Nesta tese são estudados alguns tópicos sobre otimização ergódica e formalismo termodinâmico que generalizam resultados, de Contreras, Lopes e Thieullen (2006), Garibaldi e Lopes (2008) no primeiro caso e Baraviera, Lopes e Thieullen (2006), Bissacot, Mengue e Pérez (2006) no segundo, para contextos onde não existem medidas de Gibbs, ou, em outras palavras, não é satisfeita a propriedade BIP. É demonstrada a existência de subações calibradas para potenciais coercivos de variação finita em espaços shift transitivos de alfabeto enumerável. O método usado é a construção da barreira de Peierls nesse contexto. Provam-se algumas das propriedades da barreira de Peierls e, como consequência das construções, é mostrada uma classificação dos shifts que possuem subações calibradas e limitadas. Posteriormente é realizado um estudo do formalismo termodinâmico para potenciais somáveis de variação finita e pressão finita com medida maximizante única f em shifts topologicamente mixing. Fazendo uso dos resultados de Freire e Vargas (2015), são estudadas a famlia de estados de equilbrio correspondente com f e a famlia de funções 1/B log h_ B , onde h_B são auto vetores do operador de Ruelle para Bf . É demonstrado que os pontos de acumulação quando B vai para infinito são subações uniformemente contnuas. Finalmente é provada uma propriedade dos grandes desvios para a famlia de estados de equilbrio \\mu_B com hipóteses sobre a convergência de uma famlia de funções g_B que normaliza o operador de Ruelle para cada B> 1 (Veja seção 4.4) / In this thesis, the study of topics on ergodic optimization and thermodynamic formalism for countable Markov shifts is presented. It provides a generalization of the previous results, in Contreras, Lopes and Thieullen (2006), Garibaldi and Lopes (2008) for the first subject and Baraviera, Lopes and Thieullen (2006), Bissacot, Mengue e Pérez (2006) for the second one, to situations where there are no Gibbs measures, ie, the BIP property is not verified. The existence of calibrated subactions for coercive potentials with finite variation over transitive countable Markov shifts is proved. The method is based on the construction of the Peierls barrrier in this context. Some properties of the Peierls barrier are proved and, as consequence of the proof, a classification of the Markov shifts which support calibrated and limited subactions is shown. Subsequently, the thermodynamic formalism for topologically mixing Markov shift and summable potentials with finite variation, finite pressure and unique maximizing measure f is studied. Using results in Freire and Vargas (2015), the class of equilibrium states corresponding with f and the class of functions 1/ log h_B are studied where h_B are the eigenfunctions for the Ruelle operator. It is proved that its accumulation points, as goes to infinity, are uniformly continuous subactions. Finally, it is proved a large deviation principle for the equilibrium states family \\mu_B , assuming a hypothesis about the convergence in a family of functions that normalizes the Ruelle operator (See section 4.4 for more details).

Nesta tese são estudados alguns tópicos sobre otimização ergódica e formalismo termodinâmico que generalizam resultados, de Contreras, Lopes e Thieullen (2006), Garibaldi e Lopes (2008) no primeiro caso e Baraviera, Lopes e Thieullen (2006), Bissacot, Mengue e Pérez (2006) no segundo, para contextos onde não existem medidas de Gibbs, ou, em outras palavras, não é satisfeita a propriedade BIP. É demonstrada a existência de subações calibradas para potenciais coercivos de variação finita em espaços shift transitivos de alfabeto enumerável. O método usado é a construção da barreira de Peierls nesse contexto. Provam-se algumas das propriedades da barreira de Peierls e, como consequência das construções, é mostrada uma classificação dos shifts que possuem subações calibradas e limitadas. Posteriormente é realizado um estudo do formalismo termodinâmico para potenciais somáveis de variação finita e pressão finita com medida maximizante única f em shifts topologicamente mixing. Fazendo uso dos resultados de Freire e Vargas (2015), são estudadas a famlia de estados de equilbrio correspondente com f e a famlia de funções 1/B log h_ B , onde h_B são auto vetores do operador de Ruelle para Bf . É demonstrado que os pontos de acumulação quando B vai para infinito são subações uniformemente contnuas. Finalmente é provada uma propriedade dos grandes desvios para a famlia de estados de equilbrio \\mu_B com hipóteses sobre a convergência de uma famlia de funções g_B que normaliza o operador de Ruelle para cada B> 1 (Veja seção 4.4) / In this thesis, the study of topics on ergodic optimization and thermodynamic formalism for countable Markov shifts is presented. It provides a generalization of the previous results, in Contreras, Lopes and Thieullen (2006), Garibaldi and Lopes (2008) for the first subject and Baraviera, Lopes and Thieullen (2006), Bissacot, Mengue e Pérez (2006) for the second one, to situations where there are no Gibbs measures, ie, the BIP property is not verified. The existence of calibrated subactions for coercive potentials with finite variation over transitive countable Markov shifts is proved. The method is based on the construction of the Peierls barrrier in this context. Some properties of the Peierls barrier are proved and, as consequence of the proof, a classification of the Markov shifts which support calibrated and limited subactions is shown. Subsequently, the thermodynamic formalism for topologically mixing Markov shift and summable potentials with finite variation, finite pressure and unique maximizing measure f is studied. Using results in Freire and Vargas (2015), the class of equilibrium states corresponding with f and the class of functions 1/ log h_B are studied where h_B are the eigenfunctions for the Ruelle operator. It is proved that its accumulation points, as goes to infinity, are uniformly continuous subactions. Finally, it is proved a large deviation principle for the equilibrium states family \\mu_B , assuming a hypothesis about the convergence in a family of functions that normalizes the Ruelle operator (See section 4.4 for more details).

Nous avons réalisé des calculs de structure électronique ab initio, basés sur la théorie de lafonctionnelle de la densité (DFT), pour étudier les propriétés des dislocations vis h111i dansles métaux de transition cubiques centrés (V, Nb, Ta, Mo, W et Fe). Dans tous ces éléments,le coeur facile non-dégénéré est la configuration d’énergie minimale et la configuration de coeurdissociée a une énergie très élevée, comparable ou plus élevée que celle du coeur difficile, encontradiction avec les prédictions des potentiels interatomiques. Nous avons mis en évidence destendances de groupe marquées sur l’énergie de coeur de la dislocation facile, reliées à la positiondu niveau de Fermi par rapport au minimum du pseudo-gap de la densité d’états électroniques.Notre travail fait aussi apparaitre un comportement atypique du fer, avec une énergie relativedu coeur difficile basse, proche de celle du point col entre deux coeurs faciles, conduisant à unpotentiel de Peierls plat autour de la configuration difficile, contrairement aux autres éléments.A partir de ces calculs DFT, nous avons construit le paysage énergétique à deux dimensionsdans le plan {111} (potentiel de Peierls) et nous avons étudié plusieurs propriétés relativesau glissement des dislocations, et en particulier l’énergie de formation de la paire de décrochementset la dépendance de la contrainte de Peierls en fonction de l’orientation cristalline.Nous proposons une modification simple de la loi de Schmid, qui prend en compte la trajectoirenon rectiligne de la dislocation et qui permet d’expliquer qualitativement pourquoi l’asymétriemaclage/antimaclage est moins marquée dans Fe que dans les autres métaux cubiques centrés. / We performed electronic structure ab initio calculations based on density functional theory(DFT) to study the h111i screw dislocation properties in body-centered cubic transition metals(V, Nb, Ta, Mo, W and Fe). In all investigated elements, the nondegenerate easy coreis the minimum energy configuration and the split core configuration has a high energy nearor above that of the hard core, contrary to interatomic potential predictions. A strong groupdependence of the core energy of the easy dislocation is also evidenced, related to the positionof the Fermi level with respect to the minimum of the pseudogap of the electronic density ofstates. Our work also reveals an atypical behavior in Fe, with a low relative energy at the hardcore position, close to that of the saddle configuration between easy cores, resulting in a flatPeierls potential around the hard core configuration, at variance with other elements. Fromthese DFT calculations, the two-dimensional energetic landscape in the {111} plane (Peierlspotential) is constructed and we investigated several properties of dislocation glide and in particular,the kink-pair formation enthalpy, as well as the dependence of the Peierls stress oncrystal orientation. We proposed a simple modification to the Schmid law that takes accountof the non-straight trajectory of the dislocation and that qualitatively explains why the twinning/antitwinning asymmetry is less pronounced in Fe than in other body-centered cubic metals.

Estudamos o diagrama de fases para uma classe de modelos de Ising ferromagnéticos em $ \\mathbb^2 $, com campo magnético externo periódico. O campo externo assume dois valores: $ h $ e $ -h $, onde $ h> 0 $. Os sítios associados a valores positivos e negativos do campo externo, formam uma configuração em forma de tabuleiro de xadrez (nós chamamos de {\\it cell-board configuration}), com células retangulares de tamanho $ L_1 \\times L_2 $ sítios, de tal forma que o valor total do campo externo é zero. Como principal resultado, mostramos a presença de uma transição de fase de primeira ordem. A transição de fase existe para $ h <\\frac + \\frac $, onde $ J $ é uma constante de interação. A prova é construida usando o método de {\\it reflection positivity (RP)}. Aplicamos uma desigualdade que é normalmente referida como a estimativa de {\\it chessboard}. Além disso, incluímos uma região de unicidade da medida de Gibbs em $h>4J$, isto usando um critério baseado nas ideias de percolação em desacordo. / We study the low-temperature phase diagram for a ferromagnetic Ising model on $\\mathbb^2$, with a periodical external magnetic field. The external field takes two values: $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\\times L_2$ sites, such that the total value of the external field is zero. As a main result, we show the presence of a first-order phase transition. The phase transition holds if $h<\\frac+ \\frac$, where $J$ is an interaction constant. We use the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate. Furthermore, we prove uniqueness for Gibbs measure in $h>4J$, using a uniqueness condition obtained in terms of disagreement percolation.

In dieser Arbeit wird unter Verwendung des Su-Schrieffer-Heeger-Modells die Dynamik des einfachsten konjugierten Polymers Trans-Polyacetylen in Mean-Field-Approximation untersucht. Dazu wird erstmalig von einer dimerisierten Peierls-Kette im Grundzustand ausgegangen, die durch einen gaußförmigen Laserpuls angeregt wird. Sowohl bei subresonanter Anregung als auch bei Anregungen hoch im Band bilden sich geladene Solitonpaare mit gemischten Gap-Zuständen, für die ein Phasendiagramm in Abhängigkeit vom Mischungsparameter abgeleitet wird. Diese Zustände werden im Rahmen des Takayama-Lin-Liu-Maki-Modells erfolgreich beschrieben.Um einen Ladungstransfer zwischen benachbarten Ketten zu untersuchen, werden die Rechnungen auf zwei gekoppelte Peierls-Ketten ausgedehnt. In Abhängigkeit von der Kopplungsstärke und der Kettenlänge ergibt sich eine Reihe von Grundzustandskonfigurationen. Bei einer Anregung mit einem transversalen Laserpuls findet ein Ladungstransfer zwischen den beiden Ketten mit einer Frequenz statt, die der doppelten Kopplungsstärke entspricht. Zur Einbeziehung der Coulomb-Wechselwirkung der Pi-Elektronen wird das Pariser-Parr-Pople-Modell verwendet. Durch Anpassung an möglichst viele experimentelle Größen wird ein geeigneter Satz von Modell-Parametern berechnet. Die von der Kette aufgenommene Energie und deren Dynamik werden diskutiert. / In the framework of the Su-Schrieffer-Heeger model the dynamics of the most simple conjugated polymer trans-polyacetylene is investigated in mean field approximation. Starting with a dimerized Peierls chain in the ground state, a femtosecond laser pulse with a gaussian enveloppe is applied. For subresonant and high band excitation charged soliton pairs are created and are related to mixed gap states. For different values of the mixing strength a phase diagram can be drawn. The creation of the mixed gap states can be also explained within the continuum approximation with the Takayama-Lin-Liu-Maki model. In order to investigate charge transfer between two Peierls chains, the calculations are extended to a system with two coupled chains. Different ground state configurations occur, depending on the coupling stength and the chain length. In the case of an excitation perpendicular to the chain axis, a strong charge transfer between the chains is observed. Furthermore the Pariser-Parr-Pople model is used to take into account the Coulomb interaction between the Pi-electrons. A model parameter set can be determined by fitting the experimental values. The energy absorbed by the chain and the soliton dynamics are discussed in this model.

Estudamos o diagrama de fases para uma classe de modelos de Ising ferromagnéticos em $ \\mathbb^2 $, com campo magnético externo periódico. O campo externo assume dois valores: $ h $ e $ -h $, onde $ h> 0 $. Os sítios associados a valores positivos e negativos do campo externo, formam uma configuração em forma de tabuleiro de xadrez (nós chamamos de {\\it cell-board configuration}), com células retangulares de tamanho $ L_1 \\times L_2 $ sítios, de tal forma que o valor total do campo externo é zero. Como principal resultado, mostramos a presença de uma transição de fase de primeira ordem. A transição de fase existe para $ h <\\frac + \\frac $, onde $ J $ é uma constante de interação. A prova é construida usando o método de {\\it reflection positivity (RP)}. Aplicamos uma desigualdade que é normalmente referida como a estimativa de {\\it chessboard}. Além disso, incluímos uma região de unicidade da medida de Gibbs em $h>4J$, isto usando um critério baseado nas ideias de percolação em desacordo. / We study the low-temperature phase diagram for a ferromagnetic Ising model on $\\mathbb^2$, with a periodical external magnetic field. The external field takes two values: $h$ and $-h$, where $h>0$. The sites associated with positive and negative values of external field form a cell-board configuration with rectangular cells of sides $L_1\\times L_2$ sites, such that the total value of the external field is zero. As a main result, we show the presence of a first-order phase transition. The phase transition holds if $h<\\frac+ \\frac$, where $J$ is an interaction constant. We use the reflection positivity (RP) method. We apply a key inequality which is usually referred to as the chessboard estimate. Furthermore, we prove uniqueness for Gibbs measure in $h>4J$, using a uniqueness condition obtained in terms of disagreement percolation.

La majeure partie de cette thèse concerne l’étude de la susceptibilité diamagnétique en champ magnétique nul d’un gaz d’électrons de Bloch à température et densité fixées dans la limite de sfaibles températures. Pour les électrons libres (i.e. en l’absence de potentiel périodique), la susceptibilité diamagnétique a été calculée par L. Landau en 1930 ; le résultat est connu sous le nom de formule de Landau. Quant au cas des électrons de Bloch, E.R. Peierls montra en 1933 que dans l’approximation des électrons fortement liés, la formule pour la susceptibilité diamagnétique reste la même en remplaçant la masse de l’électron par sa ”masse effective” ; ce résultat est connu sous le nom de formule de Landau-Peierls. Depuis, de nombreuses tentatives pour clarifier les hypothèses de validité de la formule de Landau-Peierls ont vu le jour. Le résultat principal de cette thèse établit rigoureusement qu’à température nulle, lorsque la densité d’électrons tend vers zéro, la contribution dominante à la susceptibilité diamagnétique est donné par la formule de Landau-Peierls avecla masse effective de la plus petite bande d’énergie de Bloch. / The main part of this thesis deals with the zero-field diamagnetic susceptibility of a Blochelectrons gas at fixed temperature and fixed density in the limit of low temperatures. For a freeelectrons gas (that is when the periodic potential is zero), the steady diamagnetic susceptibilityhas been computed by L. Landau in 1930 ; the result is known as Landau formula. As for the Blochelectrons, E.R. Peierls in 1933 showed that under the tight-binding approximation, the formula forthe diamagnetic susceptibility remains the same but with the mass of the electron replaced by its”effective mass” ; this result is known as the Landau-Peierls formula. Since, there were very manyattempts in order to clarify the assumptions of validity of the Landau-Peierls formula. The mainresult of this thesis establishes rigorously that at zero temperature, as the density of electrons tendsto zero, the leading contribution of the diamagnetic susceptibility is given by the Landau-Peierlsformula with the effective mass of the lowest Bloch energy band.

Many materials with complex crystal structures have attractive properties, including high specific strength, good creep resistance, oxidation resistance, often through high silicon or aluminium content. This makes them of interest for high temperature structural applications, but the use of many such phases is limited by low toughness. Even outside structural applications, brittle failure is a primary cause of failure in coatings and device materials and, therefore, improved toughness is desirable. In complex crystals plasticity, and hence toughness, is limited by the energy increases that occur as linear defects, dislocations, move. This is known as the lattice resistance. By understanding the factors controlling the lattice resistance in complex crystal structures, it is hoped that a general method for tailoring the flow stress of a material might be found. Present ductile-brittle criteria are based on simple ratios of polycrystalline elastic constants and are too limited to accurately capture flow behaviour. There are complex materials which, despite such criteria predicting brittle behaviour, exhibit low flow stresses, though on a limited number of slip systems: MAX phases, Mo$_2$BC, Nb$_2$Co$_7$ and Ta$_4$C$_{3-x}$ are examples of this. Where plastic flow is limited by the lattice resistance we must consider the effect of crystal structure on dislocation motion more directly. Aspects which are lost by considering bulk polycrystalline properties are elastic heterogeneity, elastic anisotropy and contributions to the energy changes by other interactions, such as electrostatic interactions. In this work examples of each of these are presented and modelled using an adapted version of the Peierls model. A Peierls model generalised to use the entire stiffness tensor has been implemented in Python; this allows the investigation of the effect of varying anisotropy on the yield stress of materials that would not be picked up by the use of polycrystalline elastic constants. Calculations using the changing elastic tensor during hydrogen loading of cementite suggest that hydrogen loading causes a dramatic reduction in the flow stress, consistent with experiments and associated with hydrogen embrittlement of steel. Materials for which empirical potentials can provide more insight than linear elasticity are explored with the example of ionic materials. This is done with a Peierls dislocation configuration and a molecular statics energy calculation. A simple model built electrostatic and Lennard-Jones interactions was used for the rocksalt structure, this model was found to describe the hard slip system well, but was insufficient to describe the softer slip system. Local heterogeneity in elastic properties is explored in the MAX phases where local variation in chemical environment, characterised by electronegativity, produces pronounced variation in the local stiffness within the unit cell. These local variations have been modelled with density functional theory and have been shown to be consistent with the macroscopic elastic properties while also explaining the apparent scatter in the elastic properties. These non-uniform strains are shown to have a dramatic effect on the flow stress of the MAX phases. The face-centred cubic Ti$_2$Ni structure has been used to experimentally demonstrate this effect of heterogeneity softening. The slip system was characterised by micropillar compression and the slip planes were found to be the {1 1 1} planes. The hardness of a range of alloys with the Ti$_2$Ni structure was characterised by nanoindentation of the {1 1 1} faces of single crystals. The hardness was found to decrease as the chemical, and thus elastic, heterogeneity of the unit cell increased, as expected. This effect of heterogeneity softening presents a potential route to tailoring the yield stress of crystals.

Wurtzite (W)-type ZnO condensates showed preferred orientation {10 1} when deposited on glass substrate by pulsed laser ablation on Zn target in the presence of oxygen. Such an artificial epitaxy depends on the well developed {10 1} surfaces of the condensates, which enabled {10 1}-specific coalescence to form twin and single crystal regardless of the co-deposited Zn. The W-ZnO condensates have decreasing particle size with increasing oxygen flow rate and a considerable residual stress due to the combined effects of rapid heating/cooling and thermal/lattice mismatch with Zn following parallel epitaxy or (01 )W-ZnO//(01 0)Zn; [ 2 3]W-ZnO//[0001]Zn involving {10 1} slip (Part I). In addition, wurtzite (W)-type ZnO/Zn composite deposit with preferred orientation {10 1}W-ZnO and (0001)Zn respectively on glass substrate in chapter I under Isothermal (600oC) atmospheric annealing caused self-catalyzed vapor-liquid-solid growth of rod-like W-ZnO whiskers with unusual habit. Analytical electron microscopic observations indicated that the W-ZnO whiskers extend along the zone axis of the well-developed polar surfaces {10 1} for a beneficial lower electrostatic energy and surface energy. Alternatively, the whiskers extend via {11 1}-specific growth twinning and/or coalescence twinning for a beneficial fair coincidence-site lattice at the twin boundary (Part II). Furthermore, Zn particulates overlain with wurtzite (W)-type ZnO condensates having nearly orthogonal {10 1} and {11 1} facets were found to self-catalyze unusual tapered W-ZnO whiskers upon isothermal atmospheric annealing, i.e. thermal oxidation, at 600oC. Analytical electron microscopic observations indicated that such whiskers formed tapered slabs having mosaic {10 1} and {2 1} twinned domains. The tapered whiskers can be rationalized by unconventional vapor-liquid-solid growth, i.e. {hkil}-specific coalescence twinning growth from the ZnO condensates taking advantage of a partially molten bottom source of Zn and the adsorption of atoms at the whisker tips and steps under the influence of capillarity effect (Part III). Finally, Electron irradiation of nano-size wurtzite (W)-type ZnO condensates with intimate mixture of parallel epitaxial Zn caused {10 1}W slip to form a single domain of rock salt (R)-type core and W-type shell. The two polymorphs follow (1 1)R//(0 11)W; [011]R//[ 2 3]W, i.e. chair type Peierls distortion with additional 38 degree tilting (001)R along the ( 2 0)W plane for a fair match of (10 1)W/(1 1)R, the same as one of the two paths for the back-transformation of R-ZnO into a lower crystal symmetry. The martensitic nucleation of R-type ZnO can be attributed to dynamic migration of interstitials/vacancies, lattice mismatch stress, and capillarity effect.

No description available.