Nessa tese de doutorado apresentamos um estudo da estrutura eletronica de materiais isolantes topologicos. A teoria fundamental dos isolantes topologicos foi abordada atraves de invariantes topologicos Z2, assim como os seus metodos para o calculo desses invariantes topologicos e as consequencias da topologia de bandas nao-trivial. Assim como as propriedades atomisticas e energeticas, as propriedades eletronicas de alguns isolantes topologicos foram calculadas atraves de metodos de primeiros principios baseados na Teoria do Funcional da Densidade. Apresentamos nessa tese o estudo de quatro sistemas de interesse fisico: (1) Em isolantes topologicos do tipo Bi2Se3 e Bi2Te3 com falhas de empilhamentos, encontramos que o Bi2Te3 com falhas de empilhamentos apresentam estados metalicos na regiao do defeito; (2) Na interface Bi2Se3/GaAs com tratamento de Se na regiao do GaAs, encontramos que a interacao entre o cone de Dirac do Bi2Se3 com a banda de valencia do GaAs abre um gap de energia no ponto ; (3) Em nanoestradas de germaneno imersas em germanano com interfaces zigzag, encontramos que a partir de uma largura critica podemos observar o efeito Hall quantico de spin; e (4) nas ligas desordenadas hexagonais de SixGe1-x em duas dimensoes, o sistema desordenado compartilha a mesma topologia de bandas do siliceno e do germaneno, enquanto que a liga ordenada Si0.5Ge0.5 e um isolante trivial. As estruturas eletronicas desses sistemas foram investigadas no intuito de entender as consequencias fisicas da topologia de bandas nao-trivial nos estados de Bloch de bulk e de superficies/interfaces. / In this doctoral thesis we present a study of the electronic structure of topological insulators materials. The fundamental theory of topological insulators was addressed through the Z2 topological invariants, as well as their methods to calculate these topological invariants and the consequences of non-trivial band topology. Just as atomistic and energetic properties, the electronic properties of some topological insulators were calculated using first-principles methods based upon Density Functional Theory. We present in this thesis the study of four systems of physical interest: (1) In topological insulators like Bi2Se3 and Bi2Te3 with stacking faults, we found that the Bi2Te3 with stacking faults presents metallic states in the region of the defect; (2) For Bi2Se3/GaAs interface with Se-treatment in the GaAs region, we found that the interaction between the Dirac cone of the Bi2Se3 and the valence band of the GaAs opens a bandgap at the -point; (3) In germanene nanoroads embedded on germanane with zigzag interfaces/edge, we found that from a critical width we can observe the quantum spin Hall effect; and (4) For SixGe1x two-dimensional hexagonal disordered alloy, the system shares the same non-trivial band topology of the silicene and germanene, while the ordered alloy Si0.5Ge0.5 is a trivial insulator. The electronic structures of these systems were investigated in order to understand the physical consequences of non-trivial band topology in the bulk and surfaces/interfaces Bloch states.

Nessa tese de doutorado apresentamos um estudo da estrutura eletronica de materiais isolantes topologicos. A teoria fundamental dos isolantes topologicos foi abordada atraves de invariantes topologicos Z2, assim como os seus metodos para o calculo desses invariantes topologicos e as consequencias da topologia de bandas nao-trivial. Assim como as propriedades atomisticas e energeticas, as propriedades eletronicas de alguns isolantes topologicos foram calculadas atraves de metodos de primeiros principios baseados na Teoria do Funcional da Densidade. Apresentamos nessa tese o estudo de quatro sistemas de interesse fisico: (1) Em isolantes topologicos do tipo Bi2Se3 e Bi2Te3 com falhas de empilhamentos, encontramos que o Bi2Te3 com falhas de empilhamentos apresentam estados metalicos na regiao do defeito; (2) Na interface Bi2Se3/GaAs com tratamento de Se na regiao do GaAs, encontramos que a interacao entre o cone de Dirac do Bi2Se3 com a banda de valencia do GaAs abre um gap de energia no ponto ; (3) Em nanoestradas de germaneno imersas em germanano com interfaces zigzag, encontramos que a partir de uma largura critica podemos observar o efeito Hall quantico de spin; e (4) nas ligas desordenadas hexagonais de SixGe1-x em duas dimensoes, o sistema desordenado compartilha a mesma topologia de bandas do siliceno e do germaneno, enquanto que a liga ordenada Si0.5Ge0.5 e um isolante trivial. As estruturas eletronicas desses sistemas foram investigadas no intuito de entender as consequencias fisicas da topologia de bandas nao-trivial nos estados de Bloch de bulk e de superficies/interfaces. / In this doctoral thesis we present a study of the electronic structure of topological insulators materials. The fundamental theory of topological insulators was addressed through the Z2 topological invariants, as well as their methods to calculate these topological invariants and the consequences of non-trivial band topology. Just as atomistic and energetic properties, the electronic properties of some topological insulators were calculated using first-principles methods based upon Density Functional Theory. We present in this thesis the study of four systems of physical interest: (1) In topological insulators like Bi2Se3 and Bi2Te3 with stacking faults, we found that the Bi2Te3 with stacking faults presents metallic states in the region of the defect; (2) For Bi2Se3/GaAs interface with Se-treatment in the GaAs region, we found that the interaction between the Dirac cone of the Bi2Se3 and the valence band of the GaAs opens a bandgap at the -point; (3) In germanene nanoroads embedded on germanane with zigzag interfaces/edge, we found that from a critical width we can observe the quantum spin Hall effect; and (4) For SixGe1x two-dimensional hexagonal disordered alloy, the system shares the same non-trivial band topology of the silicene and germanene, while the ordered alloy Si0.5Ge0.5 is a trivial insulator. The electronic structures of these systems were investigated in order to understand the physical consequences of non-trivial band topology in the bulk and surfaces/interfaces Bloch states.

The near-Fermi-energy energy band structure of carbon nanotubes is given by cross-sections of the graphene Dirac cones near the K and K' points. Using second-order perturbation theory and a nearest-neighbor approximated tight-binding model, curvature-induced corrections to the graphene-based effective model are derived. In addition to the already known Dirac-point shift, the curvature is shown to cause not only a warping of the Dirac cone, tantamount to a slight compression and a correction to the overall Fermi velocity, but also a tilting of the Dirac cone and the associated nanotube energy bands. This tilting results in a velocity asymmetry for left- and right-going waves and two different kinds of excitations, allowing for varying degeneracy in the same sample. Previous experiments have shown irregularities in the level degeneracy and should be reconsidered in this context. / Energibandstrukturen för kolnanorör ges av tvärsnitt av grafens Dirac-koner nära K- och K'-punkterna. Medelst andra ordningens störningsteori och en tight-binding-modell med närmaste-granne-approximationen härleds de kurvaturinducerade korrektionerna till den grafenbaserade effektiva modellen. Utöver det redan kända Dirac-punkt-skiftet så visas kurvaturen orsaka inte bara en förvrängning av Dirac-konen, liktydigt med en mild kompression och en korrektion till den övergripande fermihastigheten, utan också en lutning av Dirac-konen och de associerade nanorörsenergibanden. Denna lutning resulterar i en hastighetsasymmetri för vänster- och högergående vågor och två olika sorters excitationer, vilket tillåter för varierande degeneration i samma prov. Tidigare experiment har visat oregelbundenheter i nivådegenerationen och bör omprövas i denna kontext.

Ce mémoire présente les travaux réalisés durant ces trois années de thèse. Ils se sont orientés autour de l'étude des propriétés électroniques et magnétiques des matériaux pour la spintronique. Ce domaine d'avenir doit encore trouver les matériaux adaptés qui permettraient de réaliser les concepts liés à la spintronique. Nos résultats sont les suivants. Nous avons proposé un nouveau modèle qui décrit l'arrangement magnétique dans les couches minces ferromagnétiques possédant une anisotropie uniaxiale. Le modèle apporte une meilleure description des largeurs de domaine en fonction de l'épaisseur ainsi qu'une bonne estimation de l'épaisseur critique à partir de laquelle les domaines de Weiss ne sont plus stables. L'ensemble d'équations en unité réduite découlant du modèle apporte quant à lui un outil supplémentaire aux expérimentateurs. L'étude ab-Initio fait sur les semi-Conducteurs magnétiques dilués à mis en évidence l'importance de la corrélation forte et de l'effet Jahn-Teller dans les matériaux tel que le (Ga,Mn)N et le (Zn,Cr)S. Ces calculs confirment l'ensemble des données expérimentales existantes sur le (Ga,Mn)N. La modélisation analytique apporte un complément aux calculs ab-Initio en faisant le lien entre les paramètres expérimentaux et la théorie de Vallin, très largement utilisée pour interpréter les mesures optiques du (Zn,Cr)S. Nos calculs ab-Initio ont montré que le métacinabre est un isolant topologique robuste, qui se distingue par la présence d'un cône de Dirac extrêmement anisotrope. Les effets de passivation à l'hydrogène influencent les états de surface des isolants topologiques de la série HgX (X : S, Se, Te). / This thesis contains the scientific work of three years. The main topic can be defined as follows : study of electronic and magnetic properties of materials for spintronics. That technology of the future has still to find the necessary materials to realize new concepts. Our results are the following : We propose a new model to describe the magnetic configuration in thin ferromagnetic film with uniaxial anisotropy. It gives a better description of domain widths in function of film thickness and permits to obtain a good evaluation of the critical thickness where domains of Weiss type are no longer stable. The set of equations in reduced units is a useful tool to analyze experimental data. The ab-Initio study of diluted magnetic semiconductors has demonstrated the combined effect of strong correlations and Jahn-Teller distortion in (Ga,Mn)N and (Zn,Cr)S. The calculations confirm the experimental results of (Ga,Mn)N. We develop an analytical model that is complementary to the ab-Initio calculations and permits to create a link between several experimental parameters, the ab-Initio calculations and the former theory of Vallin. It has been used to interpret the optical measurements of (Zn,Cr)S. Our ab-Initio studies show that metacinnabar is a strong topological insulator with one peculiarity, it has a highly anisotropic Dirac cone. The passivation with hydrogen atoms influences the surface states of the topological insulator in the series HgX (X : S, Se, Te). When the dangling bonds are saturated by hydrogen, the trivial surface states disappear.

Cette thèse porte sur la caractérisation et l'étude du magnéto-transport sur les structures de type HgTe/CdTe sous contraintes développant un transport de surface topologique tout en étant isolant en volume ; on nomme cette nouvelle classe de matériau isolant topologique 3D.Je développerai dans cette thèse la caractérisation et définition d'un isolant topologique 2D/3D pour ensuite me focalise plus particulièrement sur les systèmes II-VI HgTe/CdTe.Une partie de la thèse développe les conditions de croissance réalisées au CEA/Leti ainsi que la caractérisation du matériau par rayon X. La structure de bande des surfaces est caractérisée par ARPES.Une troisième partie traite de la fabrication des barres de Hall nécessaires à la caractérisation du comportement topologique des surfaces. La partie développement expérimentale est également fournie.La dernière partie traite du magnétotransport réalisé avec ces barres de Hall à faible et fort champ magnétique. Le comportement ambipolaire, une phase de Berry non triviale, l'antilocalisation faible et l'effet Hall quantique entier dans ces structures sont abordés tout tentant de fournir une interprétation des résultats obtenus. / This report deal with caracterisation of magnetotransport in HgTe/CdTe structures bulk strained in that a topological surface transport is predicted. This new kind of material is a 3D topological insulator.In this thesis, I will explain what means 3D/2D topological insulator before focusing on II-VI system lijke HgTe/CdTe.Next, I will discuss about growing conditions performed in CEA/Leti and then material caracterisation by X-ray. Surfaces band structures were also, observed by ARPES, underligned in the report.A third part deal with Hall bars design and conception in order to emphasize topological behavior of these surfaces.The last part shows the results obtained on these Hall bars with magnetotransport at low and high magnetic field. Ambipolaire behaviour, non trivial Berry phase, weak antilocalization and the interger quantum hall effect in HgTe/CdTe structures are studied and a possible interpretation of these results are given.

Ce travail de thèse porte sur l’étude de la propagation d’ondes de volume et d’ondes de surface dans des cristauxphononiques granulaires en régime linéaire. Différents aspects sont développés dans ce manuscrit. L’effet de la prise encompte des degrés de liberté en rotation des particules sur la structure de bande de différents cristaux phononiquesgranulaires est étudié. En effet, l’introduction de ces degrés de liberté additionnels rend possible l’existence de modes derotation qui interagissent fortement avec les modes transverses. Ce travail s’intéresse aussi à l’existence d’ondeslocalisées et d’ondes de surface dans des cristaux phononiques granulaires et en particulier à la comparaison des théoriesdéveloppées avec les prédictions de la théorie de Cosserat. Dans un premier temps, l’étude d’une chaîne phononique granulaire monoatomique est présentée. En considérant lachaîne semi-infinie avec une condition aux limites appliquée à son extrémité, le modèle analytique démontre l’existence demodes localisés, chaque mode étant composé de deux modes évanescents. Ensuite, une description théorique des modes se propageant dans un cristal phononique granulaire en deux dimensions est présentée. Les particules possèdent trois degrés de liberté, deux en translation et un en rotation. L’analyse des interactions entre ondes de translation et ondes de rotation permet de mettre en évidence une grande richesse de structure de bandes ainsi que des phénomènes particuliers (bandes interdites complètes, cône de Dirac, modes non-monotones, phénomène de double réfraction). Dans un dernier temps, une analyse de l’existence d’ondes de Rayleigh et de cisaillement horizontal dans un cristal phononique granulaire en trois dimensions est effectuée. Les limites de la théorie de Cosserat dans la description d’ondes acoustiques de surface dans les milieux micro- et nano-inhomogènes sont établies. / This work is devoted to the analysis of propagating and surface acoustic waves in granular phononic crystals in thelinear regime. First, the propagation and localization of transversal-rotational waves in a two-dimensional granularchain of equal masses are analyzed. By considering the semi-infinite chain with a boundary condition applied at itsbeginning, the analytical study demonstrates the existence of localized modes, each mode composed of two evanescentmodes. Secondly, the phononic properties of a two-dimensional discrete phononic crystal, made of circular cross-section, infinitely long contacting elastic cylinders arranged on a simple cubic lattice, are described analytically. The theoretical analysis provides a clear physical explanation for the existence of a zero-group velocity point of the lowest-energy acoustic mode in particular directions of the phononic crystal and demonstrates the birefraction phenomenon. Finally, the existence of surfaces elastic waves at mechanically free surface of granular phononic crystals is presented. Depending on the degrees of freedom of the particles, different types of surface waves exist in the structure. First, Rayleigh type surface waves aredemonstrated in a granular phononic crystal with particles possessing two translational and one rotational degrees offreedom; and secondly, shear-horizontal surface waves are studied in a granular phononic crystal with particlespossessing two rotational and one translational degrees of freedom. A comparison with surface waves predicted by theCosserat theories is made in order to establish the limitations of the Cosserat theories.

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.

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.

Since its inception, the field of nonlinear optics has only increased in importance as a result of a growing number of applications. The efficiency of all parametric nonlinear optical processes is limited by challenges associated with phase-matching requirements. To address this constraint, a variety of approaches, such as quasi-phase-matching, birefringent phase matching, and higher-order-mode phase matching have historically been used to phase-match interactions. However, the methods demonstrated to date suffer from the inconvenience of only being phase-matched for one specific arrangement of beams, typically co-propagating along the same axis. This stringency of the phase-matching requirement results in cumbersome optical configurations and large footprints for integrated devices. In this thesis, we show that phase-matching requirements in parametric nonlinear optical processes may be satisfied for all orientations of input and output beams when using zero-index media: a condition of omnidirectional phase matching. To validate this theory, we perform experimental demonstrations of phase matching for five separate FWM beam configurations to confirm this phenomenon. Our measurements constitute the first experimental observation of the simultaneous generation of a forward- and backward-propagating signal with respect to the pump beams in a medium longer than a free-space optical wavelength, allowing us to determine the coherence length of our four-wave-mixing process. Our demonstration includes nonlinear signal generation from spectrally distinct counter-propagating pump and probe beams, as well as the excitation of a parametric process with the probe beam's wave vector orthogonal to the wave vector of the pump beam. By sampling all of these beam configurations, our results explicitly demonstrate that the unique properties of zero-index media relax traditional phase-matching constraints, and provide strong experimental evidence for the existence of omnidirectional phase matching in zero-index media. This property can be exploited to facilitate nonlinear interactions and miniaturize nonlinear devices, and adds to the established exceptional properties of low-index materials.