Itinerant Spin Dynamics in Structures of Reduced Dimensionality A dissertation presented by Paul Thomas Wenk in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Thesis Committee: Prof. Dr. Stefan Kettemann [thesis advisor] (Jacobs University Bremen, Germany & Pohang University of Science and Technology, South Korea) Prof. Dr. Georges Bouzerar (Institut N´eel, France & Jacobs University Bremen, Germany) Prof. Dr. Ulrich Kleinekath¨ofer (Jacobs University Bremen, Germany) Date of Defense: August, 29 2011 ——————————— Jacobs University Bremen School of Engineering and Science © 2011 - Paul Thomas Wenk All rights reserved. Itinerant Spin Dynamics in Structures of Reduced Dimensionality Abstract In the present thesis results of the study of spin dynamics and quantum trans- port in disordered semiconductor quantum wires with spin-orbit coupling are presented. Starting from basic spin dynamics we derive the dependence of the weak localization cor- rection to the conductance on the strength and the kind of spin-orbit interaction (linear and cubic Dresselhaus, as well as Rashba coupling), the width of the quantum wires as well as the mobility, temperature and Zeeman term. Furthermore, we exploit the connec- tion found between the microscopic picture given by the Cooperon and the spin diffusion equation to extract the spin relaxation rate which shows the same wire dependencies as the weak localization correction. We also show how the result depends on the smoothness and the direction of the transverse confinement of the quantum wires. In this context we have addressed the question concerning long persisting or even persistent spin states in spintronic devices, presenting the corresponding optimal adjustment of spin orbit couplings of different kind and optimal alignment of the wire direction in semiconductor crystals. Experiments[HSM6 +0 , HSM7 +0 , KKN09, LSK7 +0 , WGZ6 +0 , SGB9 +0 ] which report the dimensional reduction of the spin relaxation rate in agreement with previous results were raising new questions, in particular as regarding the crossover from diffusive to ballistic wires, which we answer using modified Cooperon equation. In addition, we focus on the intrinsic spin Hall effect, which is only due to spin-orbit coupling. Having shown the ba- sic features with analytical calculations, we solve the spin Hall conductivity in presence of binary and block-distributed impurities (Anderson model). At this we apply the Kernel Polynomial Method, which allows for a finite size analysis of the metal-insulator transi- tion and the calculation of spin Hall conductivity in large systems compared with those addressable with exact diagonalization. iii Contents TitlePage........................................ i Abstract......................................... iii TableofContents.................................... iv CitationstoPreviouslyPublishedWork . ..... vii Acknowledgments.................................... viii Dedication........................................ ix 1 Introduction 1 2 Spin Dynamics: Overview and Analysis of 2D Systems 6 2.1 Short Reminder on the Origin of Spin Orbit Coupling . ...... 6 2.2 DynamicsofaLocalizedSpin . 8 2.3 Spin Dynamics of Itinerant Electrons . ..... 8 2.3.1 BallisticSpinDynamics . 8 2.3.2 SpinDiffusionEquation . 9 2.3.3 Spin Orbit Interaction in Semiconductors . .... 11 2.3.4 Spin Diffusion in the Presence of Spin-Orbit Interaction ....... 15 2.4 SpinRelaxationMechanisms . 18 2.4.1 D’yakonov-Perel’ Spin Relaxation . .. 19 2.4.2 DP Spin Relaxation with Electron-Electron and Electron-Phonon Scattering................................. 20 2.4.3 Elliott-Yafet Spin Relaxation . .. 21 2.4.4 Spin Relaxation due to Spin-Orbit Interaction with Impurities . 21 2.4.5 Bir-Aronov-Pikus Spin Relaxation . .. 22 2.4.6 MagneticImpurities . 22 2.4.7 NuclearSpins............................... 23 2.4.8 Magnetic Field Dependence of Spin Relaxation . .... 23 3 WL/WAL Crossover and Spin Relaxation in Confined Systems 25 3.1 Introduction.................................... 25 3.1.1 One-DimensionalWires . 25 3.1.2 Wires with W > λF ........................... 26 3.2 QuantumTransportCorrections . .. 27 3.2.1 DiagrammaticApproach. 27 iv Contents v 3.2.2 Weak Localization in Quantum Wires . 34 3.3 TheCooperonandSpinDiffusionin2D . 37 3.4 Solution of the Cooperon Equation in Quantum Wires . ...... 42 3.4.1 Quantum Wires with Spin-Conserving Boundaries . .... 42 3.4.2 Zero-ModeApproximation. 44 3.4.3 Exact Diagonalization . 46 3.4.4 Other Types of Boundary Conditions . 53 3.5 Magnetoconductivity with Zeeman splitting . ....... 57 3.5.1 2DEG ................................... 57 3.5.2 Quantum Wire with Spin-Conserving Boundary Conditions . 60 3.6 Conclusions .................................... 64 4 Direction Dependence of Spin Relaxation and Diffusive-Ballistic Crossover 67 4.1 Introduction.................................... 67 4.1.1 Example.................................. 69 4.2 Spin Relaxation anisotropy in the (001) system . ....... 70 4.2.1 2Dsystem................................. 70 4.2.2 Quasi-1Dwire .............................. 71 4.3 Spin relaxation in quasi-1D wire with [110] growth direction......... 75 4.3.1 Special case: without cubic Dresselhaus SOC . .... 76 4.3.2 WithcubicDresselhausSOC . 77 4.4 WeakLocalization ................................ 77 4.5 Diffusive-Ballistic Crossover . .... 78 4.5.1 Spin Relaxation at Q W 1 ..................... 80 SO ≪ 4.6 Conclusions .................................... 82 5 Spin Hall Effect 83 5.1 Introduction.................................... 83 5.1.1 About the Definition of Spin Current . 84 5.2 SHE without Impurities: Exact Calculation . ...... 85 5.3 NumericalAnalysisofSHE . 90 5.3.1 Exact Diagonalization . 90 5.3.2 KernelPolynomialMethod . 92 5.3.3 SHCcalculationusingKPM. 101 6 Critical Discussion and Future Perspective 106 List of Symbols 108 List of Figures 110 List of Tables 115 Bibliography 116 A SOC Strength in the Experiment 129 vi Contents B Linear Response 132 B.1 KuboFormulaforWeakDisorder . 134 C Cooperon and Spin Relaxation 137 C.1 SumFormulafortheCooperon . 137 C.2 Spin-ConservingBoundary . 139 C.3 RelaxationTensor ................................ 140 C.4 Weak Localization Correction in 2D . ... 141 C.5 ExactDiagonalization . 142 D Hamiltonian in [110] growth direction 146 E Summation over the Fermi Surface 148 F KPM 150 Citations to Previously Published Work Large portions of the chapters have appeared in the following four papers: 1. Wenk P., Kettemann S. et al. Spin Polarized Transport and Spin Relaxation in Quantum Wires. In: NanoScience and Technology. Springer-Verlag Berlin Heidelberg; (2010). 2. Wenk P., Kettemann S. Spin Relaxation in Quantum Wires. In: Sattler K, Francis& Taylor Handbook on Nanophysics.; (2010) p.49. 3. Wenk P., Kettemann S. Dimensional Dependence of Weak Localization Corrections and Spin Relaxation in Quantum Wires with Rashba Spin Orbit Coupling. Phys. Rev. B. 81 125309 (2010). 4. Wenk P., Kettemann S. Direction Dependence of Spin Relaxation in Confined Two- Dimensional Systems. Phys. Rev. B. 83 115301 (2011). vii Acknowledgments Completing this doctoral work has been a wonderful experience. It is with pleasure that I acknowledge the advise and support by the people with whom I was working during the time in Hamburg and Bremen. In particular I owe my interest in the fascinating world of condensed matter to my thesis advisor Professor Stefan Kettemann. It started all by a small and seemingly simple task concerning ballistic spin transport. During the induction into the field of spintronics his advice helped me to see on the one hand the richness of effects which were often counter- intuitive, and on the other hand he showed me how to get a feeling for the nature of the complex problem we were working on by application of fundamental rules. He gave me plenty of scope to chose my direction of interest and always encouraged me in many fruitful discussions. Professor Georges Bouzerar I owe not only the rediscovery of my interest in programming and a boost in my rusty C++ skills from school time, but also a better understanding of how to squeeze the essential of an phenomenon into the computer and extract meaningful physics out of it. His questions furthered my insight of the subject matter. Throughout my years as a PhD student, I was supported for many semesters by the DFG- SFB508 B9 at the Institute for Theoretical Physics in Hamburg and by the DFG CC5120 at the Jacobs University Bremen, through the generosity of my advisor. I thank the Conrad Naber foundation for providing my scholarship for my first year at the Jacobs University Bremen. In addition I had the opportunity to take part in many interesting and productive conferences (some including even singing) where some were founded by the World Class University program through the Korea Science and Engineering Foundation. Finally, I also would like to acknowledge the support of my colleagues at the Hamburg and Jacobs University, creating a stimulating environment. viii Dedicated to Linda, my parents and my brother Daniel. Stephanie: So, how was your day? Leonard: Y’know, I’m a physicist - I thought about stuff. Stephanie: That’s it? Leonard: I wrote some of it down. From “The Big Bang Theory“ episode The Panty Pinata Polarization. ix x Acknowledgments Chapter 1 Introduction Structure of this thesis This thesis falls into four parts, Fundamentals of spin dynamics and spin relaxation mechanisms (Chapter 1 and 2), • Spin Dynamics in Quantum