In this thesis, we study an isolated vortex in an s-wave superconductor by solving the Bogoliubov-de Gennes equations self-consistently on a disc. We calculate the order parameter and supercurrent profiles, as well as the distribution of quasiparticle states. In contrast to quasi-classical treatments, the ratio Δ∞/EF between the order parameter and the Fermi energy is not assumed negligible. We study a regime where this ratio is on the order of 10-1, relevant to high-temperature superconductors. In this regime, we find a Friedel-like oscillation in the order parameter profile at low temperatures. This oscillation is attributed to an increased level spacing of the quasiparticle states, causing a decrease of the number of states present inside the superconducting energy gap. The results are in good agreement with previously published works. In future studies, the method used in this thesis will be generalized to d-wave superconductors. / I detta examensarbete studeras en ensam virvel i en s-vågssupraledare genom att självkonsistent lösa Bogoliubov och de Gennes' ekvationer på en cylinderskiva. Vi beräknar ordningsparameter- och superströmsprofiler, samt fördelningen av kvasipartikeltillstånd. Till skillnad från i kvasiklassiska metoder så antas inte kvoten Δ∞/EF mellan ordningsparametern och Fermi-energin vara negligerbar. Vi studerar en regim där denna kvot är av storleksordningen 10-1, vilket är fallet i högtemperatur-supraledare. Vid låga temperaturer finner vi i denna regim en Friedelliknande oscillation i ordningsparameterprofilen. Denna oscillations förklaras genom att separationen mellan kvasipartikeltillstånd ökar, vilket får som effekt att färre tillstånd ryms innanför det supraledande energigapet. Våra resultat överensstämmer väl med tidigare publicerade artikler. I framtida studier kommer metoden vi använder i detta examensarbete att generaliseras till d-vågssupraledare.

The focus of this report is the derivation of the Bogoliubov-de Gennes equations for superconductors from a tight-binding model, restricting ourselves to the case of s-wave superconductors.

Recent developments in the study of Majorana fermions through braid theory have shown that there exists a set of interchanges that allow for the realization of true quantum computation. Alongside these developments there have been studies of topological superconductivity which show the existence of states that exhibit non-Abelian exchange statistics. Motivated by these developments we study the differences between Abelian and non-Abelian topological phase in the vortex state through the Bogoliubov de-Gennes (BdG) formalism. Due to our interests in low-energy states we first implement computationally efficient algorithms for calculating the mean fields and computing eigenpairs in an arbitrary energy window. We have shown that these algorithms adequately reproduce results obtained from a variety of other techniques and show that these algorithms retain spatial inhomogeneity information. Our results show topological superconductivity and vortex states can coexist; providing a means to realize zero-energy bound states, the number of which corresponds to the topological phase. With the use of our methods we present results contrasting the differences between Abelian and non-Abelian topological phase. Our calculations show that an increase in Zeeman field affects numerous parameters within topological superconductors. It causes the order parameter to become more sensitive to temperature variations in addition to a reduced rate of recovery to the bulk value from a vortex core. The increased field suppresses spin-up local density of states (LDOS) in close proximity to the vortex core for low-energy states. Further, it narrows the spectral gap at the lattice centre. Both energy spectrum and LDOS calculations confirm that trivial topological phase have no zero-energy bound states, Abelian phases have an even number, while non-Abelian phases have an odd number.

Nous avons étudié le magnétisme orbital d'un anneau hybride normal-supraconducteur balistique. Nous avons obtenu le spectre d'excitation complet en fonction du flux Aharonov-Bohm traversant l'anneau pour des longueurs arbitraires de métal normal et de supraconducteur et proposé une nouvelle méthode pour évaluer les harmoniques du courant dans un système balistique sans interaction. Nous avons ainsi décrit le passage du courant permanent de période h/e au courant Josephson de période h/2e. Dans une seconde étude, nous avons calculé le courant Josephson dans une jonction supraconducteur-ferromagnétique-supraconducteur balistique en tenant compte de la compétition entre la réflexion d'Andreev et la réflexion ordinaire. Nous avons montré que le spectre est fortement modifié par des ouvertures de gaps tandis que le courant Josephson et la transition 0-\pi sont très robustes à l'introduction de réflexion ordinaire. Enfin, nous avons analysé le contenu de certaines solutions des équations d'Usadel: supraconducteur uniforme et interface NS sans barrière tunnel. La théorie de perturbation standard habillée par des cooperons permet d'interpréter ces solutions en termes de chemins diffusifs reliant des évènements de réflexion d'Andreev.

<p>The pi-striped, <em>d</em>-wave superconducting (SC) state, which is a type of pair density wave wherein the SC order is spatially modulated, has recently been shown to generate the key ingredients for quantum oscillations consistent with experimental observations (Zelli <em>et al.</em>, 2011, 2012). This was accomplished with a phenomenological approach using non-self-consistent Bogoliubov-de Gennes (BdG) theory. The objective of this thesis is to explore two aspects of this approach: the addition of a charge density wave (CDW) order to the previous non-self-consistent calculations, and an attempt at stabilizing the pi-striped state in fully self-consistent BdG theory. It was found that the CDW order had a minimal effect on the Fermi surface characteristics of the pi-striped state, but that a sufficiently strong CDW degrades the Landau levels which are essential for the formation of quantum oscillations. The self-consistent mean-field calculations were unable to stabilize the pi-striped state under a range of modifications to the Hamiltonian. Free energy calculations with the modulated SC order treated as a parameter demonstrate that the pi-striped state is always less energetically favourable than the normal state for the scenarios which were considered. The results of this study constitute a basis for future, more comprehensive studies, using the BdG approach, of the stability of possible pi-striped SC phases.</p> / Master of Science (MSc)

Quasicrystals are systems that are ordered but not periodic. They do however still have long-range order and well-defined diffraction peaks. This leads to interesting properties, like critical states which are neither extended nor localized, and to topological invariants and edge states. We study how these peculiar properties impact superconductivity in an SNS-junction, by attaching superconducting leads to a quasicrystal nanowire. We choose to investigate proximitzed superconductivity in Fibonacci quasicrystals, since their normal state has been thoroughly studied and understood. Using the Bogoliubov-de Gennes method and solving the order parameter self-consistently, we calculate the proximity effect as well as the Josephson current. We find that quasicrystals can enhance the proximity effect and significantly enhance the Josephson currents.

Le phénomène de condensation d’un gaz de bosons lorsqu’il est refroidi à zéro degrés Kelvin futdécrit par Einstein en 1925 en s’appuyant sur des travaux de Bose. Depuis lors, de nombreux physiciens,mathématiciens et numériciens se sont intéressés au condensat de Bose-Einstein et à son caractère superfluide. Nous proposons dans cette étude des méthodes numériques ainsi qu’un code informatique pour la simulation d’un condensat de Bose-Einstein en rotation. Le principal modèle mathématique décrivant ce phénomène physique est une équation de Schrödinger présentant une non-linéarité cubique,découverte en 1961 : l’équation de Gross-Pitaevskii (GP). En nous appuyant sur le logiciel FreeFem++,nous nous servons d’une discrétisation spatiale en éléments-finis pour résoudre numériquement cette équation. Une méthode d’adaptation du maillage à la solution et l’utilisation d’éléments-finis d’ordre deux nous permet de résoudre finement le problème et d’explorer des configurations complexes en deux ou trois dimensions d’espace. Pour sa version stationnaire, nous avons développé une méthode de gradient de Sobolev ou une méthode de point intérieur implémentée dans la librairie Ipopt. Pour sa version instationnaire, nous utilisons une méthode de Time-Splitting combinée à un schéma de Crank-Nicolson ou une méthode de relaxation. Afin d’étudier la stabilité dynamique et thermodynamique d’un état stationnaire, le modèle de Bogoliubov-de Gennes propose une linéarisation de l’équation de Gross-Pitaevskii autour de cet état. Nous avons élaboré une méthode permettant de résoudre ce système aux valeurs et vecteurs propres, basée sur un algorithme de Newton ainsi que sur la méthode d’Arnoldi implémentée dans la librairie Arpack. / The phenomenon of condensation of a boson gas when cooled to zero degrees Kelvin was described by Einstein in 1925 based on work by Bose. Since then, many physicists, mathematicians and digitizers have been interested in the Bose-Einstein condensate and its superfluidity. We propose in this study numerical methods as well as a computer code for the simulation of a rotating Bose-Einstein condensate.The main mathematical model describing this phenomenon is a Schrödinger equation with a cubic nonlinearity, discovered in 1961: the Gross-Pitaevskii (GP) equation. By using the software FreeFem++ and a finite elements spatial discretization we solve this equation numerically. The mesh adaptation to the solution and the use of finite elements of order two allow us to solve the problem finely and to explore complex configurations in two or three dimensions of space. For its stationary version, we have developed a Sobolev gradient method or an internal point method implemented in the Ipopt library. .For its unsteady version, we use a Time-Splitting method combined with a Crank-Nicolson scheme ora relaxation method. In order to study the dynamic and thermodynamic stability of a stationary state,the Bogoliubov-de Gennes model proposes a linearization of the Gross-Pitaevskii equation around this state. We have developed a method to solve this eigenvalues and eigenvector system, based on a Newton algorithm as well as the Arnoldi method implemented in the Arpack library.