therrien, A. Smith, S. Rao, and R. Twesten, “Observation of a magic discrete family of ultrabright Si nanoparticles”, Appl. Phys. Lett. 80, 841 (2002) [68] M. C. Beard, K. P. Knutsen, P. Yu, J. M. Luther, Q. Song, W. K. Metzger, R. J.
Ellingson, and A. J. Nozik, “Multiple Exciton Generation in Colloidal Silicon Nanocrystals”, Nano Letters 7, 2506 (2007)
[69] Lok C. Lew Yan Voon, Electronic and Optical Properties of Semiconductors:A Study Based on the Empirical Tight Binding Model, WORCESTER POLYTECHNIC INSTITUTE, Degree of Doctor of Philosopy, p.25 , 1993
[70] J. Jancu, R. Scholz, F. Beltram, and F. Bassani, “Empirical spds* tight-binding
calculation for cubic semiconductors:General method and material parameters”, Phys.
Rev. B 57, 6493 (1998)
[71] D. M. Bylander and L. Kleinman, “Self-consistent energy bands and formation energy of the (GaAs)(A1As)(001) superlattice”, Phys. Rev. B 34, 5280 (1986)
[72] M.D. Jaffe, J. Singh, “Inclusion of spin-orbit coupling into tight binding bandstructure calculations for bulk and superlattice semiconductors”, Solid State Communications 62, 399 (1987)
[73] 許克銘,“應變對砷化銦塊材能帶結構及傳輸性質之影響”，國立交通大學，碩士 論文，民國99 年
[74] 林以理,“以緊束縛模型使用 supercell 方法計算半導體中的雜質能態與奈米線能 帶結構”，國立交通大學，碩士論文，民國100 年
附錄 A:奈米晶體程式與文獻驗證之參數
表 A-2 Si 在緊束縛 sp3s*模型的材料參數 [60]
表 A-3 CdTe 在緊束縛 sp3s*模型且考慮電子自旋效應的材料參數[47]
表 A-4 Si 在緊束縛 sp3d5 模型的材料參數 [16]
表 A-5 Si 在緊束縛 sp3d5s*模型的材料參數與氫原子參數[16]
0.9998400 3.999720 1.697700 4.251750 2.105520 unit:eV
H HSi HSi HSi HSi
Es ss
ss
sp
sd
附錄 B:Slater-Koster 有效鍵結參數表