Analysis of the resistive transition in Bi2Sr2CaCu2O8+δ epitaxial thin films

ELSEVIER

PhysicaC 226 (1994) 320-324

Analysis of the resistive transition in Bi2Sr2CaCu208+ epitaxial thin films D.V. Livanov

1 G. Balestrino *, M. Montuori

Dipartimento di Ingegneria Meccanica, Universita" di Roma, "'Tot Vergata", v.O.Raimondo 8, 1-00173 Roma, Italy

Received 19 November 1993;revised manuscript received31 March 1994

Abstract

Electrical resistivities of epitaxial Bi2Sr2CaCu2Oa÷6films near the superconductingtransition have been studied. The analysis of the data was performed self-consistently in terms of two-dimensional Aslamazov-Larkin theory for temperatures above the transition and within the context of the Kosterlitz-Thouless theory at temperatures below the mean-fieldtransition temperature. A quantitative agreement between experimental data and theory was found confirming the picture of a highly two-dimensional behavior of macroscopic fluctuations connected with the superconducting transition in Bi2Sr2CaCu2Oa+6with no measurable effect of interplane coupling on the in-plane electrical-transport properties. Much effort is presently devoted to the study of the features of the resistive transition in high-T¢ superconductors in connection with both the possible technological applications and the understanding of the nature and mechanisms of high-temperature superconductivity. As better and better samples are available, the intrinsic behavior of the superconducting transition in these materials becomes manifest. The first important effect which should be mentioned is the pronounced order-parameter fluctuations above the critical temperature. Such fluctuations are due to the large spatial anisotropy and the very short coherence length and have been observed in a large number of kinetic and thermodynamical properties of high-Tc oxides. It is worth noting that the studies of thermodynamical fluctuations above the Ginzburg-Landau (mean-field) transition tem* Correspondingauthor. 1On leave of absence from Department of Theoretical Physics, Moscow Institute of Steel and Alloys, Leninskiy pr. 4, 117936 Moscow, Russia.

perature provide valuable information about the dimensionality of the order parameter. The second effect is the broadening of the transition itself which has been observed even in very good single crystals and epitaxial films. It has been proposed [ 1,2 ] that such a behavior can be attributed to the dissipation due to dissociation of vortex-antivortex pairs resulting from thermal fluctuations above the temperature of the Kosterlitz-Thouless (KT) phase transition (Tio-), which is an intrinsic feature of two-dimensional (2D) superconductors [ 3,4 ]. The question of the applicability of strictly two-dimensional KT theory to high-To materials still remains open because even thin films which are several unit cells thick consist of many conducting layers and are not actually a 2D system. Properties of systems which are geometrically three-dimensional can exhibit a two-dimensional behavior only if the effect of coupling between the layers is negligible. The dimensionality of the superconducting transition can be easily probed through order-parameter fluctuations above the Ginzburg-Landau transition

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D.V. Livanov et al. /Physica C 226 (1994) 320-324

temperature (TEL). Results form a large number of papers demonstrate that the fluctuation enhancement of conductivity in Bi and T1 based oxides is wellconsistent with 2D Aslamazov-Larkin (AL) theory in a wide temperature range and that there is no crossover to the three-dimensional regime in the range of gaussian fluctuations [ 5-7 ]. This conclusion applies both to thin films and bulk single crystals indicating that the interlayer coupling is negligibly small, at least near the transition, and that a strictly 2D behavior of in-plane electrical transport properties is an intrinsic feature of these materials. This result supports strongly the possibility of explaining the resistivity behavior at the edge of the superconducting transition in Bi and T1 based oxides within the context of KT theory. On the other hand for YBaECU307_x material the fluctuation conductivity was shown to have a more complex behavior with a 2D-3D crossover which seems to depend on the nature of the sample, whether it is a polycrystalline pellet, a bulk single crystal or thin film [8 ]. Therefore the study of the resistive transition in this material requires a comprehensive theory which takes into account the essential interlayer interaction. In the recent past there has been considerable effort to investigate experimentally the occurrence of the KT transition in high-temperature superconductors. KT properties have been examined in bulk single-crystal samples of YBa2Cu307 (YBCO) [ 9 ] and Bi2Sr2CaCu20 s (BSCCO) [ 1 ], oriented T1EBa2CaCu2Os (TBCCO) and YBaECU307_x thin films [10,11 ], and YBaECU307_x/PrBa2Cu307_x multilayers [ 12 ]. In these papers the resistivity data just below the mean-field critical temperature were compared with the predictions of the KT theory. The agreement between theory and experiment varied somewhat from paper to paper. For the reasons we have pointed out above, the data on YBCO can be compared only qualitatively with the predictions of KT theory whereas the analysis of data on high-quality BSCCO and TBCCO samples can provide reliable information about the nature of the resistive transition as well as an estimate of some important microscopic parameters. In this paper we present the results of resistivity measurements in Bi2Sr2CaCu208 + 6 epitaxial films at the edge of the superconducting transition and their interpretation in terms of KT theory for the case of a

clean superconductor. We show the good agreement between theory and experiment in the range of temperatures between TKa-and TeL. The exponential inverse-square-root reduced-temperature dependence of the resistance just above TKx is shown to be also consistent with the parametrization of the resistance data well above the transition by the 2D AL theory. The resistance measurements were carried out on high-quality monocrystalIine Bi2Sr2CaCu208 + 6 ! Ixm thick films grown by the liquid phase epitaxy (LPE) technique on NdGaOa substrates. LPE has proved to be a suitable technique to grow monocrystalline Bi2Sr2CaCu208 films truly epitaxial relative to the substrate with a very narrow mosaic spread (less than 0.1 degree). (001) oriented NdGaO3 slices provide a satisfactory in-plane match between the a and b lattice parameters of the film (a=5.41 /~, b=5.43 •, c=30.81 /~) and substrate ( a = 5 . 4 0 / ~ , b=5.50/i,, c = 7.71/k) and, therefore, can be used to grow ab oriented Bi2SrECaCuEOs films. The sample preparation as well as the structural and transport characterizations have been reported elsewhere [ 13 ]. The resistance was measured by the standard four-probe technique on a narrow strip 50 ~tm wide and 1 mm long. The strip was obtained by a standard photolithographic technique. Resistivity as a function of temperature is represented in Fig. 1 in the range 80-100 K. Measurements were performed on a number of samples showing that the shape of the superconducting transition remains exactly the same from sample to sample [ 7 ]. First we compared our data with the predictions o f fluctuation theory above TGL to determine precisely the value of the mean-field critical temperature and the effective interplane distance and to confirm the 2D character of the transition in our samples. The 4 .t:

~2 ¢.

21 08;

.f TKT-- 82.4 K .oo"

IjS i '

TGL='85"9 K

85 90 9; 100 Temperature (K) Fig. 1. Resistancevs. temperaturecurve.

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D.V. Livanov et al. / Physica C226 (1994) 320-324 -1.5

theoretical functional form for excess conductivity in the case of 2D fluctuation is [ 14 ] e2 TeL Atrfl= p - l (T) - p f f i (T) = 16h2d T - TeL '

where T is the temperature, p is the measured resistivity, PN is the normal-state resistivity extrapolated from high temperatures and d is the effective interplane distance. In the process of fitting the Eq. ( 1 ) to our data we have taken into account that Eq. ( 1 ) is valid only in the temperature range of gaussian (noninteracting) fluctuations, i.e. at temperatures where the thermal energy of the fluctuations is lower than their free energy. The size of the critical region in zero magnetic field can be estimated by means of the Ginzburg criterion, which for clean 2D superconductors is [ 15 ]: I T - T e L l
TeL

-2

(1)

1 (AC/ka)~2d '

where AC is the specific-heat j u m p and ~o is the zerotemperature BCS coherence length. Assuming the value coming from Ginzburg-Landau theory: A C = 0.35(pl, m/h3)T¢k 2 (Pv and m are Fermi momentum and quasiparticle effective mass, respectively) one obtains: Gi,~0.08. Notice, that the estimate of Gi from the experimental value of the specific-heat jump gives a Gi value of the same order, but somewhat lower. Hence in any case to be sure that data are out of the critical-fluctuations region we should consider temperatures above at least TGL-~-8 K. Results of fitting in the temperature range 94- l 10 K are shown in Fig. 2. The normal-state resistivity was extrapolated from the temperature range 200-250 K, where the effects of fluctuations are negligible. One can see, that in the region of gaussian fluctuations there is no visible evidence of a crossover to the 3D regime. From the best fit of the experimental data we found the following values: TeL= 85.90 +_0.05 K, d = 20__ 1 A. The obtained value of TGL corresponds to the ratio P ( T e L ) / P N ( T e L ) = 0.42. Results obtained support strongly our assumption that the resistive transition in our sample is of a strictly 2D character. Our next step is to fit the resistance data below TeL using the KT theory. Let us briefly discuss the main ideas. Since BSCCO epitaxial films seem to be within the clean limit (that is l> ~o, where ! is electron mean

-2.5

-3..

-2.5

-2

-1'.5

-1

-o.s

ln((T-TGL)/T~L) Fig. 2. Excess conductivity normalized for the normal-state conductivity at zero temperature p Kt (0) vs. ( T - TGL)/TGL in a lnIn scale with TOL=85.90 K. The solid line represents 2D Aslamazov-Larkin theory.

free path), we present a modification of KT theory for the clean superconductors. As was shown [ 16-18 ] the KT transition in a twodimensional superconductor involves dissociation of vortex-antivortex pairs into free vortices above the characteristic temperature TI> ;t the interaction energy falls off as 1/r. Here Ec is the vortex core energy, K i s the renormalized stiffness constant, r is the separation of the vortex pair and 2 is the two-dimensional magnetic penetration depth. The temperature range of our interest is Tio-< T < TeL. It was shown by Nelson and Kosterlitz [ 19 ], that a universal relation between TKT and two-dimensional superfluid density ns holds:

TKT-- nnsh2 8m

c2h2 _ 16e22,

(2)

where ns and ;t are themselves functions of the temperature. To evaluate explicitly Tio- for a clean superconductor we notice that for this case [ 15 ] 2=1.33

tanh\

,

(3)

where ;rE is the London penetration depth. By substitution of Eq. (3) into Eq. (2) one can obtain the implicit equation for TKx:

TKT A_~(TKxI =3.93 ~ hd TOE \TEL/ e2pN

'

(4)

D. V. Livanov et al. / Physica C 226 (1994) 320-324 where A (T/TOE) is the temperature-dependent function contained within the brackets in Eq. (3). For temperatures near TK-r Eq. (3) reduces to the simple form [3] r¢=0.23 1 eEpN ~o hd ' Zc=

323

10 o

8 6 6

TGL--TKT TKT

'

(5) 2

which is valid when Zc<< 1. One can estimate from Eq. ( 5 ) the expected value of % using the typical data for BSCCO films: l / ~ , 5, d ~ 15 A and PN (85 K) ,~ 70 ~tf~ cm. One finds: Zc~ 0.03. Under an external electric current above TKT freevortex motion results in a flux-flow resistivity proportional to ns which in the low-current limit for a clean superconductor is given by p=8.351~lns .

(6)

At temperatures just above Tr.Tn~ is proportional to the inverse square of the average distance between thermally induced free vertices ~+ [ 4 ]: ns = 2 ~ C ~ 2 ,

[

~ + = a ~ ( T ) exp b

,

(7)

where a, b and C are constants of order unity. Finally one finds: P =A l PN ~exp

[

-2b

.

(8)

Here A is a nonuniversal constant of order unity which depends on the sample characteristics. Notice, that the difference with the dirty case is only the appearance of a factor l/~o in the right sides of Eqs. (5) and (8). It is worth emphasizing that, as is seen from Eq. (5), in very clean samples with l/~o >> 1 the interval between TGL and TKT can increase substantially and can be much larger than the value predicted by the dirty-limit theory. In Fig. 3 we have plotted the behavior of - I n (p/ PN) as a function of reduced temperature [ ( TGL-- T) / ( T - T K T ) ] 1/2. A good agreement with the expected behavior for 6×10-3~

00

0.5

1 1.5 2 2.5 3 3.5 ((TGL_T)/(T.TKT))1/2 Fig. 3. Temperaturedependence of the resistivitybelow TaL in a In(PN/P) VS. [ ( TOL-- T) / ( T - TKX) ] t/2 plot. The solid line is the fit with TKr= 82.41 K over the temperature range 82.5 < T< 85.7 K. TKT=82.41+0.05 K, b = 1 . 3 5 + 0 . 0 1 , l / ~ 2 . 5 (we assumed A ~ 1 ). Such a value of TKx corresponds to a value rc=0.042 in reasonable agreement with the estimate from Eq. (5). Let us discuss the results obtained. The first important issue has to do with the dimensionality of the order parameter in BSCCO films. The KosterlitzThouless theory is associated with the establishment of long-range order and phase fluctuations below the transition point, while 2D AL theory is connected with thermodynamical fluctuations of the order,parameter amplitude above the transition. As both theories appear to be applicable to the resistive transition and a 2 D - 3 D crossover in the excess conductivity was not found, we can conclude that the 2D character of the transition holds throughout the transition including the region of critical fluctuations which was out of our consideration. Of course, this conclusion relates to the in-plane electrical-transport properties only, because actually there exists a finite probability for an electron to move between layers. The magnitude of the corresponding hopping integral can be evaluated from the anisotropy of normal-state conductivities or upper critical fields and more precisely from the analysis of the out-of-plane fluctuation conductivity. In Ref. [21] the hopping integral for BSCCO epitaxial films was estimated to be w~ 1015 K. This leads to the temperature of 2 D - 3 D crossover Tcr-TGL ~ 1-2 K, which lies in the region of critical fluctuations in accordance with our results. Another essential point is our estimation of the ratio between the electron mean free path and the correlation length, I / ~ 2.5. This value can be consid-

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D. V. Livanov et al. / Physica C 226 (1994) 320-324

ered only as a quite rough estimate, because the fitting parameter from which it was obtained involves the constant A of order unity which is not defined precisely in the theory. Nevertheless we can conclude that the films are in the clean limit rather than in the dirty one, but in fact an intermediate case is realized in agreement with an independent estimate of this value [20]. If we assume that in our high-quality monocrystalline films the mean free path is associated with disorder due to excess oxygen atoms, there should be a correlation between l/~o determined by the method shown above and the oxygen content of the sample. Results of such an analysis will be presented elsewhere [ 22 ]. In summary, we have performed an analysis of the shape of the resistive transition in high-quality epitaxial Bi2Sr2CaCu2Os+ 6 thin films n terms of the Kosterlitz-Thouless transition theory below the threshold of superconductivity and Aslamazov-Larkin fluctuation theory well above the transition. An exponential inverse-square-root reduced-temperature dependence of resistivity was found above the Kosterlitz-Thouless temperature TKT~ 82.41 K and below the Ginzburg-Landau critical temperature TOL~ 85.90 K. The use of the clean-limit theory allowed us to improve the agreement between theory and experiment. The parameters obtained from the fitting of the experimental data with the KT theory are consistent with the analysis of the resistivity behavior above the mean-field transition temperature by means of two-dimensional AL theory. These resuits confirm the scenario of a highly two-dimensional behavior of macroscopic fluctuations connected with the superconducting transition in Bi2Sr2CaCu2Os+ ~ films in which there is no measurable effect of the interaction between the layers.

Acknowledgements One of us (DVL) would like to thank the cooperation program between University of Roma "Tor

Vergata" and the Moscow Institute of Steel and Alloys for supporting his stay at the University of Rome "Tot Vergata" during this work.

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