The study of cuprate high-temperature superconductors has undergone a recent resurgence due to the discovery of charge order in several families of cuprate materials. While its existence is now well established, little is known about its microscopic origins or its relationship to high-temperature superconductivity and the pseudogap. The aim of the research presented in this thesis is to address these questions. In this thesis I will report on the use of spectroscopic-imaging scanning tunnelling microscopy (SI-STM) to visualise the short-ranged charge density wave (CDW) in Bi₂Sr₂CaCu₂O₈₊ₓ and NaxCa₂₋ₓCuO₂Cl₂. Building on previous measurements of the intra unit-cell electronic structure of cuprates, I introduce sub-lattice segregated SISTM to individually address the atomic sub-lattices in the CuO₂ plane with spatial phase sensitivity. Using this technique I establish that the CDW in Bi₂Sr₂CaCu₂O₈+x and NaxCa₂₋ₓCuO₂Cl₂ has a previously unobserved d-symmetry form factor, where a breaking of rotational symmetry within the unit cell is modulated periodically in space. Towards identifying a mechanism of CDW formation, I establish that the amplitude of CDW modulations in the electronic structure are maximal at the pseudogap energy-scale and that these modulations exhibit a spatial phase difference of π between filled and empty states. Together with the doping evolution of the CDW wave-vector this highlights the role of the low-energy electronic structure of the pseudogap regime in CDW formation. To elucidate the relationship between the CDW and the superconducting condensate I will introduce nanometer resolution scanned Josephson tunnelling microscopy (SJTM). In this approach the Cooper pair (Josephson) tunnelling current between a Bi₂Sr₂CaCu₂O₈₊ₓ sample and a scan-able Bi₂Sr₂CaCu₂O₈₊ₓ nano-flake STM tip is used to directly visualise the superconducting condensate. I will report the observation of a periodic modulation in the Cooper pair condensate at the same wave-vector as the CDW, the first direct detection of a periodically modulating condensate in any superconductor.

<p><strong>ABSTRACT </strong> Approximating the kinetic energy as a functional of the electron density is a daunting, but important, task. For molecules in equilibrium geometries, the kinetic energy is equal in magnitude to the total electronic energy, so achieving the exquisite accuracy in the total energy that is needed for chemical applications requires similar accuracy for the kinetic energy functional. For this reason, most density functional theory (DFT) calculations use the Kohn-Sham method, which provides a good estimate for the kinetic energy. But the computational cost of Kohn-Sham DFT calculations has a direct dependence on the total number of electrons because the Kohn-Sham method is based on the orbital picture, with one orbital per electron. Explicit density functionals, where the kinetic energy is written explicitly in terms of the density, and not in terms of orbitals, are much faster to compute. Unfortunately, the explicit density functionals in the literature had disappointing accuracy. This dissertation introduces several new approaches for orbital-free density functional methods. One can try to include information about the Pauli principle using the exchange hole. In the weighted density approximation (WDA), a model for the exchange hole is used to approximate the one-electron density matrix, which is then used to compute the kinetic energy. This thesis introduces a symmetric, normalized, weighted density approximation using the exchange hole of the uniform electron gas. Though the key results on kinetic energy are not accurate enough, an efficient algorithm is introduced which, with a more sophisticated hole model, might give better results. The effects of electron correlation on the kinetic energy can be modeled by moving beyond the one-electron distribution function (the electron density) to higherorder electron distributions (k-electron DFT). For example, one can model electron correlation directly using the pair electron density. In this thesis, we investigated two different functionals of the pair density, the Weizsäcker functional and the March-Santamaria functional. The Weizsäcker functional badly fails to describe the accurate kinetic energy due to the N-representability problem. The March-Santamaria functional is exact for a single Slater determinant, but fails to adequately model the effects of electron correlation on the kinetic energy. Finally, we established a relation between Fisher information and Weizsäcker kinetic energy functional. This allowed us to propose generalisations of the Weizsäcker kinetic energy density functional. It is hoped that the link between information theory and kinetic energy might provide a new approach to deriving improved kinetic energy functionals. <strong> Keywords: </strong><em>Kinetic energy functional, Density functional theory (DFT), von-Weizsäcker</em> <em> functional, March-Santamaria functional, Thomas-Fermi model, density matrix, Twopoint normalization, Pair-density functional theory (PDFT). </em></p> / Doctor of Science (PhD)

<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)

En la literatura sobre mecànica quàntica és freqüent trobar descriptors basats en la densitat de parells o la densitat electrònica, amb un èxit divers segons les aplicacions que atenyin. Per tal de que tingui sentit químic un descriptor ha de donar la definició d'un àtom en una molècula, o ésser capaç d'identificar regions de l'espai molecular associades amb algun concepte químic (com pot ser un parell solitari o zona d'enllaç, entre d'altres). En aquesta línia, s'han proposat diversos esquemes de partició: la teoria d'àtoms en molècules (AIM), la funció de localització electrònica (ELF), les cel·les de Voroni, els àtoms de Hirshfeld, els àtoms difusos, etc.L'objectiu d'aquesta tesi és explorar descriptors de la densitat basats en particions de l'espai molecular del tipus AIM, ELF o àtoms difusos, analitzar els descriptors existents amb diferents nivells de teoria, proposar nous descriptors d'aromaticitat, així com estudiar l'habilitat de totes aquestes eines per discernir entre diferents mecanismes de reacció. / In the literature, several electronic descriptors based in the pair density or the density have been proposed with more or less success in their pratical applications. In order to be chemically meaningful the descriptor must give a definition of an "atom" in a molecule, or instead be able to identify some chemical interesting regions (such as lone pair, bonding region, among others). In this line, several molecular partition schemes have been put forward: atoms in molecules (AIM), electron localization function (ELF), Voronoi cells, Hirshfeld atoms, fuzzy atoms, etc. The goal of this thesis is to explore the density descriptors based on the molecular partitions of AIM, ELF and fuzzy atom, analyze the existing decriptors at several levels of theory, propose new aromaticity descriptors, and study its ability to discern between different mechanisms of reaction.