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Modeling Nonlocality in Quantum Systems A MODELING NONLOCALITY IN QUANTUM SYSTEMS A Dissertation Submitted to the Faculty of Purdue University by James A. Charles In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 2018 Purdue University West Lafayette, Indiana THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF COMMITTEE APPROVAL Dr. Tillmann Kubis, Co-chair Department of Electrical and Computer Engineering Dr. Gerhard Klimeck, Co-chair Department of Electrical and Computer Engineering Dr. Sabre Kais Department of Chemistry Dr. Zhihong Chen Department of Electrical and Computer Engineering Approved by: Dr. Pedro Irazoqui Head of the Graduate Program ii ACKNOWLEDGMENTS I would like to express my gratitude to my co-advisor, Professor Tillmann Kubis, for his support during these years. His direction and continuous support during my PhD has been invaluable in learning how to conduct research. I would also like to thank my other co-advisor, Professor Gerhard Klimeck for his helpful discussions and suggestions. He helped guide the research by looking at the big picture. Additionally, thank you to my other committee members Professor Zhihong Chen and Professor Sabre Kais. Professor Chen made sure that the experimental side of things was always considered. Professor Kais gave his advice and insights as a expert in chemistry. They were invaluable members of my PhD committee and helped guide the research. Thanks also to my preliminary committee member Dmitri Nikinov for his questions and insights from an industrial perspective. I would also like to thank my friends and family for their support both profes- sionally and personally. They were always there for me when I needed it. Finally, I would like to thank my wife, Alicia Charles, for her never-ending support during my PhD. She made the five year journey much easier and more enjoyable. iii TABLE OF CONTENTS Page LIST OF FIGURES ::::::::::::::::::::::::::::::::: vi SYMBOLS :::::::::::::::::::::::::::::::::::::: ix ABSTRACT ::::::::::::::::::::::::::::::::::::: x 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 1.1 Overview ::::::::::::::::::::::::::::::::::: 1 1.1.1 Non-Equilibrium Green's Function Method :::::::::::: 1 1.1.2 Outline of this Thesis :::::::::::::::::::::::: 1 1.2 Motivation Part I: Modeling of Nonlocal Scattering in Quantum Systems 2 1.2.1 Scattering in Ultra-Scaled Devices ::::::::::::::::: 2 1.2.2 The important of non-local scattering ::::::::::::::: 3 1.3 Motivation Part II: Towards NEGF in liquid systems :::::::::: 4 1.3.1 Modeling molecules in a liquid ::::::::::::::::::: 4 2 PART I: STATE OF THE ART OF MODELING SCATTERING IN QUAN- TUM SYSTEMS ::::::::::::::::::::::::::::::::: 8 2.1 Dyson and Keldysh Equations ::::::::::::::::::::::: 8 2.2 Local Electron-Phonon Scattering Self-energies :::::::::::::: 9 2.2.1 Electron Density and Current ::::::::::::::::::: 17 2.2.2 Algorithm Flow ::::::::::::::::::::::::::: 18 2.3 How to solve the Keldysh and Dyson Equations? ::::::::::::: 18 2.3.1 Traditional Recursive Green's Function Algorithm :::::::: 20 3 LOCAL SCATTERING RESULTS ::::::::::::::::::::::: 23 3.1 Silicon TFET :::::::::::::::::::::::::::::::: 23 3.2 Resonance TFET :::::::::::::::::::::::::::::: 24 4 NEW NONLOCAL RECURSIVE GREEN'S FUNCTION ALGORITHM :: 30 iv Page 4.1 Validation : Nonlocal Recursive Green's Function Algorithm :::::: 30 4.2 Nonlocal Recursive Green's Function Derivation and Details :::::: 39 4.2.1 General concept for the retarded Green's function :::::::: 39 4.2.2 Recursive Green's Function Algorithm Overview ::::::::: 40 4.2.3 Forward recursive Green's function :::::::::::::::: 40 4.2.4 Backward recursive Green's function ::::::::::::::: 44 4.3 Non-local Recursive Green's Function Results :::::::::::::: 45 5 PART II: OVERVIEW OF THE STATE OF THE ART OF MODELING CHEMICAL SYSTEMS IN LIQUIDS :::::::::::::::::::::: 47 5.1 Overview of Density Functional Theory and its Applications :::::: 47 5.1.1 Thermal and Excited State Density Functional Theory ::::: 48 5.1.2 Density Functional Theory Applied to Water ::::::::::: 49 5.1.3 Implicit Solvent Models :::::::::::::::::::::: 50 5.2 Molecular Electronics :::::::::::::::::::::::::::: 51 6 IMPLICIT, EXPLICIT AND HYBRID SOLVENT METHODS ::::::: 53 6.1 General Ingredients for Implicit Solvent Methods :::::::::::: 53 6.2 General Ingredients for Explicit Solvent Methods :::::::::::: 59 6.3 Hybrid Methods ::::::::::::::::::::::::::::::: 60 7 HAMILTONIAN - DESCRIPTION OF THE SYSTEM :::::::::::: 64 7.1 Overview of Different Tight-Binding Theories :::::::::::::: 64 7.1.1 Density Functional Tight-Binding ::::::::::::::::: 66 7.2 DFTB Bandstructure and Density of States ::::::::::::::: 70 8 NON-EQUILIBRIUM GREEN'S FUNCTION OPEN BOUNDARY CON- DITION SOLUTION METHODS :::::::::::::::::::::::: 73 8.1 Sancho Rubio :::::::::::::::::::::::::::::::: 73 8.2 Transfer Matrix ::::::::::::::::::::::::::::::: 74 8.3 Scattering in the Leads ::::::::::::::::::::::::::: 77 8.4 General Leads :::::::::::::::::::::::::::::::: 77 v Page 9 EXTENSION OF NON-EQUILIBRIUM GREEN'S FUNCTION FOR AR- BITRARY THREE-DIMENSIONAL LIQUID LEADS :::::::::::: 81 9.1 Verification ::::::::::::::::::::::::::::::::: 81 9.2 Nonequilibrium Green's Function with Crystalline Lead Results :::: 83 9.3 Hamiltonian for Liquid Leads ::::::::::::::::::::::: 86 9.4 Extension of Thermodynamic Integration for Quantum Mechanical Calculations ::::::::::::::::::::::::::::::::: 88 9.5 Future Outlook ::::::::::::::::::::::::::::::: 92 10 CONCLUSION :::::::::::::::::::::::::::::::::: 94 REFERENCES :::::::::::::::::::::::::::::::::::: 97 PUBLICATIONS :::::::::::::::::::::::::::::::::: 105 VITA :::::::::::::::::::::::::::::::::::::::: 109 vi LIST OF FIGURES Figure Page 1.1 Contribution of different scattering mechanisms to the mobility of a single layer MoS2 device for different temperatures. The data is obtained from [27]. 5 1.2 Contribution of different scattering mechanisms to the mobility of a two- dimensional electron gas of GaAs. The data is obtained from [28]. ::::: 6 2.1 Overview of the Poisson and Self-consistent Born approximation iterative approach used ::::::::::::::::::::::::::::::::: 19 2.2 Recursive Green's function Algorithm Overview ::::::::::::::: 20 3.1 Schematic of a PIN circular nanowire TFET that is simulated. :::::: 24 3.2 Comparison of scattered to ballistic transfer characteristics for a 3nm cir- cular nanowire TFET with a Vds =1.0V. Two scattered cases are shown. With and without energy renormalization. Including energy renormaliza- tion increases the phonon-assisted tunneling and thus increases current. :: 25 3.3 Comparison of scattered to ballistic DOS near the transition region at x = 8 nm for the IV in figure 5 at Vgs = 0.6 V. The energy shift between scattered and ballistic is about 16 meV. ::::::::::::::::::: 26 3.4 Cross-sectional schematics of a triple-HJ TFET. Schematic design ob- tained from [53]. :::::::::::::::::::::::::::::::: 27 3.5 IV Characteristics for the Resonance TFET comparison ballistic to local scattering with different scattering strengths. Optical C is used the Sil- icon non-polar deformation constant value of 110 eV/nm. Optical B is approximately two times this value and Optical A is four times optical C. : 28 3.6 Subthreshold Slope comparison for the different configurations. Optical C is used the Silicon non-polar deformation constant value of 110 eV/nm. Optical B is approximately two times this value and Optical A is four times optical C. [53]. :::::::::::::::::::::::::::::: 29 4.1 Current-voltage comparison for traditional RGF and nonlocal RGF. ::: 31 4.2 Numerical comparison of GR calculated with non-local RGF and with full inversion. Note: Same atom orbitals are summed for clarity. :::::::: 32 vii Figure Page R 4.3 The anti-diagonal of ΣP OP comparison of full inversion and NL-RGF for different screening lengths. The lines are full-inversion and the symbols are NL-RGF. :::::::::::::::::::::::::::::::::: 33 4.4 Timing for NL-RGF as a function of non-locality in the transport direction. For comparison, the timing is plotted against the traditional (local) RGF. 34 4.5 Memory for NL-RGF as a function of non-locality in the transport direc- tion. For comparison, the memory is plotted against the traditional (local) RGF. :::::::::::::::::::::::::::::::::::::: 35 4.6 Timing for NL-RGF as a function of non-locality in the transport direction for three different sizes of slabs. A horizontal line at the ratio of non-local to local time of 150 is added to guide the eye. :::::::::::::::: 36 4.7 Memory for NL-RGF as a function of non-locality in the transport direc- tion for three different sizes of slabs. ::::::::::::::::::::: 37 4.8 The theoretical comparison of complexity of NL-RGF to full inversion. The break even point is marked with a red dashed line to guide the eye. : 38 4.9 IV characteristics for a 2nm diameter InGaAs wire in effective mass. The ballistic result is also shown for comparison. ::::::::::::::::: 46 5.1 Typical molecular electronic simulation setup. The molecule is placed between two ideal copper contacts. The delineation between the device and the infinite, ideal, and equilibrium leads is shown with a vertical dashed line on both sides. ::::::::::::::::::::::::::::::: 52 6.1 A schematic of the different components that makeup the implicit solvent method. :::::::::::::::::::::::::::::::::::: 56 6.2 Example of Lennard Jones Potential including the different contributions. 60 6.3 A schematic of the structure in the explicit solvent method. Acetone is surrounded by water. The dashed line represents the mathematical boundary conditions.
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