Solid State Communications, Vol. 71, No. I, pp. 25-28, 1989. Printed in Great Britain.
HIGH T E M P E R A T U R E S U P E R C O N D U C T I V I T Y
0038-1098/8953.00+.00 Maxwell Pergamon Macmillan plc
AND VORTEX FLUCTUATIONS
Petter Minnhagen Dept. of Theoretical Physics, Ume£ University, S-901 87 Ume£, Sweden (Received 31 March 1989 by L. Hedin) Analysis of recent data for Bi2Sr2CaCu2Os-crystals shows that the resistive tail just above the superconductive transition is in good agreement with the vortex-fluctuation description given by the two-dimensional Ginzburg-Landau Coulomb gas model. This suggests that the superconductivity just above the (three dimensional) superconducting transition for these crystals is effectively two-dimensional, associated with individual pairs of strongly coupled CuO2-planes, and well described by a standard Ginburg-Landan theory for a two-component orderparameter.
In a recent letter, Martin et al.  interpret the resistance parallel to the CuO2-planes close to the threedimensional superconducting transition for crystals of Bi2Sr2CaCu2Os in terms of vortex fluctuations associated with the CuO2-planes. In the present paper we reanalyse their data in a more consistent way and find that the connection between the data and the vortex fluctuation interpretation can be considerably sharpened. We find that the vortex fluctuations near the transition are well described by the two-dimensional Ginzburg-Landau Coulomb gas model  and that on this description level the vortex fluctuations are in fact indistinguishable from those of 'dirty' type II thin superconducting films. This suggests that for the Bi~Sr2CaCu2Os-crystals the superconductivity just above the three-dimensional superconducting transition is associated with the individual pairs of closely coupled CuOz- planes contained in the crystal structure and is well described by a standard GinzburgLandau theory for a two-component orderparameter. We will in the following describe how the data can be tested against the interpretation in terms of vortex fluctuations described by the two-dimensional GinzburgLandau Coulomb gas associated with the CuOTplanes. The connection between the resistance data and the vortex fluctuations is based on the Bardeen-Stephen formula for the flux-flow resistance [2,3]
R/RN = 2~r~2nF
less quantity of this model can only be a function of the Coulomb gas temperature variable. This property translates into scaling relations for the corresponding quantities of the superconductor . The resistance ratio R / R N is such a dimensionless quantity and the corresponding scaling variable X can be expressed in terms of two phenomenological temperatures i.e. the Ginzburg-Landau temperature T~0 and the Kosterlitz-Thouless temperature
To as  T X = Too - T
It follows that the two temperatures T~0 and Tc play a crucial role in the connection between vortex-fluctuations and resistance data. T~0 can be obtained from the Aslamasov-Larkin formula  for the superconducting fluctuations just above T~0. In the present case the Aslamasov-Larkin formula in two-dimensions give T~o -- 86.80 4- 0.5K . Tc can be determined from the dependence of the flux-flow resistance R on a magnetic field B applied perpendicular to the superconducting planes. At Tc this dependence is linear whereas for T > Tc (T < To) one has dl~(S) < 1 (d~_.m.~ 41a(B) > 1). This determination of T¢ for the data by Martin et al  gives Tc -- 84.16K . The remaining quantity needed to establish the connection is the normal-state resistance which can be readily extracted from measurements of the resistance somewhat above T~0 . The key feature, which our analysis is is based on, is that the resistance ratio R/RN as a function of X is calculable within the two-dimensional Ginzburg-Landan Coulomb gas model. Next we review what is known about this function at present. From Kosterlitz renormalization group equations  one knows that close to X -- 1, when X goes to 1 from above, the function goes to zero as
Here R is the flux-flow resistance, RN is the normal state resistance, ~ is the Ginzburg-Landau coherence length, and nF is the density of free vortices created by thermal fluctuations in the superconducting plane (in the present case the individual pairs of closely coupled CuO2-planes). Vortex fluctuations for a superconducting plane are described by a two-dimensional Coulomb gas and, when the underlying superconductivity is described by a Ginzburg-Landau theory with a two-component order parameter, the corresponding Coulomb gas is the Ginzburg-Landan Coulomb gas . The GinzburgLandau Coulomb gas has the property that a dimension-
R/RN ~ exp( 25
SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
where A is a constant which is non-universal in the renormalization group sense but has a unique value within the Ginzburg-Landau Coulomb gas model . This connection between the resistance ratio and the vortex-fluctuations was first given by Halperin and Nelson  and has unfortunately been the cause of many too rash conclusions in the literature . The reason for this unfortunate circumstance is that the form given by eq.(3) is an expansion in X - 1 and that consequently eq.(3) can only be motivated for X - 1 < < 1 which usually implies an extremely small temperature region above Tc . Another way of expressing the limitation is that eq.(3) is only valid for very large length-scales and these length-scales are in practice usually masked by non-universal finite size effects . Practically all good agreement reported so far between resistance data and eq.(3) in the literature is based on data well outside the range where eq.(3) can be justified from the Kosterlitz renormalization group equations . To large extent such good agreement must hence be attributed to too little structure in the data in relation to too many free parameters . There is, however, yet another reason which to some extent helps to explain the purportly good agreement with eq.(3) which we will come to later. First we will comment on the analysis by Martin et al  in the ligth of the above remarks. The resistance data for the Bi2Sr2CaCu2Os-crystal by Martin et al  is in the range 1.5 < X < 3.5 and consequently the significance of the obtained good agreement with eq.(3) is weakened by the fact that the data is well outside the validity range for the derivation of eq.(3). Their analysis is also weakened by the fact that they approximate X by X = ~ in their analysis. Such an approximation is only valid for fJ0 < < 1 whereas for their data ~0 > 0.98. This means that the temperature dependence of the underlying superconductivity is not properly taken into acount in their analysis. The good agreement reported can hence to large extent be ascribed to the fact that Tc is used as a free parameter in their fit to the data. The main point of the present paper is that the function R/RN[X] is by now rather well-known also for larger values of X and that this offers a more consistent way of comparing data with theory. The function R/RN[X] has been determined from resistance measurements for 'dirty' type II superconducting films . It is given by the full drawn curve in figure 3. That this experimentally determined scaling function R/RN[X] is indeed the one given by the Ginzburg-Landau Coulomb gas has been corroborated through Monte Carlo simulations . Fig.1 gives a direct comparison between the data by Martin et al for the Bi2Sr2CaCu2Os-crystal (solid dots) and the scaling function determined for 'dirty' type II superconducting films (full line). The data is plotted against 1 / v / - X - 1. The striking thing to note is the good agreement; the Bi2Sr2CaCu2Os-data is everywhere within a factor of 2 of the scaling function. This is quite significant since the comparison involves no free parameter. We will assess the significance further in connection with figs 2 and 3. A second striking point to note in fig.1 is that the scaling function is, in an average sense,
71, No. 1
/ . . . .
1 (X- 1)-1/2
Fi~.l. Comparison between the scaling function R/RN[X] (full drawn curve) and the data for Bi2Sr2CaCu2Oscrystals from ref. (solid dots). The comparison involves no adjustable parameter. In the figure ln(R/RN) is plotted against 1 / ( k / ~ - 1). The data is everywhere within a factor of 2 of the scaling function. The data and the sealing function are well represented by two parallel lines in the region closest to the transition (dashed lines in the figure) which means that the functional dependence in both cases are well approximated by eq.(3) with the same value of the (non-universal) constant A.
well described by the functional form given by eq.(3) i.e. it is well represented by a straight line in fig.1. However, this is unrelated to the universal critical properties described by Kosterlitz renormalization group equations. The result is a property of the Ginzburg-Landau Coulomb gas itself for Coulomb gas temperatures well outside the critical region of the model. However, the coincidence between the functional form given by eq.(3) and the non-critical (and hence non-universal) properties of the Ginzburg-Landau Coulomb gas model partly helps to explain why fitting to the resistance data outside the validity range of eq.(3) from a Kosterlitz renormalization group point of view, has nevertheless often been successful. As seen in fig.1 the Bi2Sr2CaCu2Os-data is also well represented by a straight line and that the line representing this data is in fact parallel to the one representing the scaling function. This further emphasises the close relation between the scaling function and the Bi2Sr2CaCu2Os-data. Fig.2 presents an alternative way of establishing the relation between the scaling function R/RN[X] and resistance data . In this case the measured resistance ratio R(T)/RN is converted into a function X(T) by using the scaling function R/RN[X] i.e. the X(T)-value corresponding to a measured R(T)/Rg-value is read off from the scaling function R/RN[X]. In fig.2 the obtained function X(T) is plotted as T/X(T) against T. According to eq.(2) T/X(T) should be proportional to Tco-T. As seen in fig.2 this is togood approximation true for the data closest to the transition i.e. the data falls on a straight line (full drawn line in the figure). From the crossing point with the T-axis we read off To0 = 86.7K from
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SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
' ' L±~'-'-
Fig.2. The quantity T/X(T) for the Bi2Sr2CaCu2Os-data from ref. as obtained by assuming that the data falls on the scaling function R/RN[X] (see text). The data is plotted as T / X against T (solid dots). The data closest to the transition falls on a straight line (full drawn line). The crossing of this straight line with the T-axis gives T~0 = 86.7K. The dashed line in the figure is the function T / X = 1. The crossing between this line and the full drawn line gives T~ - 84.02K. These values for T~0 and T~ are in very good agreement with the a priori experimental determinations of these temperatures (see text).
this construction. This is in excellent agreement with the determination from the Aslamasov-Larkin formula T~0 = 86.80 4- 0.5K . The dashed line in fig.2 corresponds to X(T) = 1 or T/X(T) = T. The crossing point between this line and the full drawn line hence gives a determination of T~. This construction gives T~ = 84.02K, again in good agreement with the T~ = 84.16K which was obtained from the dependence of the resistance on a perpendicular applied magnetic field. An even more suggestive way of establishing the connection is shown in fig.3. The full drawn curve in this figure is the scaling function R/RN[X] and the solid dots is the data for the Bi2Sr2CaCu2Os-crystals. The other four data sets shown in the figure (i.e. circles, squares, diamonds, and crosses) are data for four granular indium/indium oxide composite films . The T~0 and the Tc for the indium/indium oxide films have been determined in precisely the same way as for the Bi2Sr2CaCu2Os-crystal i.e. 2/~0 from the AslamasovLarkin formula and T~ from the dependence of the resistance on a perpendicular magnetic field . Thus with the parameters determined in precisely the same way the data for the indium/indium oxide films and the Bi2Sr~CaCu2Os-erystal is virtually indistinguishable when plotted in the scaling variables corresponding to the Ginzburg-Landau Coulomb gas model. Furthermore all the data sets are in good agreement with the scaling function R/RN[X] in spite of the fact that the corn-
X Fi~.3. Comparison between the Bi2Sr2CaCu2Os-data from ref. (solid dots), the indium/indium oxide film data froin ref. (crosses, circles, diamonds, and squares, corresponding to the samples $2, $3, $4, and $5 of ref., respectively), and tile Ginzburg-Landau Coulomb gas scaling function (full drawn curve). The quantity ln(R/RN) is plotted against the scaling variable X (compare eq.(2)). The comparison involves no adjustable parameter. The a priori experimental determination of To0 and Tc are indentical for the Bi2Sr2CaCu2Os-data and the indium/indium oxide film data. As seen in the figure the Bi2Sr2CaCu2Os-data and the indium/indium oxide film data are virtually indistinguishable and in very good agreement with the Coulomb gas scaling function.
parisons involve no adjustable parameter. Fig.3 hence demonstrate the common origin of the resistive tail for Bi2Sr2CaCu2Os-crystals and thin 'dirty' type II superconducting films. In addition it demonstrates that the resistive tail in both cases are well represented by the two-dimensional Ginzburg-Landau Coulomb gas scaling function R/ RN[X]. In summary we find that the resistive tail for the Bi2Sr2CaCu~Os-crystals and for thin 'dirty' type II superconducting films are remarkably similar when analysed according to the two-dimensional Ginzburg-Landau Coulomb gas model. This clearly suggests that the resistive tail for Bi2Sr2CaCu2Os-erysta]s are due to twodimensional vortex-fluctuations. Since the resistance data was outside the critical region described by the Kosterlitz renormalization group equations, we were in fact able to link the data to the explicit non-universal properties of the Cinzburg-Landau Coulomb gas. This means that a stronger conclusion can be drawn about the underlying superconductivity of the CuO2-planes. This underlying superconductivity is, in the region just above the (three-dimensional) superconducting transition, well described by a standard two-dimensional Cinzburg-Landau theory for a two-component orderparameter.
SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
REFERENCES 1. S. Martin, A.T. Fiory, R.M. Fleming, G.P. Espinosa, and A.S. Cooper, Phys. Rev. Lett. 62, 677 (1989). 2. For a review see e.g.P. Minnhagen, Rev. of Mod. Phys. 59, 1001 (1987). 3. J. Bardeen and M.J. Stephen, Phys. Rev. 140, 1197A (1965). 4. L.G. Aslamasov and A.I. Larkin, Phys. Lett. A26, 238 (1968). 5. This T~-value is obtained from fig.2a of ref. by notd In(R) ing that the vertical axis in the figure gives _,) din(B)" 6. J.M. Kosterlitz, J. Phys. C 7, 1046 (1974).
7. B.I. Halperin and D.R. Nelson, J. Low Phys. 36, 599 (1979). 8. P. Minnhagen and H Weber, Phys. Rev. B 32, 3337 (1985); H. Weber and P. Minnhagen, Phys. Rev. B 33, 8730, (1988). 9. P. Minnhagen, Phys. Rev. B 24, 6758 (1983); P. Minnhagen, Phys. Rev. B 27, 2807 (1983); P. Minnhagen, Phys. Rev. B 28, 2463 (1983); P. Minnhagen, Phys. Rev. B 29, 1440 (1984). 10. J.C. Garland and Hu Jong Lee, Phys. Rev. B 36, 3638 (1987).