Linear and nonlinear conductivity of a superconductor nearTc

Abstract

We consider the scaling properties of the linear ac and nonlinear dc conductivities of a superconductor near the transition temperature ${\mathit{T}}_{\mathit{c}}$. We first review a scaling theory of the conductivity near a critical point recently proposed by Fisher, Fisher, and Huse. By combining these scaling assumptions with causality arguments, we show that the phase ${\mathrm{\phi }}_{\mathrm{\sigma }}$ of the linear ac conductivity takes on a universal value near the critical point, ${\mathrm{\phi }}_{\mathrm{\sigma }}$=π(2-d+z)/2z, where d is the spatial dimension and z is the dynamic critical exponent. Hence, a measurement of the phase lag between the current and the voltage near the critical point provides a measurement of the exponent z. Then, using relaxational dynamics, we calculate these conductivities for 2<d<4, using (a) a Gaussian approximation, in which the quartic term in the Ginzburg-Landau Hamiltonian is neglected, and (b) an O(2n) generalization of the Ginzburg-Landau Hamiltonian, which for n→∞ produces a Hartree approximation for the quartic term. These results agree with the scaling theory, and provide explicit expressions for the universal scaling functions. In addition, it is possible to study the crossover behavior from Gaussian to critical fluctuations as the critical temperature is approached. The prospects for going beyond the mean-field level by using renormalization-group methods are discussed, along with possible applications to the high-${\mathit{T}}_{\mathit{c}}$ superconductors.