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Fluctuations in the microwave conductivity

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Fluctuations in the microwave conductivity of YBa2Cu3O72dsingle crystalsin zero dc magnetic fieldSteven M. Anlage, J. Mao, J. C. Booth, Dong Ho Wu, and J. L. PengCenter for Superconductivity Research, Department of Physics, University of Maryland, College Park, Maryland 20742-4111~Received 16 June 1995; revised manuscript received 5 October 1995!We present a quantitative analysis of finite frequency fluctuation conductivity above and below Tcin cupratesuperconductors in zero dc magnetic field. In a YBa2Cu3O72dcrystal showing a linear in temperature increaseof the penetration depth at low temperatures, we find that two-dimensional ~2D! finite frequency Gaussianconductivity fluctuations above Tccross over into a slower divergence of the conductivity as Tcis approachedfrom above. We find that the critical regime above Tcis less than about 0.6 K wide. At and below Tc,3Dfluctuations dominate the conductivity, with evidence of 3D XY critical scaling of the imaginary part of theconductivity down to 5 K below Tc.Cuprate superconductors differ from their low-temperature counterparts in a number of significant and dra-matic ways. Their small superconducting coherence length,reduced dimensionality imposed by the CuO2planes, lowcarrier density and high transition temperature lead to situa-tions where superconductivity can be strongly affected by thelarge thermal energies available. The cuprates offer the op-portunity to establish the nature and universality class ofsuperconducting fluctuations.It has been known for some time that the normal state dcconductivity of the cuprates is strongly enhanced well abovethe thermodynamic transition temperature, Tc.1,2Specificheat measurements on highly homogeneous single crystalsshow evidence of a peak at Tc, where BCS ~mean field!theory says that only a discontinuity should appear.2Theresemblance of this peak to the lambda transition in super-fluid4He suggests that the cuprates may be in the same uni-versality class, namely that of the three-dimensional ~3D!XY model. Recently, experimental evidence obtained from rfpenetration depth measurements also suggests that 3D XYcritical fluctuations exist down to 9 K below Tc.3In this paper we shall examine the microwave conductiv-ity in the normal state to identify further evidence for en-hanced fluctuations in the cuprates. We shall focus on micro-wave surface impedance measurements on a high qualitysingle crystal of YBa2Cu3O72d~YBCO! which shows alinear-in-T increase of the penetration depth at lowtemperatures,4and 3D XY critical behavior just below Tc.We find that this crystal has no corresponding range of criti-cal fluctuations in the microwave conductivity above Tc.Growth and characterization of the YBCO crystals hasbeen discussed in detail elsewhere.4,5The twinned YBCOcrystals ~;1mm31mm330–50mm! typically show zerodc resistance at 92–93 K, with a normal state resistivity at100 K of about 40–50mV cm, similar to that of the Illinoiscrystals.6ac- and dc-magnetization measurements also con-firm the transition temperature to be 92–93 K with the tran-sition width approximately 0.2 K. Although several crystalswere measured for this work, we shall concentrate on theanalysis of the crystal showing the greatest degree of homo-geneity in surface impedance measurements.The surface impedance measurements were made by acavity perturbation method which is discussed elsewhere.4,5,7Briefly, the surface resistance Rs(T) and change in surfacereactance DXs(T) of the samples are measured simulta-neously, from 4.2 K continuously up to well above Tc.Asakey step in the analysis of the data, we convert DXs(T) intoan absolute magnitude: Xs(T), following the method of Refs.4 and 5. We then determine the full complex conductivitys5s12is2, from the definition Rs1 iXs5[ivm0/(s12is2!#1/2in the local limit. Note that the data used in thispaper has no contribution from shielding currents which flowin the c direction4~unlike Ref. 3!, and that no residual resis-tance is subtracted from Rs2 abbefore calculatings1.The inset of Fig. 1 shows results for the temperature de-pendence of the penetration depth and conductivitys1of asingle crystal of YBCO. The magnetic penetration depth,l~Xs5m0vl!, increases from its low temperature value lin-FIG. 1. Complex conductivity,s1~T!~circles!,s2~T!~dashedline! in the vicinity of Tcof a YBCO crystal at 9.6 GHz, along with2D fit mean field and total conductivities ~solid line! above Tc.Inset shows the penetration depth ands1~T! over the temperaturerange from 4.2 to 110 K, as well as the BCSs1~T!~dashed line!.PHYSICAL REVIEW B 1 FEBRUARY 1996-IVOLUME 53, NUMBER 5530163-1829/96/53~5!/2792~5!/$06.00 2792 © 1996 The American Physical Societyearly with temperature, with a slope of approximately 5 Å/Kfor5K,T,35 K.4,5,8As Tcis approached from below, thepenetration depth increases very abruptly, suggestive of un-conventional behavior. On the other hand, as Tcis ap-proached from above,s1~T! is considerably enhanced, show-ing a sharp peak in the vicinity of Tcat a temperature wedefine as Tcs~see Fig. 1!. This enhancement ofs1~T! nearTcin microwave measurements of YBCO has been notedbefore4,9–11but has not yet received a detailed analysis.When the same measurements are done on a Nb crystal, noenhancement ofs1~T! above Tcis seen, and there is no peakat Tc, although a BCS coherence peak is observed at lowertemperatures5,12~similar to the dashed line in the inset of Fig.1!.When analyzing the data above Tcshown in Fig. 1, onemakes the assumption that the total conductivity is the sum:s1~T,v!5s1MF~T!1s1fluc~T,v!, wheres1MF~T!51/~r01bT! isthe ‘‘mean field’’ conductivity ~T. Tc!, which we take to befrequency independent, ands1flucis due to superconductingfluctuations above Tc. Far above Tc, the fluctuation contri-bution tos1should be small, and fits tos1~T! for 100 K,T,150 K ~Ref. 13! giver~T!'26.8mV cm10.47 ~mV cm/K!T, which is consistent with the measured dc resistivity ofcrystals from the same batch. The enhanced conductivityabove Tchas been measured by many groups at dc,6,14and asummary of their best fit results is given in Table I. The meanfield Tcvalues of all these dc fits are several kelvin belowthe measured zero resistance temperature. This is manifestedin the data by a conductivity which diverges more quicklythan the theoretical predictions as Tcis approached fromabove.6As a result, the dc data is only fit relatively far abovethe transition, excluding a great deal of potentially importantdata


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