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A Topological Explanation of the Urbach Tail a Thesis Presented To A Topological Explanation of the Urbach Tail A thesis presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Master of Science Dale J. Igram April 2016 © 2016 Dale J. Igram. All Rights Reserved 2 This thesis titled A Topological Explanation of the Urbach Tail by DALE J. IGRAM has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by David A. Drabold Distinguished Professor of Physics and Astronomy Robert Frank Dean, College of Arts and Sciences 3 ABSTRACT IGRAM, DALE J., M.S., April 2016, Physics A Topological Explanation of the Urbach Tail Director of Thesis: David A. Drabold The Urbach tail is considered as one of the most significant properties of amorphous structures because of its almost universal characteristics. Several theoretical attempts have been used to explain the existence of the Urbach tail only to fail. Utilizing the best models, it has been shown that Urbach tails are associated with topological filaments [11]. The original work in this thesis is presented in two parts. The first part involves the effects of thermal statistics on eigenvalues and electronic density of states (EDOS) for a 512 atom a-Si model, which revealed a significant variation of the EDOS and energy gap as a function of temperature; thus, showing how a semiconducting state can change to a conducting state. The second part deals with the comparison of two 4096 atom a-Si models (unrelaxed and relaxed), in which physical and electronic structures were compared, revealing that the connectivity of the short and long bonds for both models indicated that short bonds prefer short bonds and similarly for long bonds, and that short bonds of the relaxed model germinated and proliferated faster as compared to the unrelaxed model. The electronic structure calculations for both models were performed using an ab initio code SIESTA. 4 DEDICATION To my wife, Esperanza 5 ACKNOWLEDGEMENTS First of all, I would like extend my deepest gratitude to Dr. David A Drabold for being my advisor and mentor during my research endeavor, providing patient support and valuable suggestions on my thesis. Second, my appreciation goes to Drs. Charlotte Elster, David Drabold, and Alexander Neiman for being a committee member and providing helpful suggestions for improving the thesis. Third, a special thank you goes to my student colleagues, Anup Pandey and Bishal Bhattarai, for assistance with Unix and SIESTA, and Kiran Prasai, for valuable discussions regarding modifications of SIESTA and general support. I would also like to thank all the faculty and staff of the Physics and Astronomy Department and students that provided support in one form or another. Most of all, I would like to express my deepest appreciation to my wife, Esperanza, who supported me during good times and bad times, and was very patient and understanding. 6 TABLE OF CONTENTS Page Abstract.......…………………………………………………….......................................3 Dedication..........................................................................................................................4 Acknowledgements.............…………………………………….......................................5 List of Figures……………………………………........………........................................8 1. Introduction………………………………………………………….........................14 2. The WWW Model..………………………………………………….........................21 2.1 Modeling Methodology and Theory............……….………........................22 2.2 Discussion of Model Results ….........................................…..…................31 2.3 Summary................……………...................................................................37 3. Mathematical Description of SIESTA…………………………...............................38 3.1 Basic Assumptions..............……………………..........................................39 3.2 Pseudopotentials.................................……..................................................39 3.3 LCAO Basis Sets...........……………………...............................................41 3.4 Matrix Elements of the Hamiltonian Operator….......……..........................45 3.5 Total Energy.....………………………………………………....................48 3.6 Harris Functional..........................................................................................49 3.7 Summary.......................................................................................................50 4. An Explanation of the Urbach Tail………………......………...................................51 4.1. Optical Absorption……………………………………..……......................51 4.2 Electronic Density of States....………………………........….....................55 4.2.1 Electronic structure for amorphous materials.........................................55 4.2.2 Electronic density of states for amorphous materials.............................58 4.3 Urbach Tails (A Theoretical Review)...........................................................61 7 4.3.1 Research from other research groups......................................................61 4.3.2 Research from our group.........................................................................64 4.4 Summary.......................................................................................................76 5. Results and Discussion…........……………....……………………...........................77 5.1 The 512 Atom a-Si Structure........................................................................77 5.2 The 4096 Atom a-Si Structure......................................................................81 5.2.1 Physical structure information.................................................................81 5.2.2 Electronic density of states......................................................................84 5.2.3 Bond-center to bond-center correlation distribution................................93 5.2.4 Connectivity of shortest and longest bonds.............................................96 5.3 Summary.....................................................................................................100 6. Summary and Future Work......................................................................................102 6.1 Summary.....................................................................................................102 6.2 Future Work................................................................................................104 References……………………………………………........…………..........................105 Appendix A. Derivation of Equation 2.2.......................................................................111 8 LIST OF FIGURES Figure Page 1.1. Disorder Types: (a) topological (no long-range order), (b) spin (on regularlattice), (c) substitutional (on regular lattice), (d) vibrational (about equilibrium positions)[5]................................................................................. 15 1.2. A pictorial representation of the structural origin of special featuresin the RDF. An oscillatory behavior is present for the density function ρ(r) as a function of r. The shaded area under a given peak gives the effective coordination number[1]................................................................................... 16 1.3 The straight lines represent what is now known as the Urbach tails. These lines were produced from absorption measurement of silver bromide crystals [101]................................................................................................... 18 2.1. Local bond switching used to create random networks from the diamond cubic structure, where (a) represents the orientation of bonds in the diamond cubic structure and (b) the relaxed orientation of the atoms for a pair bond switch [23]....................................................................................... 24 2.2. Potential energy variation for two alternative configurations which are related by a bond switch. The dashed line indicates the smooth transition that defines the barrier [34]............................................................................. 25 2.3. Procedure for creating continuous random networks...................................... 27 2.4. The Keating interactions [34].......................................................................... 29 2.5. A pictorial of the progression of the spectrum 푛(∆퐸) at different steps during the building process for various bond switch energy, ∆퐸. The bond switch as shown in Fig. 1 is represented by ∆퐸 = 퐸표 [34] ............................ 30 2.6. Correlation function for various amounts of randomization. (a) perfect diamond cubic structure; (b) – (f), density of bond-pair switches per atom d = 0.06,0.12, 0.18, 0.24, and 0.30, respectively [34]........................................ 33 2.7. Correlation function for three different locations in the annealing process for the rms angular deviation of 15.6˚ (top), 13.2˚ (middle), and 12.6˚ (bottom), respectively [34].............................................................................. 35 9 2.8. A comparison of the correlation functions between experimental data for a- Ge and calculated data for a-Si [23]................................................................ 36 3.1. Local pseudopotential and neutral atom pseudopotential for silicon. The dashed line represents −푍⁄푟 and 푉푁퐴 the screened local part of the pseudopotential from an electron charge distribution produced by filling the first-ζ basis orbitals with the free-atom valence occupations [55].................
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