Establishing Quantum Monte Carlo and Hybrid Density Functional Theory as benchmarking tools for complex solids DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Kevin P. Driver, B.S., M.S. Graduate Program in Physics The Ohio State University 2011 Dissertation Committee: John W. Wilkins, Advisor Richard J. Furnstahl Ciriyam Jayaprakash Arthur J. Epstein c Copyright by Kevin P. Driver 2011 Abstract Quantum mechanics provides an exact description of microscopic matter, but predictions require a solution of the fundamental many-electron Schr¨odingerequation. Since an ex- act solution of Schr¨odinger'sequation is intractable, several numerical methods have been developed to obtain approximate solutions. Currently, the two most successful methods are density functional theory (DFT) and quantum Monte Carlo (QMC). DFT is an exact theory which, which states that ground-state properties of a material can be obtained based on functionals of charge density alone. QMC is stochastic method which explicitly solves the many-body equation. In practice, the DFT method has drawbacks due to the fact that the exchange-correlation functional is not known. A large number of approximate exchange-correlation functionals have been produced to accommodate for this deficiency. Conceptual systematic improve- ments known as \Jacob's Ladder" of functional approximations have been made to the stan- dard local density approximation (LDA) and generalized gradient approximation (GGA). The traditional functionals have many known failures, such as failing to predict band gaps, silicon defect energies, and silica phase transitions. The newer generation functionals in- cluding meta-GGAs and hybrid functionals, such as the screened hybrid, HSE, have been developed to try to improve the flaws of lower-rung functionals. Overall, approximate functionals have generally had much success, but all functionals unpredictably vary in the quality and consistency of their predictions. Often, a failure of one type of DFT functional can be fixed by simply identifying another DFT functional that best describes the system under study. Identifying the best functional for the job is a challenging task, particularly if there is no experimental measurement to ii compare against. Higher accuracy methods, such as QMC, which are vastly more compu- tationally expensive, can be used to benchmark DFT functionals and identify those which work best for a material when experiment is lacking. If no DFT functional can perform adequately, then it is important to show more rigorous methods are capable of handling the task. QMC is high accuracy alternative to DFT, but QMC is too computationally expensive to replace DFT. Hybrid DFT functionals appear to be a good compromise between QMC and standard DFT. Not many large scale computations have been done to test the feasibility or benchmark capability of either QMC or hybrid DFT for complex materials. This thesis presents three applications expanding the scope of QMC and hybrid DFT to large, scale complex materials. QMC computes accurate formation energies for single-, di-, and tri- silicon-self-interstitials. QMC combined with phonon energies from DFT provide the most accurate equations of state, phase boundaries, and elastic properties available for silica. The HSE DFT functional is shown to reproduce QMC results for both silicon defects and high pressure silica phases, establishing its benchmark accuracy compared to other functionals. Standard DFT is still the most efficient and useful for general computation. However, this thesis shows that QMC and hybrid DFT calculations can aid and evaluate shortcomings associated the exchange-correlation potential in DFT by offering a route to benchmark and improve reliability of standard, more efficient DFT predictions. iii To my family and friends for guidance, help, and love. iv Acknowledgments The research and underlying educational enlightenment represented by this thesis are most notably a product of the continuous support and encouragement of my advisor, John Wilkins. John provided impeccable guidance and maintained a high standard of excellence in developing my scientific career. Other than my advisor, a few others deserve specific mention for their critical guidance and support. Richard Hennig was an excellent mentor and source of scientific inspiration through out my entire graduate career. My office mate and group partner, William Parker, provided invaluable amounts of feedback and support throughout my entire graduate career as well. I am also indebted to the help and guidance of Ronald Cohen, whose training made significant portions of this research (silica) possible. There are many other people have taught me or played some role in my scientific ed- ucation. I would like to thank Cyrus Umrigar, Burkhard Militzer, Hyoungki Park, Amita Wadhera, Mike Fellinger, Jeremy Nicklas, Ken Esler, Neil Drummond, Yaojun Du, Jeong- nim Kim, Thomas Lenosky, Shi-Yu Wu, Chakram Jayanthi, David Brown, P. J. Ouseph, and my high school physics teacher { Robert Rollings for general support and advice. Many departmental staff members offered important assistance me in some manner while carrying out this work. I'd like to thank Trisch Longbrake, Shelly Palmer, Carla Allen, Tim Randles, Brian Dunlap, and John Heimaster. I owe much gratitude to my family: Gerald and Patricia and Silvia Driver, Diane and Gerald Link, Jeremy and Leah Driver, Betty and Tom Wells, Robert and Dorothy Driver, and Irene Muller. Thanks for all of the love and support, and the opportunities provided that made my academic career possible. v I also want to acknowledge many important friends that have helped me personally and/or academically persevere in somewhat of a chronological order: Roseanne Cheng, Chuck and Danna Pearsall, Jeff Stevens, Sheldon Bailey, Julia Young, Grayson Williams, Nick and Barbara Harmon, Jake and Nichole Knepper, Brandon and Ester Parks, Chad and Nikki Morris, James Morris, Charlie Ruggiero, Greg Mack, Yuhfen Lin, Becky Daskalova, Kevin Knobbe, Mark and Sara Murphey, Matt Fisher, Yi Yang, Louis Nemzer, Kaden Haz- zard, Shawn Walsh, Justin Link, Chen Zhao, Qiu Weihong, Jia Chen, David Daughton, Kent Qian, Eric Jurgenson, Daniel Clark, Taeyoung Choi, Kerry Highbarger, Anthony Link, Emily Sistrunk, Mike Boss, Mike Hinton, Fred Kuehn, Iulian Hetel, Jen White, Va- lerie Bossow, Jim Potashnik, Deniz Duman, Greg Sollenberger, Patrick Smith, Anastasios Taliotis, Steven Avery, Aaron Sander, Eric Cochran, Hayes Lara, Adam Hauser, Luke Corwin, Srividya Iyer Biswas, Colin Schisler, Borun Chowdhury, Reni Ayachitula, Neesha Anderson, Rakesh Tivari, Nicole De Brabandere, Jim Davis, Rob Guidry, Lee Mosbacker, Don Burdette, Mehul Dixit, Dave Gohlke, Alex Mooney, Greg Vieira, James and Veronica Stapleton, John Draskovic, Mariko Mizuno, Yuval DaYu, Emily Harkins, Sabine Shaikh, Angie Detrow, The two Ashers, The HCGs, Meghan Ruck, Chiaki Ishikawa, April Brown, Kim Pabilona, Eumie Carter, Liesen Parkus, Cassandra Plummer, Nadia Ahmad, Natalie, Emma Brownlee, Claudia Veltze, Savannah Laurel-Zerr, Tom Steele, Alex Gray, Michelle Oglesbee, Kimberly Rousseau, Carlos Rubio, JC Polanco and all my friends from La Fogata, Patrick Roach, Heather Doughty, and many more whose names I've forgotten. I also have much appreciation for several financial agencies that supported me and my work. I was supported for two years at The Ohio State physics department as a Fowler fellow and further supported mostly by the DOE. I'd also like to thank the NSF for supporting my stay at the Carnegie Institution of Washington at the Geophysical Laboratory during the summers of 2007 and 2008. This work was also made possible by generous computational resources from OSC, NERSC, NCSA, and CCNI. vi Vita April 14, 1980 . Born|New Albany, Indiana, USA 2003 . B.S., University of Louisville, Louisville, Kentucky 2003-2005 . .Fowler Fellow, Department of Physics, Ohio State University, Columbus, Ohio 2006 . M.S., Ohio State University, Columbus, Ohio 2005-Present . Graduate Research Associate, Depart- ment of Physics, Ohio State University, Columbus, Ohio Publications K. P. Driver, R. E. Cohen, Zhigang Wu, B. Militzer, P. L´opez R´ıos,M. D. Towler, R. J. Needs, and J. W. Wilkins, Quantum Monte Carlo computations of phase stability, equations of state, and elasticity of high-pressure silica, Proc. Natl. Acad. Sci. USA, 107, 9519 (2010). R. G. Hennig, A. Wadehra, K. P. Driver, W. D. Parker, C. J. Umrigar, and J. W. Wilkins, Phase transformation in Si from semiconducting diamond to metallic beta-Sn phase in QMC and DFT under hydrostatic and anisotropic stress, Phys. Rev. B, 82, 014101 (2010). M. Floyd, Y. Zhang, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, Nanometer- scale composition variations in Ge/Si(100) islands, Appl. Phys. Lett. 82, 1473 (2003). Y. Zhang, M. Floyd, K. P. Driver, Jeff Drucker, P.A. Crozier, and D.J. Smith, Evolution of Ge/Si(100) island morphology at high temperature, Appl. Phys. Lett. 80, 3623 (2002). P. J. Ouseph, K. P. Driver, J. Conklin, Polarization of Light By Reflection and the Brewster Angle, Am. J. Phys. 69, 1166 (2001). vii Fields of Study Major Field: Physics Studies in quantum Monte Carlo claculations of solids: J. W. Wilkins viii Table of Contents Page Abstract . ii Dedication . iv Acknowledgments . v Vita . vii List of Figures ...................................... xii List of Tables ....................................... xiv Chapters 1 Introduction