Effect of reduction in the density of states on fluctuation
conductivity in Bi 2 Sr2 CaCu 2 O 8qx single crystals
P. Chowdhury, S.N. Bhatia
Department of Physics, Indian Institute of Technology, Bombay 400 076, India
Abstract
The in-plane Ž r ab . and out-of-plane Ž rc . resistivities of BSCCO single crystals have been measured by six terminals
technique. The r ab and rc are well described by the fluctuation theory developed by Dorin et al. The main effect of these
fluctuations is to cause a reduction in the quasi-particle density of states ŽDOS., leading to a negative contribution in the
fluctuation conductivity Lawrence–Doniach ŽLD. and Maki–Thompson ŽMT. contributions. We have analyzed paraconductivity by adding this DOS contribution to LD and MT contributions. The analysis shows that approaches based on the
conventional LD model alone cannot explain the paraconductivity along ab-plane and c-axis, even when the MT
contribution is included.
PACS: 74.25.Fy; 74.40.q k; 74.72.Hs; 74.72.Bk
Keywords: Fluctuation conductivity; BSCCO; Superconductivity
1. Introduction
With the availability of good quality single crystals of high-Tc superconductors, it has become possible to study experimentally the anisotropic properties
of these materials. Such studies are important to
understand the intrinsic mechanisms responsible for
electric transport in these materials. The measurement of resistivities of such materials shows that the
out-of-plane transport is characteristically different
from the in-plane transport. Whereas the in-plane
resistivity r a b ŽT . decreases linearly with temperature
up to approximately the zero field transition tempera-
ture Tc0 , the out-of-plane resistivity rc ŽT . shows a
maximum before dropping to zero at Tc0 .
In the mean field region ŽŽT y Tcmf . < Tcmf ., the
in-plane fluctuation conductivity Ž D sa b ŽT .. for layered materials can be presented by the well known
Lawrence–Doniach ŽLD. w1x formula
D sa b Ž T . s
e2
16"s
y1
e
1q
4 jc Ž 0.
e
ž
s
2 y1 r2
/
Ž 1.
where e s ŽTrTcmf . y 1, Tcmf is the mean field critical temperature, s is the interlayer spacing and j c Ž0.
is the coherence length along the c-axis at T s 0.
In the vicinity of the mean field critical temperature, where j c ŽT . s j c Ž0.re 1r2 f s, this equation
predicts three-dimensional Ž3D. fluctuations to take
151
place and farther away from Tcmf where j c ŽT . F s,
their dimensionality reduces to two and their conductivity to the well known two-dimensional Ž2D. Aslamazov–Larkin ŽAL. w2x paraconductivity
D sa b Ž T . s
e2
16"s
ey1
Ž 2.
with 2D to 3D crossover occurring at a temperature
T U , where T U s Žw2 j c Ž0.rs x 2 q 1.Tcmf . In YBCOŽ123., spacing between the superconducting layers s
is quite small, so the crossover temperature T U can
be observed quite far away from Tcmf . But in BSCCO,
a large value of interlayer spacing pushes the T U
very close to Tcmf and leads to the observation of a
dominant 2D fluctuation behavior in the mean field
region. In-plane fluctuation conductivity D sa b ŽT . has
been investigated in BSCCO single crystals by several authors w3–7x and analyzed within the framework of LD model in the mean field region. Pradhan
et al. w6x found a 2D behavior for the fluctuation
conductivity D sa b ŽT . of BSCCO within the entire
range of temperature y4 F lnŽ e . F y1 investigated
by them. Han et al. w8x do observe a crossover but
they find its temperature T U to be strongly dependent on the choice of Tcmf . More effort appears to
have been concentrated on determining the exponent
n of the fluctuation conductivity rather than on its
quantitative comparison with the predictions of Eq.
Ž1.. Such a comparison in Ref. w6x yields the unit cell
˚ which is almost half the value
length to be ; 15 A,
obtained from the structure data. Romallo et al. w9x
have extended the LD theory by considering n superconducting layers in a unit cell interacting with
varying strengths g j with the layers of the neighboring cell. They find the LD expression Ž1. to be still
valid but the conductivity D sa b ŽT . enhanced by a
number NeŽ e . denoting the effective number of independently fluctuating superconducting layers per unit
cell-length. In particular, for BSCCO, they find NeŽ e .
to be given by
Ne Ž e . s 2
ž
1 q rre
1 q 4 rre
/
Ž 3.
for g 1rg 2 s 1, where g 1 and g 2 are the coupling
strengths between the CuO 2 planes and r ŽTc . s
Ž2 j c Ž0.rs . 2 measures the anisotropy in these materials. For j c Ž0. s 0.04 nm as they have reported for
BSCCO single crystals, NeŽ e . varies from 2.1 to 2.4
with e values varying from 0.01 to 0.1, i.e., in the
mean field region, there are effectively two layers
˚ The data on thin films
within a distance of 15 A.
with their w10x c-axis oriented perpendicular to the
substrate also agrees well with the LD model, showing a dominating 2D behavior within the range y5.0
F lnŽ e . F 0.7.
In all these measurements, a linear in-plane background r B ŽT . resistivity has been subtracted from
the measured resistivity r T even in the mean field
region. With the formation of Cooper pairs, the
electron gas within the plane does get depleted. This
reduced number of electrons will increase the magnitude of r B ŽT . which will in turn change the value of
fluctuation conductivity as estimated above. Such a
fluctuation reduction in single electron density of
states ŽDOS. and its effect on fluctuation conductivity in both directions has been calculated by Dorin et
al. w11x.
A number of mechanisms have been proposed to
explain the peculiar behavior of rc but none has
been found to be satisfactory w12x. They include
anisotropic weak localisation, c-axis charge confinement due to a non-Fermi liquid state in the layers,
renormalisation of the interlayer hopping rate by
in-plane scattering or the presence of a physical
tunneling barrier associated with the blocking layer
ŽBaO–CuO layers. between the superconducting CuO
layers. The model under consideration is another
attempt in explaining this behavior and attributes the
peak in rc to the thermal fluctuations. A recent
study on the magnetoconductivity on the BSCCO
thin films by Livanov et al. w13x shows a good
agreement of their experimental data in both direc˚ Axnas et
tions with this theory with s s 15 " 2 A.
al. w14x have measured the rc ŽT . of YBCO single
crystal in field up to 12T. They find the fluctuation
conductivity D sc Ž H . to change from positive to
negative values at a temperature Ts above Tc . The
present theory w11x is found to explain their results
satisfactorily including the change in sign of D sc Ž H .
and the temperature at which it occurs.
We have grown single crystals of BSCCOŽ2212.
by self flux method and have measured r T by six
terminal techniques to determine the in-plane and
152
Table 1
EDAX analysis of BSCCO single crystals
Serial no.
Bi
Sr
Ca
Cu
Average ŽBi:Sr:Ca:Cu.
1
2
3
1.75
1.79
1.87
1.82
1.77
2.06
1.18
1.16
1.21
2.19
2.19
1.82
1.80:1.88:1.18:2.06
out-of-plane resistivities for the same crystal. Our
measurements show a linear behavior of resistivity
along the ab-plane in the normal state before dropping to zero at Tc0 and a ‘semiconductor’ type
behavior along the c-axis with a peak close to Tc0 .
We have analyzed our experimental fluctuation conductivity in both the directions with LD conductivity
Ž1. together with the DOS contribution and with the
indirect MT contributions calculated by Dorin et al.
w11x. Our calculations show that this theory predicts a
quantitative agreement with our experimental data
for both the directions.
2. Experimental details
We have grown BSCCO single crystals by the
self flux method using Li 2 CO 3 as the charge w15x.
High-purity Bi 2 O 3 , SrCO 3 , CaCO 3 , CuO and
Li 2 CO 3 were used as starting materials. The starting
composition of approximately 15 to 20 g was taken
according to the formula Bi 2 Sr2yy CaCu 2yy Li y Ž y s
0.5.. The mixture, in a 30 ml alumina crucible, was
heated rapidly to 8508C and soaked for 5–10 h at
this temperature to allow it to melt properly. The
melt was then cooled slowly at a rate f 0.58Crh to
temperature 8008C for crystallisation to take place.
Then, it was cooled to room temperature at a faster
rate f 508Crh. Platelet crystals typically having
dimensions 2 mm = 1 mm = 0.02 mm were formed
inside the melt-mixture. These crystals were removed mechanically from the crucible.
These crystals were characterised by X-ray
diffraction ŽXRD., scanning electron microscopy
ŽSEM., and EDAX analysis. The XRD patterns for
the crystals were recorded after crushing them into
powders. The exclusive presence of Ž00 l . reflection
indicates that the crystals are oriented with the c-axis
normal to the large surface of the crystal. SEM
shows that the platelets consist of agglomerates of
thin plate-like crystals having edges of about 2 mm
and thickness 0.02 mm. The crystals show smooth
surfaces. EDAX analysis gives the composition of
the elements inside the crystals. We have chosen
some crystals randomly from the melt. The results at
different positions of a particular crystal are given
below ŽTable 1., which depict the homogeneity of
the elements inside the sample.
3. Resistivity measurements
Measurements of in-plane and out-of-plane resistivities were made by employing the six terminal
techniques, first introduced by Busch et al. w16x. On
a rectangular shape crystal, six contacts were made
on the two large opposite faces of the sample by
means of indium micrometer soldering. As shown in
Fig. 1, out of the four contacts made on the top
surface, two were used as current leads and the other
two as the voltage leads to record potential difference called ‘‘Vtop’’. The other two contacts were
made on the bottom surface to record potential difference on the bottom face called ‘‘V bot’’. The contact resistances were less than 3 V. A DC current of
1 mA was passed and the appropriate voltages were
monitored by Keithley nanovoltmeters. The variation
of the Vtop and V bot with temperature for two arbitrarily chosen crystals are shown in Fig. 2. It is seen
that the Vtop and V bot vary differently for each
sample.
Fig. 1. Shows the geometry of the contact pads.
153
Fig. 3. Experimental ab-plane resistivity vs. temperature and its
comparison with linear behaviour from Eq. Ž6..
Fig. 2. Temperature dependence of Vtop and V bot for samples 1
and 2.
Since our crystals show a linear behavior along
the ab-plane, we fit the data to the equation
The following relations were developed by Busch
et al. w16x, taking into account the nonuniform distribution of current in the crystal,
Ž rcrra b .
L
f
pD
1r2
Arccosh
Vtop
V bot
sin
sin
1r2
f
2 I sin
2L
p Ž x 2 y x1 .
2L
pD
Vtop btanh
Ž rcrra b .
ž
ž
p Ž xX2 y xX1 .
ž
Ž rcrra b .
L
p Ž x 2 y x1 .
2L
/
/
,
Ž 4.
1r2
r a b Ž T . s r a b Ž 0 . q BT
Ž 6.
for T ) 130 K. Here, r a b Ž0. is the linearly extrapolated resistivity at zero temperature and the slope B
denotes the temperature coefficient of resistivity. The
solid lines in Fig. 3 show these fits for each sample.
The parameters obtained are strongly sample dependent. From Fig. 3, we observe that for sample 1,
below 130 K measured r a b deviates from linearity.
This decrease in resistivity is due to the onset of
thermal fluctuations. However, sample 2 shows an
upturn before fluctuation effects set in. This weak
, Ž 5.
/
where L, b and D are the length, width and thickness of the sample, Ž x 1 , x 2 . and Ž xX1 , xX2 . are the
abscissa of the contact pads on the top and bottom
surfaces of the sample, respectively.
Combining Eqs. Ž4. and Ž5., the voltages Vtop and
V bot may be transformed into the resistivities r a b
and rc . The results of these calculations for r a b of
samples 1 and 2 are presented in Fig. 3. As observed
for many high-Tc superconductors, r a b ŽT . decreases
linearly with the decreasing temperature in the normal state above ; 130 K and rc ŽT . shows a nonmetallic behavior in the normal state below ; 130 K
as shown in Fig. 4.
Fig. 4. Experimental c-axis resistivity vs. temperature and its
comparison with Eq. Ž7..
154
dependence of r a b on temperature, i.e., turning upwards at temperature below ; 150 K has earlier
been explained as arising from the misalignment of
the Cu–O planes with respect to each other w17x.
However, it could also be due to a slight deficiency
of oxygen inside the crystal w18x.
The shapes of the rc ŽT . curves for both the
samples are similar to those observed for most high-Tc
cuprates. They reveal metallic resistivities Žd rcrdT
- 0. at higher temperatures and a non-metallic
Žd rcrdT ) 0. behavior at lower temperatures. These
two regions are naturally separated by a minimum
rcmin at temperature Tmin . We fit rc ŽT . to the equation
rc Ž T . s rc Ž 0 . q BX T q CXrT
Ž 7.
X
where C is a constant. The last term proposed by
Anderson and Zou w19x arises due to the tunneling of
electrons between the Cu–O layers. The corresponding fitted curves are shown in Fig. 4.
4. Theory and analysis
For all high-Tc materials, below a so-called on-set
temperature, the resistivity deviates from the usual
behavior before approaching zero. This depression in
resistivity or the enhancement in conductivity is
known as the excess conductivity w20x. The enhancement in conductivity arises due to the formation of
short-lived Cooper pairs by thermal fluctuations.
The excess conductivity D sex ŽT . is obtained from
the measured resistivity r T ŽT . Žs 1rs T ŽT .. as
D sex Ž T . s s T Ž T . y sn Ž T .
Ž 8.
where snŽT . s 1rrnŽT .. The background resistivity
rnŽT . is calculated using Eq. Ž6. for the ab-plane
and Eq. Ž7. for the c-axis resistivities.
ductors considering the fluctuations to be mainly
confined within the two-dimensional superconducting layers, the layers being coupled to each other by
Josephson tunneling.
Recently, Dorin et al. w11x have proposed a model
for the fluctuation conductivity for high-Tc materials
taking into account the layered structure of these
materials. Thermodynamic fluctuations in the order
parameter above Tc , cause a formation of nonequilibrium Cooper-pairs near the Fermi level. According to
them, these nonequilibrium Cooper-pairs produce a
reduction in the quasi-particle DOS. This reduced
DOS produces a negative contribution to the fluctuation conductivity.
Another correction to AL contribution was earlier
proposed by Maki and Thompson w21x. The electrons
forming a Cooper pair decouple after a certain relaxation time forming quasi-particles of equal and opposite momenta. By time reversal symmetry, the
quasi-particles remain in a phase locked state and
continue to accelerate for a time tf till they scatter
from the impurity potential and decay into the normal electrons or recombine into superconducting
pairs. It has two contributions; one is negative and
called MT regular, the other is positive and called
MT anomalous term. Earlier the first term ŽMTŽreg..
was usually neglected because it had a smaller magnitude in comparison to other terms. But in the clean
limit Ž4p k Bt Tcr" ) 1. these are comparable in
magnitude as we will see below.
In zero magnetic field, all the four terms can be
written as w11x
D saLD
b s
e2
1
16 s"
S
D saDO
sy
b
e Ž eqr .
e 2k
4 s"
2ln
1r2
,
Ž 9.
2
e 1r2 q Ž e q r .
,
Ž 10 .
1r2
, Ž 11 .
1r2
4.1. In plane fluctuation conductiÕity
The theory to study the effect of thermodynamic
fluctuations on the conductivity was initially developed for low-Tc superconductors. The fluctuation
conductivity arising from the accelerated pairs was
calculated by Aslamazov and Larkin w2x for the
isotropic superconductors. It was modified by
Lawrence and Doniach w1x for anisotropic supercon-
Žreg .
D saMT
sy
b
Žan .
D saMT
s
b
e 2k
4 s"
2ln
e2
8 s" Ž e y g .
2
e 1r2 q Ž e q r .
2ln
e 1r2 q Ž e q r .
1r2
g 1r2 q Ž g q r .
1r2
Ž 12 .
155
where r ŽT . s 7z Ž3.p 2 J 2r8T 2 k B2 is the anisotropy
parameter characterising the dimensional crossover
in fluctuation behavior, with J being the effective
quasi-particle nearest-neighbor interlayer hopping
energy and Õ F the Fermi velocity of the electrons
parallel to the layer.
The constants k and k X are determined by the
impurity concentration and in the clean limit are
defined as k s 9.384Žt T . 2 k B2 r" 2 , k s 0.5865 and
g s 7z Ž3. " 2r16p 2ttf k B T 2 where tf is the phase
breaking time.
sc
MT Žreg .
e 2 srk
sy
Ž eqr .
1r2
y e 1r2
r 1r2
16h "
2
, Ž 15 .
scMT Žan .
e2s
eqgqr
s
16h "
e Ž eqr .
1r2
q g Žgqr .
1r2
y1
Ž 16 .
where h s 7z Ž3. Õ F2 " 2r32p 2 T 2 k B2 .
4.3. Data analysis
4.2. Out-of-plane conductiÕity
The experimental data for rc is shown in Fig. 4.
It shows a peak at a temperature just above Tc0 . Such
a peak cannot be explained by the LD contributions
only, because for T ™ Tc , both sn and s LD increase. Dorin et al.’s w11x model was put forward
mainly to explain this behavior. The peak results
because of the anisotropy in the materials. Along the
c-axis in the normal state, the conduction is due to
the hopping of the quasi-particles along this axis. But
in the superconducting state, due to the suppression
of the one electron DOS at Fermi level in the layers,
lesser number of electrons will be available to tunnel
along this axis. On the other hand, coherent tunneling of Cooper pairs gives a direct contribution Ži.e.,
LD term. along this axis, but its magnitude is reduced by Ž P12t GL . in comparison to the in-plane
fluctuation conductivity given by Eq. Ž1.. Here, P1
is the one electron interlayer hopping probability and
P12t GL is the conditional probability that the pair will
tunnel coherently within its ‘lifetime’ t GL . There is a
competition between quasi-particle tunneling associated with the DOS and this LD term. The former is
less singular than the latter far away from Tc . This
competition gives rise to the peak in rc w11x.
Along the c-axis, all the four contributions were
given as w11x
scLD s
e2s
32h "
scDO S s y
e q rr2
e Ž eqr .
e 2 srk
16h "
ln
1r2
y1 ,
Ž 13 .
2
2
e 1r2 q Ž e q r .
It is worth mentioning that the above equations
describe the leading contributions to the conductivity
arising from the thermodynamic fluctuations of the
order parameter above the mean field temperature
ŽT y Tcmf . F Tcmf in the Gaussian approximation and
below this temperature the fluctuations deviate from
the Gaussian behavior.
The first step of our analysis is to compare the
in-plane fluctuation conductivity data with the LD
theory only. During this fitting, we have varied the
parameters r, s and the mean field temperature Tcmf .
We have varied Tcmf around the inflection point of
the d rrdT curve and find Tcmf to coincide with this
temperature within the experimental errors. We found
that a qualitative agreement as shown in Fig. 5 with
RMS deviation ŽRMSD. s 0.50 can be obtained with
1r2
,
Ž 14 .
Fig. 5. Experimental values of the excess conductivity D sa b . The
dashed curve is calculated from theory taking LD contribution
only.
156
the following set of parameters: Tc0 s 92.06 K, r ŽTc .
˚ The calculated curve does
s 0.006 and s s 27 A.
not pass symmetrically through the data points. In
particular, for T ) 100 K, it lies consistently above
the experimental value.
By adding DOS term to the LD contribution, the
agreement improves as shown in Fig. 6 and the value
of RMSD drops to 0.14, which is almost one third of
its earlier value. This large drop in RMSD signifies
that the LD contribution alone was not sufficient and
that the gap remaining between this term and the
data is being filled by the DOS contribution. The
parameters obtained are: r ŽTc . s 0.018 " 0.006, s s
˚ Tc0 s 92.06 " 0.1 K, t s 1.71 " 0.30 =
16 " 1 A,
10y1 4 s at 100 K. The values reported here are for
sample 1. The error bars here indicate the range over
which the values of the parameters varied for five
other samples from three different batches which
were studied. The agreement is excellent over the
temperature range 93–110 K which corresponds to
0.01 F e F 0.20. Using the relation for J as defined
earlier, its value works out to be k B Tc0rJ s 1.88,
which is very close to the value 2.1 as reported in
Ref. w22x for BSCCO thin films. The value of s s 16
Å is also close to the distance between the nearest
CuO 2 planes of a unit cell.
Fig. 7 shows the agreement of the experimental
excess conductivity with the theory including all the
four terms of Eqs. Ž9. – Ž12.. Adding the two MT
contributions, the agreement further improves but
Fig. 6. Experimental excess conductivity data are shown together
with contributions calculated from LD and DOS terms.
Fig. 7. Experimental excess conductivity data. The calculated
curve includes the LD, DOS, MTŽreg. and MTŽan. terms. Inset
shows the MTŽreg. and MTŽan. terms.
only marginally with RMSD value becoming 0.13%.
The parameters obtained were r ŽTc . s 0.019 " 0.006,
t s 2.34 " 0.3 = 10y1 4 s at 100 K, tf s 4.93 " 0.4
= 10y1 4 s at 100 K, Tc0 s 92.06 " 0.1 and s s 16 "
˚ The value of k B Tc0tfr" s 5.27 is close to the
1 A.
value reported earlier in Ref. w22x in magnetoconductivity calculations for BSCCO thin films and shows a
moderate pair breaking. The ratio tfrt works out to
be almost 2, which means that the quasi-particles
formed after the decoupling of the pair face two
scatterings in order to relax to the normal state.
Thus, we have shown that the LD contribution
alone is unable to explain the experimental data
within the ab-plane. DOS and both MT contributions
are essential to get a quantitative agreement between
the experiment and the theory.
Fig. 8 shows the experimental c-axis fluctuation
conductivity together with the values calculated from
theory, taking into account all the four contributions.
The values of s, Tc0 , r ŽTc ., t and tf obtained from
the ab-plane fluctuation conductivity were used. t
and tf were allowed to change independently but
the agreement did not improve significantly. Above
; 100 K, the magnitude of the LD fluctuation conductivity here has reduced in comparison with its
in-plane value D sa b ŽLD. as predicted by the theory
w11x. This magnitude is smaller than the DOS term
leading to a negative fluctuation conductivity as
observed experimentally. Just below 100 K, D sc ŽLD.
157
Fig. 8. Experimental c-axis excess conductivity data are shown
together with values calculated from theory including LD, DOS,
MTŽan. and MTŽreg. terms. Inset shows the MTŽreg. and MTŽan.
terms.
starts increasing in magnitude in accordance to its
singular behavior. It overtakes the DOS term giving
rise to the peak in rc . Further, it is observed from
Fig. 8 Žinset. that MTŽreg. Žnegative in magnitude.
and MTŽan. Žpositive in magnitude. contributions are
comparable in magnitude, therefore, the net MT
contribution becomes small and has little influence
on the total conductivity.
5. Conclusion
We have presented the experimental results of
anisotropy in resistivity in superconducting
BSCCOŽ2212. single crystals. The ab-plane resistivity varies almost linearly with temperature above the
onset temperature.
To fit the experimental fluctuation conductivity
we have considered different direct and indirect contributions associated with Cooper pairs created by
thermal fluctuations above the superconducting transition. The currently used Lawrence–Doniach approach for the in-plane fluctuation conductivity D sa b
cannot explain the experimental D sa b by itself. The
presence of the DOS contribution recently proposed
by Dorin et al. along with the LD term explains the
fluctuation conductivity to a quantitative level in the
reduced temperature region 0.02 - e - 0.2. DOS
term is small in magnitude and varies very slowly
with temperature. It does not significantly change the
dependence of fluctuation conductivity on temperature from that given by Eq. Ž1.. Because of this, the
need for an additional term was never felt in the
analysis of polycrystalline and single crystal data
where the emphasis was to determine the exponent n
w20,23x. Earlier the anomalous terms Eqs. Ž12. and
Ž16. were the only contributions calculated from the
Maki–Thompson process. Dorin et al. have added
the regular term which makes the net MT contribution a small fraction of the total fluctuation conductivity and shows why an unacceptably small value
tf - 10y1 5 s for pair breaking time was obtained for
the polycrystalline samples w20,23x. The peak observed in the out-of-plane resistivity is a natural
consequence of thermal fluctuations in anisotropic
medium.
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