對於Rashba自旋軌道耦合與Dresselhaus自旋軌道耦合在一個具有任意形狀的曲面中，其精確的哈密頓函數被嚴謹地推導獲得。我們發現兩個正交的主曲率可以控制電子的自旋傳輸，而且在曲面正交方向的局限位勢的漸近行為則是可以忽略的。另外我們也發現高階的動量項在大曲率的曲面中發揮了重要的作用。曲面中的線性自旋軌道耦合只誘導產生額外的虛位勢項，然而曲面中的非線性自旋軌道耦合則會誘導產生額外的虛動能項、虛動量項以及虛位勢項。由於額外的曲率誘導項以及關聯虛磁場的作用，曲面中的自旋傳輸是不相同於在平面中的。我們也明確地推導獲得在柱面或球面中的自旋軌道耦合的哈密頓函數，而且在奈米圓環中的自旋進動以及關聯本徵態也被詳細地分析研究。因此我們推論曲率會顯著影響彎曲結構中的自旋軌道耦合與自旋傳輸。

The exact Hamiltonians for Rashba and Dresselhaus spin-orbit couplings on a curved surface with an arbitrary shape are rigorously derived. Two orthogonal principal curvatures dominate the electronic spin transport, and the asymptotic behavior of the normal confined potential on a curved surface is insignificant. For a curved surface with a large curvature, the higher order momentum terms play an important role in controlling spin transport. The linear spin-orbit coupling on a curved surface only induces the extra pseudo-potential term, and the cubic spin-orbit coupling on a curved surface can induce the extra pseudo-kinetic, pseudo-momentum, and pseudo-potential terms. Because of the extra curvature-induced terms and the associated pseudo-magnetic fields, spin transport on a curved surface is very different from that on a flat surface. The spin-orbit Hamiltonians on a cylindrical or spherical surface are explicitly derived here, and the spin precession and the associated eigenstates on a nanoring are analyzed in detail. We can conclude that the curvature has a significant influence on the spin-orbit coupling and spin transport in curved structures.

口試委員會審定書 i
謝辭 ii
摘要 iii
Abstract iv
Table of Contents v
List of Figures vi
Chapter 1 Introduction 1
Chapter 2 Hamiltonian of Spin-Orbit Coupling for Electron System on a Curved Surface 6
Chapter 3 Hamiltonian Formalism on the Nanotube and the Nanobubble 16
Chapter 4 Hamiltonian of Spin-Orbit Coupling for Hole System on a Curved Surface 30
Chapter 5 Hamiltonian Formalism on the Nanoring 36
Chapter 6 Summary and Discussion 50
Appendix A Asymptotic Behavior for the Confined Potential of an Ultrathin Film 54
Appendix B Definitions of Basic Mathematics on a Curved Surface 56
Appendix C Tensor Transformation of a Hamiltonian without SOC 58
Appendix D Tensor Transformation of Linear Rashba Spin-Orbit Coupling 60
Appendix E Tensor Transformation of Cubic Dresselhaus Spin-Orbit Coupling 62
Appendix F Tensor Transformation in a Cylindrical Nanotubular System 65
Appendix G Tensor Transformation in a Spherical Nanobubble System 68
Appendix H Tensor Transformation of Cubic Rashba Spin-Orbit Coupling 71
Appendix I Tensor Transformation in a Nanoring System 74
Bibliography 78

[1] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001).
[2] I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).
[3] P. Rabl, S. J. Kolkowitz, F. H. L. Koppens, J. G. E. Harris, P. Zoller, and M. D. Lukin, Nature Phys. 6, 602 (2010).
[4] G. C. Han, J. J. Qiu, L. Wang, W. K. Yeo, and C. C. Wang, IEEE Trans. Mag. 46, 709 (2010).
[5] M. G. Zeng, L. Shen, Y. Q. Cai, Z. D. Sha, and Y. P. Feng, Appl. Phys. Lett. 96, 042104 (2010).
[6] A. Saffarzadeh, and R. Farghadan, Appl. Phys. Lett. 98, 023106 (2011).
[7] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997).
[8] S. Sahoo, T. Kontos, J. Furer, C. Hoffmann, M. Graber, A. Cottet, and C. Schonenberger, Nature Phys. 1, 99 (2005).
[9] L. Meier, G. Salis, I. Shorubalko, E. Gini, S. Schon, and K. Ensslin, Nature Phys. 3, 650 (2007).
[10] D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B 74, 155426 (2006).
[11] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Nature 452, 448 (2008).
[12] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
[13] N. Levy, S. A. Burke, K. L. Meaker, M. Panlasigui, A. Zettl, F. Guinea, A. H. Castro Neto, and M. F. Crommie, Science 329, 544 (2010).
[14] F. de Juan, A. Cortijo, M. A. H. Vozmediano, and A. Cano, Nature Phys. 7, 810 (2011).
[15] T. C. Cheng, J. Y. Chen, and Ching-Ray Chang, Phys. Rev. B 84, 214423 (2011).
[16] J. S. Jeong, J. Shin, and H. W. Lee, Phys. Rev. B 84, 195457 (2011).
[17] J. S. Jeong, and H. W. Lee, Phys. Rev. B 80, 075409 (2009).
[18] E. Zhang, S. Zhang, and Q. Wang, Phys. Rev. B 75, 085308 (2007).
[19] M. H. Liu, J. S. Wu, S. H. Chen, and Ching-Ray Chang, Phys. Rev. B 84, 085307 (2011).
[20] A. Chuvilin, U. Kaiser, E. Bichoutskaia, NA Besley, and AN Khlobystov, Nature Chem. 2, 450 (2010).
[21] S. H. Chen, and Ching-Ray Chang, Phys. Rev. B 77, 045324 (2008).
[22] J. S. Yang, X. G. He, S. H. Chen, and Ching-Ray Chang, Phys. Rev. B 78, 085312 (2008).
[23] R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
[24] A. Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica (CRC Press, Boca Raton, FL, 1997).
[25] EI Rashba, Sov. Phys. Solid State 2, 1109 (1960).
[26] YA Bychkov, and EI Rashba, J. Phys. C 17, 6039 (1984).
[27] G. Dresselhaus, Phys. Rev. 100, 580 (1955).
[28] MI D’yakonov, and VI Perel’, Sov. Phys. JETP 33, 1053 (1971).
[29] V. Ya Prinz, VA Seleznev, AK Gutakovsky, AV Chehovskiy, VV Preobrazhenskii, MA Putyato, and TA Gavrilova, Physica E 6, 828 (2000).
[30] N. Shaji, H. Qin, R. H. Blick, L. J. Klein, C. Deneke, and O. G. Schmidt, Appl. Phys. Lett. 90, 042101 (2007).
[31] K.-J. Friedland, R. Hey, H. Kostial, A. Riedel, and K. H. Ploog, Phys. Rev. B 75, 045347 (2007).
[32] T. Georgiou, L. Britnell, P. Blake, R. V. Gorbachev, A. Gholinia, A. K. Geim, C. Casiraghi, and K. S. Novoselov, Appl. Phys. Lett. 99, 093103 (2011).
[33] C. L. Chen, S. H. Chen, M. H. Liu, and Ching-Ray Chang, J. Appl. Phys. 108, 033715 (2010).
[34] M. Trushin, and J. Schliemann, New J. Phys. 9, 346 (2007).
[35] R. Winkler, H. Noh, E. Tutuc, and M. Shayegan, Phys. Rev. B 65, 155303 (2002).
[36] D. V. Bulaev and D. Loss, Phys. Rev. Lett. 95, 076805 (2005).
[37] C. Brune, A. Roth, E. G. Novik, M. Konig, H. Buhmann, E. M. Hankiewicz, W. Hanke, J. Sinova, and L. W. Molenkamp, Nature Phys. 6, 448 (2010).
[38] E. I. Rashba, Phys. Rev. B 68, 241315(R) (2003).
[39] Q.-F. Sun, X. C. Xie, and J. Wang, Phys. Rev. Lett. 98, 196801 (2007).
[40] J. Splettstoesser, M. Governale, and U. Zulicke, Phys. Rev. B 68, 165341 (2003).
[41] F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, Phys. Rev. B 66, 033107 (2002).
[42] D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310 (2004).
[43] C. Ortix and J. van den Brink, Phys. Rev. B 81, 165419 (2010).
[44] C. Ertler, S. Konschuh, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 041405(R) (2009).
[45] A. H. Castro Neto, and F. Guinea, Phys. Rev. Lett. 103, 026804 (2009).
[46] D. R. Herschbach, J. Avery, and O. Goscinski, Dimensional scaling in chemical physics (Kluwer, Dordrecht, 1993).