arXiv:cond-mat/9404002 · Physical review. B, Condensed matter

Scaling in high-temperature superconductors

Ian D. Lawrie

Published Oct 1, 1994• 53 citations

Abstract

A Hartree approximation is used to study the interplay of two kinds of scaling which arise in high-temperature superconductors, namely critical-point scaling and that due to the confinement of electron pairs to their lowest Landau level in the presence of an applied magnetic field. In the neighborhood of the zero-field critical point, thermodynamic functions scale with the scaling variable [T-${\mathit{T}}_{\mathit{c}2}$(B)]/${\mathit{B}}^{1/2\ensuremath{\nu}}$, which differs from the variable [T-${\mathit{T}}_{\mathit{c}}$(0)]/${\mathit{B}}^{1/2\ensuremath{\nu}}$ suggested by the Gaussian approximation. Lowest-Landau-level (LLL) scaling occurs in a region of high field surrounding the upper critical-field line but not in the vicinity of the zero-field transition. For ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ in particular, a field of at least 10 T is needed to observe LLL scaling. These results are consistent with a range of recent experimental measurements of the magnetization, transport properties, and, especially, the specific heat of high-${\mathit{T}}_{\mathit{c}}$ materials.

DOI: 10.1103/physrevb.50.9456

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Critical Behavior of High Temperature Superconductors

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Dynamic scaling and two-dimensional high- Tc superconductors

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arXiv:cond-mat/9404002 · Physical review. B, Condensed matter

Scaling in high-temperature superconductors

Ian D. Lawrie

Published Oct 1, 1994• 53 citations

arXiv HTML PDF

Abstract

{\mathit{T}}_{\mathit{c}2}

(B)]/

{\mathit{B}}^{1/2\ensuremath{\nu}}

, which differs from the variable [T-

{\mathit{T}}_{\mathit{c}}

(0)]/

{\mathit{B}}^{1/2\ensuremath{\nu}}

suggested by the Gaussian approximation. Lowest-Landau-level (LLL) scaling occurs in a region of high field surrounding the upper critical-field line but not in the vicinity of the zero-field transition. For

{\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}

in particular, a field of at least 10 T is needed to observe LLL scaling. These results are consistent with a range of recent experimental measurements of the magnetization, transport properties, and, especially, the specific heat of high-

{\mathit{T}}_{\mathit{c}}

materials.

DOI: 10.1103/physrevb.50.9456

Related papers

Critical-point scaling function for the specific heat of a Ginzburg-Landau superconductor

Dominic J. Lee, Ian D. Lawrie

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 $\mathrm{XY}$ 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 $1/N$ 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 $1/N$ expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.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 $\mathrm{XY}$ 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 $1/N$ 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 $1/N$ expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.

arXiv:cond-mat/0107403Physical review. B, Condensed matterOct 16, 20015 citations

Scaling critical behavior of superconductors at zero magnetic field

C. de Calan, Flavio S. Nogueira

We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where ${m}_{h,0}$ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with ${m}_{h,0}$ having the scaling behavior of the renormalized photon mass.We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The first part of the paper is concerned with the Ginzburg-Landau model in the zero-magnetic-field Meissner phase. We discuss the scaling behavior of the superfluid density and we give an alternative proof of Josephson's relation for a charged superfluid. This proof is obtained as a consequence of an exact renormalization group equation for the photon mass. We obtain Josephson's relation directly in the form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{t}^{\ensuremath{\nu}};$ that is, we do not need to assume that the hyperscaling relation holds. Next, we give an interpretation of a recent experiment performed in thin films of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}.$ We argue that the measured mean-field-like behavior of the penetration depth exponent ${\ensuremath{\nu}}^{\ensuremath{'}}$ is possibly associated with a nontrivial critical behavior and we predict the exponents $\ensuremath{\nu}=1$ and $\ensuremath{\alpha}=\ensuremath{-}1$ for the correlation length and specific heat, respectively. In the second part of the paper we discuss the scaling behavior in the continuum dual Ginzburg-Landau model. After reviewing lattice duality in the Ginzburg-Landau model, we discuss the continuum dual version by considering a family of scalings characterized by a parameter $\ensuremath{\zeta}$ introduced such that ${m}_{h,0}^{2}\ensuremath{\sim}{t}^{\ensuremath{\zeta}},$ where ${m}_{h,0}$ is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with ${m}_{h,0}$ having the scaling behavior of the renormalized photon mass.

arXiv:cond-mat/9903247Physical review. B, Condensed matterAug 1, 199920 citations

Critical scaling of the ac conductivity for a superconductor above Tc

Robert A. Wickham, Alan T. Dorsey

We consider the effects of critical superconducting fluctuations on the scaling of the linear ac conductivity, $\ensuremath{\sigma}(\ensuremath{\omega}),$ of a bulk superconductor slightly above ${T}_{c}$ in zero applied magnetic field. The dynamic renormalization-group method is applied to the relaxational time-dependent Ginzburg-Landau model of superconductivity, with $\ensuremath{\sigma}(\ensuremath{\omega})$ calculated via the Kubo formula to $O({\ensuremath{\epsilon}}^{2})$ in the $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion. The critical dynamics are governed by the relaxational $\mathrm{XY}$ -model renormalization-group fixed point. The scaling hypothesis $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{2\ensuremath{-}d+z}S(\ensuremath{\omega}{\ensuremath{\xi}}^{z})$ proposed by Fisher, Fisher, and Huse is explicitly verified, with the dynamic exponent $z\ensuremath{\approx}2.015,$ the value expected for the $d=3$ relaxational $\mathrm{XY}$ model. The universal scaling function $S(y)$ is computed and shown to deviate only slightly from its Gaussian form, calculated earlier. The present theory is compared with experimental measurements of the ac conductivity of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ near ${T}_{c},$ and the implications of this theory for such experiments is discussed.We consider the effects of critical superconducting fluctuations on the scaling of the linear ac conductivity, $\ensuremath{\sigma}(\ensuremath{\omega}),$ of a bulk superconductor slightly above ${T}_{c}$ in zero applied magnetic field. The dynamic renormalization-group method is applied to the relaxational time-dependent Ginzburg-Landau model of superconductivity, with $\ensuremath{\sigma}(\ensuremath{\omega})$ calculated via the Kubo formula to $O({\ensuremath{\epsilon}}^{2})$ in the $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion. The critical dynamics are governed by the relaxational $\mathrm{XY}$-model renormalization-group fixed point. The scaling hypothesis $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{2\ensuremath{-}d+z}S(\ensuremath{\omega}{\ensuremath{\xi}}^{z})$ proposed by Fisher, Fisher, and Huse is explicitly verified, with the dynamic exponent $z\ensuremath{\approx}2.015,$ the value expected for the $d=3$ relaxational $\mathrm{XY}$ model. The universal scaling function $S(y)$ is computed and shown to deviate only slightly from its Gaussian form, calculated earlier. The present theory is compared with experimental measurements of the ac conductivity of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ near ${T}_{c},$ and the implications of this theory for such experiments is discussed.

arXiv:cond-mat/9906124Physical review. B, Condensed matterMar 1, 200018 citations

Linear and nonlinear conductivity of a superconductor near Tc

Alan T. Dorsey

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.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.

Physical review. B, Condensed matterApr 1, 1991106 citations

Critical Behavior of High Temperature Superconductors

C. de Calan

We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The Josephson's relation for a charged superfluid is proved without assuming the hyperscaling relation. On the other hand we discuss the dual Ginzburg-Landau model. In this dual model, due to the presence of two mass scales, a continuous family of non equivalent scalings can be defined. The relevant critical regimes are identified, and the corresponding critical exponents are predicted.We consider the scaling behavior in the critical domain of superconductors at zero external magnetic field. The Josephson's relation for a charged superfluid is proved without assuming the hyperscaling relation. On the other hand we discuss the dual Ginzburg-Landau model. In this dual model, due to the presence of two mass scales, a continuous family of non equivalent scalings can be defined. The relevant critical regimes are identified, and the corresponding critical exponents are predicted.

Brazilian Journal of PhysicsJun 1, 20010 citations

Dynamic scaling and two-dimensional high- Tc superconductors

Douglas R. Strachan, Christopher J. Lobb, Richard S Newrock

There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high ${T}_{c}$ 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.There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high ${T}_{c}$ 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.

arXiv:cond-mat/0302370Physical review. B, Condensed matterMay 22, 200320 citations

{\mathit{T}}_{\mathit{c}2}

(B)]/

{\mathit{B}}^{1/2\ensuremath{\nu}}

, which differs from the variable [T-

{\mathit{T}}_{\mathit{c}}

(0)]/

{\mathit{B}}^{1/2\ensuremath{\nu}}

{\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}

{\mathit{T}}_{\mathit{c}}

materials.

\mathrm{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/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/N

expansion is not sufficiently reliable at small values of N for a quantitative fit to be feasible.

{m}_{h,0}

is the bare mass of the magnetic induction field. We discuss the difficulties in identifying the renormalized magnetic induction mass with the photon mass. We show that the only way to have a critical regime with ${\ensuremath{\nu}}^{\ensuremath{'}}=\ensuremath{\nu}\ensuremath{\approx}2/3$ is having $\ensuremath{\zeta}\ensuremath{\approx}4/3,$ that is, with

{m}_{h,0}

having the scaling behavior of the renormalized photon mass.

We consider the effects of critical superconducting fluctuations on the scaling of the linear ac conductivity, $\ensuremath{\sigma}(\ensuremath{\omega}),$ of a bulk superconductor slightly above

{T}_{c}

in zero applied magnetic field. The dynamic renormalization-group method is applied to the relaxational time-dependent Ginzburg-Landau model of superconductivity, with $\ensuremath{\sigma}(\ensuremath{\omega})$ calculated via the Kubo formula to $O({\ensuremath{\epsilon}}^{2})$ in the $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion. The critical dynamics are governed by the relaxational

\mathrm{XY}

-model renormalization-group fixed point. The scaling hypothesis $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\sim}{\ensuremath{\xi}}^{2\ensuremath{-}d+z}S(\ensuremath{\omega}{\ensuremath{\xi}}^{z})$ proposed by Fisher, Fisher, and Huse is explicitly verified, with the dynamic exponent $z\ensuremath{\approx}2.015,$ the value expected for the

d=3

relaxational

\mathrm{XY}

model. The universal scaling function

S(y)

is computed and shown to deviate only slightly from its Gaussian form, calculated earlier. The present theory is compared with experimental measurements of the ac conductivity of ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ near

{T}_{c},

and the implications of this theory for such experiments is discussed.

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.

There has been ongoing debate over the critical behavior of two-dimensional superconductors; in particular for high

{T}_{c}

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.