臺灣博碩士論文加值系統

即使在沒有外加磁場之下，因為自旋-軌道耦合效應(spin-orbit coupling)使得半導體塊材中能產生垂直於電流方向的自旋流(spin-current)。此種效應稱為自旋霍爾效應（spin Hall effect）。我們知道電子除了帶有電荷外還帶有自旋。有別於傳統的電子電路學，自旋電子學（spintronics）利用電子的自旋做應用，如儲存資訊。藉由產生沒有熱消秏的自旋電流，自旋霍爾效應在自旋電子學扮演很重要的角色。直覺上，自旋流可定義為自旋的通量。但在有自旋-軌道耦合效應的系統中，這樣定義的自旋流不遵守通量的連續方程式(continuity equation)。我們利用有效的自旋流定義導出了在具有Rashba 自旋-軌道耦合效應的二維電子系統。同樣地，我們也重新定義了軌道角動量流(orbital angular momentum current)。在具Rashba 自旋-軌道耦合效應的二維電子系統中，我們也得到了一個不為零的軌道角動量霍爾流，此現象稱為軌道角動量霍爾效應。
除了Rashba 系統在，我們還多考慮了Dresselhaus 自旋-軌道耦合效應。Dresselhaus 這一項來自於塊材本身的反向不對稱性(bulk inversion symmetry)。在有些半導體量子井(quantum well)中，這兩項的大小是差不多的，如砷化鎵(GaAs)。假設外加電場很小，我們考慮線性項，發現軌道角動量流的通率(OAM Hall conductivity)只與兩項自旋-軌道耦合效應的耦合強度的比值有關
係而且與載子的濃沒有關係。在有Dresselhaus 項的系統中，系統的總角動量是不守恆的，所以我們得到的角動量霍爾流與自旋流不會恆抵消。於是就產生一個角動量霍爾流。進一步我們發現在兩者自旋-軌道耦合效應相當的時候，軌道角動量霍爾流是發散的。

The spin Hall effect means that an electric field can induce a transverse spin current even in the absence of magnetic field. By generating the dissipationless spin current, the spin Hall effect has important applications in
spintronics. The effective definition of spin current in a spin-orbit system is introduced while the conventional spin current is intuitively thought of as the spin flux. In the linear response limit, we rederive the spin Hall conductivity in a two dimension electron system(2DES) with Rashba spin-orbit system.Similarly, we also define an effective orbital angular momentum(OAM) cur-rent in a spin-orbit system by the same way and discover it existed non-zero OAM Hall current in a pure Rashba system.

Besides Rashba system, we also consider Dresselhuas spin-orbit coupling into the 2DES. The Dresselhaus term stemmed form the bulk inversion sym-metry and the term is comparable with the Rashba term(e.g. GaAs). We predict that the OAM Hall conductivity is relative to the ratio of the two couplings and independent on the carrier density. Due to the non-conservation of total angular momentum in the presence of Dresselhaus term, the spin Hall current doesn’t cancel with the OAM Hall current. It causes a finite magnetization Hall current(or angular momentum current ). When the two couplings are comparable, the OAM Hall conductivity is divergent.

Contents
1 Introduction 5
2 HamiltonianModels and effective current operators in a spin-orbit system 9
2.1 Spin-Orbit Effect in Semiconductors . . . . . . . . . . . . . . . 9
2.1.1 Luttinger Model . . . . . . . . . . . . . . . . . . . . . . 10
2.1.2 Rashba Model and Dresselhaus Model . . . . . . . . 14
2.2 Continuity Equation in Spin-Orbit System . . . . . . . . . . . 16
2.3 Effective Spin Current Operator . . . . . . . . . . . . . . . . . 19
3 Spin and OAM Hall effect 22
3.1 Brief Review of Hall effect . . . . . . . . . . . . . . . . . . . . 22
3.2 Spin Hall Effect in Rashba System . . . . . . . . . . . . . . . 24
3.3 Spin and OAM Hall Effect in 2DES with Rashba and Dresselhaus
couplings . . . . . . . . . . . . . . . . . . . . . . . 28
4 Discussion and Conclusions 32
A Appendix 34
A.1 Derivation of the Spin Hall Conductivity . . . . . . . . . . . . 34
A.2 Derivation of OAM Hall Conductivity . . . . . . . . . . . . . . 37

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