本論文的主旨為探討二端及三端導線Aharonov-Cashser (AC) 量子環在不同自旋軌道交互作用(SOI)強度下的自旋傳輸特性。研究方法利用一維量子波導理論以及轉移矩陣來完成反射及穿透分析。在適當環半徑以及電場的選擇之下，二端環型結構在特定的導線夾角之下可以做為一系列的量子閘，包含相位偏移閘、泡立 閘、泡立 閘、泡立 閘以及阿達瑪閘。在創新的部分，本研究中會提出這些閘的完整解析解，並且構成量子閘所使用的環數比起以往的研究將會更為優化。三端AC環可以將輸入端的非極化電子，於輸出的兩端轉換為不同方向完全極化的自旋電子狀態，為本文研究的重點之一，同時也探討了三端AC環導線夾角對於自旋極化和無反射的影響。除此之外，在二、三端導線AC環的搭配之下，可以用於實現量子漫步，提供一種固態元件的實現方案。

The main purpose in this thesis is to investigate spin transport characteristics of two- and three-terminal Aharonov-Cashser (AC) quantum rings under different spin-orbit interaction (SOI) strength. Reflection and transmission analysis completed by using one-dimensional quantum waveguide theory and transfer matrix. Via selecting ap-propriate radius of ring and electric field, the two-terminal ring structure can be used as a series of quantum gates on specific angle of the leads, including phase-shift gate, Pau-li- gate, Pauli - gate, Pauli- gate, and Hadamard gate. In the innovative part, the complete analytical solution of these gates is proposed in this study, and amounts of rings applied in quantum gates are more optimized than previous studies. Three-terminal AC rings can make unpolarized incoming electrons become to fully polarized spin states with different directions at two output leads, which is the main key of this paper. At the same time, the influence of the three-terminal AC rings with different angle of leads on spin polarization and reflectionless is also discussed. In addition, the device can be used to realize quantum walk in the combination with the two- and three-terminal AC rings, providing a scheme of solid-state device.

摘要 i
Abstract ii
目錄 iii
圖目錄 v
表目錄 viii
符號表 ix
第一章 導論 1
1.1 背景與研究動機 1
1.2 歷史文獻回顧 3
1.3 論文架構 6
第二章 量子環理論與模型 7
2.1 AC相位與自旋軌道交互作用 7
2.2 Rashba SOI二端量子環模型 12
2.3 Rashba SOI三端量子環模型 17
2.4 二端AC環傳輸分析 20
2.5 三端AC環傳輸分析 22
第三章 二端AC量子環自旋極化與應用 27
3.1 二端環導線夾角為任意角之分析 27
3.2 二端環量子閘之實現 32
第四章 三端AC量子環之自旋極化與應用 40
4.1 三端導線夾角皆為2*pi/3之極化分析 40
4.2 三端環導線夾角分別為pi/2、pi、pi/2之極化分析 49
4.3 三端環量子漫步之實現 57
第五章 結論與未來展望 61
5.1 結論 61
5.2 未來展望 62
參考文獻 63

[1]H. Akinaga and H. Ohno, “Semiconductor Spintronics,” IEEE Trans. Nanotechnol. 1, 19 (2002).
[2]Žutić, J. Fabian, and S. D. Sarma, “Spintronics: Fundamentals and applications, ” Rev. Mod. Phys. 76, 323 (2004).
[3]S. A. Wolf and D. D. Awschalom, “Spintronics: A spin-based electronics vision for the future,” Science. 294, 1488 (2001).
[4]Leslie L. Foldy and Siegfried A. Wouthuysen, “Mesoscopic Stern-Gerlach spin filter by nonuniform spin-orbit interaction,” Phys. Rev. 78, 29 (1950).
[5]J. Ohe and M. Yamamoto, “On the Dirac theory of spin 1/2 particles and its non-relativistic limit,” Phys. Rev. B. 72, 041308 (2005).
[6]M. Oestreich, J. Hübner, and D. Hägele, “Spin injection into semiconductors,” Appl. Phys. Lett. 74, 1251 (1999).
[7] S. P. Dash, S. Sharma, and R. S. Patel, “Electrical creation of spin polarization in silicon at room temperature,” Nature. 462, 491 (2009).
[8] S. Sahoo, T. Kontos, and J. Furer, “Electric field control of spin transport,” Nature. Physics. 1, 99 (2005).
[9] X. Lou, and C. Adelmann “Electrical detection of spin transport in lateral ferromagnet–semiconductor devices,” Nature. Physics. 3, 197 (2007).
[10]Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in quantum theory,” Phys. Rev. 115, 485 (1959).
[11] Y. Aharonov and A. Casher, “Topological quantum effects for neutral particles,” Phys. Rev. Lett. 53, 319 (1984).
[12] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki,“Gate control of spin-orbit interaction in an inverted heterostructure,” Phys. Rev. Lett. 78, 1335 (1997).
[13] D. D. Awschalom, M. E. Flatté, and N. Samarth, “Spintronics-microelectronic devices that compute with the spin of the electron may lead to quantum microchips,” Sci. Am. 286,66 (2002).
[14]M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information,Cambridge University Press, Cambridge (2000).
[15] R. Ionicioiu and I. D’Amico, “Mesoscopic Stern-Gerlach device to polarize spin currents,” Phys. Rev. B 67, 041307 (2003).
[16] P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26, 1484 (1997).
[17] D. Kielpinski, C. Monroe, and D. J. Wineland, “Architecture for a large-scale ion-trap quantum computer,” Nature. 417, 709 (2002).
[18] M. Martinis, S. Nam, J. Aumentado, and C. Urbina, “Rabi oscillations in a large josephson-junction qubit,” Phys. Rev. Lett. 89, 117901 (2002).
[19] M. K. Vandersypen, M. Steffen, and G. Breyta, “Experimental realization of Shor''s quantum factoring algorithm using nuclear magnetic resonance,” Nature. 414, 883 (2001).
[20] P. Földi, B. Molnár, M. G. Benedict, and F. M. Peeters, “Spintronic single-qubit gate based on a quantum ring with spin-orbit interaction,” Phys. Rev. B 71, 033309 (2005).
[21] S.J. Gong and Z.Q. Yang, “Flying spin-qubit gates implemented through Dresselhaus and Rashba spin–orbit couplings,” Phys. Lett. A 367, 369 (2007).
[22] P. Földi, O. Kálmán, M.G. Benedict, and F.M. Peeters, “Quantum rings as electron spin beam splitters,” Phys. Rev. B 73 155325 (2006).
[23] O. Kálmán, T. Kiss, and P. Földi, “Quantum walk on the line with quantum rings,” Phys. Rev. B 80 035327 (2009).
[24] J. Nitta, F. E. Meijer, and H. Takayanagi, “Spin-interference device,” Appl. Phys. Lett. 75, 695 (1999).
[25]Y. Sato, T. Kita, S. Gozu, and S. Yamada, “Large spontaneous spin splitting in gate-controlled two-dimensional electron gases at normal heterojunctions,” J. Appl. Phys. 89, 8017 (2001).
[26] A. A. Kiselev and K. W. Kim, “T-shaped ballistic spin filter,” Appl. Phys. Lett. 78, 755 (2001).
[27] D. Frustaglia, M. Hentschel, and K. Richter, “Quantum transport in nonuniform magnetic fields: Aharonov-Bohm ring as a spin switch,” Phys. Rev. Lett. 87, 256602 (2001).
[28] M. Governale, D. Boese, U. Zu¨licke, and C. Schroll, “Filtering spin with tunnel-coupled electron wave guides,” Phys. Rev. B 65, 1440403 (2002).
[29]J. Levy, “Universal quantum computation with spin-1/2 pairs and heisenberg ex-change,” Phys. Rev. Lett. 89, 147902 (2002).
[30] R. Ionicioiu and I. D’Amico, “Mesoscopic Stern-Gerlach device to polarize spin currents,” Phys. Rev. B 67, 041307 (2003).
[31] B. Molnár, F. M. Peeters, and P. Vasilopoulos, “Spin-dependent magnetotransport through a ring due to spin-orbit interaction,” Phys. Rev. B 69,155335 (2004).
[32] D. Frustaglia and K. Richter, “Spin interference effects in ring conductors subject to Rashba coupling,” Phys. Rev. B 69, 235310 (2004).
[33] P. Földi, B. Molnár, M. G. Benedict, and F. M. Peeters, “Spintronic single-qubit gate based on a quantum ring with spin-orbit interaction,” Phys. Rev. B 71, 033309 (2005).
[34] O. Kálmán, P. Földi, and M.G. Benedicta, “Spatial interference induced spin polarization in a three-terminal quantum ring,” Physica E, 40, 567 (2008).
[35] O. Kalman, P. Foldi, M. G. Benedict, and F. M. Peeters, “Magnetoconductance of rectangular arrays of quantum rings,” Phys. Rev. B 78, 125306 (2008).
[36] P. Foldi, O. Kalman, F.M. Peeters, “Stability of spintronic devices based on quantum ring networks,” Phys. Rev. B 80, 125324(2009).
[37] L. G. Wang, Kai Chang, and K. S. Chan, “Charge and spin currents in a three-terminal mesoscopic ring,” J. Appl. Phys. 105, 013710 (2009).
[38] V. Moldoveanu and B. Tanatar, “Spin splitter regime of a mesoscopic Rashba ring,” Phys. Lett. A 375, 187 (2010).
[39] F. A. Chegeni and E. Faizabadi, “Quantum conductance of three-terminal Nanoring in the Presence of Rashba Interaction and an Impurity,” Int. J. Appl. Phys. Math. 1, 155 (2011).
[40] F. Fallah and M. Esmaeilzadeh, “Spin transport in a quantum ring in the presence of Rashba spin–orbit interaction using the S-matrix method,” J. Appl. Phys. 111, 043717 (2012).
[41] D. C Leo and F. Mireles, “Quantum-ring spin interference device tuned by quantum point contacts,” J. Appl. Phys. 114, 193706 (2013).
[42] A. S. Naeimi, L. Eslami, M. Esmaeilzadeh, and M. R. Abolhassani, “Spin transport properties in a double quantum ring with Rashba spin-orbit interaction,” J. Appl. Phys. 113, 014303 (2013).
[43] L. X. Zhai, Y. Wang, and J. J. Liu, “Zero-conductance resonances and spin polarizations in three-terminal rings in the presence of spin-orbit coupling,” J. Appl. Phys. 116, 203703 (2014).
[44] Y. Wang, L. Z. Duan, and L. X. Zhai, “Spin conductances and spin polarization components in a three-terminal ring subject to Rashba coupling,” AIP. Advance 5, 077169 (2015).
[45] E. Faizabadi, M. Molavi, “Radius effect on the spintronic properties of a triangular network of quantum nanorings in the presence of Rashba spin-orbit interaction, ” Curr. Appl. Phys. 17, 207 (2017).
[46] Li. X. Zhai, Y. Wang, and Z. An, “Effects of Zeeman splitting on spin transportation in a three-terminal Rashba ring under a weak magnetic field,” AIP. Advance 8, 055120 (2018).
[47] F. E. Meijer, A. F. Morpurgo, and T. M. Klapwijk, “One-dimensional ring in the presence of Rashba spin-orbit interaction: Derivation of the correct Hamiltonian,” Phys. Rev. B 66, 033107 (2002).
[48] T. Ihn, Semiconductor Nanostructures, Oxford, Zurich (2010).
[49] A. Manchon, H. C. Koo, and J. Nitta, “New perspectives for Rashba spin–orbit coupling,” Nat. Mater. 14, 871 (2015).
[50] J. Kempe ,“Quantum random walks - an introductory overview” Contemp. Phys. 44, 307 (2003).