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Solid State Communications, Vol. 71, No. I, pp. 2528, 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
00381098/8953.00+.00 Maxwell Pergamon Macmillan plc
AND VORTEX FLUCTUATIONS
Petter Minnhagen Dept. of Theoretical Physics, Ume£ University, S901 87 Ume£, Sweden (Received 31 March 1989 by L. Hedin) Analysis of recent data for Bi2Sr2CaCu2Oscrystals shows that the resistive tail just above the superconductive transition is in good agreement with the vortexfluctuation description given by the twodimensional GinzburgLandau Coulomb gas model. This suggests that the superconductivity just above the (three dimensional) superconducting transition for these crystals is effectively twodimensional, associated with individual pairs of strongly coupled CuO2planes, and well described by a standard GinburgLandan theory for a twocomponent orderparameter.
In a recent letter, Martin et al. [1] interpret the resistance parallel to the CuO2planes close to the threedimensional superconducting transition for crystals of Bi2Sr2CaCu2Os in terms of vortex fluctuations associated with the CuO2planes. 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 twodimensional GinzburgLandau Coulomb gas model [2] 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~Sr2CaCu2Oscrystals the superconductivity just above the threedimensional 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 twocomponent orderparameter. We will in the following describe how the data can be tested against the interpretation in terms of vortex fluctuations described by the twodimensional GinzburgLandau Coulomb gas associated with the CuOTplanes. The connection between the resistance data and the vortex fluctuations is based on the BardeenStephen formula for the fluxflow 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 [2]. 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 GinzburgLandau temperature T~0 and the KosterlitzThouless temperature
To as [2] T X = Too  T
T~oTc Tc
(2)
It follows that the two temperatures T~0 and Tc play a crucial role in the connection between vortexfluctuations and resistance data. T~0 can be obtained from the AslamasovLarkin formula [4] for the superconducting fluctuations just above T~0. In the present case the AslamasovLarkin formula in twodimensions give T~o  86.80 4 0.5K [1]. Tc can be determined from the dependence of the fluxflow 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 [1] gives Tc  84.16K [5]. The remaining quantity needed to establish the connection is the normalstate resistance which can be readily extracted from measurements of the resistance somewhat above T~0 [1]. 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 twodimensional GinzburgLandan Coulomb gas model. Next we review what is known about this function at present. From Kosterlitz renormalization group equations [6] one knows that close to X  1, when X goes to 1 from above, the function goes to zero as
(1)
Here R is the fluxflow resistance, RN is the normal state resistance, ~ is the GinzburgLandau 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 CuO2planes). Vortex fluctuations for a superconducting plane are described by a twodimensional Coulomb gas and, when the underlying superconductivity is described by a GinzburgLandau theory with a twocomponent order parameter, the corresponding Coulomb gas is the GinzburgLandan Coulomb gas [2]. The GinzburgLandau Coulomb gas has the property that a dimension
R/RN ~ exp( 25
A
Xvr2~_~)
(3)
26
SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
where A is a constant which is nonuniversal in the renormalization group sense but has a unique value within the GinzburgLandau Coulomb gas model [2]. This connection between the resistance ratio and the vortexfluctuations was first given by Halperin and Nelson [7] and has unfortunately been the cause of many too rash conclusions in the literature [2]. 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 [2]. Another way of expressing the limitation is that eq.(3) is only valid for very large lengthscales and these lengthscales are in practice usually masked by nonuniversal finite size effects [2]. 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 [2]. To large extent such good agreement must hence be attributed to too little structure in the data in relation to too many free parameters [2]. 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 [1] in the ligth of the above remarks. The resistance data for the Bi2Sr2CaCu2Oscrystal by Martin et al [1] 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 wellknown 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 [2]. 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 GinzburgLandau Coulomb gas has been corroborated through Monte Carlo simulations [8]. Fig.1 gives a direct comparison between the data by Martin et al for the Bi2Sr2CaCu2Oscrystal (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 Bi2Sr2CaCu2Osdata 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,
Vol.
71, No. 1
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Fi~.l. Comparison between the scaling function R/RN[X] (full drawn curve) and the data for Bi2Sr2CaCu2Oscrystals from ref.[1] (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 (nonuniversal) 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 GinzburgLandau 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 noncritical (and hence nonuniversal) properties of the GinzburgLandau 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 Bi2Sr2CaCu2Osdata 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 Bi2Sr2CaCu2Osdata. Fig.2 presents an alternative way of establishing the relation between the scaling function R/RN[X] and resistance data [9]. 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)/Rgvalue 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 TcoT. 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 Taxis we read off To0 = 86.7K from
Vol. 71, No.
i
27
SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
' ' L±~''
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83
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85 T
86
87
Fig.2. The quantity T/X(T) for the Bi2Sr2CaCu2Osdata from ref.[1] 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 Taxis 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 AslamasovLarkin formula T~0 = 86.80 4 0.5K [1]. 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 Bi2Sr2CaCu2Oscrystals. 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 [10]. The T~0 and the Tc for the indium/indium oxide films have been determined in precisely the same way as for the Bi2Sr2CaCu2Oscrystal i.e. 2/~0 from the AslamasovLarkin formula and T~ from the dependence of the resistance on a perpendicular magnetic field [10]. Thus with the parameters determined in precisely the same way the data for the indium/indium oxide films and the Bi2Sr~CaCu2Oserystal is virtually indistinguishable when plotted in the scaling variables corresponding to the GinzburgLandau 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
14
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X Fi~.3. Comparison between the Bi2Sr2CaCu2Osdata from ref.[1] (solid dots), the indium/indium oxide film data froin ref.[10] (crosses, circles, diamonds, and squares, corresponding to the samples $2, $3, $4, and $5 of ref.[10], respectively), and tile GinzburgLandau 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 Bi2Sr2CaCu2Osdata and the indium/indium oxide film data. As seen in the figure the Bi2Sr2CaCu2Osdata 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 Bi2Sr2CaCu2Oscrystals and thin 'dirty' type II superconducting films. In addition it demonstrates that the resistive tail in both cases are well represented by the twodimensional GinzburgLandau Coulomb gas scaling function R/ RN[X]. In summary we find that the resistive tail for the Bi2Sr2CaCu~Oscrystals and for thin 'dirty' type II superconducting films are remarkably similar when analysed according to the twodimensional GinzburgLandau Coulomb gas model. This clearly suggests that the resistive tail for Bi2Sr2CaCu2Oserysta]s are due to twodimensional vortexfluctuations. 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 nonuniversal properties of the CinzburgLandau Coulomb gas. This means that a stronger conclusion can be drawn about the underlying superconductivity of the CuO2planes. This underlying superconductivity is, in the region just above the (threedimensional) superconducting transition, well described by a standard twodimensional CinzburgLandau theory for a twocomponent orderparameter.
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SUPERCONDUCTIVITY AND VORTEX FLUCTUATIONS
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No. 1
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.[1] 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).