Na busca de um melhor entendimento das propriedades eletrônicas e magnéticas dos isolantes topológicos nos deparamos com uma das suas caraterísticas mais marcantes, a existência de estados de superfície metálicos com textura helicoidal de spin os quais são protegidos de impurezas não magnéticas. Na superfície estes canais de spin possuem um potencial enorme para aplicações em dispositivos spintrônicos. Muito há para se fazer e o tratamento via cálculos de primeiros princípios por simulações permite um caráter preditivo que corrobora na elucidação de fenômenos físicos via análises experimentais. Nesse trabalho analisamos as propriedades eletrônicas de isolantes topológicos tais como: (Bi,Sb)$_2$(Te,Se)$_3$, Germaneno e Germaneno funcionalizado. Cálculos baseados em DFT evidenciam a importância das separações entre as camadas de Van der Waals nos materiais Bi$_2$Se$_3$ e Bi$_2$Te$_3$. Mostramos que devido a falhas de empilhamento, pequenas oscilações no eixo de QLs (\\textit{Quintuple Layers}) podem gerar um desacoplamento dos cones de Dirac, além de criar estados metálicos na fase \\textit{bulk} de Bi$_2$Te$_3$. Em se tratando do Bi$_2$Se$_3$ um estudo sistemático dos efeitos de impurezas de metais de transição foi realizado. Observamos que há quebra de degenerescência do cone de Dirac se houver magnetização em quaisquer dos eixos. Além disso se a magnetização permanecer no plano, além de uma pequena quebra de degenerescência, há um deslocamento do mesmo para outro ponto da rede recíproca. No entanto, se a magnetização apontar para fora do plano a quebra ocorre no próprio ponto $\\Gamma$, porém de maneira mais intensa. Importante enfatizar que além de mapear os sítios com suas orientações magnéticas de menor energia observamos que a quebra da degenerescência está diretamente relacionada com a geometria local da impureza. Isso proporciona imagens de STM distintas para cada sítio possível, permitindo que um experimental localize cada situação no laboratório. Estudamos ainda a transição topológica na liga (Bi$_x$Sb$_{1-x}$)$_2$Se$_3$, onde identificamos um isolante trivial e topológico para $x=0$ e $x=1$. Apesar de óbvia a existência de tal transição, detalhes importantes ainda não estão esclarecidos. Concluímos que a dopagem com impurezas não magnéticas proporciona uma boa técnica para manipulação e engenharia de cone nesta família de materiais, de forma que dependendo da faixa de dopagem podemos eliminar a condutividade que advém do \\textit{bulk}. Finalmente estudamos superfícies de Germaneno e Germaneno funcionalizado com halogênios. Usando uma funcionalização assimétrica e com a avalição do invariante topológico $Z_2$ notamos que o material Ge-I-H é um isolante topológico podendo ser aplicado na elaboração de dispositivos baseados em spin. / In the search of a better understanding of the electronic and magnetic properties of topological insulators we are faced with one of its most striking features, the existence of metallic surface states with helical spin texture which are protected from non-magnetic impurities. On the surface these spin channels allows a huge potential for applications in spintronic devices. There is much to do and treating calculations via \\textit{Ab initio} simulations allows us a predictive character that corroborates the elucidation of physical phenomena through experimental analysis. In this work we analyze the electronic properties of topological insulators such as: (Bi, Sb)$_2$(Te, Se)$_3$, Germanene and functionalized Germanene. Calculations based on DFT show the importance of the separation from interlayers of Van der Waals in materials like Bi$_2$Se$_3$ and Bi$_2$Te$_3$. We show that due to stacking faults, small oscillations in the QLs axis (\\textit{Quintuple Layers}) can generate a decoupling of the Dirac cones and create metal states in the bulk phase Bi$_2$Te$_3$. Regarding the Bi$_2$Se$_3$ a systematic study of the effects of transition metal impurities was performed. We observed that there is a degeneracy lift of the Dirac cone if there is any magnetization on any axis. If the magnetization remains in plane, we observe a small shift to another reciprocal lattice point. However, if the magnetization is pointing out of the plane a lifting in energy occurs at the very $ \\Gamma $ point, but in a more intense way. It is important to emphasize that in addition to mapping the sites with their magnetic orientations of lower energy we saw that the lifting in energy is directly related to the local geometry of the impurity. This provides distinct STM images for each possible site, allowing an experimental to locate each situation in the laboratory. We also studied the topological transition in the alloy (Bi$_x$Sb$_{1-x}$)$_ 2$Se$_3$, where we identify a trivial and topological insulator for $x = 0$ and $x = 1$. Despite the obvious existence of such a transition, important details remain unclear. We conclude that doping with non-magnetic impurities provides a good technique for handling and cone engineering this family of materials so that depending on the range of doping we can eliminate conductivity channels coming from the bulk. Finally we studied a Germanene and functionalized Germanene with halogens. Using an asymmetrical functionalization and with the topological invariant $Z_2$ we noted that the Ge-I-H system is a topological insulator that could be applied in the development of spin-based devices.

The Hall resistivity in ferromagnetic materials has an anomalous contribution proportional to the magnetization, which is defined as the anomalous Hall effect (AHE). Being a central topic in the study of ferromagnetic materials for many decades, the AHE was revived in recent years by generating many new understandings and phenomena, e.g. spin-Hall effect, topological insulators. The phase diagram of the AHE was shown recently to exhibit three distinct regions: a skew scattering region in the high conductivity regime, a scattering-independent normal metal regime, and an insulating regime. While the origin of the metallic regime scaling has been understood for many decades through the expected dependence of each contribution, the origin of the surprising scaling in the insulating regime was completely unexplained, leaving the primary challenge to the last step to understand fully the AHE. In this dissertation work we developed a theory to study the AHE in the disordered insulating regime, whose scaling relation is observed to be omega_xy^AH is proportional to omega_xx^(1.40∼1.75) in a large range of materials. This scaling is qualitatively different from the ones observed in metals. In the metallic regime where kFl > > 1, the linear response theory predicts that omega_xx is proportional to the quasi-particle lifetime tau, while omega_xy^AH scales as alpha*tau beta*tau^0, indicating that the upper limit of the scaling exponent is 1.0. Basing our theory on the phonon-assisted hopping mechanism and percolation theory, we derived a general formula for the anomalous Hall conductivity (AHC), and showed that the AHC scales with the longitudinal conductivity as omega_xy^AH ~ omega_xx^gamma with gamma predicted to be 1.33 <= gamma <= 1.76, quantitatively in agreement with the experimental observations. This scaling remains similar regardless of whether the hopping process is long range type (varible range hopping) or short range type (activation E3 hopping), or is influenced by interactions, i.e. Efros-Shklovskii (E-S) regime. Our theory completes the understanding of the AHE phase diagram in the insulating regime.

Na busca de um melhor entendimento das propriedades eletrônicas e magnéticas dos isolantes topológicos nos deparamos com uma das suas caraterísticas mais marcantes, a existência de estados de superfície metálicos com textura helicoidal de spin os quais são protegidos de impurezas não magnéticas. Na superfície estes canais de spin possuem um potencial enorme para aplicações em dispositivos spintrônicos. Muito há para se fazer e o tratamento via cálculos de primeiros princípios por simulações permite um caráter preditivo que corrobora na elucidação de fenômenos físicos via análises experimentais. Nesse trabalho analisamos as propriedades eletrônicas de isolantes topológicos tais como: (Bi,Sb)$_2$(Te,Se)$_3$, Germaneno e Germaneno funcionalizado. Cálculos baseados em DFT evidenciam a importância das separações entre as camadas de Van der Waals nos materiais Bi$_2$Se$_3$ e Bi$_2$Te$_3$. Mostramos que devido a falhas de empilhamento, pequenas oscilações no eixo de QLs (\\textit{Quintuple Layers}) podem gerar um desacoplamento dos cones de Dirac, além de criar estados metálicos na fase \\textit{bulk} de Bi$_2$Te$_3$. Em se tratando do Bi$_2$Se$_3$ um estudo sistemático dos efeitos de impurezas de metais de transição foi realizado. Observamos que há quebra de degenerescência do cone de Dirac se houver magnetização em quaisquer dos eixos. Além disso se a magnetização permanecer no plano, além de uma pequena quebra de degenerescência, há um deslocamento do mesmo para outro ponto da rede recíproca. No entanto, se a magnetização apontar para fora do plano a quebra ocorre no próprio ponto $\\Gamma$, porém de maneira mais intensa. Importante enfatizar que além de mapear os sítios com suas orientações magnéticas de menor energia observamos que a quebra da degenerescência está diretamente relacionada com a geometria local da impureza. Isso proporciona imagens de STM distintas para cada sítio possível, permitindo que um experimental localize cada situação no laboratório. Estudamos ainda a transição topológica na liga (Bi$_x$Sb$_{1-x}$)$_2$Se$_3$, onde identificamos um isolante trivial e topológico para $x=0$ e $x=1$. Apesar de óbvia a existência de tal transição, detalhes importantes ainda não estão esclarecidos. Concluímos que a dopagem com impurezas não magnéticas proporciona uma boa técnica para manipulação e engenharia de cone nesta família de materiais, de forma que dependendo da faixa de dopagem podemos eliminar a condutividade que advém do \\textit{bulk}. Finalmente estudamos superfícies de Germaneno e Germaneno funcionalizado com halogênios. Usando uma funcionalização assimétrica e com a avalição do invariante topológico $Z_2$ notamos que o material Ge-I-H é um isolante topológico podendo ser aplicado na elaboração de dispositivos baseados em spin. / In the search of a better understanding of the electronic and magnetic properties of topological insulators we are faced with one of its most striking features, the existence of metallic surface states with helical spin texture which are protected from non-magnetic impurities. On the surface these spin channels allows a huge potential for applications in spintronic devices. There is much to do and treating calculations via \\textit{Ab initio} simulations allows us a predictive character that corroborates the elucidation of physical phenomena through experimental analysis. In this work we analyze the electronic properties of topological insulators such as: (Bi, Sb)$_2$(Te, Se)$_3$, Germanene and functionalized Germanene. Calculations based on DFT show the importance of the separation from interlayers of Van der Waals in materials like Bi$_2$Se$_3$ and Bi$_2$Te$_3$. We show that due to stacking faults, small oscillations in the QLs axis (\\textit{Quintuple Layers}) can generate a decoupling of the Dirac cones and create metal states in the bulk phase Bi$_2$Te$_3$. Regarding the Bi$_2$Se$_3$ a systematic study of the effects of transition metal impurities was performed. We observed that there is a degeneracy lift of the Dirac cone if there is any magnetization on any axis. If the magnetization remains in plane, we observe a small shift to another reciprocal lattice point. However, if the magnetization is pointing out of the plane a lifting in energy occurs at the very $ \\Gamma $ point, but in a more intense way. It is important to emphasize that in addition to mapping the sites with their magnetic orientations of lower energy we saw that the lifting in energy is directly related to the local geometry of the impurity. This provides distinct STM images for each possible site, allowing an experimental to locate each situation in the laboratory. We also studied the topological transition in the alloy (Bi$_x$Sb$_{1-x}$)$_ 2$Se$_3$, where we identify a trivial and topological insulator for $x = 0$ and $x = 1$. Despite the obvious existence of such a transition, important details remain unclear. We conclude that doping with non-magnetic impurities provides a good technique for handling and cone engineering this family of materials so that depending on the range of doping we can eliminate conductivity channels coming from the bulk. Finally we studied a Germanene and functionalized Germanene with halogens. Using an asymmetrical functionalization and with the topological invariant $Z_2$ we noted that the Ge-I-H system is a topological insulator that could be applied in the development of spin-based devices.

Quelques particularités des polaritons, (quasi) particules-modes normaux du système d'excitons en interaction avec des photons en régime de couplage dit fort, sont théoriquement et numériquement analysés dans les systèmes de dimensionnalité basse. Dans le chapitre 1 est donné un bref aperçu en structure 0D, 1D et 2D semi-conductrices avec une introduction générale au domaine des polaritons. Le chapitre 2 est consacré aux micro / nano fils. Les modes de galerie sifflants sont étudiés dans le cas général d'un système anisotrope ainsi que la formation des polaritons dans les fils de ZnO. Le modèle théorique est comparé à l’expérience. Dans le chapitre 3 la dynamique de type Josephson pour les condensats de Bose-Einstein des polaritons est analysé en prenant en compte le pseudospin. Le chapitre 4 commence par une introduction à l'effet Aharonov-Bohm, qui est la phase géométrique la plus connue. Une autre phase géométrique - phase de Berry, qui existe pour une large classe de systèmes en évolution adiabatique sur un contour fermé, est l'objet principal de cette section. Nous avons examiné une proposition d'un interféromètre en anneau avec exciton-polaritons basé sur l'effet phase de Berry. Le chapitre 5 concerne un système 0D: un exciton d’une boîte quantique fortement couplé avec des photons dans une cavité optique. Nous avons discuté de la possibilité d'obtenir des états intriqués à partir d'une boîte quantique embarquée dans un cristal photonique en régime polaritonique. / Some special features of polaritons, quasi-particles being normal modes of system of excitons interacting with photons in so called strong coupling regime, are theoretically and numerically analyze in low dimensional systems. In Chapter 1 is given a brief overview of 0D, 1D and 2D semiconductor structures with a general introduction to the polariton field. Chapter 2 is devoted to micro / nano wires. The so called whispering gallery modes are studied in the general case of an anisotropic systems as well as polariton formation in ZnO wires. Theoretical model is compared with an experiment. In the Chapter 3 Josephson type dynamics with Bose-Einstein condensates of polaritons is analyzed taking into account pseudospin degree of freedom. Chapter 4 start with an introduction to Aharonov-Bohm effect, as the best known represent of geometrical phases. An another geometrical phase – Berry phase, occurring for a wide class of systems performing adiabatic motion on a closed ring, is main subject of this section. We considered one proposition for an exciton polariton ring interferometer based on Berry phase effect. Chapter 5 concerns one 0D system : strongly coupled quantum dot exciton to cavity photon. We have discussed possibility of obtaining entangled states from a quantum dot embedded in a photonic crystal in polariton regime.

Optical microcavities (OMs) are receiving increasing attention owing to their potential applications ranging from cavity quantum electrodynamics, optical detection to photonic devices. Recently, rolled-up structures have been demonstrated as OMs which have gained considerable attention owing to their excellent customizability. To fully exploit this customizability, asymmetric and topological rolled-up OMs are proposed and investigated in addition to conventional rolled-up OMs in this thesis. By doing so, novel phenomena and applications are demonstrated in OMs. The fabrication of conventional rolled-up OMs is presented in details. Then, dynamic mode tuning by a near-field probe is performed on a conventional rolled-up OM. Next, mode splitting in rolled-up OMs is investigated. The effect of single nanoparticles on mode splitting in a rolled-up OM is studied. Because of a non-synchronized oscillating shift for different azimuthal split modes induced by a single nanoparticle at different positions, the position of the nanoparticle can be determined on the rolled-up OM. Moreover, asymmetric rolled-up OMs are fabricated for the purpose of introducing coupling between spin and orbital angular momenta (SOC) of light into OMs. Elliptically polarized modes are observed due to the SOC of light. Modes with an elliptical polarization can also be modeled as coupling between the linearly polarized TE and TM mode in asymmetric rolled-up OMs. Furthermore, by adding a helical geometry to rolled-up structures, Berry phase of light is introduced into OMs. A -π Berry phase is generated for light in topological rolled-up OMs so that modes have a half-integer number of wavelengths. In order to obtain a deeper understanding for existing rolled-up OMs and to develop the new type of rolled-up OMs, complete theoretical models are also presented in this thesis.

This report discusses higher differential forms with applications in the study of topological phenomena. The integer quantum Hall effect is first discussed, demonstrating a connection between models on a lattice and quantum field theories bridged by a topological invariant, namely the Chern number. Next, for parametrized models on a lattice, the higher Berry curvature is described. This is a rank-(d + 2) differential form on a (d + 2)-dimensional parameter manifold which provides a relation between models in a bulk and on a lower-dimensional interface. Finally, a family of quantum field theories connected to a (d + 1)-dimensional manifold, termed a target space, is constructed. This connection is realized through the incorporation of a set of classical fields, and the effective action of the full field theories all contain a Wess-Zumino-Witten term given by the pullback of a rank-(d + 1) differential form from the target space to spacetime. By performing an extension of spacetime, a (d + 2)-form on a (d + 2)-dimensional target space is constructed in a similar way. Extending a theory in d dimensions thus yields a form on a target space of the same dimension as that of a (d + 1)-dimensional theory without extension, defining a dimensional hierarchy. The dimensional relations inherent in the two higher forms studied indicate the possibility of a relation between them. / Denna rapport beskriver högre ordningens differentialformer med tillämpningar inom topologiska fenomen. Den heltaliga kvantmekaniska Halleffekten beskrivs först, som ett exempel på ett samband mellan modeller på ett gitter och kvantfältteorier som förbindas av topologiska invarianter, specifikt Chern-talet. För parametriserade modeller på ett gitter beskrivs därefter den högre Berrykrökningen. Detta är en differentialform av ordning (d + 2) definierad på en (d + 2)-dimensionell parametermångfald som ger en koppling mellan modeller i en kropps inre och på dens gränsskikt, som är i en lägre dimension. Slutligen konstrueras en familj av kvantfältteorier som är kopplade till en (d + 1)-dimensionell mångfald kallad modellens målrum. Denna koppling realiseras genom introduktionen av ett antal klassiska fält, och den effektiva verkan för den fullständiga teorin innehåller en Wess-Zumino-Witten-term som ges av en tillbakadragen (d + 1)-form från målrummet till rumtiden. Genom att utvidga rumtiden kan även en (d + 2)-form på en (d + 2)-dimensionellt målrum konstrueras på ett motsvarande sätt. Utvidgningen av en teori i d dimensioner ger därmed en differentialform på ett målrum med samma dimension som målrummet för en (d + 1)-dimensionell teori utan utvidning, vilket definierar en dimensionell hierarki. Dimensionsrelationerna inbyggda i dessa två differentialformer indikerar den möjliga existensen av en relation mellan dem.

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