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Real Time Density Functional Simulations of Quantum Scale Real Time Density Functional Simulations of Quantum Scale Conductance by Jeremy Scott Evans B.A., Franklin & Marshall College (2003) Submitted to the Department of Chemistry in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2009 c Massachusetts Institute of Technology 2009. All rights reserved. Author............................................... ............... Department of Chemistry February 2, 2009 Certified by........................................... ............... Troy Van Voorhis Associate Professor of Chemistry Thesis Supervisor Accepted by........................................... .............. Robert W. Field Chairman, Department Committee on Graduate Theses This doctoral thesis has been examined by a Committee of the Depart- ment of Chemistry as follows: Professor Robert J. Silbey.............................. ............. Chairman, Thesis Committee Class of 1942 Professor of Chemistry Professor Troy Van Voorhis............................... ........... Thesis Supervisor Associate Professor of Chemistry Professor Jianshu Cao................................. .............. Member, Thesis Committee Associate Professor of Chemistry 2 Real Time Density Functional Simulations of Quantum Scale Conductance by Jeremy Scott Evans Submitted to the Department of Chemistry on February 2, 2009, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract We study electronic conductance through single molecules by subjecting a molecu- lar junction to a time dependent potential and propagating the electronic state in real time using time-dependent density functional theory (TDDFT). This is in con- trast with the more common steady-state nonequilibrium Green’s function (NEGF) method. We start by examining quantum scale conductance methods in both the steady state and real-time formulations followed by a review of computational quan- tum chemistry methods. We then develop the real-time density functional theory and numerical solution techniques and use them to examine transport in a simple trans-polyacetylene wire. The remaining chapters are devoted to examining real-time transport behavior of various systems and model chemistries. Open-shell calculation of the polyacetylene wire reveal that, in agreement with various correlated model calculations, charge and spin behave as separate quasiparticles with different rates of transport. However, the transport of charge, and especially spin are highly dependent upon the amount of exact exchange included in the approximate exchange-correlation energy functional. This functional dependence is further illustrated when we demon- strate that the conductance gap of a device imperfectly coupled to wires varies based upon the non-local exchange and correlation. We also study the dynamic transport behavior of benzene-1,4-dithiol (BDT) coupled to gold leads and find that both the transient current and device charge density fluctuate with time,. This suggests that the steady-state assumption of the NEGF method may not be accurate. Thesis Supervisor: Troy Van Voorhis Title: Associate Professor of Chemistry 3 4 Acknowledgments I first thank my advisor Troy Van Voorhis, whose continued guidance and leadership made this work possible. Troy’s mentoring was instrumental in allowing me to develop as a scientist and in bringing me to a place to make meaningful contributions to the scientific community. His door was always open, and he showed me how to make the best use of the time I devoted to my research. I thank the members of my thesis committee, Dr. Robert J. Silbey and Dr. Jianshu Cao, for their advice through the years and in developing this thesis. I thank Dr. Richard Moog who first taught me quantum mechanics. I thank as well my undergraduate research advisor, Dr. Ronald Musselman for giving me a taste of research and the confidence to pursue my degree here at MIT. My research colleagues were integral in making my time here interesting and enjoy- able. I thank Steve Presse, Jim Witkoskie, Qin Wu, Indranil Rudra, Xiaogeng Song, Seth Difley, Eric Zim´anyi, Aiyan Lu, Carter Lin, Lee-Ping Wang, Tim Kowalczyk and Ben Kaduk for intersting conversations and for maintaining a friendly atmosphere in the ”zoo”. I would espescially like to thank Chiao-Lun Cheng and Oleg Vydrov who collaborated on some of the research included in this thesis. I thank the members of Community of Faith Christian Fellowship who have truly been a family to me in Boston. They have provided support and continued friendship. While I cannot possibly list every member who has touched my life, I specifically thank Pastors Sean Richmond, and Jeff Bianchi for their spiritual leadership. Thank you also to the past and current members of the sound team. Our efforts together have given me an outlet outside of MIT and allowed me to retain my sanity. Finally, I would like to express my utmost love and appreciation to my family. I thank my parents, Richard Jr. and Linda Evans and my grandparents Richard Sr. and Ruth Evans. I thank my wife’s parents, Bill and Jane Ingram as well. Thank you also to my brothers, Josh and Ryan Evans, and my sister, Krista Evans. Foremost, I thank my wife Julia. Her love and support have been essential to me since before I began my graduate studies. This work is dedicated to her. 5 6 Contents 1 Introduction 17 1.1 Single Molecule Electron Transport . 17 1.1.1 LandauerFormula . 19 1.2 Schr¨odingerEquation. 21 1.3 ConductionSystemDefinitions . 23 1.4 Non-Equilibrium Green’s Function Method . 25 1.4.1 Scattering Theory and Green’s Functions . 25 1.4.2 Dyson Equation and Self Energies . 27 1.4.3 Calculating Transmission . 28 1.5 RealTimePropagationMethod . 30 1.5.1 H¨uckelMethodExample . 34 1.5.2 Advantages and Disadvantages . 36 1.6 ThesisFormat............................... 38 2 Quantum Chemistry Methods 39 2.1 BasisSets ................................. 39 2.2 VariationalPrinciple . 40 2.3 ElectronicHamiltonian . 42 2.3.1 All Electron Hamiltonian . 42 2.3.2 Effective Core Potentials . 43 2.4 TheHartree-FockMethod . 44 2.4.1 Single-determinant Solution Space . 44 2.4.2 FockEquation........................... 45 7 2.5 CorrelatedMethods. 48 2.6 DensityFunctionalTheory . 50 2.6.1 Hohenberg-Kohn Theorems . 50 2.6.2 Kohn-ShamMethod . 52 2.6.3 Spin Density Functional Theory . 54 2.6.4 Functionals ............................ 55 2.6.5 DFTInaccuracies. 59 2.6.6 Time-DependentDFT . 63 2.7 Pariser-Parr-Pople Model Hamiltonian . 64 3 Real Time TDDFT Propagation 67 3.1 TDKSPropagationTheory . 67 3.2 NumericalPropagationMethods. 69 3.3 MolecularWireConductance . 71 3.3.1 AverageCurrents . 72 3.3.2 Comparison to Long Wire Results . 75 3.3.3 ComparisontoNEGF . .. .. ... .. .. .. .. .. ... 76 3.3.4 Comparison to Voltage Bias . 78 3.4 Conclusions ................................ 79 3.5 Acknowledgements ............................ 79 4 Spin-Charge Separation in Open-Shell Propagations 81 4.1 Methods.................................. 83 4.1.1 Systems .............................. 83 4.1.2 Real-Time Density Functional Conductance Simulations ... 83 4.1.3 Empirical Model Hamiltonians . 85 4.2 Results................................... 87 4.2.1 Real-Time TDDFT Currents . 87 4.2.2 Obtaining Model Hamiltonian Parameters . 89 4.2.3 ModelHamiltonianCurrents. 91 4.3 Discussion................................. 93 8 4.4 Conclusions ................................ 97 4.5 Acknowledgements ............................ 98 5 Importance of Non-local Exchange and Correlation 99 5.1 SystemandMethods........................... 100 5.2 Real-Time Density Functional Conductance Simulations . ...... 103 5.3 Model Hamiltonian Conductance Simulations . 107 5.3.1 ThePPPModelHamiltonian . 107 5.3.2 Non-localExchangeinthePPPmodel . 108 5.3.3 Correlated Conductance of the PPP model . 110 5.4 Conclusions ................................ 115 5.5 Acknowledgements ............................ 118 6 Dynamic Coulomb Blockade in a Metal-Molecule-Metal Junction 119 6.1 SystemandMethod............................ 120 6.2 ResultsandDiscussion .......................... 122 6.3 Conclusions ................................ 129 7 Conclusions 131 A Geometry of the Gold-BDT-Gold System 133 9 10 List of Figures 1-1 From Ref. [1]. Reprinted with permission from AAAS. Schematic of the experimental system and resulting current and conductance profile for a Gold-BDT-Gold conductance measurement (BDT=benzene-1,4- dithiol). In this experiment, the junction was formed by adsorption of BDT from solution into a self-assembled monolayer in the gap of a fracturedgoldwire.. 18 1-2 Schematic of the conducting molecular contact system. The current is positive when particles travel from the source to the drain, and negative whentheytravelintheotherdirection. 19 1-3 I-V calculation results for the H¨uckel example method. The plots de- pict (a) n (t) n (t) as calculated with the voltage potential method D − S for several potentials (b) n (t) n (t) as calculated with the chemi- D − S cal potential method at a potential of 1.2β (only one line is shown for visual clarity), and (c) the resulting I-V curves. 35 2-1 Flow chart depicting the self-consistent field procedure. ....... 48 + 2-2 Dissociation energy curves of H2 calculated using Hartree-Fock, LDA, and B3LYP. The HF calculation is exact within the basis set (6-311+G**) approximationforthissystem.. 61 2-3 Dissociation energy curves of H2 calculated using CI, Hartree-Fock, LDA, and B3LYP. CI is exact within the chosen basis set (6-31+G**)
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