Evolution of the excitation spectrum of cuprate superconductors with doping and temperature

Understanding the mechanism by which d-wave superconductivity in the cuprates emerges and is optimized by doping a Mott insulator is one of the major outstanding problems in physics. A key unresolved question in this field is how the strength of electron pairing evolves as a function of doping and temperature and whether pairing strength and the Tc of the sample are related, as they are in simple BCS superconductors. To address these questions, we have developed several new experimental techniques with the scanning tunneling microscope to measure the excitation spectra in the cuprates on the atomic scale as a function of doping and temperature. In this thesis, we will describe these techniques as well as a series of new experiments that reveal a surprisingly simple picture of how superconductivity in the cuprates is optimized. We will first show that over a wide range of doping (optimal to overdoped), the pairing gaps in these systems nucleate in nanoscale regions at temperatures above Tc unlike in the conventional superconductors where the superconducting order parameter sets in at the bulk Tc. These regions proliferate as the temperature is lowered, resulting in a spatial distribution of gap sizes in the superconducting state. Analysis of our data shows no correlation between the inhomogeneous pairing gaps and either the energy scale of the boson modes or the strength of the local electron-boson coupling as measured by the local excitation spectra. This spatially inhomogeneous pairing strength is in fact determined by the unusual electronic excitations of the normal state, suggesting that strong electron-electron interactions rather than low-energy (