Motivated by the experimental measurement of transport properties such as electrical and hall conductivity, thermopower and Nernst, I present a study of longitudinal and transverse transport in graphene for the dilute limit of impurities. The temperature and carrier density dependence in this system display a number of anomalous properties that can be related to three effects: 1) emergence of &ldquo chirality &rdquo, 2) vanishing density of states at the chemical potential in the ideal undoped (zero gate voltage) systems and 3) nature of scattering. In an attempt to theoretically understand these anomalous transport properties, I use the theory of quantum transport in a two-dimensional system with Unitary(lattice vacancy) and screened Coulomb(charge impurity in the underlining substrate) scatterers. I show for a system such as graphene, the type of scattering potential has a profound effect on all transport properties, even though both types of potentials induce low energy states that yield a finite density of states at zero energy. My results are compared with experimental data for both types of scatterer and I show for a single set of impurity parameters all transport properties can be reproduced to agree qualitative with the features observed in experimental data. Parts of this work have been submitted to Physical Review B and are currently in the review process.

Relaxor ferroelectric crystals are characterized by a broad region of critical fluctuations of polarization: the observed dielectric constants and elastic diffuse scattering extend over a wide region of temperatures and have unusual line shapes. Despite more than 50 years since the synthesis of the first relaxors, a satisfactory theory of relaxor ferroelectricity that accounts for this broad region of fluctuations remains elusive, partly because of the various energy scales: the deviation from Curie-Weiss law, the onset of the elastic diffuse scattering and the maximum in the dielectric susceptibility.

We present a theory of the fluctuations of relaxors with a model of polarizable unit cells with dipolar forces, local anharmonic forces,

and local random fields. The usual Lorentz field approximation to the local dipolar field fails to account for the critical fluctuations of polarization at any temperature since it violates the

fluctuation-dissipation theorem of statistical mechanics. Thus, thermodynamic functions and temperature dependencies of the phonon frequencies are computed self-consistently using the Onsager field, which is the simplest necessary correction to the Lorentz field that guarantees the fluctuation-dissipation theorem. Compositional disorder is treated by a self-consistent method that relates the local polarization to the local random fields and to the susceptibility averaged over compositional disorder. Local anharmonic forces are treated within a quasi-harmonic approximation. We find that (i) arbitrarily small compositional disorder together with dipolar forces extend the region of critical fluctuations down to absolute zero temperature; (ii) the correlation functions of polarization are highly anisotropic and slowly varying with a power law component. We compare our results to the observed elastic diffuse scattering and dynamic dielectric constant.