The aim of this thesis was to identify topologically protected states in the materials SnTe and Fe3Sn2. Such states are currently receiving a large amount of interest due to their applications for spintronic devices. Recently SnTe was discovered to be a crystalline topological insulator, a state of matter where its surface is highly conducting while the bulk remains insulating. However detection of these surface states is difficult using transport measurements, since the bulk is not totally insulating but still contains a large number of free carriers. SnTe undergoes a rhombohedral structural distortion on cooling caused by a soft transverse optic phonon, with the exact Tc strongly dependent on the carrier concentration. The distortion acts to lower crystal symmetry removing some of the symmetries that protect the surface state. Single crystal samples displaying the structural transition were grown and investigated using inelastic X-ray scattering to measure the phonon softening previously reported by other authors. The soft phonon was seen to recover again after distortion indicative of a 2nd order ferroelectric transition. This is the first reported discovery of the recovery showing the distortion is ferroelectric in nature. Shubnikov de Haas quantum oscillations were measured to study the Fermi surface under ambient and high hydrostatic pressure conditions. A distortion of the Fermi surface caused by the structural transition was evident, resulting in 4 distinct oscillation frequencies. However at applied pressures above 6 kbar, the transition was suppressed and only 1 oscillation measured. A two component Hall response also becomes apparent under high pressure. The possible origin of this and its relation to possible surface states is discussed. The anomalous Hall effect was also measured in the ferromagnet Fe3Sn2 which has a bilayer Kagome structure. Previous measurements on polycrystalline Fe3Sn2 suggested a non-collinear spin rotation from the spins pointing along the c-axis at high temperature to lying in the a-b plane below 80 K. A spin glass phase is then expected below 80 K. Single crystal magnetisation measurements carried out in this thesis show the spins are in the a-b plane at high temperatures and begin to display a ferromagnetic component along the c-axis approaching 80 K. The difference is accounted for by considering the demagnetising factor in the plate shaped single crystals. For this temperature range an applied field along the c-direction however rotates the moments towards c. At intermediate fields there are strong features evident in both the anomalous Hall effect and magnetoresistance. These features may be due to a topological Hall effect caused by a non-collinear spin structure. The possible existence of Skyrmion excitations was also recently discussed theoretically in Fe3Sn2. Our data is more suggestive of static Skyrmions known to cause topological Hall effects in MnSi.

The main objective of this work is to research and obtain surface protected topological states in nano-ribbons created from the leaves of Germanene and Silicene. These sheets belong to the class of Topological Insulators and correspond to monolayers of germanium and silicon atoms in a hexagonal arrangement that is similar to the graphene sheet. For this investigation, we conducted a study of the electronic and structural properties of these sheets, as well as their respective nano-ribbons through first-principles calculations based on density functional theory (DFT). In this methodology we use the generalized gradient approximation (GGA) for estimating the exchange and correlation term, and the PAW method for the effective potential and the expansion of plane waves of the Kohn-Sham. We conducted a computer simulation with the aid of the package VASP (Vienna ab-initio Simulation Package). As a starting point for our research, we used the methodology of solid state physics in order to describe the crystalline structure of the leaves as well as their mutual space. Subsequently we analyze the band structure, from which many of its properties can be visualized. For this task, we initially proceeded to investigate the stability of these systems via total energy calculations, in turn obtaining the network parameters that minimizes the energy of the system. We also obtained the energy cutoff, ECUT used in our calculations, or in other words, determining the number of plane waves needed to expand the electronic wave functions on the DFT formalism. We continued our study, with the creation and analysis of two different configurations of nano-ribbons, one that corresponds to a straightforward cut of the sheet with the armchair termination pattern, and the other based on a reconstruction of those edges, which provide an energetically more stable system. Subsequently we obtained electronic structures, and conducted a study of its variation due to the change of the width of the nano-ribbon and ionic relaxation of its edges. In a way, we modified the above parameters in order to obtain a system that would give us a zero gap, or at least insignificant, as well as a specific configuration for the spin texture, in order to verify the evidence of surface protected topological states in these nano-ribbons. / O objetivo principal deste trabalho é a investigação e obtenção dos estados topologicamente protegidos de superfície em nano-fitas criadas a partir das folhas de Germaneno e Siliceno. Estas folhas pertencem a classe dos Isolantes Topológicos e correspondem a monocamadas de átomos de Germânio e Silício, em um arranjo hexagonal que se assemelha a folha do Grafeno. Para esta investigação, realizamos um estudo das propriedades eletrônicas e estruturais destas folhas, bem como de suas respectivas nano-fitas, através de cálculos de primeiros princípios fundamentados na teoria do funcional da densidade (DFT). Nesta metodologia utilizamos a aproximação do gradiente generalizado (GGA) para a estimativa do termo de troca e correlação, e o método PAW para o potencial efetivo e a expansão em ondas planas dos orbitais de Kohn-Sham. Realizamos a simulação computacional com o auxílio do pacote VASP (Vienna ab-initio Simulation Package). Como ponto de partida para nossa pesquisa, utilizamos a metodologia da física do estado sólido com o intuito de descrever a estrutura cristalina das folhas, bem como seu espaço recíproco. Posteriormente analisamos as estruturas de bandas, a partir das quais muitas de suas propriedades podem ser visualizadas. Para esta tarefa, inicialmente procedemos à investigação da estabilidade destes sistemas via cálculos de energia total, obtendo o parâmetro de rede a que minimiza a energia do sistema. Obtivemos também a energia de corte ECUT utilizada em nossos cálculos, ou em outras palavras, a determinação do número de ondas planas necessárias para expandir as funções de onda eletrônicas no formalismo da DFT. Prosseguimos nosso estudo, com a criação e análise de duas distintas configurações de nano-fitas, uma que corresponde a um corte simples e direto da folha com terminação no padrão armchair, e a outra baseada em uma reconstrução destas bordas, que acaba por fornecer um sistema mais estável energeticamente. Posteriormente obtivemos as estruturas eletrônicas, e realizamos um estudo de sua variação em função da alteração da largura da nano-fita e a relaxação iônica de suas bordas. De certa maneira, modificamos os parâmetros acima, de forma a obter um sistema que nos fornecesse um gap nulo, ou pelo menos desprezível, bem como uma determinada configuração para a textura de spin, de modo a verificarmos a evidência de uma proteção topológica nos estados de superfície nestas nano-fitas. / Mestre em Física