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Th`Ese Présentée Pour Obtenir Le Grade De Docteur De L'ecole Polytechnique Spécialité : Physique

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Th`Ese Présentée Pour Obtenir Le Grade De Docteur De L'ecole Polytechnique Spécialité : Physique Th`ese pr´esent´ee pour obtenir le grade de Docteur de l’Ecole Polytechnique Sp´ecialit´e:Physique - Sciences des Mat´eriaux par Fabien BRUNEVAL Exchange and Correlation in the Electronic Structure of Solids, from Silicon to Cuprous Oxide: GW Approximation and beyond Soutenue le 7 octobre 2005 devant le jury compos´ede Pr. Antoine Georges Pr´esident Pr. Giovanni Onida Rapporteur Dr. Christophe Delerue Rapporteur Dr. Martine Gautier-Soyer Examinatrice Dr. Lucia Reining Directrice Dr. Nathalie Vast Directrice Remerciements Bienquenefaisantapparaˆıtre qu’un seul nom en premi`ere page, un travail de th`ese n’est pas un effort individuel. Je tiens `a exprimer ma gratitudea ` une multitude de personnes qui m’ont accueilli, encourag´e, conseill´e, aid´e, envoy´eenconf´erence, et tout simplement, fait confiance. Ces trois ann´ees pass´ees au Laboratoire des Solides Irradi´es de l’Ecole Polytechnique se sont d´eroul´ees dans une excellente ambiance, en partic- ulier, grˆace `a son directeur Guillaume Petite etason´ ` equipe administra- tive et technique Catherine Julien, Marie-Madeleine Lemartinel, Christine Houy, Bernard Allali et Bernard Jou´eo. Dans l’ordre chronologique, je remercie tout d’abord les deux personnes qui m’ont donn´e envie de me plonger dans la physique num´erique et m’ont accueilli au LSI pour mon stage de DEA et de fin d’´etudes : Nathalie Vast et Silvana Botti. Je remercie mes deux directrices de th`ese, Nathalie Vast et Lucia Rein- ing, pour leur encadrement, mais aussi pour la libert´e dont j’ai joui quant aux orientations scientifiques de mes travaux. Grˆace `a elles, j’ai profit´e d’excellentes conditions de travail et j’ai ´et´eencourag´e`apr´esenter mon ac- tivit´e`a des conf´erences vari´ees. Ces quelques mots sont bien faibles pour exprimer toute ma reconnaissance. Je remercie l’ensemble de l’´equipe th´eorie du LSI, ceux qui restent, ceux qui sont arriv´es et ceux qui sont partis, pour leur aide scientifique, leur com- pagnie, leur amiti´e : (par ordre approximatif d’apparition) Silvana Botti, Valerio Olevano, Francesco Sottile, Marc Hayoun, Olivier Hardouin-Duparc, Andrea Cucca, Hansi Weissker, Philippe Baranek, Virginie Trinit´e, Wojtek Welnic, Jorge Serrano, Louise Dash, Christine Georgetti, Val´erie Veiniard, Jelena Sjakste, et Xavier Andrade-Valencia. J’aurais aim´e dire quelques mots d´edi´es `a chacun, mais pour cela, il aurait fallu rajouter un chapitre complet `a ce manuscript. Je tiens notammenta ` louer la patience des diff´erents occupants de la pi`ece 05 10 29 : Tolis Marinopoulos, Madeleine Meyer, Matteo Gatti et Fran¸coise Maury-Levesque, qui ont support´ema symphonie quotidienne en clavier mineur. Mes coll`egues exp´erimentateurs m’ont permis d’´elargir mon horizon. Je remercie particuli`erement Manuel Izquierdo, Nick Barrett et Fausto Sirotti. Les r´eseaux europ´eens NANOPHASE et NANOQUANTA sonta ` l’origine de rencontres et de collaborations fructueuses avec d’autres chercheurs eu- ii rop´eens, en particulier : Rodolfo Del Sole, Patrick Rinke, Gian-Marco Rig- nanese et Xavier Gonze. Merci ´egalement `a Brice Arnaud pour ses calculs sur Cu2O, mais aussi pour sa sympathie. Je tiens `a exprimer ma reconnaissance au jury de cette th`ese : les rap- porteurs, Christophe Delerue et Giovanni Onida, qui ont accept´e de lire ce manuscript en plein cœur de l’´et´e ; Martine Gautier-Soyer qui a permis de teinter ce jury d’une touche exp´erimentale ; Antoine Georges, enfin, qui m’a fait l’honneur de pr´esider ce jury. La lisibilit´e et la correction de ce manuscript ont ´et´e grandement am´elior´ees par les lectures et relectures de Miguel Marques, Silvana Botti, Lucia Rein- ing et Nathalie Vast. La rigueur et la pertinence de leurs remarques m’ont beaucoup aid´e, aussi bien en ce qui concerne la grammaire anglaise que l’´energie d’ionisation des syst`emes finis. Ces remerciements seraient incomplets, si je ne soulignais le rˆoledema famille, mes parents et mes soeurs. Pour finir, mes pens´ees vont `aAgn`es qui m’a soutenu durant toute ma th`ese, qui a fait l’effort d’apprendre ce qu’´etaient un photon et un electron. Contents Contents iii Notations ix Feynman diagrams xi Introduction 1 I Theoretical background 5 1 Presentation of the tremendous problem of electrons in solids 7 1.1 Electronic Hamiltonian in solids within Born-Oppenheimer approximation.......................... 8 1.2 Wavefunction methods or “integrated” variable methods . 9 1.3Structuralexperiments..................... 10 1.4 Photoemission and inverse photoemission experiments . 12 1.5 Optical measurements and energy-loss experiments . 14 2 Approach to the Many-Body Problem by means of Green’s functions 17 2.1IntroductionofGreen’sfunctions............... 17 2.2 Lehmann representation and physical interpretation . 19 2.3Definitionofthespectralfunction............... 21 2.4Spectralfunctionandphotoemission............. 23 2.5Dysonequationandself-energy................ 24 2.6Non-interactingandinteractingGreen’sfunctions...... 26 2.7Quasiparticlesandspectralfunctions............. 27 2.8Hartree-Fockself-energy.................... 28 3 Hedin’s equations, Bethe-Salpeter equation, and GW ap- proximations 33 3.1 Green’s functions in presence of an external perturbation . 33 iv CONTENTS 3.2Hedin’sequations........................ 35 3.3 GW approximation....................... 39 3.4 Static screened exchange and COHSEX approximation . 40 3.5 Bethe-Salpeter equation for the polarizability ........ 43 3.6 Approximations to the kernel of the Bethe-Salpeter equation 45 4 Approach to the Many-Body Problem by means of the electronic density 49 4.1 Ground-state density as a basic variable ........... 49 4.1.1 Hohenberg-Kohntheorems............... 50 4.1.2 Kohn-ShamapproachtoDFT............. 51 4.1.3 Exchange-correlationpotentials............ 52 4.2Time-dependentdensityasthebasicvariable......... 53 4.2.1 Runge-Gross theorem and time-dependent Kohn-Sham 54 4.2.2 Linear-responseregime................. 55 4.2.3 Exchange-correlationkernels.............. 57 5 Practical implementation of standard GW 61 5.1 Evaluation of W ........................ 61 5.1.1 Representation of ε−1 .................. 61 5.1.2 RPA polarizability χ0 .................. 62 5.1.3 Plasmon-polemodel.................. 64 5.2EvaluationofΣ......................... 64 5.2.1 First-orderperturbation................ 64 5.2.2 Linearizationoftheenergydependence........ 65 5.2.3 Bare exchange part Σx ................. 66 5.2.4 Correlation part Σc ................... 67 6 Connecting DFT and Green’s function theory 69 6.1 Exchange-correlation potentials derived from MBPT . 70 6.1.1 Sham-Schl¨uterequation................ 70 6.1.2 Optimizedeffectivepotentials............. 71 6.2Exchange-correlationkernelsderivedfromMBPT...... 71 6.2.1 Time-dependent optimized effective potentials . 72 6.2.2 Direct comparison between BSE and TDDFT . 72 6.2.3 Perturbative approach of the TDDFT/BSE comparison 74 II Methodological developments 77 7 GW with no plasmon-pole model 79 7.1Methodusingspectralfunctions................ 80 7.1.1 Formula for the imaginary part of the self-energy . 80 7.1.2 Practicalimplementation................ 82 7.1.3 Results for Silicon .................... 83 CONTENTS v 7.2Methodusingcontourintegrals................ 84 7.2.1 Descriptionofthemethod............... 84 7.2.2 Practicalimplementation................ 86 7.2.3 Results for bulk Silicon ................. 89 8 Self-consistent quasiparticle calculations 91 8.1 Looking for a better starting point for GW .......... 91 8.2RestrictedLDAbasisset.................... 92 8.3 Testing the code against published data ............ 95 8.4Somehistoricalapproximations................ 97 8.5Effectofself-consistencyonoperators.............100 8.6 COHSEX+GW .........................101 8.7 Self-consistent GW bandstructure..............103 III Theoretical developments 105 9 Derivation of TDDFT kernels from MBPT 107 9.1 Time-dependent Sham-Schl¨utercondition...........107 9.2 Integral equation for δΣ/δρ ..................109 9.3First-orderTDDFTkernel...................111 9.3.1 LinktoTD-OEP:exact-exchange...........111 9.3.2 Linktomorerealisticapproximations.........113 9.4Second-orderTDDFTkernels.................114 10 How to include vertex corrections using density-functional concept 119 10.1Density-functionalconcept...................120 vLDA 10.2 Practical applications: initiating with Σ = xc .......122 10.3 Practical applications: initiating with Σ = ΣCOHSEX .....123 IV Electronic excitations of Cu2O 127 11 Presentation of Cu2O 129 11.1Structure............................129 11.2Cu-Cubonding.........................130 11.3Excitonseries..........................131 12 DFT study of Cu2O 135 12.1Issueofsemicorestatesofcopper...............135 12.2DFTtheoreticalstructure...................137 12.3Kohn-Shambandstructure...................140 12.3.1Technicalchecks....................140 12.3.2Symmetryanalysis...................142 vi CONTENTS 12.3.3Projectedandtotaldensities-of-states.........143 12.3.4 Band structure features and overview of other works 147 12.4DFTchargedensity.......................149 13 Standard GW study of Cu2O 153 13.1 Importance of semi-core states for the GW Self-energy . 153 13.2 GW quasiparticlebandstructure...............155 13.3 Failure of GW approximation or failure of additional assump- tions?..............................156 13.3.1 Reliability of pseudo eigenvalues and eigenvectors . 157 13.3.2 Reliability of the plasmon-pole model .........159 13.3.3 Reliability of first-order perturbation technique . 161 13.3.4 Reliability of LDA
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