Density Functional Theory Techniques for Electron Energy Loss Analysis of Lithium Materials Quentin Stoyel, Mining and Materials Engineering, McGill University, Montreal November 2018 A thesis submitted to McGill University in partial fulfillment of the requirements of the degree of Masters of Science ©Quentin Stoyel 2018 Contents Table of Contents....................................i Abstract......................................... iii Abr´eg´e.......................................... iv Acknowledgements...................................v Contribution of Authors................................ vi 1 Introduction1 2 Literature Review4 2.1 Electron Energy Loss Spectroscopy (EELS)...................4 2.1.1 Electron Microscopy...........................4 2.1.2 EELS in Experiment........................... 10 2.1.3 Beam Damage............................... 13 2.1.4 Lithium in EELS............................. 15 2.1.5 Preprocessing............................... 16 2.1.6 Other Experimental Techniques..................... 19 2.2 Density Functional Theory (DFT)........................ 21 2.2.1 DFT Background............................. 22 2.2.2 Formulation................................ 22 2.2.3 WIEN2k.................................. 28 2.2.4 Quantum Theory of Atoms in Molecules................ 29 2.2.5 Lithium in DFT.............................. 30 i 2.3 EELS Calculations................................ 30 2.3.1 Time Dependent Density Functional Theory (TDDFT)........ 32 2.3.2 Cross Section Approach......................... 34 2.3.3 Beth Salpeter Equations (BSE)..................... 34 2.3.4 Core Hole Approximations........................ 35 3 Methods 40 3.1 Improvement to Core Hole Shielding Calculations............... 40 3.2 Implementation.................................. 43 3.3 Calculation Details................................ 43 3.4 Experimental Details............................... 44 4 Results and Discussion 45 4.1 Lithium Oxide................................... 45 4.2 Metallic Lithium................................. 47 4.3 Lithium Fluoride................................. 50 4.4 Li-LiF Mixture.................................. 51 4.5 Discussion..................................... 53 5 Conclusion 56 A Calculation Details 58 A.1 Crystal Structures................................ 58 A.2 DFT Parameters................................. 59 ii Abstract Electron energy loss spectroscopy (EELS) is an electron microscopy technique ideal for char- acterizing lightweight elements. The technique relies on using reference spectra to fingerprint and analyze results. These references can be obtained either through experiments on stan- dards or through simulation. Simulations provide the ability to produce spectra for previ- ously unstudied or sensitive materials, but rely on numerous approximations. Central to the simulation of EELS are the approximations treating core holes and shielding. In this work, a new method of calculating the magnitude of core hole screening in the case of lithium ma- terials is developed and implemented in density functional theory calculations. The method is used to verify the validity of performing electron energy loss spectroscopy at 30 keV to reduce beam damage. EELS fine structure are calculated for metallic lithium, Li2O, and LiF and marked improvements in agreement between calculation and experiment are observed. The technique uses linear response theory to relate the electron density to the core hole shielding contribution. This contribution is then implemented via a non integer core hole in final state rule density functional theory calculations. The improvements enabled by the technique open the possibility of conducting increasingly quantitative EELS analysis. iii Abr´eg´e La spectroscopie par perte d'´energiedes ´electrons(EELS) est une technique de caract´erisation sensible aux ´el´ements de faible masse. L'analyse EELS requiert des spectres de r´ef´erences pour valider les r´esultats obtenus. Les simulations de spectres offrent une faon d'obtenir des r´ef´erencespour des nouveaux mat´eriaux. Bien que les simulations soient utiles, elles requi`erent des ajustements de param`etres. Un de ces param`etre est la fa¸conde g´ererun trou d'´etatde cœur. Une nouvelle technique pour calcul´eel'intensit´ede masquage des trous d'´electronsest propos´ee,ce qui a pour effet d'augmenter la pr´ecisiondes spectres EELS des mat´eriauxcontenant du lithium. La m´ethode est appliqu´eepour v´erifierla pertinence de conduire des exp´eriencesEELS `a30 keV, ce qui a pour effet de minimiser l'endommagement des ´echantillons induits par de le faisceau d'´electron.La m´ethode est bas´eesur la th´eoriede la r´eponse lin´eaire,qui relie la densit´edes ´electronsaux effets de masquage. Ces effets sont par la suite repr´esent´espar un trou partiel dans les calculs par la th´eoriede la fonctionnelle de la densit´ed'´etatsd'´electrons. En utilisant la technique propos´eedans ce travaille, les spectres EELS calcul´eespour le Li2O, le LiF et le lithium m´etallique sont en meilleur accor- dance avec les r´esultatsexp´erimentaux de ces mat´eriaux. Ces r´esultatsouvrent la porte `a des m´ethode quantitatives bas´eesur la spectroscopie EELS. iv Acknowledgements Many people need to be acknowledged for helping make this work possible. Foremost are my supervisors, Professor Raynald Gauvin and Professor George P. Demopoulos for all their support, guidance, and valuable discussions. Equally essential to this work are Fr´ed´eric Voisard and Nicolas Brodusch who provided all the experimental results needed to verify the theoretical ones presented here. I would also like to thank Motoki Nakamura for taking the time to read and help edit my thesis and Fr´ed´ericVoisard for assistance in writing the French abstract. Acknowledgement is also due to the agencies who helped fund this work: NSERC and IREQ. Support also came from a multitude of places in ways that were not always appreciated in the moment. To this end, I would like to thank everyone who helped instill me with love and excitement for physics and science. Whether it was in a classroom, a pub, or wandering the streets, those willing to seek out and discuss the world's interesting problems inspired me into making this work happen. Finally, I would like to thank my parents for knowing that this was my path well before I did, and supporting me every step of the way. v Contribution of Authors In this thesis, the author developed the calculation method presented in Chapter3 and performed all of the density functional theory and Bader theory calculations for the results presented in Chapter4. The author also performed all of the analysis and processing of the experimental spectra. The lithium K edge experimental spectra in Chapters2 and4, as well as the carbon K edge spectra in Figure 2.7 were obtained by Nicolas Brodusch and Fr´ed´eric Voisard. Prof. Raynald Gauvin and Prof. George P. Demopoulos supervised the research project. vi Chapter 1 Introduction The study of lithium materials has become increasingly relevant in recent years [1]. In par- ticular, the field has been driven by the need to develop improved and more cost effective battery materials [1]. This drive has come from increasing demands for electric vehicles and portable electrical devices demanding longer lifetimes and faster charging. Many aspects of batteries can be improved as the theoretical limits have not yet been achieved in fundamental areas such as capacity, charge density, and charge/discharge rates. The theoretical limits in the case of lithium ion batteries are especially high, as lithium offers a range of advantages relative to other charge carriers. As the third element on the periodic table, lithium is extremely lightweight, allowing lithium ion batteries to be smaller and lighter without sacrificing lifetime [2]. Lithium's small ion size makes it highly mobile, which leads to superior discharge rates [2]. Furthermore, as an alkali earth metal with a single weakly bound valence electron, lithium is highly electropositive. As a result, lithium ion batteries can achieve higher operating voltages than alternatives such as nickle-cadmium or lead-acid batteries [2]. Some of these comparative performance advantages to these alter- natives are illustrated in Fig 1.1[2]. The pursuit of further improving the performance of lithium batteries has shifted analysis 1 Figure 1.1: Plot illustrating the the power and energy densities achievable by three types of battery, as well as the minimum requirements for various types of electric vehicle, including hybrid electric vehicles (HEV), plug in hybrids (light vehicle). From Etacheri, 2011 [2]. of these materials increasingly towards measuring microstructure properties [3{5]. Identify- ing microscale features such as crystal structure, diffusion mechanics, and composition is an essential component of characterizing new battery materials [6]. These features are being increasingly analyzed through electron microscopy, which offers the high spatial resolution needed to analyze them and is becoming increasing accessibility [7{9]. Lithium materials however, present a range of challenges to electron microscopy. Battery materials are becoming increasingly intricate and lithium's light weight and ionizable nature make it particularly susceptible to electron beams [10]. This sensitivity limits analysis largely to low dosage techniques to avoid damaging the samples long enough to acquire results. One such technique that is targeted for light elements, such as lithium, is electron energy