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Efficient Theoretical and Numerical Approaches Towards a Dynamical Many-Body Theory

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Efficient Theoretical and Numerical Approaches Towards a Dynamical Many-Body Theory Damping, satellites and multiple excitations in oxides and nanostructures: efficient theoretical and numerical approaches towards a dynamical many-body theory Lorenzo Sponza To cite this version: Lorenzo Sponza. Damping, satellites and multiple excitations in oxides and nanostructures: efficient theoretical and numerical approaches towards a dynamical many-body theory. Materials Science [cond-mat.mtrl-sci]. Ecole Polytechnique (EDX), 2013. English. tel-01883527 HAL Id: tel-01883527 https://hal.archives-ouvertes.fr/tel-01883527 Submitted on 15 Oct 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Thèse présentée pour obtenir le grade de DOCTEUR DE L’ÉCOLE POLYTECHNIQUE par LORENZO SPONZA Damping, satellites and multiple excitations in oxides and nanostructures: efficient theoretical and numerical approaches towards a dynamical many-body theory Thèse soutenue le 30 Octobre 2013 Silke BIERMANN Président Angel RUBIO Rapporteur Valerio OLEVANO Rapporteur Odile STÉPHAN Examinatrice Mark VAN SCHILFGAARDE Examinateur Christine GIORGETTI Co-directrice Lucia REINING Directrice Table of Contents 1 Introduction 1 1.1 An introduction to spectroscopy . .2 1.2 Many-body interactions and dynamical effects . .3 1.3 Dynamical effects in real systems . .4 1.3.1 Photoemission and plasmon satellites . .4 1.3.2 Light absorption and multiple exciton generation . .7 1.4 Multiple exciton generation and solar energy production . .8 I Ab initio theoretical spectroscopy 11 2 Dielectric theory and theoretical spectroscopy 13 2.1 An introduction to dielectric theory . 13 2.1.1 The microscopic dielectric constant . 13 2.1.2 Longitudinal-longitudinal framework . 14 2.1.3 Macroscopic average of microscopic quantities . 15 2.1.4 Micro-macro connection . 16 2.1.5 Why longitudinal framework? . 17 2.2 Connection with some spectroscopic technique: EELS and absorption . 17 2.2.1 Angle-resolved electron energy loss spectroscopy . 17 2.2.2 Light absorption . 18 2.3 Summary . 19 3 Density-Functional Theory 21 3.1 Solving the many-body problem . 21 3.2 Foundations of Density-Functional Theory . 22 3.2.1 The energy functional . 23 3.3 The Kohn-Sham system and the exchange-correlation potential . 24 3.3.1 The Local Density Approximation . 24 3.3.2 Kohn-Sham eigenvalues and eigenfunctions . 25 3.4 Excited states with DFT . 26 3.5 Summary . 27 4 Neutral excitations in Time-Dependent DFT 29 4.1 Spectroscopy and evolution in time . 29 4.1.1 The time-dependent many-body problem . 30 4.2 The foundations of TDDFT . 30 4.2.1 Functionals of the TD-density . 30 4.2.2 A time-dependent Kohn-Sham system . 31 i 4.3 Linear response to a perturbation . 32 4.3.1 The density-density response function . 32 4.4 From the susceptibility to the spectra . 34 4.5 Practical calculations and approximations . 35 4.5.1 Tamm-Dancoff Approximation . 36 4.5.2 Exchange-correlation kernels . 36 4.5.3 The Local Field Effects . 38 4.5.4 Spectra of isolated systems . 39 4.6 Summary . 39 5 Many Body Perturbation Theory 41 5.1 Charged excitations with MBPT . 41 5.1.1 The equations of motion of the Green's function . 43 5.1.2 The quasi-particle equation . 46 5.1.3 Hedin's equations and the GW approximation . 47 5.1.4 GW in practice . 49 5.1.5 A focus on dynamical effects on the one-particle Green's function: GW and the cumulant expansion . 55 5.2 Neutral excitations with MBPT . 58 5.2.1 Approximations to the BSE . 59 5.2.2 Solving the BSE in practice . 60 5.2.3 Going beyond the static screening approximation . 63 II Application to real systems and analysis 69 6 Optical properties of SrTiO3 71 6.1 Preface to SrTiO3 ................................. 71 6.2 Geometry and Crystal field splitting . 72 6.3 DFT and GW band structure . 73 6.3.1 Distribution of quasiparticle corrections . 73 6.3.2 State assignment . 75 6.3.3 Quasiparticle effects on electrons of different character . 76 6.4 Optical absorption . 78 6.4.1 The role of the local field effects . 78 6.4.2 Analysis of the transitions . 80 6.4.3 Localized excitations and the screening of the electron-hole pair . 85 6.4.4 Excitons via TDDFT . 87 6.5 Conclusion . 88 7 Graphite allotropes 91 7.1 Carbon allotropes . 91 7.2 Ground state properties . 94 7.3 Electron energy loss spectra . 95 7.4 Spectral functions . 100 7.5 Conclusion . 105 ii 8 From bulk graphite to isolated graphene 107 8.0.1 The transition from graphite to graphene . 109 8.1 Ground state properties . 111 8.2 Electron energy loss spectra . 113 8.2.1 Parallel momentum transfer . 113 8.2.2 Perpendicular momentum transfer . 115 8.2.3 Conclusions on the EEL spectra . 116 8.3 Spectral functions . 117 8.3.1 Comparison with EEL spectra . 121 8.3.2 Conclusions on the spectral functions . 122 III Developments 125 9 Isolated 2D-systems: Coulomb cutoff 127 9.1 Modifying the interaction . 127 9.1.1 The cutoff Coulomb interaction in 2D . 128 9.1.2 Practical implementation and approximations . 131 9.2 EELS of graphite and graphene using the CC . 136 9.2.1 Parallel momentum transfer . 137 9.2.2 Perpendicular momentum transfer . 138 9.3 Spectral functions of isolated graphene . 139 9.3.1 G0W0 spectra . 140 9.3.2 Conclusions . 142 10 A different(ial) approach to the dynamics of absorption 145 10.1 The linear integro-differential functional equation . 146 10.1.1 Mutual cancellations in the Bethe-Salpeter Equation . 147 10.2 Solving the differential equation . 149 10.3 The bare Coulomb equation (BCE) . 152 10.3.1 Solution of the bare Coulomb equation . 152 10.3.2 Spectrum of the bare Coulomb solution . 153 10.4 The Differential equation . 156 10.4.1 From the differential operator to the dynamical screening . 157 10.4.2 Solution of the Dynamical equation . 157 10.4.3 Discussion of the dynamical contributions . 158 10.4.4 Where dynamical effects are important? . 161 10.5 Conclusions and perspectives . 161 11 Conclusions 165 A Solution by iteration of the bare Coulomb equation 171 B Proof of the solution 173 C Singularity at the origin of the Coulomb cutoff matrix elements of W 175 D Dynamical effects in the Green's function of real materials 177 iii Chapter 1 Introduction This thesis is about dynamical correlation between electrons effects in light absorption. This study is conducted in the framework of theoretical ab-initio spectroscopy. I present here the analysis of real materials spectra as well as the development of new technical and theoretical approaches. When light is absorbed by a material, its energy gives rise to a big variety of different phenomena and is converted into excitations of the system, e.g., the creation of electron- hole pairs, the excitations of collective vibrations (electrons and nuclei), or the emission of particles. Each excitation may be the result of the direct absorption of light, or may be induced by other excitations occurring in the system. One strong point of theoretical spectroscopy is that it allows to select which physics to include in the theory, to neglect at will some process or to focus specifically on others. Ab initio spectroscopy therefore constitutes an indispensable tool to interpret experimental data and to predict the behaviour of a system. Simple models, accounting only for few phenomena, lead to simple interpretations, but are most often expected to give only qualitative trends. Usually a better quantitative agreement comes with much more complicated models which account for the physics of many processes taking place at the same time. Dynamical effects describe the coupling between different excitations of the system and are therefore a clear manifestation of the many-body nature of the electronic processes. They are included in nowadays theoretical approaches only to a limited extent. In the absorption spectrum, dynamical effects are responsible for the reduction of the spectral weight of some structures, for the renormalization of energies, but also they may create new features at higher energy which in many cases constitute a challenge for state-of-the-art theories. Moreover they are potentially important for technological applications, hence a quantitative theory to explain and predict their properties is more and more needed. One example is the multiple exciton generation (MEG), where the absorption of a single high-energy photon is converted into the creation of multiple electron-hole pairs (excitons). Its possible applications are in the photovoltaic technology since the full exploitation of this effect could ensure high currents even at low intensity. This process can be correctly described only by including a dynamical coupling at the level of the electron-hole interaction. In the present work I try to analyse successes and flaws of present theories, and I propose some steps towards more efficient and precise calculations based on methodological and theoretical advances. 1 1.1 An introduction to spectroscopy The general framework of a spectroscopic experiment is depicted in Fig. 1.1. It is composed by a source, a scatterer and an analyser. A perturbing (or external) field created by the source interacts with the scatterer (for us an N-electron system). By effect of the interaction, the field exchanges energy with the system, which consequently undergoes some excitations.
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