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Thesis (doctoral)--Göteborg University, 2008. / Includes bibliographical references.

Neste trabalho usamos a teoria de Thomas-Fermi para estudar as propriedades eletrônicas de semicondutores planarmente dopados, ou delta-dopados, com densidade de dopantes de moderada a alta. O principal objetivo do trabalho é a verificação de que esta teoria apresenta muito bons resultados com os do método auto-consistente na aproximação de Hartree quando aplicada a este tipo de problema. Verificamos que muitas situações físicas relacionadas a semicondutores delta-dopados podem ser descritas de uma maneira simples e com muito bons resultados. Estudamos o problema de um poço isolado e o problema da super-rede, comparando os resultados de Thomas-Fermi e Hartree. / In this work we have used the Thomas-Fermi theory to study the electronic properties of planar doped semiconductors, or delta-doping, with a moderate to high density of dopants. The main aim of this work is to verify that this theory gives very good results when compared with the self-consistent method in the Hartree aproximation. We have checked that many physical situations related to delta-doping can be described in a simple manner and also with very good results. We have studied the single delta problem and the superlattice problem and we have compared the Thomas-Fermi´s and Hartree´s results.

Neste trabalho usamos a teoria de Thomas-Fermi para estudar as propriedades eletrônicas de semicondutores planarmente dopados, ou delta-dopados, com densidade de dopantes de moderada a alta. O principal objetivo do trabalho é a verificação de que esta teoria apresenta muito bons resultados com os do método auto-consistente na aproximação de Hartree quando aplicada a este tipo de problema. Verificamos que muitas situações físicas relacionadas a semicondutores delta-dopados podem ser descritas de uma maneira simples e com muito bons resultados. Estudamos o problema de um poço isolado e o problema da super-rede, comparando os resultados de Thomas-Fermi e Hartree. / In this work we have used the Thomas-Fermi theory to study the electronic properties of planar doped semiconductors, or delta-doping, with a moderate to high density of dopants. The main aim of this work is to verify that this theory gives very good results when compared with the self-consistent method in the Hartree aproximation. We have checked that many physical situations related to delta-doping can be described in a simple manner and also with very good results. We have studied the single delta problem and the superlattice problem and we have compared the Thomas-Fermi´s and Hartree´s results.

In the rst part of this thesis we discuss some details of properties of graphene and we explain the tight-binding approach to nd the energy spectrum in graphene. In the second part of the thesis, we solve a nonlinear integral equation for the electrostatic potential in doped graphene due to an external charge, arising from a Thomas-Fermi (TF) model for screening by graphene's electron bands. In particular, we study the e ects of a nite equilibrium charge carrier density in graphene, non-zero temperature, non-zero gap between graphene and a dielectric substrate, as well as the nonlinearity in the band density of states. E ects of the exchange and correlation interactions are also brie y discussed for undoped graphene at zero temperature. Results from the nonlinear model are compared with results from both the linearized TF model and the dielectric screening model within the random phase approximation (RPA). In addition, the image potential of the external charge is evaluated from the solution of the nonlinear integral equation and compared to the results of linear models. We have found generally good agreement between the results of the nonlinear TF model and the RPA model in doped graphene, apart from Friedel oscillations in the latter model. However, relatively strong nonlinear e ects in the TF model are found to persist even at high doping densities and large distances of the external charge.

In the rst part of this thesis we discuss some details of properties of graphene and we explain the tight-binding approach to nd the energy spectrum in graphene. In the second part of the thesis, we solve a nonlinear integral equation for the electrostatic potential in doped graphene due to an external charge, arising from a Thomas-Fermi (TF) model for screening by graphene's electron bands. In particular, we study the e ects of a nite equilibrium charge carrier density in graphene, non-zero temperature, non-zero gap between graphene and a dielectric substrate, as well as the nonlinearity in the band density of states. E ects of the exchange and correlation interactions are also brie y discussed for undoped graphene at zero temperature. Results from the nonlinear model are compared with results from both the linearized TF model and the dielectric screening model within the random phase approximation (RPA). In addition, the image potential of the external charge is evaluated from the solution of the nonlinear integral equation and compared to the results of linear models. We have found generally good agreement between the results of the nonlinear TF model and the RPA model in doped graphene, apart from Friedel oscillations in the latter model. However, relatively strong nonlinear e ects in the TF model are found to persist even at high doping densities and large distances of the external charge.

Universiẗat, Diss., 2002--München.

A new approach is developed for evaluating the mixing properties of binary solutions at high pressure. This involves solving Poisson's equation throughout three-dimensional cubic lattices, consistent with Thomas-Fermi-Dirac (TFD) theory. Zero temperature calculations are carried out for a variety of compositions and crystal structures in 3 pressure groups relevant to Jovian planetary interiors. Pseudopotentials based on the two-component-plasma model (with a uniform electron background) are fitted to the solid-state results, and are then used in liquid-state calculations using hard-sphere perturbation theory. TFD results for H-He solutions find critical temperatures (above which all compositions are soluble) to be ∿ 0, 500, and 1500°K at pressures of 10, 100, and 1000 Mbar, respectively. These temperatures are much lower than those obtained using free electron perturbation theory, where T(crit) ∿ 10,000°K at 10 Mbar. Thus, unlike the perturbation theory results, the TFD results predict that helium should be soluble in metallic hydrogen in the deep interiors of both Jupiter and Saturn, and our calculations give an indication of the degree of model-dependence in computing high pressure mixing properties. In addition, TFD calculations for H-C and H-O solutions find phase separation temperatures to be≲ 10⁴ °K for pressures ≲ 10³ Mbar. These temperatures are considerably lower than those found assuming a uniform electron distribution (where T(crit) ≳ 10⁵ °K), and suggest that H-C and H-O solutions should also be miscible in the metallic zones of Jupiter and Saturn.

Cette thèse présente l'étude de divers problèmes mathématiques en modélisation des solides, tant à l'échelle atomique qu'à l'échelle macroscopique. Les modèles correspondants sont très simplifiés, mais présentent tout de même des comportements qualitatifs acceptables, et permettent, du fait de leur simplicité, de pousser l'analyse mathématique plus loin que dans le cas de modèles plus réalistes.<br /><br />Une première partie (chapitres 2,3,4) est consacrée à l'étude de l'origine de la structure cristalline. Ce problème peut être posé de la façon suivante : les modèles étudiés ici rendent-ils compte du fait qu'à température nulle, la matière est ordonnée ? ou, de façon équivalente, l'état de minimum d'énergie de N atomes identiques ressemble-t-il, pour N grand, à une structure périodique ? Ce type de problème est relié au problème de limite thermodynamique, dont certains aspects sont également étudiés ici.<br /><br />Dans un deuxième temps, nous étudions au chapitre 5 le cas où précisément, la matière n'est pas ordonnée : dans le cas d'un système périodique, il est possible de définir l'énergie du système pour les modèles utilisés ici par le processus de limite thermodynamique. Nous étudions ce même processus dans un cas non-périodique, donnant des hypothèses générales qui permettent de mener à bien une telle étude.<br /><br />Les chapitres 6 et 7 sont consacrés à l'étude du lien possible entres des théories macroscopiques des solides et ces modèles microscopiques, le premier dans le cas de comportements mécaniques, le deuxième dans le cas du comportement en présence d'un champ électrique.<br /><br />Enfin, le dernier chapitre présente une brève introduction à certaines techniques utilisées en numérique des solides, pour des modèles beaucoup plus élaborés que ceux des chapitres précédents.

In this work, we have implemented an orbital-free density functional theory (OF-DFT) solver using the finite element method. In OF-DFT, the total ground state energy is minimized directly with respect to the electron density, rather than via orbitals like in the standard Kohn-Sham approach. For this to be possible, one needs an approximation of a universal density functional of the non-interacting kinetic energy. Presently available approximations allow for computation with very low computational expense, but which gives inaccurate energies. A stable OF-DFT code can be used as a testbed for new kinetic energy functionals and provide the necessary tool for investigating the accuracy of OF-DFT calculations for complex systems. We have implemented Thomas-Fermi theory with and without nuclear cusp condition, as well as additional exchange terms of Dirac and Amaldi. The program uses an extended version of the steepest descent in order to find the minimizing density in the variational principle. Our results include convergence tests for the hydrogen atom, weak bonding in the H2 molecule, and accurate results for the lightest noble gases (He, Ne, Ar). For heavier atoms (Kr, Xe, Rn), the results are less accurate. In addition, we consider hydrogen in the simple cubic structure without the cusp condition, which is a first attempt to use the code for periodic systems. Lastly, we discuss some possible improvements for the iterative process towards the minimizing density, as well as other possible directions for future development.