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Physical review. B, Condensed matter

Estimate of the critical region for the Kosterlitz-Thouless transition

Petter Minnhagen, Peter Olsson

Published May 1, 1992• 16 citations
Publisher

Abstract

The width of the critical region for the Kosterlitz-Thouless transition is estimated and found to be extremely narrow. The estimate is based on a numerical solution of a set of renormalization equations for the two-dimensional Coulomb gas. It is concluded that the Kosterlitz-Thouless critical behavior cannot be observed in the resistance data for quasi-two-dimensional superconductors. The implications for the quasi-two-dimensionality of high-${\mathit{T}}_{\mathit{c}}$ superconductors are discussed.

DOI
10.1103/physrevb.45.10557

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The standard treatment of the Kosterlitz-Thouless-Berezinskii (KTB) transition is supplemented here by the inclusion of free-vortex screening effects. We obtain the high-density corrections to the KTB critical behavior. The size of the KTB critical region is extracted, and is found in most cases to be quite small. The tricritical point, separating second-order (KTB) phase transitions from first-order transitions, is also determined. We make predictions concerning similar KTB-like transitions, such as in the layered cuprate superconductors.

Physical review. B, Condensed matter•Jan 1, 1996•6 citations
The width of the critical region for the Kosterlitz-Thouless transition is estimated and found to be extremely narrow. The estimate is based on a numerical solution of a set of renormalization equations for the two-dimensional Coulomb gas. It is concluded that the Kosterlitz-Thouless critical behavior cannot be observed in the resistance data for quasi-two-dimensional superconductors. The implications for the quasi-two-dimensionality of high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors are discussed.
A new set of renormalization equations for the two-dimensional Coulomb gas is derived and solved numerically. These equations give a new picture of the Kosterlitz-Thouless transition. A new temperature T∗{T}^{\mathrm{*}}T∗ is found. Above T∗{T}^{\mathrm{*}}T∗ the value of the dielectric constant at the transition is ${\ensuremath{\epsilon}}_{c}$=1/(4Tc{T}_{c}Tc​) as before whereas below T∗{T}^{\mathrm{*}}T∗ the new result ${\ensuremath{\epsilon}}_{c}1/(41/(41/(4{T}_{c}$) is obtained. The singular behavior at T∗{T}^{\mathrm{*}}T∗ is explored. The results are translated into a renormalization-group flow diagram. An interesting consequence is that the universal jump prediction for the Kosterlitz-Thouless transition turns into a nonuniversal jump prediction for critical temperatures below T∗{T}^{\mathrm{*}}T∗.
It has been suggested [Pierson et al., Phys. Rev. B 60, 1309 (1999); Ammirata et al., Physica C 313, 225 (1999)] that for two-dimensional (2D) superconductors there exists a phase transition with the dynamic critical exponent $z\ensuremath{\approx}5.6.$ We perform simulations for the 2D resistively-shunted-junction model and compare the results with the experimental data in Repaci et al. obtained for an ultrathin YBCO sample [Phys. Rev. B 54, R9674 (1996)]. We then use a different method of analyzing dynamic scaling than in Pierson et al., and conclude that both the simulations and the experiments are consistent with a conventional Kosterlitz Thouless (KT) transition in the thermodynamic limit for which z=2.z=2.z=2. For finite systems, however, we find both in simulations and experiments that the change in the current-voltage $(I\ensuremath{-}V)$ characteristics caused by the finite size shows a scaling property with an exponent $\ensuremath{\alpha}\ensuremath{\approx}1/6,$ seemingly suggesting a vanishing resistance at a temperature for which $z={\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is pointed out that the dynamic critical exponent found in Pierson et al. corresponds to the exponent ${\ensuremath{\alpha}}^{\ensuremath{-}1}.$ It is emphasized that this scaling property does not represent any true phase transition since in reality the resistance vanishes only at zero temperature. Nevertheless, the observed scaling behavior associated with $\ensuremath{\alpha}\ensuremath{\approx}1/6$ appears to be a common and intriguing feature for the finite size caused change in the $I\ensuremath{-}V$ characteristics around the KT transition.
We explore the implications of Berezinskii-Kosterlitz-Thouless (BKT) critical behavior on the two-dimensional (2D) quantum superconductor-insulator (QSI) transition driven by the tuning parameter x. Concentrating on the sheet resistance R(x,T) BKT behavior implies: an explicit quantum scaling function for R(x,T) along the superconducting branch ending at the nonuniversal critical value Rc = R(xc); a BKT-transition line T(c)(x) [proportionality] (x - x(c))(zν[overline]), where z is the dynamic exponent and ν[overline] the exponent of the zero-temperature correlation length; independent estimates of zν[overline], z and ν[overline] from the x dependence of the nonuniversal parameters entering the BKT expression for the sheet resistance. To illustrate the potential and the implications of this scenario we analyze the data of Bollinger et al (2011 Nature 472 458) taken on gate voltage tuned epitaxial films of La2-xSrxCuO4 that are one unit cell in thickness. The resulting estimates, z ~/= 3.1 and ν[overline] ~/= 0.52, indicate a clean 2D-QSI critical point where hyperscaling, the proportionality between d/λ(2)(0) and Tc, and the correspondence between the quantum phase transitions in D dimensions and the classical ones in (D + z) dimensions are violated.
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.
Fundamental properties of the Kosterlitz-Thouless-Berezinskii (KTB) transition which occurs in systems in the universality class of the two-dimensional X-Y model are reviewed here with an emphasis on the real-space renormalization-group analysis used to derive the recursion relations. A derivation of the unique temperature dependence of the correlation length of this system will be presented and the signatures of the KTB transition in electronic transport measurements will be explained. Extensions of this model and technique to layered superconductors will also be presented. The size of the three-dimensional (3D) critical region, the nature of the 3D cross-over, and the effect of a finite electric current will be discussed.
The standard treatment of the Kosterlitz-Thouless-Berezinskii (KTB) transition is supplemented here by the inclusion of free-vortex screening effects. We obtain the high-density corrections to the KTB critical behavior. The size of the KTB critical region is extracted, and is found in most cases to be quite small. The tricritical point, separating second-order (KTB) phase transitions from first-order transitions, is also determined. We make predictions concerning similar KTB-like transitions, such as in the layered cuprate superconductors.