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Studies of inhomogeneous superconducting states in novel materials

I provide detailed studies of two types of novel superconducting systems. In the first, I examine the effect of thermal (Gaussian) magnetic fluctuations on the superconducting transition of paramagnetically-limited superconductors under a Zeeman magnetic field. I consider transitions into both the uniform and the modulated (Fulde-Ferrell-Larkin-Ovchinnikov) superconducting states. I derive the Landau free energy expansion in powers of the superconducting order parameter, allowing for competition between the magnetic fluctuations and the superconducting order. I determine the order of the transition at the upper critical field and find that the fluctuations drive the transition, usually second-order, to first order at intermediate temperatures for both the uniform and modulated states. I also compute the thermodynamic signatures of the transition along the upper critical field. I use these results to help explain experiments on the heavy-fermion superconductor CeCoIn5, for which the superconducting transition is first-order at low temperatures and large magnetic fields. In the second study, I use a T -matrix approach to examine the resonant state generated by a single, non-magnetic impurity in multi-band superconducting systems. I consider extended s-wave symmetry of the superconducting gap and allow for anisotropy of the gap along the Fermi surface. I derive analytic expressions for the Green’s functions in the continuum and identify the criteria for the formation of the impurity states, emphasizing the role the band structure plays for existence of the resonant state. I then use these results to guide and explain the results of numerical studies of the impurity states on a lattice. For my numerical approach, I use dispersion relations appropriate for the description of the ferropnictides, a recently-discovered family of iron-based superconductors. I map the impurity state in realspace and emphasize how the features of these states can help identify the nodal structure of the gap on each of the Fermi surface sheets.