Topics in the physics of inhomogeneous materials

We discuss the effective macroscopic properties of inhomogeneous materials, such as composite or polycrystalline media, ferromagnetic domain structures and inhomogeneous superconductors. In particular, we use Bergman's spectral theory to analyze the effects of disorder on a periodic composite system such as a colloidal crystal; we further extend that spectral theory to describe polycrystalline media and use that extension to analyze the third-order nonlinear response of a polycrystalline material. For the problem of inhomogeneous materials in an external magnetic field, we demonstrate that in certain 3-constituent composites the Hall effect in the metallic component can lead to either saturating or non-saturating effective magnetoresistance, depending on the constituent volume fractions; we identify and study the percolation problem discriminating between those two cases. We also demonstrate that metallic ferromagnets can exhibit negative magnetoresistance originating from the changes in the magnetic domain structure; we suggest quantitative models de scribing this behavior for some domain structure geometries. For superconducting media, we demonstrate that inhomogeneities in the magnitude of the order parameter lead to an additional absorption at finite frequencies, and develop sum rules for this absorption for a number of possible models. We then discuss how the fast suppression of the superconductivity in Zn-doped YBa₂Cu ₃O_7−δ can be described by a simple percolation model based on the local suppression of the order parameter in the regions surrounding Zn atoms. We also discuss how those various changes in the magnitude of the order parameter affect the nonlinear properties such as the intermodulation current. Finally, in the last chapter we consider a problem that is somewhat complimentary to the other problems discussed in this thesis, and analyze the origin of inhomogeneities in the Mg_{1− x}Al_xB₂ at certain Al concentrations. We use the results of ab initio simulations to develop a model for the energetics of various ionic arrangements and demonstrate that the free energy profile based on this model predicts two regions of phase-separation instability, consistent with experiment.