Echange et Corrélation dans la Structure Electronique des Solides, du Silicium à l'Oxyde Cuivreux: Approximation GW et au-delà

Electronic structure of crystals is a polymorphic name that covers a wide range of properties of electrons in periodic solids. For instance, it can refer to the probability to find an electron at space point r, in other words, to the electronic density. It can refer to the energy needed to extract an electron of the material, the ionization energy, or alternatively, the energetic gain when an electron is added to the system, the electron affinity. It can also stand for the response of the electrons of the solid to an external perturbation (a photon or a fast electron). All these properties characterize the electronic structure of the solid. They describe indeed different aspects. Some of them are ground-state properties, others correspond to excited states. Some of them conserve the number of particles, others do not. As a consequence, the properties that are generically called “electronic structure” are measured with distinct experimental setups, e.g. direct and inverse photoemission, optical absorption, electron energy-loss... And analogously, the theoretical description and prediction of these properties require distinct frameworks. This thesis work will handle the issue of the electronic structure of the cuprous oxide, Cu2O. Therefore, different theoretical methods will be used and results will be compared to a wide range of experimental techniques. The present manuscript contains different parts, which may seem at firstsight independent. It is therefore instructive to draw here quickly, what has been the historical development of this work, from the original project tothe final achievements. Three years ago, I started this thesis work with the purpose to describetheoretically the electronic structure of copper oxides, begining with the simplest one Cu2O, going to the antiferromagnetic CuO, and possibly to the CuO2 planes of high Tc superconducting cuprates. The cuprous oxide, Cu2O, with its closed electronic d shell was thought as a starting point to initiate the study of the Cu–O bonding in the different oxides. Cu2O is alsointeresting because spectacular excitonic series have been measured in its optical absorption or reflectivity spectra during the sixties [1–3]. However, it has been quickly clear that the usual electronic structuremethods, as density-functional theory (DFT) or even the state-of-the-art GW approximation of the many-body perturbation theory (MBPT), were unable to give a proper description of Cu2O. Comparison with existing photoemission or optical measurements were surprisingly bad. This unexpected failure changed the aims of my thesis, which turned into the analysis andthe cure of the shortcomings of the theoretical methods applied to Cu2O. I have been motivated to develop a number of theoretical and technical toolsthat, as I hope, will also be useful in future studies of other materials. The failure of the state-of-the-art GW approximation could come from two distinct reasons: either the current implementation of the method that worked well with simpler materials uses further assumptions, that are not anymore valid with the complex oxide Cu2O; or the GW approximationitself is not enough to account for the electronic structure of Cu2O. In order to check which hypothesis was the right one, I had first to identify, analyse and avoid all further approximations used in a standard GW, as it has most often been performed for 20 years. This part required some code and method development in order to remove the technical approximations that were used to make calculations easier, in our code or in most existing codes. This is what part II of this manuscript is concerned with. Ofcourse, the implementation of new pieces of code had to be checked on simple textbook examples, before being applied to copper oxide. That is why part II provides many results on bulk silicon and solid argon. This methodological part aims at removing the single plasmon-pole approximation that models the dynamical behavior of the screened Coulomb interaction W in GW. It has also the purpose of going beyond the usual perturbative evaluation of GW and to perform real self-consistent calculations within GW or within simpler approximations. The application of these developments to cuprous oxide are postponed to part IV, where all results concerning Cu2O are gathered. Alternatively, if the failure of the GW approximation were really a breakdown of the first-order perturbation theory in W (the coupling constantof the perturbation procedure), one would have to include further correcting terms to improve the results: these terms are commonly called “vertex corrections”. Due to their complexity, there is no unique method in literature to approximate them. Having in mind the purpose of applying vertex corrections to cuprous oxide, I had, first of all, the general task to define a proper scheme to do that, and to provide meaningful approximations to it. The basic idea was to start from the earlier developmentsmade in our group concerning the comparison between time-dependent DFT (TDDFT) and MBPT [4–6] in order to simplify the otherwise untractabletask of calculating vertex corrections. Deep insight in the respective role of the two theories was required and the study finally ended with theoreticalachievements that went farther than the initial project. Part III exposes first the advances made in the understanding of the link between TDDFT and MBPT with a new simple equation that derives the crucial, but unfortunately unknown, kernel of TDDFT from the central quantity of MBPT, the self-energy. Second, part III shows how the same kind of ideas can be used in the other direction (namely, use TDDFT in order to progress within MBPT), to make the calculation of vertex corrections easier. These developments were applied in practice to simple materials: once again bulk silicon and solid argon. In fact, even though the derived vertex correction are “simpler”, the calculations remain orders of magnitude more complicated than the usual GW ones. It is still out of reach to apply these developments to cuprous oxide at present. Nevertheless, this derivation and study of vertex corrections allowed me to draw the general conclusion that strong cancellations between vertex corrections occur.The last part of the present work (part IV) presents all my results concerning Cu2O, from ground-state DFT studies to new self-consistent GW results, and also from theoretical data to experimental measurements. This part starts with a standard electronic structure study, first DFT geometricalstructure, Kohn-Sham band structure and characterization of the orbitals, then standard perturbative GW evaluation of the quasiparticle bandstructure. As said earlier, this study unexpectedly fails, in particular, for the band gap and the optical threshold. That is why the methods developpedin the previous parts are indeed needed in the study of Cu2O. Moreover, existing valence band photoemission experiments did not allow one to detect some of the states found in our and previous bandstructure calculations. However, the experiments were performed on polycrystalline samples and therefore yielded spectra resulting from an integration over the whole Brillouin zone. This was the motivation to apply for beamtime at the synchrotron Elettra in Trieste, Italy together with my collaborators (theoreticians and experimentalists). The purpose was to obtain precise angle-resolved photoemission spectra of the valence states of Cu2O.This means k-point resolved information. After a one-week experimental shift at Elettra, an important part of the present work was to compare our state-of-the-art measurements to the theoretical data, taking into account the experimental aspects, as photoemission cross-sections, evaluation of the causes of experimental uncertainties... Indeed, it turned out that a careful comparison of state-of-the-art experimental and theoretical approaches could remove all existing or seeming contradictions, within the remaining uncertainty of the respective approaches. This is a significant part of the present study of the electronic structure of Cu2O. This part ends with the calculation of energy-loss and optical absorption spectra of cuprous oxide, the latter being now also in good agreement with experiment. The theories and methods used throughout this text are extensively described in the first part of the manuscript. It seemed important to me to provide an accurate account for the theoretical background, because most of the present work deals with improvements of existing methods, or going beyond some piece of the theory. In order to explain the achievements, one first needs to have clearly in mind the existing grounds. Let me then open this text with the part concerning the theoretical background my work is based on.La structure électronique des cristaux est un nom polymorphe qui couvre une large gamme de propriétés des électrons dans les solides périodiques. Par exemple, il peut se référer à la probabilité de trouver un électron au point r de l'espace, en d'autres termes, à la densité électronique. Il peut faire référence à l'énergie nécessaire pour extraire un électron du matériau, l'énergie d'ionisation, ou encore, le gain énergétique lorsqu'un électron est ajouté au système, l'affinité électronique. Il peut aussi se lire la réponse des électrons du solide à une perturbation externe (un photon ou un électron rapide). Toutes ces propriétés caractérisent la structure électronique du solide. Ils décrivent les aspects bien différents. Certains d'entre eux sont propriétés de l'état, d'autres correspondent à des états excités. Certains d'entre eux de conserver le nombre de particules, d'autres pas. En conséquence, les propriétés qui sont généralement appelés «structure électronique» sont mesurées avec différentes configurations expérimentales, par exemple photoémission directe et inverse, l'absorption optique, de l'énergie d'électrons de perte ... Et de manière analogue, la description théorique et la prévision de ces propriétés requièrent cadres distincts. Ce travail de thèse se chargera de la question de la structure électronique de l'oxyde cuivreux, Cu2O. Par conséquent, différentes méthodes théoriques seront utilisées et les résultats seront comparés à un large éventail de techniques expérimentales