From the Bethe-Salpeter Equation to Time-Dependent Density Functional Theory
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Th`esepr´esent´eepour obtenir le grade de DOCTEUR DE L’ECOLE´ POLYTECHNIQUE par FRANCESCO SOTTILE Response functions of semiconductors and insulators: from the Bethe-Salpeter equation to time-dependent density functional theory Soutenue le 29 Septembre 2003 devant le jury compos´ede : M. Friedhelm BECHSTEDT Rapporteur M. George BERTSCH M. Xavier BLASE Rapporteur M. Fabio FINOCCHI M.me Lucia REINING Directeur M. Jean-Claude TOLEDANO Directeur Laboratoire des Solides Irradi´esUMR 7642 CNRS-CEA/DSM, Ecole´ POLYTECHNIQUE, F-91128 Palaiseau, France Francesco Sottile [email protected] Response functions of semiconductors and insulators: from the Bethe-Salpeter equation to time-dependent density functional theory Pdf file available at the page http://theory.polytechnique.fr/people/sottile/Tesi dot.pdf 2003 This manuscript has been conceived only using free software: LATEX, Ghostscript, Kile. Figures and graphs were created with Xfig, The Gimp, Grace, PStoEdit and GNUPlot. Acknowledgments Le Laboratoire des Solides Irradi´esde l’Ecole´ Polytechnique, dirig´epar Guillaume Petite, m’a accueilli avec la plus grande sympathie et je tiens `aleur exprimer `atous ma reconnaissance. I would like to express my profound gratitude to my supervisor, Lucia Reining, for supporting, guiding, and working very closely with me for the last three years. In par- ticular, I will never forget our discussions at 7.30 in the morning. This thesis owes its existence to Lucia. I am very grateful to Prof. Toledano who agreed to be my supervisor. He left me free to work as I preferred, and was always available to help me. I would like to thank all those who I have collaborated with: first of all, Valerio Ole- vano, for providing the computational codes I worked on, and for the endless discussions on kernels, fortran, latex and home-cinema; Angel Rubio, who worked with me at the very beginning of the thesis, on finite systems; Giovanni Onida, Andrea Marini and Rodolfo Del Sole, for useful discussions about kernels; Ferdi Aryasetiawan and Krister Karlsson, for a fruitful work on model kernels we have been carrying out together. I acknowledge my gratitude also to the NANOPHASE Network, to which I have had the privilege to belong. My scientific background also originated from the several meetings we have had in these years. Let me take this opportunity to thank the European Community, but also the three organisms which the laboratory depends on (CNRS, CEA, Polytechnique), for covering living and travel expenses in many occasions, and also Marie Madeleine Lemartinel and Catherine Julien for helping me to get along with all these administrative things. Je tiens `aremercier chaleureusement toute l’´equipe th´eorique du laboratoire, pour toutes nos discussions et pour le temps pass´eensemble : Silvana Botti, Apostolos Ma- rinopoulos, Nathalie Vast, Fabien Bruneval, Andrea Cucca, Louise Dash and Virginie Quequet ; Marc Hayoun et Olivier Hardouin Duparc pour avoir ´ecout´emes expos´eset m’avoir ´egalement aid´eavec le fran¸cais ; en particulier Olivier, avec qui j’ai partag´ele bureau dans une tr`esbonne ambiance. I’m deeply indebted to my university supervisor, Prof. Pietro Ballone, who taught me a lot, and introduced me into the Palaiseau group. My gratitude goes to the Rapporteurs, Friedhelm Bechstedt and Xavier Blase, who accepted the work of examining and reporting; I’m also grateful to the Examinateurs, George Bertsch for chairing the jury and Fabio Finocchi, who also gave me the possibility to teach in a DEA for the first time. Un grazie sincero a tutti gli amici “parigini”, per aver trascorso bei momenti di svago insieme, molti dei quali a tavola. Ai miei genitori, Nuccia e Gianni, devo tutta la mia gratitudine, per tutto l’aiuto e l’incoraggiamento avuto da parte loro, non importa quanto lontano stiano. Finally my deepest gratitude to my girlfriend, Daniella, for her understanding and love. Her support and encouragement has made this work possible. Francesco Contents List of abbreviations, acronyms and symbols iii Jungle of Polarizabilities iv Introduction 1 I Background 5 1 Dielectric and optical properties in solids 7 1.1 Complex dielectric function and complex conductivity . 8 1.2 Complex refraction index and absorption coefficient . 10 1.3 Electron Energy Loss . 13 1.3.1 Fundamental relations . 13 1.3.2 Energy loss by a fast charged particle . 15 1.4 Microscopic-Macroscopic connection . 17 1.5 Electronic Spectra . 18 1.5.1 Alternative formulation for the spectra . 19 2 Density Functional Theory 21 2.1 The many-body problem . 21 2.1.1 The Born-Oppenheimer approximation . 22 2.1.2 The electronic problem . 24 2.2 The Hohenberg and Kohn theorem . 25 2.3 The Kohn Sham method . 29 2.4 Local Density Approximation (LDA) . 32 2.5 Beyond LDA . 33 2.6 Excited States in DFT . 34 2.6.1 ∆SCF . 35 2.7 Electronic Spectra in KS-DFT . 36 i Contents 2.8 DFT in solids . 39 2.9 Technical details of the ground state calculations in this thesis . 40 3 Green functions approach 41 3.1 Quasi-particle formulation . 42 3.1.1 Green functions . 43 3.1.2 Hedin’s pentagon . 47 3.1.3 Real calculations and GWA . 49 3.2 Response functions . 52 3.3 Bethe-Salpeter Equation . 55 3.3.1 Effective two-particle equations . 56 3.3.2 Ingredients and approximations . 60 4 Time Dependent DFT 65 4.1 The problem . 65 4.2 TD Density Response Functional Theory . 67 4.3 Adiabatic Local Density Approximation . 69 4.4 Excited States in TDDFT . 69 4.5 Electronic Spectra in TDDFT - Application to solids . 71 4.6 Electronic Spectra in TDDFT - Finite systems . 74 5 BSE & TDDFT: 77 II Developments and Applications 81 6 Role of the Coulomb interaction v 83 6.1 The local fields:v ¯ ................................ 84 6.2 Absorption and EELS - Towards finite systems . 86 6.2.1 A simple model: “one-pole model” . 89 6.2.2 Importance of the limit q → 0..................... 90 6.3 Absorption versus EELS: Numerical analysis . 91 6.3.1 Case of silicon: an “adiabatic connection” . 93 6.4 Concluding remarks . 95 7 role of Ξ 97 7.1 Introduction . 98 7.2 The contact exciton . 99 7.2.1 Technical details of the calculation of the spectra . 99 7.2.2 BSE for variable interaction strength and radius: results . 100 7.3 Links between the Bethe-Salpeter equation and TDDFT . 103 7.3.1 The contact exciton and the TDDFT kernel . 103 ii Contents 7.3.2 Continuum excitons: numerical results for different TDDFT kernels 105 7.3.3 The TDDFT kernel and bound excitons . 105 7.4 Concluding remarks . 109 8 TDDFT: parameter-free exchange-correlation kernel from the BSE 111 8.1 The theory . 112 8.1.1 When the assumption is wrong... 116 8.1.2 ...and when it is, instead, straightforward . 117 8.2 Technical analysis . 118 8.2.1 The term T1 ...............................118 8.2.2 The term T2 ...............................120 8.2.3 The (¿useless?) kernel fxc . 120 8.3 The problems of χ0, cautions and tricks. 126 9 Absorption spectrum of semiconductors 129 9.1 Absorption spectra of Silicon ... 130 9.2 ... and of Silicon Carbide . 132 9.3 Concluding remarks . 134 10 Bound excitons in TDDFT 137 10.1 BSE result for solid argon . 137 10.1.1 k-points sampling . 139 10.2 TDDFT description of Argon . 141 10.2.1 Diagonal contribution of T2: a problem. 144 10.2.2 Comparison with experiment . 147 10.3 Results of other groups . 149 10.4 Concluding remarks . 149 Conclusions 151 Appendices 155 A Linear Response Theory 157 A.1 The full polarizability . 158 A.2 The independent-particle polarizability . 159 B Dyson-like equation for the macroscopic dielectric function εM 161 C EELS versus Abs: LFE included 163 iii Contents D Size effects in finite systems 165 E Technical details 167 E.1 Linear system solver . 167 E.2 The k sampling of the Brillouin zone . 168 List of publications 169 References and Literature 171 Index 185 iv List of abbreviations, acronyms and symbols Eq.(s) equation(s) Tab. table Fig. figure Ref.(s) reference(s) Par. paragraph exp. experiment BZ Brillouin zone (TD)DFT (time dependent) density functional theory BSE Bethe-Salpeter equation MBPT many-body perturbation theory SCF self-consistent field RPA random phase approximation (A)LDA (adiabatic) local density approximation HK Hohenberg-Kohn HF Hartree-Fock KS Kohn-Sham GWA GW approximation for the self-energy GW-RPA RPA polarizability with (GW) corrected eigenvalues x − c exchange-correlation e − h electron-hole i one-particle eigenvalues Ei quasi-particle eigenvalues φ one-particle wavefunctions Ψ quasi-particle wavefunctions ϕ many-body wave function v ¯ χ χ ¯ v v ˜ ˜ χ χ + + ˜ ˜ χ χ = = ¯ χ χ ) ω ( R A) cts RP (TD)DFT effe in ox. retarded x-c (13) (appr no (irreducible) (reducible) P = ¯  0 ↓ ˜   Â; (1133) ⇒ L ctions e ¾ ¡ } ) orr ω c } reducible ( ) irreducible Σ ω T ( T ={ no | ω ω | <{ ˜ = = 0 GW   function } } t v v  ) ) ω ω oin ( ( + ¡ R R 1 1 2-p <{ ={ to = = ½ 1 ⇐ − R R " " reduction (42) G ˜ P P (13) v A) v olarizabilities iG (2) + ¡ = G P 1 1 0 cts − L GW-RP 0 = = L effe 1 ) of ox. e-h) = ω − T T ( e-h t " " L T (irreducible) (appr (reducible) no enden ¯ 0 P MBPT ˜ ↓ ; P P in P vi (indep (irreducible) (reducible) time-ordered Jungle " ¯ ¯ P P L 0 ¯ v v ˜ L ˜ ˜ L P P L; + + ˜ ˜ P P ¯ = = L L ¯ ¯ v v function P P ˜ ˜ L L + + ˜ ˜ L L = = ¯ L L dielectric the tities tities in quan quan t t oin oin 2-p 4-p Introduction ¨î¡´ ¬îââ´¤ ¬¤¥´ from West Gate of Moria Inscription Electronic excitations, caused for example by irradiation with electrons, light or mod- ern photon sources (synchrotron, ultra-fast lasers), are key quantities for the study of materials, ranging from solids to atoms, from surface to nanoscale systems.