Effects of Disorder on Spin Hall Conductivityn Two Dimensional Electron Systems

本質自旋霍爾效應(intrinsic spin Hall effect)即在具有自旋-軌道耦合的材料當中，只外加一電場，會產生垂直於電場的自旋流。此效應不同於由於電子與自旋-軌道耦合的雜質散射所引起的外界自旋霍爾效應(extrinsic spin Hall effect)，然而雜質對於本質自旋霍爾效應的影響仍不清楚。為了能夠預測實驗的偵測結果，了解雜質會減弱此效應或是無影響是重要的課題。此外，因為電子的運動帶有軌道角動量，所以本質自旋霍爾效應總是伴隨著軌道角動量霍爾效應。在Rasha系統中，軌道角動量霍爾傳導率和自旋霍爾傳導率有相同大小，但差一負號，使得總角動量霍爾傳導率為零，加入雜質後，自旋霍爾傳導率會變成零，然而雜質對於軌道角動量霍爾傳導率的影響尚無研究。在本論文中研究雜質對於自旋及軌道角動量霍爾傳導率的影響，所研究的雜質是短程且與自旋無關(spin-independent)，主要的研究系統是Rashba系統和wurtzite系統，軌道角動量霍爾效應只研究了Rashba 系統。使用了兩個方法，其一是考慮階梯近似(ladder approximation)中的頂角修正(vertex correction)，其二是考慮動量弛豫時間(momentum relaxation time)。使用有效的守恆自旋流定義，可以使我們了解自旋力矩(spin torque)對本質自旋霍爾效應的貢獻。我們發現，利用考慮動量弛豫時間的方法，在有雜質的系統中，自旋力矩比慣用定義自旋流的貢獻小了一個數量級；用有效守恆自旋流定義所算出來的自旋霍爾傳導率從本質系統到加入雜質後會變號，若是用慣用自旋流定義則得不到這個結果。由這兩個方法所得到的結果，使我們預測自旋力矩的貢獻可能是來自於費米海中的電子，而不同於慣用定義的自旋流，是來自費米面的電子的貢獻。軌道角動量霍爾傳導率的頂角修正在直流電近似下有一發散項，此發散項是因為位置算符在動量空間下的定義是一個微分算符，使得期望值與本徵態的相位有關。但是若我們考慮電子只有在不同能帶間的貢獻(interband contribution)，就沒有發散項，則軌道角動量霍爾傳導率在本質系統中回到 ，在有雜質系統中則為零。這個發散項的解決方法仍待更多的研究。The novel phenomenon of intrinsic spin Hall e ect (ISHE) is that a dissipationless transverse spin-polarized current being induced only by applying an external electric field in a spin-orbit coupling material. It is unlike the extrinsic SHE, which arises from beingcattered by the spin-orbit coupling impurities. However, the interplay between disorder and ISHE is still unclear. To understand whether the presence of disorder reduces oras no effect on the intrinsic value is important in predicting experimental detections. On the other hand, since the lateral movement of electrons in spin Hall effect can be recognized as the orbital motion, ISHE always accompanies the orbital angular momentumOAM) Hall effect. The OAM Hall conductivity is the same in magnitude, but opposite in sign with the spin Hall conductivity (SHC) in the Rashba model. The suppression ofHC in the disordered Rashba model has already been studied by several authors, but the disorder effect on the OAM Hall e ect has not yet been known. In this thesis, theffects of short-ranged and spin-independent disorder on spin Hall conductivity in the Rashba, Dresselhaus and wurtzite model and on OAM Hall conductivity in the Rashbaodel and the Dresselhaus model have been studied in the linear response regime. Two methods were employed, the vertex correction within the ladder approximation and byhe momentum relaxation time approximation. The effective conserved spin current gives the insights into the spin torque contribution to the ISHC. We found that by momentum relaxation time approximation, the spin torque contribution was one order smaller than the conventional contribution in the disordered system. Moreover, the SHCalculated from the effective conserved spin Hall current changed sign from the intrinsic system to the disordered system, while the SHC calculated from the conventional spinall current did not change sign. Results from the two methods imply that the spin torque contribution to the SHC may be from the electrons in the Fermi sea, unlike theonventional contribution to the SHC, which was from the electrons at the Fermi level.he results of the vertex correction of OAM Hall conductivity had a divergent term in the d.c. limit. The divergent term was due to the gauge dependent property of the ill-defined position operator in momentum space. Nonetheless, the OAM Hall conductivities recovered e/8pi in the ballistic limit and vanished in the diordered limit when only the interband contribution was taken into account. How to solve this ambiguity is still under study.1 Introduction...1 Models of Hamiltonian and the Definition of Effective Conserved Spin Current...3.1 Models of Hamiltonian ...3.1.1 The Rashba Model ...4.1.2 The Dresselhaus Model ...6.1.3 Wurtzite Model ... 7.2 Definition of Spin Current ... 8 The Momentum Relaxation Time Approximation ...10.1 The Momentum Relaxation Time ...10.2 The Rashba Model ...11.2.1 The Conventional Spin Hall Conductivity ...11.2.2 The Spin Torque Contribution to the Spin Hall Conductivity ...11.3 Wurtzite Model ...13.3.1 The Conventional Spin Hall Conductivity ...13.3.2 The Spin Torque Contribution to the Spin Hall Conductivity ...14.4 Results and Discussion ...15 Vertex Correction in the Rashba Model and the Dresselhaus Model ...17.1 Impurity Ensemble Average ...17.1.1 Impurity Ensemble Average of the Green’s Function ...184.1.2 Impurity Ensemble Average of the Self-Energy ...22.2 The Green’s Function and the Self-Energy due to δ - potential Impurities ...23.2.1 The Rashba model ...24.2.2 The Dresselhaus Model ...25.3 The Vertex Correction of Spin Hall Conductivity ...26.4 Spin Hall Conductivity in the Rashba Model ...27.4.1 The Conventional Spin Hall Conductivity ...27.4.2 The Spin Torque Contribution to the Spin Hall Conductivity ...31.5 Spin Hall Conductivity in the Dresselhaus Model ...32.5.1 The conventional Spin Hall Conductivity ...32.5.2 The Spin Torque Contribution to the Spin Hall conductivity ...33.6 Orbital Angular Momentum Hall Conductivity in the Rashba Model andhe Dresselhaus Model ...34.6.1 The Rashba Model ...35.6.2 The Dresselhaus Model ...36.7 Results and Discussion ...38 Discussion and Conclusions ...40.1 Comparisons between Two Methods ...40.2 The Effect of Disorder on SHE and OAM Hall Effect ...41.3 Future Work ...42 The Current Vertex Correction in the Rashba Model ...43 The Current Vertex Correction in the Dresselhaus Model ...45ibliography ...4