Mustafa Ismagambetov

We study localization effects in Josephson junctions with two superconductors connected by a strongly disordered metallic wire of length LL . The conventional description of the Josephson effect in such systems, based on the quasiclassical Usadel equation, neglects electron interference and is only applicable when LL is shorter than the localization length \xiξ in the wire. We develop a more general theory for the Josephson effect using the non-linear sigma model that fully accounts for electron interference, and hence localization. We show that for L \gg \xiL≫ξ , three qualitatively different regimes of the Josephson current arise depending on the ratio of the superconducting order parameter \DeltaΔ and the mean level spacing in the localization volume \Delta_\xiΔξ . We derive the average supercurrent as a function of the phase difference for all three regimes. Quite unexpectedly, we observe that the Ambegaokar-Baratoff relation between the average critical current and the normal-state conductance still holds in the strongly localized state when \Delta_\xi \gg \DeltaΔξ≫Δ and \xi \ll L \ll (\xi/\pi^2) \ln^2(\Delta_\xi/\Delta)ξ≪L≪(ξ/π2)ln2(Δξ/Δ) .

Weyl semimetal is a solid material with isolated touching points between conduction and valence bands in its Brillouin zone—Weyl points. Low-energy excitations near these points exhibit a linear dispersion and act as relativistic massless particles. Weyl points are stable topological objects robust with respect to most perturbations. We study effects of weak disorder on the spectral and transport properties of Weyl semimetals in the limit of low energies. We use a model of Gaussian white-noise potential and apply dimensional regularization scheme near three dimensions to treat divergent terms in the perturbation theory. In the framework of self-consistent Born approximation, we find closed expressions for the average density of states and conductivity. Both quantities are analytic functions in the limit of zero energy. We also include interference terms beyond the self-consistent Born approximation up to the third order in the disorder strength. These interference corrections are stronger than the mean-field result and nonanalytic as functions of energy. Our main result is the dependence of conductivity (in units $e^2/h$) on the electron concentration $\sigma ={\sigma }_0-0.891n^{1/3}+0.115(n^{2/3}/{\sigma }_0)ln\left|n\right|$.