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>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)