Mathematical models and numerical simulation of photovoltaic devices

Abstract : This thesis includes two independent parts, both motivated by mathematical modeling and numerical simulation of photovoltaic devices. Part I deals with cross-diffusion systems of partial differential equations, modeling the evolution of concentrations or volume fractions of several chemical or biological species. We present in Chapter 1 a succinct introduction to the existing mathematical results about these systems when they are defined on fixed domains. We present in Chapter 2 a one-dimensional system that we introduced to model the evolution of the volume fractions of the different chemical species involved in the physical vapor deposition process (PVD) used in the production of thin film solar cells. In this process, a sample is introduced into a very high temperature oven where the different chemical species are injected in gaseous form, so that atoms are gradually deposited on the sample, forming a growing thin film. In this model, both the evolution of the film surface during the process and the evolution of the local volume fractions within this film are taken into account, resulting in a cross-diffusion system defined on a time dependent domain. Using a recent method based on entropy estimates, we show the existence of weak solutions to this system and study their asymptotic behavior when the external fluxes are assumed to be constant. Moreover, we prove the existence of a solution to an optimization problem set on the external fluxes. We present in Chapter3 how was this model adapted and calibrated on experimental data. Part II is devoted to some issues related to the calculation of the electronic structure of crystalline materials. We recall in Chapter 4 some classical results about the spectral decomposition of periodic Schrödinger operators. In text of Chapter 5, we try to answer the following question: is it possible to determine a periodic potential such that the first energy bands of the associated periodic Schrödinger operator are as close as possible to certain target functions? We theoretically show that the answer to this question is positive when we consider the first energy band of the operator and one-dimensional potentials belonging to a space of periodic measures that are lower bounded in certain ness. We also propose an adaptive method to accelerate the numerical optimization procedure. Finally, Chapter 6 deals with a greedy algorithm for the compression of Wannier functions into Gaussian-polynomial functions exploiting their symmetries. This compression allows, among other things, to obtain closed expressions for certain tight-binding coefficients involved in the modeling of 2D materials
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Athmane Bakhta. Mathematical models and numerical simulation of photovoltaic devices. Dynamical Systems [math.DS]. Université Paris-Est, 2017. English. ⟨NNT : 2017PESC1046⟩. ⟨tel-01789637⟩

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