Localized Spectroscopy of Anisotropic Superconductors

Both the experimental and theoretical aspects of high-temperature superconductivity have challenged condensed matter physics. In order to find the mechanism behind the pairing interaction responsible for superconductivity in the cuprates, one of the most useful pieces of information is the symmetry of the superconducting order parameter. To date the determination of this vital piece of information has not reached any clear consensus despite much effort. One means of probing the order parameter symmetry of a superconductor is through the energy gap that quasiparticles experience near the Fermi surface. By obtaining information on how the the gap varies as a function of momentum, one acquires the ability to compare the different proposed mechanisms. This dissertation offers a new perspective on probing the anisotropy of the superconducting gap via localized tunneling spectroscopy. The advent of scanning tunneling microscopy/scanning tunneling spectroscopy (STM/STS) and nanofabrication has opened up new possibilities for directly exploring the physics of systems at short length scales and doing so with sub-meV energy resolution typical of tunneling experiments. STM imaging of a superconducting surface does not probe the superconducting gap anisotropy but rather the angle-averaged gap, however, if translational symmetry is broken by an impurity or nanofabricated object then the spatial correlations around the site are shown to reveal the anisotropy of the superconducting gap. The basic theory for localized tunneling spectroscopy of a superconductor both in and out of equilibrium is presented via the Keldysh nonequilibrium Green's function technique. Numerical calculations are done for the case of spatial correlations around an impurity on the surface of a superconductor with an anisotropic gap. A detailed calculation is presented for the transport and interference effects between two nanofabricated point contacts on the surface of a grounded anisotropic superconductor.