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Physical review. B, Condensed matter

Linear and nonlinear conductivity of a superconductor near Tc

Alan T. Dorsey

Published Apr 1, 1991• 106 citations
Publisher

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 ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$ of the linear ac conductivity takes on a universal value near the critical point, ${\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}$=\ensuremath{\pi}(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 2d4, 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\ensuremath{\rightarrow}\ensuremath{\infty} 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.

DOI
10.1103/physrevb.43.7575

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We consider the scaling properties of the linear ac and nonlinear dc conductivities of a superconductor near the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​. 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 \ensuremathφ\ensuremathσ{\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}\ensuremathφ\ensuremathσ​ of the linear ac conductivity takes on a universal value near the critical point, \ensuremathφ\ensuremathσ{\mathrm{\ensuremath{\varphi}}}_{\mathrm{\ensuremath{\sigma}}}\ensuremathφ\ensuremathσ​=\ensuremath{\pi}(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 2d4, 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\ensuremath{\rightarrow}\ensuremath{\infty} 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-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors.
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For High-Tc superconductors the fluctuation regime is much wider than conventional superconductors due to the short coherence length and high transition temperature. AC measurements can probe the conductivity at a wide range of frequencies, and therefore provide a test of the scaling properties predicted from theory. The measured critical exponents and width of the fluctuation peak are sample dependent, which suggests that the sample inhomogeneities may play an important role in the critical region. In this thesis we study the effects of disorder on the fluctuation conductivity in the critical regime of zero-field normal-superconducting transition by the time-dependent Ginzburg-Landau theory. We set up a discretized model of the superconductor and calculate the two-dimensional and three-dimensional scaling function without disorder above Tc. The result of the discretized model deviates slightly from previous theoretical studies of the continuous model, which can be explained by a finite short wavelength cutoff in Ginzburg-Landau theory. Our results agree well with other theoretical investigations on cutoff effects from an analytical approach. Disorder in a superconductor is modeled by a distribution of Tc's at the lattice sites. We add random disorder to a two-dimensional lattice and calculate the scaling functions averaged over 1000 disorder configurations. When disorder is weak, the scaling functions are increased when T is close Tc . At strong disorder anomalous behavior may occur at the region close to Tc.
There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high Tc{T}_{c}Tc​ superconductors. The conventional view is that a Kosterlitz-Thouless-Berezinskii transition occurs as long as finite size effects do not obscure the transition. However, there have been recent suggestions that a different transition actually occurs which incorporates aspects of both the dynamic scaling theory of Fisher, Fisher, and Huse and the Kosterlitz-Thouless-Berezinskii transition. Of general interest is that this modified transition apparently has a universal dynamic critical exponent. Some have countered that this apparent universal behavior is rooted in a newly proposed finite-size scaling theory; one that also incorporates scaling and conventional two-dimensional theory. To investigate these issues we study dc voltage versus current data of a 12-\AA{}-thick ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ film. We find that the newly proposed scaling theories have intrinsic flexibility that is relevant to the analysis of the experiments. In particular, the data scale according to the modified transition for arbitrarily defined critical temperatures between 0 and 19.5 K, and the temperature range of a successful scaling collapse is related directly to the sensitivity of the measurement. This implies that the apparent universal exponent is due to the intrinsic flexibility rather than some real physical property. To address this intrinsic flexibility, we propose a criterion which would give conclusive evidence for phase transitions in two-dimensional superconductors. We conclude by reviewing results to see if our criterion is satisfied.
If the zero-field transition in high-temperature superconductors such as ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ is a critical point in the universality class of the three-dimensional XY\mathrm{XY}XY model, then the general theory of critical phenomena predicts the existence of a critical region in which thermodynamic functions have a characteristic scaling form. We report an attempt to calculate the universal scaling function associated with the specific heat, for which experimental data have become available in recent years. Scaling behavior is extracted from a renormalization-group analysis, and the 1/N1/N1/N expansion is adopted as a means of approximation. The estimated scaling function is qualitatively similar to that observed experimentally, and also to the lowest-Landau-level scaling function used by some authors to provide an alternative interpretation of the same data. Unfortunately, the 1/N1/N1/N expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.
Using the functional-integral method applied to the BCS Hamiltonian we have calculated the partition function, off-diagonal correlation function, specific heat, and conductivity of a super-conductor in the transition region. The partition function is obtained in the form $\ensuremath{\int}D\ensuremath{\psi}{e}^{\ensuremath{-}\ensuremath{\beta}F[\ensuremath{\psi}]}$, where FFF is of the Ginzburg-Landau form but with a time-dependent order parameter. An examination of the coefficient bbb of the fourth-order term in FFF, however, shows that bbb is sharply peaked at zero frequency, thus justifying the use of a time-independent order parameter. We then apply a self-consistent mean-field approximation similar to that of Mar\ifmmode \check{c}\else \v{c}\fi{}elja to the fourth-order term. The self-consistency condition determines the mean-square fluctuation in the order parameter and a renormalized temperature shift in terms of which the nature of the phase transition (or lack of it) can be understood. We find that only in three dimensions does a phase transition strictly occur, at a slightly lowered transition temperature with critical indices $\ensuremath{\gamma}=2$, $\ensuremath{\nu}=1$. In two, one, and zero dimensions the mean-square fluctuation in the order parameter is bounded for T>0T>0T>0 and the mean value of the order parameter vanishes, indicating that the transition is completely suppressed in samples of reduced dimensionality. This behavior is reflected in the off-diagonal correlation function which manifests off-diagonal long-range order only in three dimensions. The specific heat in three dimensions is found to be finite at Tc{T}_{c}Tc​, with critical index $\ensuremath{\alpha}=\ensuremath{-}1$. In two, one, and zero dimensions the specific-heat transition is rounded relative to the BCS result. We find an anomalous factor of 2 in the specific-heat calculation deriving from our treatment of the fourth-order term in FFF. As a result we conjecture that the Hartree approximation of Mar\ifmmode \check{c}\else \v{c}\fi{}elja may be better than the Hartree-Fock approximation of Tucker and Halperin. The equations for the conductivity are identical to those of Mar\ifmmode \check{c}\else \v{c}\fi{}elja.
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