Theoretical study of spin currents in semiconductors

Theoretical Study of Spin Currents in Semiconductors by Takashi Fujita B.C.M./B.Eng.(Hons.), University of Western Australia A Thesis Submitted for the Degree of Doctor of Philosophy Department of Electrical and Computer Engineering National University of Singapore 2010 Acknowledgements I am indebted to my supervisor Prof Mansoor Jalil for his guidance and encouragement throughout my scholarship. I feel extremely fortunate to have worked under such a passionate and understanding research leader. I am also grateful to Dr Tan Seng Ghee for his support, patience and invaluable advice during our countless discussions. Thanks must also go to my fellow colleagues and friends at DSI including Nyuk Leong, Minjie, Bala, Gabriel, Joel, Saidur and others, as well as my friends Allan, Michael, David, Magius, Guizel and others who have made Singapore feel like home away from home. Last, but not least, I am grateful to my family and friends back in Australia from whom I have received overwhelming support. i Contents Acknowledgements i Summary vi List of Tables viii List of Figures ix Publications, Conferences and Awards xvi List of Abbreviations xix Introduction 1.1 Background and Motivation . . . . . . . . . . . . . . . 1.1.1 Spin-Orbit Coupling . . . . . . . . . . . . . . . 1.1.2 Generating Spin Currents and Polarization . . 1.1.3 Spin Manipulation and Precession . . . . . . . 1.1.4 Spin Transport and Spin-Dependent Transport 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Organization of Thesis . . . . . . . . . . . . . . . . . . 1.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 5 Review of Relevant Topics 2.1 Spin-Orbit Coupling . . . . . . . . . . . . . . . 2.1.1 Dresselhaus Effect . . . . . . . . . . . . 2.1.2 Rashba Effect . . . . . . . . . . . . . . . 2.1.3 Spin Dynamics in the Presence of SOC 2.2 Spintronic Devices . . . . . . . . . . . . . . . . 2.2.1 Spin Field-Effect Transistor . . . . . . . 2.2.2 Spin Filters . . . . . . . . . . . . . . . . 2.2.3 Subband Filters in SOC Systems . . . . 2.3 Spin-Dependent Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10 10 11 13 13 15 16 22 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii CONTENTS 2.4 2.3.1 Spin-Orbit Gauge Field . . . 2.3.2 Berry Gauge Field . . . . . . Spin-Hall Effect . . . . . . . . . . . . 2.4.1 Systems and Mechanisms . . 2.4.2 Effects of Impurity Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 23 25 25 28 Spin Polarization in Semiconductors 30 3.1 Spin Polarization of Tunneling Electrons Through SOC Barriers Induced via Asymmetries in Momentum Space . . . . . . . . . . . . . . . . . . . 31 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.1.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 35 3.1.3.1 k -Dresselhaus SOC Barriers . . . . . . . . . . . . . . . 35 3.1.3.2 RTD Barrier Structures with Combined Rashba and Dresselhaus SOC . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2 Spin Polarization Induced by a Magnetic Field and Harmonic Oscillator Potential Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.2.1 Model, Hamiltonian and Eigenstates . . . . . . . . . . . 52 3.2.2.2 Spin-Dependent Transport . . . . . . . . . . . . . . . . 56 3.2.3 Results and Discussions of Spin Polarization . . . . . . . . . . . . 57 3.2.3.1 Effect of Magnetic Field Strength . . . . . . . . . . . . 57 3.2.3.2 Dependence on Landau Level Index . . . . . . . . . . . 62 3.2.3.3 Effect of Structure Geometry . . . . . . . . . . . . . . . 63 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 Spin Polarization of Landau Levels in the Presence of Rashba SOC . . . 68 3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3.2.1 Hamiltonian and Eigenstates without Spin . . . . . . . 70 3.3.2.2 Hamiltonian and Eigenstates with Spin . . . . . . . . . 73 3.3.2.3 Gauge Invariance of Eigenstate Solutions . . . . . . . . 77 3.3.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . 80 3.3.3.1 Spin Polarization of Landau Levels . . . . . . . . . . . . 80 3.3.3.2 Proposal for Experimental Measurement . . . . . . . . 85 3.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Multichannel Spintronic Transistor 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Theory . . . . . . . . . . . . . . . . . . . . . 4.2.1 Model, Hamiltonian and Eigenstates 4.2.2 Calculation of Transport Parameters 4.2.3 Verification of Flux Conservation . . 4.3 Results and Discussions . . . . . . . . . . . 4.3.1 Transistor Action of Device . . . . . 4.3.2 Multichannel Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 89 92 92 95 96 97 98 99 iii CONTENTS 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Spin Separation Arising from Gauge Fields in Two-Dimensional Spintronic Systems 103 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.2.1 Model, Hamiltonian and Assumptions . . . . . . . . . . . . . . . 105 5.2.2 Spin-Dependent Gauge Fields . . . . . . . . . . . . . . . . . . . . 106 5.2.2.1 Non-Abelian Spin-Orbit Gauge Field . . . . . . . . . . 107 5.2.2.2 Berry Gauge Field due to Nonuniform Magnetic Fields 108 5.2.2.3 Combined Scenario . . . . . . . . . . . . . . . . . . . . 110 5.2.3 Spin-Dependent Force Operators . . . . . . . . . . . . . . . . . . 111 5.2.4 Equations of Motion Describing Spin Separation . . . . . . . . . 114 5.3 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . 115 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 Intrinsic Spin-Hall Effect of Collimated Electrons in Zincblende conductors 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Model and Hamiltonian . . . . . . . . . . . . . . . . . . . . 6.2.2 Electron Collimator Source . . . . . . . . . . . . . . . . . . 6.2.3 Appearance of Berry Gauge Field in Momentum Space . . . 6.2.4 Equations of Motion Describing Spin Separation . . . . . . 6.2.5 Spin-Hall Conductivity . . . . . . . . . . . . . . . . . . . . 6.2.6 Quantum Adiabaticity Criterion . . . . . . . . . . . . . . . 6.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Spin-Hall Conductivity in GaAs . . . . . . . . . . . . . . . 6.3.2 Effects of Impurity Scattering . . . . . . . . . . . . . . . . . 6.3.3 Proposal for Experimental Detection . . . . . . . . . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semi117 . . . 117 . . . 118 . . . 118 . . . 119 . . . 121 . . . 122 . . . 126 . . . 127 . . . 128 . . . 128 . . . 129 . . . 133 . . . 133 Unified Description of Intrinsic Spin-Hall Effect Mechanisms 134 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.2.1 SHE in the Presence of Berry Curvature in Momentum Space of SOC systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.2.1.1 Luttinger System . . . . . . . . . . . . . . . . . . . . . 136 7.2.1.2 Rashba SOC System . . . . . . . . . . . . . . . . . . . . 138 7.2.2 Time Component of Gauge Field in SOC Systems . . . . . . . . 139 7.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.1 The SHE in Rashba SOC Systems as a Time-Space Gauge Field Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.3.1.1 Adiabaticity and Transverse Spin Separation . . . . . . 143 7.3.1.2 Berry Phase . . . . . . . . . . . . . . . . . . . . . . . . 145 7.3.1.3 Effects of Impurities . . . . . . . . . . . . . . . . . . . . 145 7.3.2 Unification of SHE Mechanisms . . . . . . . . . . . . . . . . . . . 146 iv CONTENTS 7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Intrinsic Spin-Hall Effects due to Time Component of Gauge Field in Spintronic, Optical, and Graphene Systems 149 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 8.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.2.1 Calculation of Spin-Hall Current and Conductivity . . . . . . . . 150 8.3 Results and Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 8.3.1 Combined Rashba and Dresselhaus SOC . . . . . . . . . . . . . . 153 8.3.2 n-doped Bulk Semiconductors . . . . . . . . . . . . . . . . . . . . 154 8.3.3 Holes in III-V Semiconductor Quantum Wells with Rashba SOC 154 8.3.4 Bilayer Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 8.3.5 Rayleigh Scattering of Polaritons . . . . . . . . . . . . . . . . . . 157 8.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Conclusions and Recommendations 9.1 Spin Current and Polarization Generation . . 9.2 Spintronic Transistor Devices . . . . . . . . . 9.3 Intrinsic Spin-Hall Effect . . . . . . . . . . . . 9.4 Recommendations for Future Work . . . . . . 9.4.1 Nonuniform SOC . . . . . . . . . . . . 9.4.2 Edge States in Magnetic Systems . . . 9.4.3 Edge States in Nonmagnetic Systems . 9.4.4 Formal Calculations of Spin Current . 9.4.5 Competing Intrinsic SHE Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 159 159 160 161 161 162 163 163 163 A Gauge Transformations and Invariance 164 A.1 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 164 A.2 Spin-Dependent Gauge Fields . . . . . . . . . . . . . . . . . . . . . . . . 167 B Equations of Motion Arising From Gauge Fields in §5.2.4 169 C Classical Derivation of B⊥ in §7.2.2 172 Bibliography 174 v Summary Spintronics in semiconductors (SCs) offers a promising avenue for future information technologies. At the very heart of this technology is the widely known spin-orbit coupling (SOC) effect, which affords us the attractive prospect of spintronics without magnetism. The value of SOC is quickly realized through its ubiquity in nearly all aspects of SC spintronics, from the generation of spin polarized currents to the all-electric spin manipulation it permits in SC spintronic devices. It also drives the remarkable spin-Hall effect (SHE) which is a promising source of dissipationless spin currents. In this Thesis, we theoretically study several critical aspects of SC spintronics, with a focus on spin currents in the presence of SOC. These aspects include spin current generation, spin manipulation, and spin-dependent transport. Firstly, methods to generate spin currents in SCs are proposed. These range from purely nonmagnetic, SOC-based systems to those which utilize external magnetic fields. Generally, nonmagnetic approaches are preferred as stray magnetic fields can adversely affect spins. Highly spin polarized currents (approaching 100% polarization) are predicted under certain conditions in both nonmagnetic and magnetic approaches. Next, two spintronic transistor devices are proposed, which exploit the electronic tunability of the SOC in SC heterostructures. The first modifies the seminal Datta-Das device by including the effect of external magnetic fields. This is found to considerably relax transport constraints (namely single channeled transport) in the original model. vi SUMMARY The second device exhibits a gate bias modulation of spin current through the action of two spin-dependent gauge fields. Generally, such fields can be physically interpreted as effective magnetic fields, which affect the trajectory of carriers in a spin-dependent manner. These inevitably drive spin currents and are therefore of great importance to spintronics research. An in-depth study of gauge fields constitutes the second-half of this Thesis. In particular, we closely examine the intrinsic spin-Hall effect (SHE), in which dissipationless spin currents flow (these transport zero net charge) normal to an applied charge current in generic SOC systems. First, we propose a SHE of collimated conduction electrons in zincblende crystals. Important issues including calculation of the spin current and its robustness to impurities are discussed. Next, motivated by open questions, we divert our attention to the physical mechanisms which drive the SHE. Two mechanisms are known, but their relationship (if any) has hitherto been unclarified. One mechanism arises from the spin-dependent trajectory of carriers due to gauge fields in momentum space. The second results from a momentum-dependent polarization of spins. We succeed in formulating a gauge field description (in time space) of the latter mechanism. Moreover, we show that the two mechanisms are simply distinct manifestations of a common time-resolved process in SOC systems. Lastly, we discuss the ubiquity of the latter mechanism in SC spintronic and optical systems, and propose an analogous flow of pseudospin current in bilayer graphene. vii List of Tables 8.1 List of systems in which the intrinsic spin-Hall effect is analyzed. H is the Hamiltonian, D is the system dimension, α, β, η and λ are the respective SOC strengths, σ l (l = x, y, z) are the Pauli spin matrices, kl are the wavevectors, σ± = σ x ± iσ y and k± = kx ± iky . B(k) is the momentumdependent effective magnetic field, and sz (k) is the zˆ-spin polarization of carriers resulting from an electric field applied in the x ˆ-direction, obtained from Eq. (8.3). The SHE arises because the spin polarizations sz (k) are odd functions of the transverse wavevector, ky . For the case of bilayer graphene, τ z represents the pseudospin polarization, describing the probability of finding an electron on either of the two monolayers. . 153 viii List of Figures 1.1 2.1 2.2 2.3 Branch diagram of spin-related phenomena relevant to this Thesis, and how they relate to the spin-orbit coupling effect. Topics are categorized under three main sections, which represent the major blocks of work in this Thesis. Dashed lines denote dependences across sections. . . . . . . Illustration of SFET device proposed by Datta and Das. . . . . . . . . . (left) Layout of a generic magnetic barrier system, which entails deposition of a stripe above a 2DEG. (right) Various stripe configurations and their resulting magnetic field distributions, assuming h/d and h/z 1. (a) Ferromagnetic stripe with perpendicular magnetic anisotropy and (b) in-plane magnetic anisotropy. (c) Conducting stripe through which a current flows (into the page), and (d) Superconducting (S) plate interrupted by a stripe (N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chart of various spin filtering devices utilizing magnetic field barriers (green). (a) Lithographic patterning of FM materials on 2DEGs, having in-plane magnetization gives rise to spatially confined fringing fields (see Fig. 2.2 (a)), which can be approximated as magnetic delta barriers. This structure, however, does not possess spin filtering properties. (b) A symmetric configuration of delta barriers exhibits finite spin polarization. (c) Periodic array of symmetric barriers. (d) Periodic array of asymmetric barriers; a finite polarization can be attained only when the number of barriers is odd. (e) An asymmetric barrier can induce a finite polarization, when the magnetic fields are modeled as rectangles with different widths. (f) Spin filtering under the influence of magnetic barriers and Rashba and Dresselhaus SOC leads, in general, to spin polarization which is tunable via a gate bias. (g) Lithographic patterning of FM materials on 2DEGs with perpendicular magnetization gives rise to Mexican-hat type fields which can be modeled as rectangular. (h) A more accurate model of those barriers. . . . . . . . . . . . . . . . . . . . 14 17 18 ix EQUATIONS OF MOTION ARISING FROM GAUGE FIELDS IN §5.2.4 We now calculate the expectation value of the transverse displacement of a Gaussian wavepacket in the 2DEG prepared in the spin-up state. This has representation r|Ψ = r|ψ d √ 2π π /2d2 d2 ke−(k−k0 )     eik·r   , (B.8) where d is the width of the wavepacket in k-space, centered about k0 . When computing the expectation value of the transverse position operator for the (1, 0)T spin state, only the (11) matrix components of y(t) in Eq. (5.35) have a finite contribution. This eliminates the σ x and σ y terms of y(t), Ψ|y(t)|Ψ = ψ y0 + py t et2 2α2 em2 + p x Bz − + 2 m 2m 2eR ψ . (B.9) For the operators kj , the expectation value for a Gaussian wavepacket are ψ|kj |ψ = d2 4π d2 r /2(q−k d2 qe−d 0) e−iq·r /2(k−k d2 kkj e−d 0) = kj0 , eik·r , (B.10) which follows from the fact that for any Gaussian wavepacket centered about k = k0 , we have ψ|p|ψ = k0 . Thus, for the spin-up wavepacket in Eq. (B.8), the transverse displacement is Ψ|y(t)|Ψ = y0 + ky0 t e kx0 t2 + m 2m2 Bz − 2eR2 + 2α2 em2 . (B.11) . (B.12) Similarly, for a spin-down wavepacket, the corresponding displacement is Ψ|y(t)|Ψ = y0 + ky0 t e kx0 t2 + m 2m2 Bz + 2eR2 − 2α2 em2 171 APPENDIX C Classical Derivation of B⊥ in §7.2.2 We consider the dynamics of the spin vector s(t) in a time-dependent magnetic field B(t), ˙ = g s(t) × B(t) , s(t) where g is the coupling factor. (C.1) To solve the above equation, we freeze the time- dependence by transforming to a rotated coordinate frame at each point in time, such that the reference zˆ-axis is aligned with the instantaneous magnetic field. A spin vector s defined relative to the coordinate frame at time t, is expressed as the vector s = s + s × ω(t)dt in the coordinate frame at time t + dt, where ω(t) is the instantaneous angular velocity of the coordinate frame (see Fig. C.1). The choice of ω(t) is not unique; however, specifically choosing ω(t) = z˙ × z where z is the unit vector n = B/|B| as seen in the rotated frame, coincides with the parallel transport of the coordinate frames [285, 314]. Suppose we have a spin vector, s = s(t), in the rotated frame at time t. At time t+dt, this vector becomes [relative to frame t+dt] s(t+dt)+s(t+dt)×ω(t)dt, 172 CLASSICAL DERIVATION OF B⊥ IN §7.2.2 Figure C.1: (left) The classical spin vector s(t) precesses about a magnetic field which is along the z direction at some instant t. Because of the time-dependence of the magnetic field, the spin is also subject to a rotation about ω(t) = z˙ × z which transforms it from the frame at time t (left) to the frame at time t + dt (right). Here, ω(t) acts as an additional magnetic field which governs the overall spin dynamics. where s(t + dt) ≈ s(t) + g s(t) × |B|z dt. For infinitesimally small dt, we may write s(t + dt) [in frame t] ≈ s(t + dt) + s(t + dt) × ω(t)dt [in frame t + dt]. (C.2) The right-hand-side of the resulting equation can be expanded as s + g s × |B|z dt + s × ω(t)dt + O(dt2 ). Rearranging Eq. 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D 35, 2597 (1987). 186 [...]... devices proposed in the literature 1.1.4 Spin Transport and Spin- Dependent Transport Spin transport: Once a population of spins is created, they naturally diffuse via spin dephasing mechanisms (§2.1.3) These effectively randomize the spins resulting in a loss of the spin encoded information or state of the system In any practical spintronic device, it is important to be able to transport spins coherently... array of spin- related phenomena in SCs Fig 1.1 shows a selective branch diagram of topics relevant to this Thesis and how they depend on/are linked by SOC These topics are discussed below 1.1.2 Generating Spin Currents and Polarization Spin currents are electronic currents with a finite spin polarization, i.e comprising of unequal numbers of spin- up and spin- down carriers In pure semiconductors, currents. .. affects spin dynamics is given We then provide a survey of spintronic devices proposed in the literature including spin transistors and spin filters, which make use of SOC as well as external magnetic fields Next we review the origin of gauge fields and their important role in spintronics Lastly, we discuss developments of the intrinsic SHE including the two known distinct mechanisms, the spectrum of systems... new spintronic transistor devices based on the tunability of the Rashba SOC effect in SC heterostructures • Gain a better understanding of intrinsic SHE mechanisms, by studying the physical significance of gauge fields in SOC systems 1.3 Organization of Thesis We begin in Chapter 2 with a review of relevant topics in SC spintronics A general discussion of the SOC effect, including the most common types of. .. currents are inherently unpolarized The most direct way to generate spin polarized currents within a SC is via current injection from ferromagnets Alternatively, externally applied magnetic fields can be used as spin filters in which unpolarized input currents result in spin- polarized output currents However, nonmagnetic means of generating spin currents 2 INTRODUCTION Datta-Das device (Ch 4) Spin relaxation... 4) Spin relaxation (DP) Current induced spin polarization Spin precession (Ch 4,5) Zitterbewegung (Ch 5) Spin polarization (Ch 3) Spin- orbit coupling Spin transverse force (Ch 5) Spin- dependent transport (Ch 5-8) Non-Abelian gauge fields (Ch 5) Subband filtering (Ch 3.1) Spin accumulation Spin- Hall effect (Ch 6-8) AharanovCasher phase Figure 1.1: Branch diagram of spin- related phenomena relevant to... Description of Intrinsic Spin- Hall Effect Mechanisms, New J Phys 12, 013016 (2010) T Fujita, M B A Jalil, and S G Tan, Unified Model of Intrinsic Spin- Hall Effect in Spintronic, Optical, and Graphene Systems, J Phys Soc Jpn 78, 104714 (2009) T Fujita, M B A Jalil, and S G Tan, Spin- Hall effect of collimated electrons in zincblende semiconductors, Ann Phys 324, 2265 (2009) M B A Jalil, S G Tan and T Fujita, Spintronics... spin helix pseudospin-Hall effect parabolic quantum well quasi-one dimensional quantum adiabatic theorem quantum Hall effect quantum point contact quantum spin- Hall effect quantum well resonant tunneling diode semiconductor spin field-effect transistor spin- Hall conductivity spin- Hall effect structural inversion asymmetry spin- orbit coupling special unitary group of degree n time reversal transmitted spin. .. expanded below But first, we introduce the spin- orbit coupling (SOC) effect which plays a central role in this work 1.1.1 Spin- Orbit Coupling The ubiquity of the SOC effect in SC spintronics studies lends itself to the attractive possibility of “spintronics without magnetism” [8] SOC describes the inevitable coupling between the motion and spin of carriers in systems exhibiting low spatial symmetries The best... spin conductance unitary group of degree n xix CHAPTER 1 Introduction 1.1 Background and Motivation Spintronics is the study of the quantum mechanical spin degree of freedom and its usefulness in technology It is of great importance and interest to both engineering and condensed matter physics After all, it was the rapid development of spintronics in magnetic multilayers in the early 90s that shaped . theoretically study several critical aspects of SC spintronics, with a focus on spin currents in the presence of SOC. These aspects include spin current generation, spin manipulation, and spin- dependent. all-electric spin manipulation it permits in SC spintronic devices. It also drives the remarkable spin- Hall effect (SHE) which is a promising source of dissipationless spin currents. In this Thesis, we theoretically. research. An in- depth study of gauge fields constitutes the second-half of this Thesis. In par- ticular, we closely examine the intrinsic spin- Hall effect (SHE), in which dissipationless spin currents