臺灣博碩士論文加值系統

藉由薛丁格方程式，使用有限體積法計算任意形狀之異質結構量子點的離散能態。我們透過虛數時間去積分與時間相關的薛丁格方程式，然後疊代規一化的波函數以得到離散能態的波函數(特徵函數)與其對應的能量(特徵值)。為了處理複雜的任意幾何形狀之異質結構量子點，我們利用階梯式幾何近似將量子點形狀嵌入至直角座標系統中。這樣的幾何近似會使得二階精確度降為超線性(一至二階間)精確度，但是卻換來容易處理的幾何。為了維持通過異質介面之通量的連
續性，介面條件被併入離散方法中。已完成1、2、3 維的測試例子證明我們所提出的方法具有超線性精確度。更進一步，我們提出精細的方法以提升精確度。同樣地，我們也模擬豆子形狀的雙重量子線與截頭的金字塔形量子點，與文獻比較，有一致的結果。

A finite volume method is used to calculate the discrete energy states of hetero-structure quantum dots in arbitrary shape via Schrödinger equation. We integrated time-dependent Schrödinger equation through imaginary time, and iterated the normalized wave function to obtain discrete energy-state wave functions (eigenfunctions) and corresponding energies (eigenvalues). To deal with complicated
geometry of hetero-structure quantum dots in arbitrary shape, we embedded the shape of quantum dots in Cartesian coordinate system by stepwise geometry approximation. This approximation will downgrade the second order accuracy to a superlinear one, but gain the easy handling in geometry. The interface conditions were incorporated into the discretization scheme to maintain the continuous flux across the heterojunctions. Benchmarks have been completed in one, two, and three dimensions to verify the superlinear accuracy. Further, we proposed a refinement to enhance the order of accuracy. We also simulated the bean-shaped double quantum wires and truncated pyramidal quantum dot and got the consistent results compared to the references.

誌謝...................................i
摘要..................................ii
Abstract.............................iii
Table of Contents.....................iv
List of Figures.......................vi
List of Tables.........................x
1. Introduction........................1
2. Governing equation and geometry approximation...............................7
2.1 Schrödinger equation...............7
2.2 Non-dimensional time-dependent Schrödinger equation.......................11
2.3 Energy diminishing................11
2.4 Stepwise geometry approximation...15
3. Numerical method...................19
3.1 Discretization of Schrödinger equation....................................19
3.2 Seeking the discrete energy states.22
4. Numerical results..................26
4.1 Error analysis....................26
4.1.1 Benchmark for quantum well......26
4.1.2 Benchmarks for quantum wire and quantum dot.............................28
4.1.3 Enhancement of accuracy via refinement..................................33
4.2 Two examples of arbitrary shape...36
4.2.1 Double quantum wires with bean-shaped cross-section.....................38
4.2.2 Truncated pyramidal quantum dot..39
4.3 Non-parabolic band approximation..46
4.4 Continuous normalized gradient flow (CNGF)................................55
5. Conclusion and future work.........61
References............................64

[1] L. Esaki and R. Tsu, IBM J. Res. Dev. 14 (1970) 61.
[2] H. Sakaki, Jpn. J. Appl. Phys. 19 (1980) L735.
[3] W. J. Skocpol, L. D. Jackel, R. E. Howard, E. L. Hu, and L. A. Fetter, Physica (Utrecht) 117&;118 (1983) 667.
[4] R. G. Wheeler, K. K. Choi, A. Goel, R. Wisnieff, and D. E. Prober, Phys. Rev. Lett. 49 (1982) 1674.
[5] K. S. Raals, W. J. Skocpol, L. D. Jackel, R. E. Howard, L. A. Fetter, R. W. Epworth, and D. M. Tennant, Phys. Rev. Lett. 52 (1984) 228.
[6] T. J. Thornton, M. Pepper, H. Ahmed, D. Andrews, and G. J. Davies, Phys. Rev. Lett. 56 (1986) 1198.
[7] W. J. Skocpol, P. M. Mankiewich, R. E. Howard, L. D. Jackel, D. M. Tennant, and A. D. Stone, Phys. Rev. Lett. 56 (1986) 2865.
[8] A. I. Ekimov, A. A. Onushchenko, V. A. Tzehomskii, Sov. Phys. Chem. Glass (Fiz. Khim. Stekla) 6 (1980) 511.
[9] A. I. Ekimov, A. A. Onushchenko, Sov. JETP Lett. 34 (1981) 345.
[10] M. A. Reed , J. N. Randall, R. J. Aggarwal, R. J. Matyi, T. M. Moore, and A. E. Wetsel, Observation of discrete electronic states in a zero-dimensional semiconductor nanostructure, Phys. Rev. Lett. 60 (1988) 535-537.
[11] Lei Zhuang, Lingjie Guo, and Stephen Y. Chou, Silicon single-electron quantum-dot transistor switch operating at room temperature, Appl. Phys. Lett. 72 (1998) 1205.
[12] Heck, M.J.R., Renault, A., Bente, E.A.J.M., Yok-Siang Oei, Smit, M.K., Eikema, K.S.E., Ubachs, W., Anantathanasarn, S., and Notzel, R., Passively mode-locked 4.6 and 10.5 GHz quantum dot laser diodes around 1.55 μm with large operating regime, IEEE J. Sel. Top. Quantum Electron. 15(3) (2009) 634-643.
[13] G. Konstantatos, I. Howard, A. Fischer, S. Hoogland, J. Clifford, E. Klem, L. Levina, and Edward H. Sargent., Ultrasensitive solution-cast quantum dot photodetectors, Nature 442 (2006) 180-183.
[14] Hamid Reza Pouretedal, Abbas Norozi, Mohammad Hossein Keshavarz, and Abolfazl Semnani, Nanoparticles of zinc sulfide doped with manganese, nickel and copper as nanophotocatalyst in the degradation of organic dyes, J. Hazard. Mater. 162 (2009) 674-681.
[15] Andrew M Smith, Shivang Dave, Shuming Nie, Lawrence True, and Xiaohu Gao, Multicolor quantum dots for molecular diagnostics of cancer, Expert Rev. Mol. Diagn. 6 (2006) 231-244.
[16] Paul Harrison, Quantum Wells, Wires and Dots: Theoretical and Computational Physics of Semiconductor Nanostructures, John Wiley and Sons, Ltd., 2005.
[17] Y. Hirayama, J. H. Smet, L. H. Peng, C. G. Fonstad, and E. P. Ippen, Feasibility of 1.55 μm intersubband photonic devices using InGaAs/AlAs pseudomorphic quantum well structures, Jpn. J. Appl. Phys. 33 (1994) 890-895.
[18] A. Raymond, J. L. Robert, and C. Bernard, The electron effective mass in heavily doped GaAs, J. Phys. C: Solid State Phys. 12 (1979) 2289-2293.
[19] H. Asai and Y. Kawamura, Intersubband absorption in In_{0.53}Ga_{0.47}As/In_{0.52}Al_{0.48}As multiple quantum wells, Phys. Review B 43 (1991) 4748-4759.
[20] Lok C. Lew Yan Voon and Morten Willatzen, The k⋅p Method - Electronic Properties of Semiconductors, Springer-Verlag Berlin Heidelberg, 2009.
[21] T.M. Hwang, W.W. Lin, W.C. Wang, and W.C. Wang, Numerical simulation of three dimensional pyramid quantum dot, J. Comput. Phys. 196 (2004) 208-232.
[22] Heinrich Voss, Numerical calculation of the electronic structure for three dimensional quantum dots, Comput. Phys. Commun. 174 (2006) 441-446.
[23] Yiming Li, J.L. Liu, O. Voskoboynikov, C.P. Lee, and S.M. Sze, Electron energy level calculations for cylindrical narrow gap semiconductor quantum dot, Comput. Phys. Commun. 140 (2001) 399-404.
[24] Yiming Li, O. Voskoboynikov, C.P. Lee, and S.M. Sze, Computer simulation of electron energy levels for different shape InAs/GaAs semiconductor quantum dots, Comput. Phys. Commun. 141 (2001) 66-72.
[25] W.C. Wang, T.M. Hwang, W.W. Lin, and J.L. Liu, Numerical methods for semiconductor heterostructures with band nonparabolicity, J. Comput. Phys. 190 (2003) 141-158.
[26] W.C. Wang, T.M. Hwang, and J.C. Jang, A second-order finite volume scheme for three dimensional truncated pyramidal quantum dot, Comput. Phys. Commun. 174 (2006) 371-385.
[27] T.M. Hwang, W.C. Wang, and W.C. Wang, Numerical schemes for three-dimensional irregular shape quantum dots over curvilinear coordinate systems, J. Comput. Phys. 226 (2007) 754-773.
[28] Greenstein, George, Zajonc, Arthur, The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics, Jones and Bartlett Publishers, Inc., 2006.
[29] P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon Press, Oxford, Fourth edition, 1967.
[30] G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures, John Wiley and Sons, New York, 1991.
[31] A. Aftalion and Q. Du, Vortices in a rotating Bose-Einstein condensate: Critical angular velocities and energy diagrams in the Thomas-Fermi regime, Phys. Rev. A 64 (2001), article 063603.
[32] Weizhu Bao and Qiang Du, Computing the ground state solution of Bose-Einstein condensates by a normalized gradient flow, SIAM J. Sci. Comput. 25(3) (2004) 1674-1697.
[33] J.R. Dormand and P.J. Prince, A family of embedded Runge-Kutta formulae, J. Comp. Appl. Math. 6 (1980) 19-26.