PHYSICAL REVIEW B
VOLUME 49, NUMBER 17
1
MAY 1994-I
Excess conductivity and pair-breaking effects in YBazCu307 and Bi2SrzCa, „Y„Cu20s+s systems
S. N. Bhatia and C. P. Dhard
of Physics, Indian Institute of Technology, Bombay 400 076, India
Department
(Received 27 September 1993)
To explain the behavior of thermal fluctuations, we have measured the electrical resistivity of five samples: thin-film YBa2Cu307, polycrystalline YBa2Cu307, and Bi&Sr2Ca& Y Cu208+z with x =0, 0.05, and
0.20. The whole temperature range of fluctuations consists of three regions. Within the mean-field re(3D) and
gion, YBa2Cu307 and Bi2Sr&Ca, Y Cu&08+& samples show predominantly three-dimensional
2D behaviors with the exponents 2 and 1, in the critical region the two systems display a behavior analogous to 3D-XY and 2D-XY models, respectively. Above the mean-field region [1n(e) —1.0] exponents
have values larger than those predicted by Reggiani et al. Both the systems favor strong pair-breaking
effects and integral dimensions for the thermal fluctuations.
,
)
I.
u:
INTRODUCTION
In order to unravel the mechanism of conduction of
high-T, superconducting materials, the study of electrical
resistivity has been the subject of intense investigations.
In well-oxygenated samples (polycrystalline as well as single crystals) the resistivity drops linearly with temperature and below a so-called on-set temperature deviates
from the linear behavior before approaching zero. This
depression in resistivity or the enhancement in conductivity is also known as paraconductivity
(o~) and is
affected by fluctuations of small regions of the sample
where Cooper pairs are produced. Aslamazov and Larkin' (AL) have shown that this excess conductivity in the
mean-field approximation to the BCS theory is given by
0 AL
C AL g
—v
2
—
e (D 4)/2
32~
v= —,' and s =g, (0) for
the three-dimensional (D =3)
isotropic conduction and s =d/2, v= 1 for thin-film samples where the conduction is restricted within the plane
of the film (D =2). Here d is the film thickness, g, (0) the
amplitude of the coherence length perpendicular to the
film plane and e = ( T —T, ) /T, with T, the mean-field
critical temperature. For strongly anisotropic superconDoniach
Lawrence
and
ductors
(LD) used an
anisotropic-mass formulation of the AL theory to derive
with
where
This
A,
e
l
16iilde
[1+k(e)]
(e)=[2/, (
equation
)/de)
predicts
2
=(2a/e)
a
cross
(trLo=1/e) to a 3D (oLD=1/v'e)
T„given by
with
a=2[(, (0)/d]
.
over from a 2D
behavior to occur at a
temperature
T„=([[2$,(0)/d]] +1)T,
(3)
Anisotropies in the crystal structure of the cuprates affect
the transport properties and possibly cause the reduced
dimensional nature of conduction in these systems. o~ is
also expected to show a similar behavior. Two factors
influence strongly the determination of the dimensionali0163-1829/94/49(17)/12206(10)/$06. 00
49
the procedure used to separate the measured
ty of
conductivity into the normal (tT„) and the paraconductivity parts and the determination of the T, to be used in
the calculation of tr [Eqs. (1) and (2)].
Out of the four families of the oxide superconductors
(LBCO),
(YBCO),
„Ba„CuO~
YBa2Cu307
(BSCCO),
T12BazCaCu20s+ s
Bi2Sr2CaCu20s+s
(TBCCO), YBCO has been studied most extensively. In
polycrystalline YBCO, it has been possible to fit 0 to the
3D AL model
within
the
temperature
range
—5. 0(ln(e) & —2. 8 with an effective g (0) around 2.0
A. Above 1n(e)) —
2. 0, tr~ fell rapidly to zero with absence of any 2D region, and was attributed to the
of short-wavelength
fluctuations falling
predominance
outside the purview of Ginsburg-Landau (GL) and hence
the Al theory. The data on thin films with their c-axis
oriented perpendicular to the substrate, agreed well with
the LD model within —5. 0~In(e)(0. 7 except that a
qualitative factor C = —,' had to be introduced to match
the magnitude of 0. with that predicted by the model.
Measurements of Hagen, Wang, and Ong on the single
crystals agreed well within the 2D AL model but gave a
T, 3 —5 K below T, (p=O). Friedmann et al. from their
measurements on the single crystals showed that it was
possible to fit both the models mentioned above, even including the Maki-Thomson term (see below) with almost
the same precision over very wide temperature range extending to temperatures as high as 200 K. The T, obtained here had no correlation with the inflection point of
the resistivity curve or with T, (p=O) and the efFective
correlation length g', (0) and the interlayer spacing d had
Veira and
values too small to be physically meaningful.
Vidal from their extensive studies on polycrystalline
samples of YBCO have shown the individual grains in
these samples to exhibit o. similar to those in single crystals. The factors like C or the unusually small values of
g, (0) and d observed in the polycrystalline samples result
due to random orientations of the a-b planes of the grains
and are accounted for by introducing a temperatureindependent factor p which accounts for the reduction of
the effective cross-sectional area of the samples due to the
various orientations of the a-b planes. In YBCO samples
La&
12 206
1994
The American Physical Society
EXCESS CONDUCTIVITY AND PAIR-BREAKING EFFECTS IN.
they find large regions of 3D behavior with a cross over
to 2D, whereas the mean-field region in BSCCO to be
dominated by 2D behavior only. Later Bhatia et al. observed a cross over with a small 3D region in these samples also. It is essential to estimate the o. „by an independent method. Fitting the data to an expression for the total conductivity containing 5 to 6 adjustable parameters,
leads only to an erroneous evaluation of o. and therefore,
of the parameters involved therein. With cr evaluated
the next parameter required to analyze o is T, . Choosing a criteria for T, is as good as fixing it. In Veira's
analysis of YBCO, both choices of T, (T, and T„) give
exponent v=0. 48, corresponding to a 3D AL behavior
for YBCO and 0.97 for BSCCO, leaving no room for the
inclusion of any other contribution to cr . Alternative
methods not requiring a prior knowledge of T, such as
the Testardi plots' or the logarithmic derivative plots"
may prove useful here.
After the decay the quasiparticles of the pair continue
to exist in phase locked state for a time ~& till further collisions destroy this phase relationship. This enhances the
conductivity further and this anomalous contribution,
calculated by Maki and Thompson
(MT) and later
modified by Hikami and Larkin' has been the subject of
recent investigations.
The earlier analysis of o.z comMost of the present
pletely ignored this contribution.
efforts to detect this term appear to have been concentrated on YBCO only. Thus Matsuda et al. ' find strong
' sec—
pair-breaking effects (r&
to
) corresponding
negligible conductivity contribution by this effect to their
zero-field data of polycrystalline
YBCO. Hikita and
Suzuki' also find similar effects in the monocrystals of
YBCO, but they appear to have overestimated o.z as they
find significant value of o. to be present even at 200 K
(T~2T, ). Kim et al. ' in their analysis of YBCO thin
films could accommodate a substantial contribution of
MT term with ~&—10 ' sec. Veira and Vidal' rule out
(v&~10 ' sec) from this
any important contribution
effect in YBCO as well as Bi~ ~Pb, SrqCaCuqOs+g even
after allowing the quality of the fit to deteriorate.
It has been proposed that the thermal fluctuations develop in a space having fractal topology. In that case,
Char and Kapitulnik'
have shown the exponent v to be
given by
~
-10
-
v=2 —d /2,
"
12 207
squares fitting procedure as mell as by Testardi and the
logarithmic derivative plots and find the MT contribution
to be present below the detection level in zero field and
the fluctuations to grow in integral dimensions as the
temperature is reduced towards T, .
II. EXPERIMENTAL
YBCO bulk samples were prepared by the solid state
reaction method by reacting Yz03, BaCO3 and CuO at
930'C for 24 hrs. After two intermediate grindings the
samples were annealed in the Oz atmosphere at 900'C
and cooled down to 600'C at the rate 1 'C/min and maintained at that temperature for 12 hrs. Temperature was
further reduced to 300'C at the rate 0. 5'C/min and then
furnace cooled to room temperature.
YBCO thin-film
samples were grown by the laser ablation technique on
SrTi03 substrate. The BSCCO samples were prepared by
matrix
method
precursor
by reacting
Bi&03 to
in the presence
of Oz at
precursor
Sr& CaCuzOs
900-950'C. X-ray diffraction analysis confirmed the
polycrystalline samples to be of single phase and films to
be aligned with c axis perpendicular to the substrate surface. dc resistivity measurements were performed on the
bar shaped samples (1 X 2 X 8 mm ) by the standard four
probe method. Current leads were fixed on the side faces
of the bars with Ag paste curing at 100'C (approx. ). On
the films, pads were deposited by silver evaporation, and
then Ag wires were attached with the paste. Current
density used was typically 0. 1 A/cm, which is much less
than the critical current density in bulk. Keithley 224
current source and 181 Nanovoltmeter
programmable
were used as current source and voltage measuring device. The sample temperature was measured by a calibrated Si diode (LakeShore DT500 recalibrated against
Pt thermometer) which was in contact with the sample in
a Cu sample holder. The temperature was controlled by
LakeShore DRC-91C temperature controller. The sample temperature was lowered with the cooling rate =0. 5
K/hr and the data were taken at the interval of 10 mK
within the critical and mean-field regions and 0.5 K in
the normal region.
III. RESULTS AND
(4)
where d is the fractal dimension of the fluctuations. A
' or 1 has been interpretdeviation in the value of v from —,
ed as the evidence for the presence of the fractals but
there is no consistency in these results. Thus Clippe
et al. ' obtain integral values for d in ceramic sgmples of
YBCO whereas Pureur et al. observe a transition from
to 3 in same materia 1. In the Bi series the resu 1ts
3
are equally controvertial. Fractional dimension of —
', has
been observed by Clippe et al. ' whereas Mori, Wilson,
and Ozaki' observed the integral value of 2 in BSCCO.
We have measured the resistivity of five samples:
thin-film
and
YBCO
and
polycrystalline
Y CuzOs+s doped with x =0, 0.05, and 0.20
Bi&Sr&Ca,
and have analyzed the data in the conventional least-
..
DISCUSSIOiV
—
The resistivity of all the five samples
thin-film and
YBCO and the three samples of
polycrystalline
BSCCO decreases linearly with temperature thereby
showing a metallic behavior in the normal state. The
width of the transition for both types of samples is
around 1 K. Upon numerically differentiating the data
by linear least-squares fitting three points in a walking
window, the d pz /d T obtained showed temperatureindependent
straight line in the normal state region
( T 2T, ) and a single peak in the transition region which
shows the samples to consist of single phase only. The
behavior of YBCO in the transition region is shown in
Fig. 1. Similar behavior has been found in other samples
—
also.
S. N. BHATIA AND C. P. DHARD
12 208
8-0
2
YBCO
0 8
E
FIG. 1. Experimental resistivity (circles) and its temperature derivative (squares) plotted
against temperature in the transition region for the YBCO polycrystalline sample. Single symmetric dpT/dT peak shows the
sample to be of single phase.
E
4
0—
0 4
0 2 O
EA
U
N
2
0—
0 0
-0
0 0
92 0
93 5
93 0
92 5
94. 5
94 0
Temperature
95 0
96 0
95 5
(K)
and the lower end was varied from 200 K down to 120 K.
The rms deviation (rmsd) remained constant (at 0.077%
for polycrystalline YBCO, e.g. ) till a temperature =170
K, below which the deviation increased rapidly. This
lower temperature end of the normal part is =2T, for all
samples. Both the expressions for p ~ gave the same
rmsd but the value of D' in the p„(AZ) comes out to be
very small refiecting the data to follow the linear metallic
behavior. We find D' to be six orders of magnitude
smaller than C' for both the series (see Table I) giving a
nonlinear term (=D'/T) 10 ' times smaller than the
(=C'T) at 100 K. This is also
linear contribution
reflected by the temperature independence of d pr /d T for
A. Background conductivity Sttings
Data in the normal state were fitted to the expression
proposed by Veira and Vidal
pa=pga/p+pc
2
~
where p ~ is the normal state resistivity of an individual
grain, p averages the inclinations of the a-b planes of individual grains to the direction of the current How and p,
gives the intergrain resistivity. pgz has been observed to
follow either the linear metallic [pgz(L) = A+BT] or the
Anderson-Zou
(AZ) expression [pgz(AZ)=CT+D/TJ.
To estimate this resistivity, and the width of the normal
state region, the data were fitted to pz for both the expressions of pgs, i.e., ps=(A +BT)/p+p, = A'+B'T
for the linear expression with A ' = A /p + p, and
B'=B/p
and
ps =CT/p+D/pT+p,
=C'T+D'/T+p, with C'=C/p and D'=D/p for the
AZ expressions. To estimate the width of the normal
state region, the upper end of the data was fixed at 270 K
T + Tonset
'
B. Selection of T, '
Excess conductivity ho, „was calculated by subtracting
the background conductivity
tivity o z =1/pz as
b,
from the measured conduc-
o,„=1/PT —1/pz .
TABLE I. Normal state fitted parameters and T, values for various samples. Both the forms of pg&
fits with the same rms deviation.
Parameters
p,
(275 K) qnm
T, (8=0) K
Transition
Width K
T, (dpT/dT)max
K
Normal Part:
Fitt. temp. range K
A' pram
B' pQm/K
%%uo
rmsd
C' pQm/K
D' (10 pQ m K)
p, pQm
%%uo
rmsd
BSCCO
YBCO
YBCO
Polycrys.
film
x=0
14.33
92. 12
1.11
86.4
60. 16
85.7
BSCCO
=0.05
x
BSCCO
=0.20
x
1
1
4
6
93.5
88.5
88.5
89.0
36. 17
50.33
9
76.0
250-168
250-160
0. 15
260-170
260-150
260-140
12.36
1748.0
0.06
1748.0
3.26
12.36
5.80
8.88
591.88
0.06
591.88
2.600
981.34
0.09
981.34
0.055
5.80
8.87
0.06
0.06
0.09
3.77
385.48
.07
385.20
0.016
3.76
0.09
35.87
0.02
35.87
0.200
0. 15
0.03
22.20
63.58
EXCESS CONDUCTIVITY AND PAIR-BREAKING EFFECTS IN
49
These high-T, materials exhibit an anisotropic behavior.
The conduction is predominant in the a-b planes at the
temperatures away from T, . At temperatures near the
T, when correlation length along c axis becomes of the
order of interplanar distance d, the excess conductivity
shows a cross over from 2D to 3D. The proper selection
of T, (mean-field transition temperature) is very important in order to see the dimensionality of the fluctuations.
Generally this is taken either as the temperature corresponding to the half of the resistivity value at the T, onset (T, &) or the temperature
„) corresponding to the
maximum of dpr/dT.
We have chosen the later criterion.
„also cannot be determined unambiguously:
there are errors associated with the flatness of the peak
and also with the numerical differentiation of pT to obtain dpr/dT. We have tried to vary the T, in small
steps around the value of
„. We have plotted
ln(Acr, „) vs ln(e) for various values of T, as shown in
Fig. 2. For some value of T, these plots are straight
over a wide range of e, indicating thereby that the excess
conductivity is trying to obey a power law, her, „=e
There is a break in the slope of each one of these plots
giving v=1 for larger values of e and v= —,' for the smaller values. The T, obtained together with the extent
over which the exponent v has the values —,' and 1 are listed in Table II. The width of the plot can be adopted as
the criterion for the selection of T, . But a selection of
T, on this basis precludes other contributions to 50,„
which may have different dependences on e. The other
contribution relevant in this region is that due to Maki
and Thompson (MT). In zero applied field this is given
b 12
14 . 0
capo
b~ z~
0—
12
0o
c
&
o
oa
X
X
o
b
12 209
~
YBCO
0p
0
0
p
X
vvvvvv~v
10.0—
. .
X
v
a
c
8 0
(T,
T,
T,
e
8A d e(1
5
—a/5)
I+a+&1+2a
a 1+5+ &1+25
where 5=16/, (0)kTr&/(meed ). We note that this ex
pression can be simplified. The phase breaking time r&
has been seen 13 to vary as Te 'To/T wher'e ro is a con-
4 0
-9
I
0
-80
I
-70
I
-60 -5.0
I
I
-40
I
-3 0
Parameters
T
K
This makes 5 independent of T. For the values
and ro -10 ' se—
c typical
of the YBCO and BSCCO systems we obtain
for
the entire width of the mean-field region (MFR}. Even
though a depends upon e, because of the logarithm involved the coefficient within the parenthesis in 0 Mr becomes a very slowly varying function of T. Further since
a/5»1, (1 —a/5) = —a/5 and the prefactor in Eq. (7)
therefore, e /[8fide(1 —
a/5)]= —e kro(mk d) for the
values of a and 5 defined above. This means o~r is
essentially temperature independent ( = CMr } or atmost a
slowly varying function of T within the entire MFR.
This is shown in Fig. 4 for the typical values of the parameters involved.
The excess conductivity which is equal to the sum of
0 AL and 0 MT can be written as
g, (0}=2 A, d =11.7 or 30.7 A
a»5
(8)
Critical region K
Parameters
YBCO
Film
x=0
x =0.05
x =0.20
93.4+0. 1
94.2-99. 8
88. 4+0. 1
88.6+0. 1
89. 2-90. 2
90.2 —100.5
89. 1+0. 1
90. 8-92
92 —100
76.0+0. 1
11.3
0.6
9.2
1.7
BSCCO
BSCCO
90-97.3
97. 8-105
78-79. 8
79. 8-96
10.8
15
0.8
1.6
94-98
90-95
93.4+0. 2
88. 5+0.2
90. 5-100.5
88. 5+0.2
89.0+0.2
79. 5-94. 5
76.0+0.2
94 —100
90-98
98-105
89-90
90-101
91-92
92-101
78-80
80-98
88.5
88.54
89.0
76. 1
18
2.0
obtained from Fig. 3
3D Temp. range K
2D Temp. range K
Tmf
Parameters
BSCCO
YBCO
Polycrys.
100-105
K
91.6-99
obtained from Fig. 5
3D Temp. range K
2D Temp. range K
Tmf
100-104
93.45
00
stant.
obtained from Fig. 2
3D Temp. range K
2D Temp. range K
MFR K
-10
FIG. 2. The plot of ln(her, „) vs ln(e) for YBCO sample for
various values of T, '. 93.0 K for (a), 93.2 K for (b), 93.4 K for
(c), 93.6 K for (d), and 93.8 K for (e).
TABLE II. Value of the T, ' and 2D and/or 3D temperature ranges by choosing T, for the largest
2D and/or 3D widths (Fig. 2), from 1/b O.,„vs T plots (Fig. 3) and from the Testardi plots (Fig. 5).
Parameters
I
-2 0
12 210
S. N. BHATIA AND C. P. DHARD
for polycrystalline
samples
the Veira model,
within
where
1/F =(1 —
p, /pr )(1 —
p, /p&) .
(9)
In general F is also a slowly varying function of the temperature and within the same interval, it also will not
Hence over the 2D and the 3D
vary with temperature.
region,
0,„=
v=1 or
~V
+ CMT
(10)
'
for the respective regions. Thus plots of
' vs T in the 2D
region and
vs T in the 3D region should
give straight lines with the intercept of the lines on the T
axis defining the value of T, .
These plots for the undoped BSCCO samples are
shown in Fig. 3(a). The values of CMz have been calculated from Eq. (7) for different values of ~o. Values of p
and F have been taken from the nonlinear least-squares
fitting procedure described below but their exact values
do not appear to have any bearing on the present arguwith
—,
F2=[he, „(p/F—
)CMz]
Y3 = [ho, „(p/F—
)CMz]
ment. The curves (a) through (c) in this figure have been
drawn for successively decreasing values of ~0. As can be
seen they are curved and appear to stretch over to
straight lines with decreasing values of wo. Curve (d) is a
plot of 1/Ao„vs T, i.e. , with CMT=O and shows the
longest straight portion =(90—107 K). This points towards a negligible contribution from the Maki-Thompson
process. This temperature range corresponds with the
2D portion of the MFR determined above. Similar
behavior was observed in all Y-doped samples. Straight
line graphs could only be obtained by assuming vanishing
values of CMT In Y2'
In YBCO samples also Y3 vs T plots were straight only
for CM~=O [Fig. 3(b)] and here the straight portions
matched well with the 3D parts of the MFR. In principle
plots of Y3 vs T for BSCCO over the 3D region and Y2 vs
T for YBCO over the 2D region can also be used to gain
information about CMT but their widths were found to be
too small for them to be useful.
This procedure puts an upper limit of 10 ' sec for ~Q.
A value of vQ larger than this produces curvatures in the
plots that can be detected by the eye. We observe from
Fig. 2 that a change of +0. 1 K in T, changes v from
0.44 to 0.56. b, v=0. 06 is suffIcient to include the MT
5 Qx1Q
4 Ox10
3 Qx10
CL )L
I
2
1
0 x10
Q
-4
x10
80 0
85 0
90 0
95
Temperature
105 0
100 0
Q
(
110 0
K)
-10
2
0x10
YBCO
-10
1
5x10
a
b.
CL /u
I
-10
1-Q x 10
b
x
X
xx
x"
C
'o
X
C1
0
-10
5x10
(b)
0
0,
86. 0
90 0
92 0
Temperature
940
(K )
96. 0
98 0
„—
(a) (Acr,
pC»/F) ' vs T for the
BSCCO sample with CM& decreasing from
FIG. 3.
0. 0,
100-0
plots a to d. CM+=0 for plot d. Dotted
straight line here shows the 2D region in this
system. The inset shows the expended view of
the region around the intercept which gives
vs T for the
pCM+/F)
T, . (b) (ho, „—
YBCO sample with CM& decreasing from plots
a to d. CMz =0 for plot d. Dotted straight line
here shows the 3D region in this system.
EXCESS CONDUCTIVITY AND PAIR-BREAKING EFFECTS IN. . .
49
term due to the slow variation of the latter. T, determined by this method is listed in Table II together with
the 2D and 3D regions observed for each sample.
For further analysis, recourse to nonlinear leastsquares fitting of the data was taken. p, p„g', (0} and rp
were taken as adjustable parameters. The fittings were
performed in the 2D and 3D ranges separately. For the
bulk YBCO sample, using d =11.7 A as the c-axis parameter in the 3D range (i.e. 94 —100 K), we get p =0.25,
sec. The
p, =3.5 pQm, g, (0)=1.98 A and ro=2X10
fit is of good quality as rmsd is low (0. 13%) as well as
from the point of view that the parameters obtained (except T, ) have reliable values. The same procedure for
the film samples gave p, =0. 17 pQ m and p =0.76. An
increase in p and decrease in p, is expected as the grains
in the film are larger and aligned better than in the bulk
YBCO. The T, was allowed to vary within small limits
around Tm,
However the value of ~p obtained =10
sec implies again a near absence of the MT contribution.
Similar results are obtained from the fitting in the 2D region or from o z D when both the regions are combined together. (We obtain fits of inferior quality when O„D is
used as compared to the fits over the two separate regions
and feel that the expression of cr&D does not change over
in the dimensionality as fast as the data requires. ) Further
a larger value of ~p deteriorates the fits drastically. We
find the fit quality to remain unefFected for ~p 10 ' sec
and the rrnsd to jump abruptly to 1% or more for larger
values of 1p. Results are found to be similar for all five
with ro = 2 X 10 '
sec (corresponding
to
samples
' sec at 100 K} as the maximum value of
X 10 —
ro
that can be tolerated without any increase in the rmsd
value. (In general the BSCCO samples require smaller
values of Tp in fits over the 2D than those over the 3D regions. ) The fits are shown in Fig. 4 along with the individual contributions.
The values of the parameters obtained are listed in Table III.
Testardi et al. ' have devised a method to determine v
K
c3
11
0
10
0—
9
0—
8
0—
12 211
7. 0—
&
0—
5
4 0
-5
-5
5
I
I
0
-4
I
5
-4. 0
ln
-3. 5
I
-3 0
-2 5
FIG. 4. Experimental excess conductivity data (circles) are
shown with fitted theoretical expression (solid curve) in the 2D
and 3D regions for the BSCCO samples. MT contribution for
10
'Ty —
sec is shown by the dotted line.
„.
"
for the low-T, materials. The method did not require any
prior knowledge of T, . Bhatia et al. used this method
to show the MFR in TBCCO 2:2:1:2 system to be
predominantly 2D in nature. To use this method for the
present case, we combine Eq. (6) with (8}
1
1
pT
pa
F (~AL+~MT}
and difFerentiate both sides with respect to
-2
1
dPa
p dF
pa dT
F dT
dPT
pT dT
1
&
T to get
&
( +AL+ +MT}
dT
Since F and CMT are slowly varying functions of temperature dF/dT=dCMT/dT=0, and therefore,
TABLE III. Values of the various fitted parameters.
Parameters
Excess conductivity fitted
Tmf K
3D temp. range K
2D temp. range K
MFR K
4, (0} A
g, (0} A from T„
~& (3D) sec at 100 K
% rmsd in 3D
v~ (2D) sec at 100 K
% rmsd in 2D
p, pQm
T„/T,
Critical region K
Slope 3 above MFR K
YBCO
YBCO
Polycrys.
film
BSCCO
x =0.20
88.6
89.2-90. 2
90. 3-101
89.1
90.7-91.8
91.9 —100.5
10.7
78-79. 8
80-95.7
88.45
90-97
98-103
0.25
0.76
11.7
0.26
1.98
1.58
2. 2X 10
1.67
1.87
1.68
2.07
4. 1X10-"
0.18
3.53
1.07
0.55
110-140
3
2X10
0. 18
1. 1X10
0. 14
0. 17
1.09
1.55
112-138
BSCCO
x =0.05
93.45
13
BSCCO
x=0
94 —100
100.5 —104
10
0. 15
-2. 0
(E)
0.25
2.30
2.66
1.5X 10-"
76.0
17.6
0.22
3.60
3.46
1.8X10-"
0.14
2 8X1P-28
0.15
3.48
1.02
0.6
0. 10
—
2 8X1P 26
0.09
6.62
1.03
1.63
1.04
2.07
135-148
111-138
110-138
4. 9X10
0.09
8.4X 10
0.05
3.85
S. N. BHATIA AND C. P. DHARD
12 212
dPT
1
PT dT
2
1
PB
pv
dT
pa
AL
/v+1
FTmf
C
1/v
F
v
T
pCAL
1+ 1/v
P
MT
F
ex
after using Eq. (10) to eliminate e. Further, using Eq. (6)
the right-hand side (RHS) becomes
'
1/v
1+ 1/v
v
F
Tc
pCAL
'
v
F
T,
pCA
1
1
pT
pa
pCMT
F
1/v
X
where X represents the quantity
Thus a plot of ln( Y) where
dpT
in the square brackets.
talline as well as single crystals.
Figures 6(a) and 6(b) display these plots for YBCO and
BSCCO samples, respectively. All BSCCO samples are
unambigous in displaying long straight portions correconsiderable scatter is
sponding to v = 1. Although
present in the derivatives, they lead to only small errors
in the determinations of v here. The limits of the 2D region and T, determined here match favorably with the
above results. The 3D region gives considerable error in
T, due to its short length. The YBCO plots are not as
unambigous.
Although straight lines with v= 1/2 and 1
can be identified, the graph appears to consist of many
short segments each with a specific v.
Thus we find that four different methods give identical
values of T, for each sample and the values of the exponents remain —,' and 1. The extent over which these exponents are found defines the mean-field region. This is
found to extend from —
5. 0 ~ ln(e) —2. 0 for all the five
samples. For both the YBCO samples, the MFR is dom-
dpi'
p~ GT
1
pT dT
12 0
Biz Srz Cao. 95 Yo o5 Cu
against ln(X) will have a slope 1+1/v=2 in the 2D region and 3 in 3D region and an intercept
=ln
v
F
Tc
pCAL
'
8
z
8
6'
3
0—
1/v
dpT ldT was obtained by numerically differentiating the
data as mentioned above. dpi'/dT=B', its value obtained from the high-temperature
part of the fit, i.e.,
T & 2T, was used. Once again straight line graphs could
only be obtained with CMT =0 for all the samples studied.
The plots are shown in Fig. 5. 2D and 3D temperature
ranges obtained are listed in Table II along with the
values of T, calculated from the intercept. The agreement is seen to be excellent. In YBCO samples a change
of slope from 3 to 2 is clearly discernible, whereas in
BSCCO this is masked by the larger width of the 2D region.
the logarithmic derivative
Recently another method
This
has been proposed by Pureur et al.
method
method also does not require any prior knowledge of T,
and is especially useful if a power law is the only function
by which the conductivity varies. Thus if CMT is actually
negligible as the above analysis suggests then Eq. (10)
—
—
"
0
C
0
o ~~~go
o oooo
o o o~~ep
4 0
0 0
o
0—
-4 0—
-8
0
5 0
6 0
7 0
8
ln
16
0—
14
0—
12
0—
0
10 0
9 0
(x)
YBCO
10 0
8
——
0--
(b)
o
p
gives
@pg
p op
d
dT
(in', o,'"„)= —— f
v
1
eT
C
v
T —T
f
(12)
C
4 0
9 0
Then a plot of
dT
10. 0
11.0
ln
lnhcr
„
against T will yield v from the slope and T, from the intercept. Additional advantage of these plots is that unlike in Eq. (11) F here does not contribute to RHS. Equation (12) will therefore, give identical plots for polycrys-
I
I
12 0
13 0
14 0
(x}
FIG. 5. (a) Testardi plots using derivative method for the
BSCCO sample showing the slopes 2 and 3 for the 2D and 3D
regions. Slope 4 at the low temperature side (large X) and 3 at
side show the presence of the critical rethe higher-temperature
gion and short-wavelength fluctuations region, respectively. (b)
Testardi plot for YBCO. Critical and short-wavelength fluctuations regions are wider compared to BSCCO.
EXCESS CONDUCTIVITY AND PAIR-BREAKING EFFECTS IN. . .
49
inated by 3D region with a cross over from 3D to 2D
3.0. This gives the cross over temoccurring at in(e} = —
perature
05T, . In all the BSCCO samples 2D
behavior was seen over most of the MFR. For Y free
cross over occurred at
samples the dimensionality
4. 0. With Y doping the cross over temperature
ln(e) = —
shifted to higher (reduced) temperatures together with a
slight increase in the width of the 3D region. Below the
MFR, i.e., ln(e) ~ —5.0 the systems pass into the critical
regions. In YBCO the exponents were seen to change
' to —
'
from —,
', and then to —, [Fig. 7(a)]. The critical region
was found to extend to around 0.5 K above T, in both
the samples. These values of the exponents follow the
predictions of Lobb ' that the phase transition in this systern is analogous to 3D-XY model.
In the BSCCO samples we did observe the exponent to
' and reduce further as the
change to —,
temperature was
lowered [Fig. 7(b)]. Unlike the YBCO system the ex', was not found immediately
ponent —
below the MFR.
The critical region increased in width from 0.4 K for
x =0 to 1 K for x =0.20 samples. In 2D-XY model,
T„=1.
15
YBCO Polycryst.
0-
10.0-
5.0-
0. 0
I
95. 0
90 0
100 0
Temperature
105 0
110 0
(K)
Bi25r2Ca~ „Y„Cu208,q
x=O 00
x=O 20
15.0-
lQ
0
50
0. 0~
85
92
99
l,
a
i
106 85
92
„/
99 106 75
Temperature
I
82
I
89
I
96
(K)
FIG. 6. (a) Logarithmic derivative plot for YBCO. Dotted
lines show the 3D and 2D regions for v= 2 and 1, respectively.
(b) Logarithmic derivative plots for BSCCO samples for x =0.0,
0.05, and 0.20. A signi6cantly wide 2D region with v=1 has
been observed in all the samples showing the absence of fractional dimensionality.
Luther and Scalapino
1/VS
=0.35) below
12 213
have found the exponent
to be
MFR and decrease further at lower
Our values of the exponent thus agree
temperatures.
with this prediction. A change of behavior from 3D-XY
like to 2D-XY like in BSCCO is expected as the interlayer
(
spacing in BSCCO has increased to three times its value
in YBCO.
We have also observed deviation in the slope from the
2D behavior outside the upper end of the MFR. Similar
departure from the 3D behavior has been observed in the
amorphous
superconductor s by Johnson, Tsuei, and
and in high-T, materials by Frietas, Tsuei,
Chaudhari,
and Plaskett. They tried to interpret this in the YBCO
system by introducing a cut off to remove the shortwavelength fluctuations, for which the GL theory did not
have
apply. Recently Reggiani, Vaglio, and Varlamov
explained the fluctuations in a wide temperature range by
taking into consideration all the boson frequencies and
momenta in the fluctuations and have shown the excess
ao ) to
conductivity in the higher-temperature
range
1/e . We have observed the exponent to
vary as b, o,
be 3 in a wide temperature range in the YBCO bulk and
thin-film samples [Fig. 7(a}]. For the bulk sample the
range is 114-150 K and for the film 113-140 K. 113 K
corresponds to the temperature where the MFR ends
while the upper end of this range (140 or 150 K) comes
close to the temperature down to which the background
conductivity part was fitted. However the behavior is
quite different in BSCCO. The exponent v varies continuously from 3 to 8 in this region. It has a value 3 in a
narrow e range only, and this range increases with increasing Y concentration. 3 is the maximum value predicted by Reggiani, Vaglio, and Varlamov
for v.
Discrepancy here or the fortuous agreement in YBCO
can be traced to two factors. The magnitude of her, „ in
'm ') and that in BSCCO
YBCO is quite high (=10
'm ') within the MFR. Consequently erlow (=10
rors associated with the extraction of 50.,„ from pT will
have more significant influence in BSCCO than in YBCO.
MT contribution as predicted by Eq. (7) is a slowly varying function of e and therefore has significant values in
this region of e. In fact the calculated value of aMT is
larger than ho, „ for BSCCO for ~0 around 10 ' sec,
which is the value that gives vanishing contribution in
the MFR. Further this term has been derived for a&1
and may not be valid in this range of e. Also it appears
that o MT must be falling off more rapidly in YBCO than
in BSCCO. These results are at variance with the conclusions of Balestrino et al.
on epitaxial films of
BSCCO. They observed smaller values of the exponent
(v 3) over an e range equivalent to that investigated in
the present case.
The values of the exponent are corroborated by Testardi plots also. Thus in Fig. 5(a) the slope increases to 4 for
ln(X)~13 and decreases to —
', on the other side of the
plot, i.e., for ln(X) ~ 11. These correspond to v= —,' near
T, and v=3 above the 2D region. However such clear
slopes were not seen for BSCGO samples, although general trends with slopes 4 on higher abscissa and —
', on
lower side could be traced [Fig. 5(b)]. Errors associated
(e~
„=
0
(
0
S. N. BHATIA AND C. P. DHARD
12 214
with numerical
differentiation
changes v from 0.8 to 1.2.
and Pureur
Both Ausloos, Clippe, and Laurent
et al. have numerically differentiated their data of excess resistivity and excess conductivity, respectively, and
display very smooth curves. Our data, on the other hand,
shows considerable scatter. It is hard to imagine even a
good quality pT vs T data giving no scatter upon
differentiation.
So also is the abrupt change of dimen'as
', to —
sionality from —
, seen by Pureur et al.
lead to large scatter of the
data.
"
Our data are not able to support the contention of
Pureur et al. about the presence of fractals from the
logarithmic plots. Their argument is based on the observation of a region of v2=0. 85+0.09 bounded by regions
of v3=0. 51+0.06 and vl =1.32+0. 15. The values vl and
', )
', and —
', (= 1+ —
vz correspond to (fractional) dimensions —
"
"
of Gaussian fluctuations
in fractal topology. Our plots
do not show discrete regions with these values of v, and
vz. As mentioned above, the exponent v changes almost
continuously from 0.5 to 1.0 in the 3D to 2D change
over. It is difFicult to identify any finite length of the
plots with discrete values 0.85 and 1.33 of the exponent.
Further the scatter in these plots is more than in the Testardi plots and it is difBcult to identify the v= 3 exponent.
We have further noted that the fractional values of the
exponent result from the incomplete subtraction of the
normal resistivity. If the intercept Ag in p„ is increased,
of p„and therefore, the
this leads to underestimation
value of ho. ,„ increases and that of v decreases. Reverse
is true when A~ is decreased. A +10% change in Az
IV. CONCLUSIONS
With the proper selection of the T, we have found
that for both the systems YBCO as well as BSCCO, the
dimensionality of the fluctuations show a cross over from
2D to 3D behavior within the MFR as suggested by
Lawrence and Doniach. We also find the critical ex'
', and —, in the critical region as proponent values to be —
posed by Lobb and observed by Veira et al. Pairbreaking effects are found to be strong in both systems
with a negligible contribution of the MT term.
The alternative methods Figs. 3, 5, and 6 used here
15 0
YBCO
C7
14 0—
)/3
2/3
13
0—
12
0—
l
0—
'b
1
C
~--
100—
FIG. 7.
8 0
-90
I
-8
0
-70
I
-6 0
-5
I
0
-4
0
-30
j
-2 0
I
-1 0
(E}
ln
~
0—
6. 0—
2. 0—
0 0
-8. 0
0.0
the critical region. Slope 3 has been observed
side of the MFR as
at the high-temperature
predicted by Reggiani et al. (b) ln(50. ,„) vs
ln(e) plot for BSCCO. Slopes 1 and ~ have
been observed within the MFR. Slope 3 has
been observed above MFR in a temperature
range narrower compared to YBCO.
l0 0
8
(a) Various values of the exponents
sample are shown.
Within the MFR the exponent changes from 1
to —,' The value —, and —,' have been observed in
for the YBCO thin-film
I
-6 0
4. 0
—
ln
(E}
I
-2-0
0.0
EXCESS CONDUCTIVITY AND PAIR-BREAKING EFFECTS IN. . .
49
have limited utility. In the Testardi and the logarithm
plots the precision of v is determined by the accuracy of
calculating dpT/dT. Large errors in determination of
this derivative can easily mask a region as is obvious from
the absence of the critical region in the BSCCO in Fig.
5(b) although this region is clearly present in Fig. 4.
These methods at best can be considered as supplementa-
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and A.
I.
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