考慮二維空間與絕對零度的環境下，利用線性自旋軌道耦合之半導體來研究守恆自旋流。基於自旋霍爾效應，我們證明自旋流的連續方程式；並且使用Kubo方程式計算自旋流的電導率。線性自旋軌道耦合的Hamiltonian中，自旋軌道矩陣β是代表自旋與軌道耦合之強度，同時我們發現自旋軌道矩陣β之行列式與自旋霍爾和縱向自旋電導率為正比關係。其中，自旋矩縱向電導率永遠為零。我們進一步發現，自旋霍爾與縱向自旋電導率之上限分別為|Q|/4π 與|Q|/8π；在費米能量曲面上，φ方向之能帶分裂大小會影響自旋霍爾與縱向自旋電導率。另一方面，利用nonmagnetic Stern-Gerlach實驗在Rashba-Dirac系統中，讓電子經過狹縫繞射，我們可從z方向之自旋繞射圖分佈得到自旋z方向的期望值為0.21ℏ。

In the two dimensional systems, we investigate the conserved spin current response to an electric field for the generic k-linear spin-orbit coupled semiconductors at zero temperature. Within the spin-Hall effect, we derive the spin continuity equation for the k-linear systems, and use the Kubo formula to calculate the total spin conductivities σ_ab^s. In the previous work, the spin-orbit matrix β was constructed and the determinant of β was shown to be related to the effective coupling of spin and orbital motion. We show that det(β) is also related to the total spin-Hall conductivity and the total longitudinal spin conductivity. Interestingly, the torque longitudinal spin conductivity is always zero. The total spin-Hall and longitudinal spin conductivities have upper bound values |Q|/4π and |Q|/8π, respectively. On the Fermi surface, the band-splitting along the angle φ (φ=0,π/2,π/4,3π/4) determines the values of the total spin-Hall and longitudinal spin conductivities. On the other hand, in the nonmagnetic Stern-Gerlach experiment and the Rashba-Dirac system, the spin distribution shows the value of spin for z component can be generated up to 0.21ℏ by electron diffraction.

Abstract i
Contents ii
List of Figures iii
chapter 1 Introduction 1
section 1.1 Spin-orbit interaction 1
section 1.2 History of spin current 4
section 1.3 Recent developments 6
chapter 2 Model Hamiltonian 9
chapter 3 Spin Continuity Equation 13
chapter 4 Derivation of Spin Conductivity 15
section 4.1 Kubo formula for conductivity 15
section 4.2 Conventional and torque spin conductivities 17
chapter 5 Calculation of Spin Conductivity 23
section 5.1 Conventional spin conductivity 24
section 5.2 Torque spin conductivity 26
section 5.3 Upper bound values of conductivity 32
section 5.4 Maximum value of conductivity 37
chapter 6 Electron Diffraction 40
chapter 7 Conclusion 47
Appendix 48
Appendix A 48
A.1 : The k-cubic Rashba system 48
A.2 : The k-cubic Dresselhaus system 52
Appendix B : Program 55
Bibliography 56

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