PHYSICAL REVIEW B
VOLUME 57, NUMBER 13
1 APRIL 1998-I
Resistive transition and fluctuation conductivity in Bi2Sr2CaCu2O81d single crystals
S. H. Han,* Yu. Eltsev, and Ö. Rapp
Department of Solid State Physics, Kungliga Tekniska Högskolan, SE 100 44 Stockholm, Sweden
~Received 14 July 1997!
Detailed resistivity measurements were made along the planes of Bi2Sr2CaCu2O81d single crystals in zero
field and in the temperature region above the superconducting transition temperature T c . Analyses of the
fluctuation conductivity Ds are presented using a careful construction of the mean-field critical temperature
T mf
c . Ds can be described by the Lawrence-Doniach model with a two-dimensional ~2D!-3D crossover close to
T mf
c . The c-axis coherence length was found to be 0.960.1 Å. For comparison Ds was also calculated from T c
defined from the peak temperature of d r /dT. It is shown that a consistent description of the 2D-3D transition
is then lost. @S0163-1829~98!07609-7#
It is well known that superconducting fluctuations play an
important role in high-T c superconductors. The high-T c itself
and the short superconducting coherence lengths, lead to a
pronounced rounding of the superconducting transition and
to favorable conditions for fluctuation studies. Such studies
are important in order to understand intrinsic properties and
dimensionality, and a large number of papers have been devoted to the subject. Examples of current issues are the temperature width of the critical region, the correct exponents
for the temperatature dependence of the excess conductivity
in the mean-field region, and the role of dimensional crossover in different materials. These questions are often connected to defining the correct temperature difference relative
to T c , a problem which increases in importance when the
critical temperature is approached.
In YBa2Cu3O72d single crystals, where anisotropy is
moderate, fluctuations were analyzed in the AzlamasovLarkin ~AL! model,1 and a dominating two-dimensional ~2D!
fluctuation contribution was found.2 However, the zero resistance transition temperature T c (R50), was higher than the
fitted mean-field temperature T mf
c , where conductivity diverges according to theory. Friedmann et al.3 found that the
best fit to their data was obtained for a Lawrence-Doniach
~LD! model,4 but T mf
c was again below T c (R50), and the
3D-2D crossover also occurred below T c . Alternatively,
avoiding the use of T mf
c as an additional fitting parameter,
and instead empirically associating the peak temperature of
d r /dT with T c , 5,6 a crossover was found in twin free single
crystals at about 1.8 K above T c , from a 3D behavior to a
critical regime closer to T c . The width of the critical region
estimated from the Ginzburg-Landau criterion is an order of
magnitude smaller than this value.7
In Bi2Sr2CaCu2O81d the larger anisotropy leads to reduced width of the 3D region, and the study of these crossovers is therefore correspondingly more difficult. 2D fluctuations in single crystals have been investigated by several
authors,8–11 and analyzed within the AL model in the meanfield regime. A crossover from 2D-3D has been reported,12
but it has also been claimed that this transition is absent due
to the intrinsic high anisotropy,9,10 which would push the
crossover into the critical regime. Some authors limited their
analyses to the region above T c 12 K, implying that a di0163-1829/98/57~13!/7510~4!/$15.00
57
mensional crossover or the transition to the critical regime
could not be investigated. The difficulty of defining T mf
c mentioned above, aggravates these problems, since temperatures
closer to T c have to be studied.
In this paper some of these problems are addressed. Measurements and analyses will be presented of the temperature
dependence of the resisitivity of single crystals of
Bi2Sr2CaCu2O81d. It is found that a careful determination of
T mf
c leads to a simple explanation of the results within the LD
model. For comparison, we also analyzed the excess conductivity, using the often employed definition of T c as the peak
temperature of d r /dT, and found that this choice leads to
some ambiguities in interpretation close to T c .
Single crystals of Bi2Sr2CaCu2O81d were grown by a selfflux method. CaO-stabilized ZrO2 crucibles were used for
carefully mixed powders of Bi2O3, CaCO3, SrCO3, and
CuO. The growth procedure was similar to previously published descriptions.13,14 The crystals were annealed at 500 °C
in air during 20 h. They were typically thin platelets ~10–50
mm! with shiny surfaces of dimensions in the range 0.5–2
mm. X-ray diffraction showed almost single phase samples.
Figure 1 is a typical example.
FIG. 1. X-ray-diffraction pattern for a Bi2Sr2CaCu2Ox single
crystal with Cu K a radiation incident on an ab-plane surface of the
crystal. With the exception of one reflection, marked with ? in the
figure, only (00l) peaks were observed.
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© 1998 The American Physical Society
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FIG. 2. Temperature dependence of the normalized electrical
resistance of two single crystals of Bi2Sr2CaCu2Ox . The straight
lines shown are extrapolations of the normal-state resistance.
Two single crystals of sizes 130.530.02 and 0.830.4
30.01 mm3 were selected for the measurements. A fourprobe in-line contact arrangement of silver strips was used
with 20 mm gold wires attached by silver paint. After heat
treatment at 500 °C for 3–5 min, the contact resistances were
less than 2 V.
The resistance was measured using a digital multimeter
~Solartron SB7081! with thermal compensation. The measuring current was 1 mA. Temperature was monitored using a
calibrated Pt resistor, with an absolute error below 50 mK
and a relative accuracy of 0.5 mK. Figure 2 shows the temperature dependence of the normalized electrical resistivity,
r (T)/ r (300 K) for the two single crystals, designated A and
B, respectively. Data taken on increasing and decreasing
temperature were found to trace the same curve. The temperature derivative d r /dT, is shown in the transition region
of both samples in Fig. 3. The fairly narrow and singlepeaked functions indicate crystals of good homogeneity.
FIG. 3. Normalized temperature derivative of the electrical resistance in the transition region. Data were taken on cooling ~open
circles! and warming ~full curve!. For clarity only a reduced set of
data is shown. Insets show the superconducting transitions on expanded temperature scales.
FIG. 4. Analyses of the fluctuation conductivity for both
samples. Mean-field critical temperature and exponents are given in
the figures and crossover temperatures are indicated by arrows. Insets: determination of T mf
c for both samples. The number of data
points have been reduced for clarity.
The excess conductivity, Ds is obtained from the measured resistivity r (T) and the linearly extrapolated normalstate resistivity r n (T) by
D s 51/r ~ T ! 21/r n ~ T ! .
~1!
The results were analyzed writing the excess conductivity as
D s 5C« x .
~2!
C is a temperature-independent parameter, and the reduced
mf
mf
temperature «5ln(T/Tmf
c )'T/T c 21 close to T c in the region where our studies are focused. In the mean-field region,
the exponent x is 21.0 in a two-dimensional system and
21/2 in three dimensions.1,4
Before performing the analysis, two important points
should be addressed; ~i! determination of the normal-state
resistivity r n , and ~ii! the definition of the mean-field critical
temperature T mf
c .
~i! r n was defined from extrapolations of the straight line
fits shown in Fig. 2. These lines could be well fitted to the
normal-state data for each sample in the region from 180 to
above 250 K.
~ii! For a strongly anisotropic system we still expect a 3D
fluctuation region close to T c when z c (T)5 z c (0)« 21/2 becomes large. With D s ;« 21/2 in this region, D s 22 vs T
should be a straight line, approaching 0 when T→T mf
c . This
condition is fulfilled over a certain temperature range as illustrated in the insets of Fig. 4, allowing T mf
c to be determined by extrapolation of straight lines. This method has the
advantage to avoid the often used, convenient, but somewhat
arbitrary definitions of T c , such as the midpoint of the transition curve, the temperature of the maximum of d r /dT, or
the zero resistance temperature. It also reduces the risk for
overflexible analyses, which could result with T mf
c as an additional fitting parameter.
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The main panels in Fig. 4 show the normalized excess
conductivity
for
both
samples
in
the
form
ln@Ds(T)/s(300 K) # vs ln«. Above about ln«>24 for both
samples, critical exponents x 1 of 21.04 and 21.00 were
found for samples A and B, respectively, in agreement with
thermodynamic fluctuations for a 2D systems. At a cusp
point T * , corresponding to « * 51.7131022 for sample A
and 1.2531022 for sample B, there is a crossover to a threedimensional behavior, with an exponent x 2 520.50 for both
samples.
A second cusp is observed at T G , with the corresponding
« G 53.9331023 and 2.8431023 for sample A and B, respectively. The width of the 3D region is small; 0.74 K for
sample A and 0.55 K for sample B. In the critical region
below T G , the critical exponents x 3 are numerically smaller
than in the 3D region. We found x 3 520.32 for sample A
and 20.27 for sample B. This shows that there is a cusp at
T G for both samples. The width of this region, T G 2T mf
c is
0.34 K for sample A and 0.25 K for sample B.
One must ask whether the observed properties are influenced by any sample deficiencies or reflect the intrinsic properties. Both sample A and B show an x-ray-diffraction pattern similar to Fig. 1 indicating single-phase samples. They
also show a linear normal-state resistivity ~Fig. 2! often used
as an empirical characterization of good sample quality. On
the other hand, the linearly extrapolated intercepts of the
resistivities at T50 are somewhat different and the temperature derivatives of the resistance in the transition region also
show some different structure ~Fig. 3!, which could be due to
small sample inhomogeneities. However, there is agreement
between the results for the two samples, both regarding the
temperature exponents as well as the various crossover temperatures, and these results are consistent with the LawrenceDoniach theory described below. These observations suggest
that sample differences do not influence the analyses and that
observed properties are intrinsic.
For comparison we also calculated the excess conductivity using the temperature T ci of the maximum of d r /dT
as an alternative definition of T c . These results are compared
in Fig. 5 to those using T mf
c . Also in this case there are two
cusps separating three temperature regions. For the region
at the highest temperatures, the critical exponents are
x 1 521.08 for sample A and 21.03 for sample B, in fair
agreement with thermodynamic fluctuations in the 2D regime.
Below this cusp, however, the critical exponents x 2 are
20.62 and 20.76 for samples A and B, respectively, significantly deviating from the value 20.5 expected for 3D fluctuations. As seen in Fig. 5 the exponents derived from the
region below a second cusp are also anomalous. The difference between T ci in this analysis, and T mf
c as used above, is
only about 0.2 K for both samples. The different crossover
temperatures are roughly equal as can be seen in Fig. 5, but
the critical exponents x 2 and x 3 have values which do not
lend themselves to a straightforward interpretation and furthermore vary appreciably between the two samples. This
illustrates the sensitivity of the estimated value of the excess
conductivity at small « (,131022 ) to the choice of the
critical temperature.
57
FIG. 5. Comparison between analyses with different critical
temperatures, T mf
c and T ci . The inflexion point of the transition
curve T ci can be seen to give approximately the same crossover
temperatures, but the exponents in the 3D region deviate from 20.5
and differ between the two samples.
The Lawrence-Doniach expression for the excess conductivity due to thermodynamic fluctuations for a layered superconductor is4
D s LD5
e2
16\s
1
A« A~ «14J !
.
~3!
Here J5 @ j c (0)/s # 2 is a measure of the interlayer coupling
strength, j c (0) is the c-axis coherence length at T50 K and
s is the distance between planes. Equation ~3! thus suggests a
crossover from 2D to 3D behavior, and a corresponding
change in the exponent of D s LD from 21 to 21/2, at a
temperature T 0 where « 0 5T 0 /T mf
c 2154J. Identifying T 0
with T * from the analysis in Fig. 4 we found J to be 4.3
31023 and 3.131023 for sample A and B, respectively.
Taking s515 Å, 15 we then obtained j c (0)50.98 Å for
sample A, and 0.84 Å for sample B.
We briefly compare these values of the coherence lengths
with results from other experiments. From careful analyses
of upper critical-field measurements, Palstra and coworkers16
found j c (0) of single crystal Bi2Sr2CaCu2Ox to decrease
from 1.6 to 0.45 Å, when T c was allowed to increase from
the zero resistance point to 50% of the normal-state resistance. Our T mf
c is in between these two estimates and our
results are thus in qualitative agreement with that analysis.
Magnetoresistance measurements may be the best method
to determine the coherence lengths. However, when a material is in the 2D limit, j c (0) cannot be determined. Since this
has often been found to be the case for Bi2Sr2CaCu2Ox ,
results for j c (0) are scarce. In a recent study of the c-axis
magnetoresistance of epitaxial films of Bi2Sr2CaCu2Ox , 17 a
formulation of the theory was used in the analyses, including
also a density of states effect in the normal-state
quasiparticles.18 j c (0) can be calculated from these results.
Within 10–15 % uncertainty given for the results of Ref. 17,
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BRIEF REPORTS
we found from their data j c (0)51.0560.12 Å, which compares favorably with our result of 0.960.1 Å for the two
samples.
The magnitude of the observed fluctuations is, in principle, a further source of information. However, due to the
small crystal size, the determination of resistivities is uncertain within a factor of 2. In addition a c factor may have to
be empirically introduced in such an analysis, which could
be larger than one even for single crystals. These uncertainties made it difficult to obtain a quantitative result, and these
attempts have not been further pursued.
We briefly summarize this work. The fluctuation-induced
excess conductivity in single crystal Bi2Sr2CaCu2Ox has
been studied. We have observed dimensional crossovers
from 2D to 3D fluctuations in the mean-field region and also
to a critical region closer to T c . A careful definition of
*Permanent address: Institute of Physics, Academy of Sciences,
Beijing 100080, China.
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1
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the mean-field critical temperature appears to be an essential
feature. This was illustrated by an alternative conventional
measure of T c as the inflexion point of the transition curve,
in which much of the consistency of the results was lost.
The important issue of sample quality was addressed to
some extent by investigating two samples of different qualities. The results were consistent in all aspects between these
two crystals, supporting that the results obtained are intrinsic. The zero-temperature c-axis coherence length was estimated from these analyses and found to be j c (0)50.9 Å
610%.
Useful discussions with Johan Axnäs are gratefully acknowledged. This work has been supported by the Swedish
Superconductivity Consortium and by the Swedish Natural
Science Research Council.
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