7 Current - voltage characteristics of break junctions
of high-Tc superconductors
D M Gokhfeld, D A Balaev, K A Shaykhutdinov, S I Popkov,
M I Petrov
L.V. Kirensky Institute of Physics SD RAS, Krasnoyarsk, 660036, Russia
E-mail: smp@iph.krasn.ru
Abstract.
The current-voltage (I-V ) characteristics of break junctions of polycrystalline
La1.85Sr0.15CuO4, Y0.75Lu0.25Ba2Cu3O7−δ, Bi1.8Pb0.3Sr1.9Ca2Cu3Ox and composite
YBa2Cu3O7−δ + Ag are investigated. The experimental I-V curves exhibit the specific
peculiarities of superconductor/normal-metal/superconductor junctions. The relation
between an I-V characteristic of network of weak links and I-V dependencies of typical
weak links is suggested to describe the experimental data. The I-V curves of typical
weak links are calculated by the Kümmel - Gunsenheimer - Nicolsky model considering
the multiple Andreev reflections.
PACS numbers: 74.25.Fy, 74.45.+c
1. Introduction
Measurements of current - voltage (I-V ) characteristics are accompanied with the heat
emission and the selfheating. The selfheating can modify dramatically the resulting I-V
curve. A heat hysteresis of I-V curve and a dependence of I-V curve on the velocity of
scanning of current are signs of selfheating.
A removal of the selfheating is very important for transport measurements of high-Tc
superconductors because their heightened temperature sensibility. By reducing the cross
section S of a bulk sample one can measure I-V curve at the fixed range of the current
density for the smaller values of the measuring current. The selfheating decreases as
well. In the case of non-tunneling break junction (BJ) technique, a significant reducing
of S is achieved by the formation of a microcrack in a bulk sample. The non-tunneling
BJ of high-Tc superconductors represents two massive polycrystalline banks connected
by a narrow bottleneck (Figure 1a). The bottleneck is constituted by granules and
intergranular boundaries which are weak links (Figure 1b). The current density in
the bottleneck is much larger than that in the banks. If the bias current I is less
than the critical current Ic of the bulk sample then the weak links in the banks have
zero resistance. Provided small transport currents, (i) Ic and the I-V curve of the BJ
http://arxiv.org/abs/0704.0694v1
Current - voltage characteristics of break junctions of high-Tc superconductors 2
are determined by the weak links in the bottleneck only, (ii) the selfheating effect is
negligible.
Figure 1. a) Break junction of polycrystalline sample. The crack 1 and the bottleneck
2 are displayed. b) Granules in the bottleneck. Filled circles mark weak links that are
intergranular boundaries. Dotted lines are the main paths for transport current. c)
Simplified circuit for the network (Sec. 3.1).
The experimental I-V curves of BJs of high-Tc superconductors have rich
peculiarities reflecting physical mechanisms of a charge transport through weak links.
It was a topic of many investigations [1, 2, 3, 4, 5, 6, 7]. Here we analyze the earlier
works of our group [4, 5, 6, 7] and the new experimental data (Section 2). The model for
description of the I-V curves is suggested in Section 3. The peculiarities observed on the
experimental I-V curves of BJs have been explained in Section 4. Also the parameters
of weak links in the investigated samples are estimated in Section 4.
2. Experiment
La1.85Sr0.15CuO4 (LSCO), Y0.75Lu0.25Ba2Cu3O7−δ (YBCO) and Bi1.8Pb0.3Sr1.9Ca2Cu3Ox
(BSCCO) were synthesized by the standard ceramic technology. The composite 67 vol.%
YBa2Cu3O7−δ + 33 vol.% Ag (YBCO+Ag) was prepared from YBa2Cu3O7−δ powder
and ultra-dispersed Ag [8]. The initial components were mixed and pressed. Then the
composite was synthesized at 925◦C for 8 h. The critical temperatures Tc are 38 K for
LSCO, 112 K for BSCCO, 93.5 K for YBCO and YBCO+Ag.
Samples with a typical size of 2 mm x 2 mm x 10 mm were sawed out from
synthesized pellets. Then the samples were glued to a sapphire substrates. The sapphire
was chosen due to its high thermal conductivity at low temperatures. The central part
of the samples was polished down to obtain a cross-sectional area S ≈ 0.2 x 1 mm2.
For such a value of S, the critical current Ic of YBCO and BSCCO has a typical value
about 2 A at 4.2 K (current density ≈ 1000 A/cm2). Further controllable decrease in S
is very difficult due to an inevitable mechanical stresses breaking the sample. In order
Current - voltage characteristics of break junctions of high-Tc superconductors 3
to obtain a contact of the break junction type, the sample with the above value of S
was bent together with the substrate with the help of screws of spring-loaded current
contacts. It led to the emergence of a microcrack in the part of sample between the
potential contacts. As a result, either a tunnelling contact (no bottleneck, the resistance
R > 10 Ohm at the room temperature) or a metal contact (R < 10 Ohm) was formed.
Only the metal contacts were selected for investigation.
The drop of Ic(4.2 K) when the sample was cracked shows that the values of
S decreased by ≈30 times for LSCO and ≈100 times for YBCO and BSCCO. For
YBCO+Ag Ic(77.4 K) decreases by ∼500 times.
The I-V curves were measured by the standard four probe technique under bias
current. A typical V (I) dependence of BJ has the hysteretic peculiarity which decreases
as temperature increases. Also there is the excess current on the I-V curves. The I-V
curves of LSCO and BSCCO BJs exhibit an arch-like structure at low temperature.
The I-V curves of BJs investigated are independent of scanning velocity of bias current.
Thus, the experimental conditions provides that the hysteretic peculiarity on these I-V
curves is not caused by the selfheating.
3. Model
3.1. I-V curve of network
A polycrystalline high-Tc superconductor is considered to be the network of weak links.
The I-V curve of a network is determined by the I-V curves of individual weak links
and their mutual disposition.
Let us consider firstly an influence of mutual disposition of weak links on the I-V
curve. For a bulk high-Tc superconductor the I-V curve resembles the one of typical
single weak link [9]. However the I-V curves of BJs are distorted usually in comparison
with the one of a single weak link. It is because the combination of finite number
of weak links remains in the bottleneck of BJ (Figure 1 a, b). So the contribution
of different weak links to the resulting I-V curve is more stronger in a BJ than in a
large network. The characteristics of a chaotic network is difficult to calculate [9]. To
simplify the calculation of resulting I-V curve of BJ we consider an equivalent network:
the simple parallel connection of a few chains of series-connected weak links (Figure 1
c). Indeed there are percolation clusters [10] in a network that are paths for current
(Figure 1 b). The each percolating cluster in the considered network is considered to be
the series-connected weak links.
The V (I) dependence of the series-connected weak links is determined as V (I) =
Vi(I), where the sum is over all weak links in the chain, Vi(I) is the I-V curve of each
weak link. The weak links and their I-V curves may be different. It is conveniently to
replace here the sum over all weak links with the sum of a few more typical weak links
multiplied by a weighting coefficient Pi. The relation for the series-connected weak links
Current - voltage characteristics of break junctions of high-Tc superconductors 4
is resulted:
V (I) = NV
PiVi (I) , (1)
where NV is the number of typical weak links, Pi shows the share of ith weak link in
the resulting I-V curve of the chain,
Pi = 1.
The parallel connection of chains is considered further. If the current I flows
through the network then the current Ij through jth chain equals IP‖j/N‖ and
Ij = I.
Here N‖ is the number of parallel chains in the network, P‖j is the weighting coefficient
determined by the resistance of jth chain,
P‖j = 1.
An addition (a subtraction) of chains in parallel connection smears (draws down)
the I-V curve of network to higher (lower) currents. It is like to the modification of
I-V curve due to the increase (the decrease) of cross section of sample. For the sake of
simplicity the difference of parallel chains can be neglected and the typical chain may
be considered only. Then the expression for I-V curve of network of weak links follows:
V (I) = NV
, (2)
where the sum is over the typical weak links with weighting coefficients Pi, NV is the
number of series-connected weak links in the typical chain in the network, I/N‖ = Ii is
the current through the ith weak link of the typical chain.
3.2. I-V curve of a typical weak link
The metal intergranular boundaries were revealed in the polycrystalline YBCO
synthesized by the standard ceramic technology [11]. The excess current and
other peculiarities on the I-V curves of the studied samples are characteristic for
superconductor/normal-metal/superconductor (SNS) junctions [12]. These facts verify
that the intergranular boundaries in the high-Tc superconductors investigated are
metallic. Therefore the networks of SNS junctions are realized in the samples.
The Kümmel - Gunsenheimer - Nicolsky (KGN) theory [13] only among theories
developed for SNS structures predicts the hysteretic peculiarity on the I-V curve of
weak link. The KGN theory considers the multiple Andreev reflections of quasiparticles.
According to the KGN model, the hysteretic peculiarity reflects a part of I-V curve with
a negative differential resistance which can be observed under bias voltage [12, 13]. The
KGN approach was used earlier to the description of experimental I-V curves of low-Tc
[14, 15] and high-Tc weak links [16, 14].
The approach based on consideration of the phase slip in nanowires [17] may
alternatively be employed to compute the hysteretic I-V curve. The model [17] is
valid at T ≈ Tc while the KGN model is appropriate at temperature range T < Tc.
We use the simplified version [14] of KGN to describe the I-V curves of individual
weak links. According to [14] the expression for the current density of SNS junction is
given by:
Current - voltage characteristics of break junctions of high-Tc superconductors 5
Table 1. Parameters of superconductors
Sample ∆0 [meV] m
∗/me kF [Å
BSCCO 25 6.5 0.61
LSCO 9 5 0.35
YBCO 17.5 5 0.65
YBCO+Ag 17.5 5 0.65
j(V ) =
em∗2d2
2π3~5
−∆+neV
∆2 − E2
1− C 2|E|
E2 −∆2
with C = π/2(1 − dm∗∆/2~2kF ) for C > 1 and C = 1 otherwise, E1 = −∆ + neV for
−∆ + neV ≥ ∆ and E1 = ∆ otherwise. Here A is the cross section area and d is the
thickness of normal layer with the inelastic mean free path l and resistance RN , e is the
charge and m∗ is the effective mass of electron, ∆ is the energy gap of superconductor,
n is the number of Andreev reflections which a quasiparticle with energy E undergoes
before it moves out of the normal layer.
One should calculate a few I(V ) dependencies by Eq.(3) for different parameters
to simulate the I-V curve of network by Eq.(2). Almost all parameters in Eq.(3) can be
dispersing for different weak links. Indeed there are some distribution functions of the
parameters of intergranular boundaries (d, A, RN) or the parameters of superconducting
crystallites (∆, the angle of orientation) in the SNS network.
4. Current - voltage characteristics
Figures 2-5 show the experimental I-V curves of BJs (circles) and the calculated I-V
curves of SNS networks (solid lines). The right scale of V -axis of all graphs is given in
the units eV/∆ to correlate the position of peculiarities on I-V curve with the value of
energy gap.
The parameters of superconductors are presented in Table 1. The mean values of
energy gap ∆0 at T = 0 known to be for high-Tc superconductors were used. Parameters
kF , m
∗ were estimated by the Kresin-Wolf model [18].
For a fitting we have calculated the I(V ) dependencies of different SNS junctions
by Eq.(3) to describe different parts of the experimental I-V curve. The parameters
varied were d and RN . Then we have substituted the arrays of I-V values to Eq.(2).
The most experimental I-V curves are satisfactory described when the sum in Eq.(2)
contains at least two members. The first member describes the hysteretic peculiarity,
Current - voltage characteristics of break junctions of high-Tc superconductors 6
the second one describes the initial part of I-V curve. The fitting is illustrated in detail
on Figure 2 were curve 1 is calculated for d = 78 Å, curve 2 is calculated for d = 400 Å
Figure 2. I-V curve of YBCO+Ag break junction at T = 77.4 K. Experiment (circles)
and computed curves (solid lines). Arrows display the jumps of voltage drop. Curve 1
that is N‖I1(V1) fits the hysteretic peculiarity. Curve 2 that is N‖I2(V2) fits the initial
part of I-V curve. Curve 3 is the dependence V (I) = NV
P1V1(I/N‖) + P2V2(I/N‖)
Figure 3. I-V curve of BSCCO break junction at T = 4.2 K. Experiment (circles)
and computed curve (solid line). Arrows display the jumps of voltage drop.
The main fitting parameters are d/l, NV , N‖A, RN/N‖, P1,2. The parameter P1
is the weighting coefficient for the stronger (with the thinner d) typical weak link,
P2 = 1− P1. Some parameters used are presented in Table 2. Values of l are estimated
from the experimental data of resistivity (2, 3, 1.6, 3.6 mOhm cm at 150 K for bulk
BSCCO, LSCO, YBCO, YBCO+Ag correspondingly) and data of works [19, 20]. The
value of l for Ag at 77 K is known to be ∼ 0.1 cm. But it is more realistic to use much
Current - voltage characteristics of break junctions of high-Tc superconductors 7
Figure 4. Temperature evolution of I-V curve of LSCO break junction. Experiment
(circles) and computed curves (solid lines). The I-V curves at 11.05 K, 23.6 K, 32.85
K are shifted up by 0.2 V, 0.4 V, 0.6 V correspondingly.
Figure 5. Temperature evolution of I-V curve of YBCO break junction. Experiment
(circles) and computed curves (solid lines). The I-V curves at 21.65 K, 41.1 K, 61.95
K are shifted up by 10 mV, 20 mV, 30 mV correspondingly.
smaller value for composite. Table 2 shows the possible different values of l and the
corresponding values of d for YBCO+Ag.
The number of the parallel paths N‖ is estimated by assuming A ≃ 10−11 cm2
for the weak links in polycrystalline high-Tc superconductors. Such choice of A is
reasonable because the cross section area of weak link should be more smaller than
D2 (Figure 1b), where D ∼ 10−4 cm is the grain size of high-Tc superconductors. This
rough estimation of N‖ is influenced by a form of the percolation clusters in the sample
[10] and imperfections of weak links.
Figures 2-5 demonstrate that the hysteretic peculiarity on the experimental I-V
curves is resulted from the region of negative differential resistance. This region is due
to the number of the Andreev reflections decreases when the voltage increases.
Current - voltage characteristics of break junctions of high-Tc superconductors 8
Table 2. Parameters of SNS junctions in the networks
Sample l [Å] d1 [Å] d2 [Å] P1 NV N‖
BSCCO 72∗ 3.5 - 1 1 1
LSCO 50∗ 4.8 20 0.905 15 20
YBCO 90∗ 2 20 0.333 3 1
YBCO+Ag 1000∗∗ 78 400 0.75 4 5
100∗∗ 7.8 40 0.75 4 5
∗ value at T = 4.2 K
∗∗ value at T = 77.4 K
The experimental I-V curves for LSCO and YBCO at different temperatures and
the corresponding curves computed are presented in figures 4 and 5. We account a
decreasing of l and ∆ to compute I-V curves at higher temperatures (for LSCO l(11.05
K) = 50 Å, ∆(11.05 K) = 0.93 meV, l(23.6 K) = 50 Å, ∆(23.6 K) = 0.69 meV, l(32.85
K) = 47 Å, ∆(32.85 K) = 0.46 meV; for YBCO l(21.65 K) = 81 Å, ∆(21.65 K) = 17.3
meV, l(41.1 K) = 70 Å, ∆(41.1 K) = 16.6 meV, l(61.95 K) = 60 Å, ∆(61.95 K) = 13.3
meV). The coincidence of computed curves and experimental I-V curves becomes less
satisfactory then T approaches to Tc. As possible, this discrepancy is due to an influence
of other thermoactivated mechanisms.
As the simulation curves demonstrate (Figs. 3 and 4), the arch-like peculiarity
on the experimental I-V curves of LSCO and BSCCO is one of the arches of the
subharmonic gap structure [13]. By using Eq.(2) we account for the arch-like peculiarity
at voltages ≫ ∆/e for LSCO (Figure 4) that should seem to contradict the KGN model
prediction for the subharmonic gap structure at V ≤ 2∆/e [13].
Also we have used Eq.(2) to estimate the number of resistive weak links in the
sample of composite 92.5 vol. % YBCO + 7.5 vol. % BaPbO3 [16]. The I-V curve of
this composite was described earlier by the KGN based approaches [16, 14]. We obtained
NV = 13, N‖ ≈ 4000 and the full number of resistive weak links is 52000. Small number
NV is the evidence that the shot narrowest part of bulk sample is resistive only.
5. Conclusion
We have measured the I-V characteristics of break junctions of polycrystalline high-Tc
superconductors. The peculiarities that are typical for SNS junctions are revealed on
the I-V curves.
The expression for I-V curve of network of weak links (Eq.(2)) was suggested to
describe the experimental data. Eq.(2) determines the relation between the I-V curve
of network and the I-V characteristics of typical weak links.
The I-V curves of SNS junctions forming the network in the polycrystalline high-
Tc superconductors are described by the Kümmel - Gunsenheimer - Nicolsky approach
[13, 14]. The multiple Andreev reflections are found to be responsible for the hysteretic
Current - voltage characteristics of break junctions of high-Tc superconductors 9
and arch-like peculiarities on the I-V curves. The shift of subharmonic gap structure
to higher voltages is explained by the connection of a few SNS junctions in series.
We believe that the expression suggested (Eq.(2)) allows to estimate the number
of junctions with nonlinear I-V curves and R > 0 in various simulated networks.
Acknowledgements
We are thankful to R. Kümmel and Yu.S. Gokhfeld for fruitful discussions. This
work is supported by program of President of Russian Federation for support of young
scientists (grant MK 7414.2006.2), program of presidium of Russian Academy of Sciences
”Quantum macrophysics” 3.4, program of Siberian Division of Russian Academy of
Sciences 3.4, Lavrent’ev competition of young scientist projects (project 52).
References
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Ripamonti G. 2001 Phys. Rev. Lett. 87 097001
[4] Petrov M I, Balaev D A, Gokhfeld D M, Shaikhutdinov K A, Aleksandrov K S 2002 Phys. Solid
State 44 1229
[5] Petrov M I, Balaev D A, Gokhfeld D M, Shaikhutdinov K A 2003 Phys. Solid State 45 1219
[6] Petrov M I, Gokhfeld D M, Balaev D A, Shaihutdinov K A, Kümmel R 2004 Physica C 408 620
[7] Gokhfeld D M, Balaev D A, Shaykhutdinov K A, Popkov S I, Petrov M I 2006 Physics of Metals
and Metallography 101 (Suppl. 1) S27 (Preprint cond-mat/0410112)
[8] Mamalis A G, Ovchinnikov S G, Petrov M I, Balaev D A, Shaihutdinov K A, Gohfeld D M,
Kharlamova S A, Vottea I N 2001 Physica C 364-365 174
[9] Haslinger R, Joynt R 2000 Phys. Rev. B 61 4206
[10] Stauffer D 1979 Physics Reports 54 1
[11] Petrov M I, Balaev D A, Gokhfeld D M 2007 Phys. Solid State 49 619
[12] Likharev K K 1979 Rev. Mod. Phys. 51 101
[13] Kümmel R, Gunsenheimer U, Nicolsky R 1990 Phys. Rev. B 42 3992
[14] Gokhfeld D M 2007 Supercond. Sci. Technol. 20 62 (Preprint cond-mat/0609541)
[15] Gokhfeld D M 2007 Physica C (materials of M2S-HTSC, Preprint cond-mat/0605427)
[16] Petrov M I, Balaev D A, Gohfeld D M, Ospishchev S V, Shaihutdinov K A, Aleksandrov K S 1999
Physica C 314 51
[17] Michotte S, Matefi-Tempfli S, Piraux L, Vodolazov D Y, Peeters F M 2004 Phys. Rev. B 69 094512
[18] Kresin V Z, Wolf S A 1990 Phys. Rev. B 41 4278
[19] Larbalestier D, Gurevich A, Feldmann D M, Polyanskii A 2001 Nature 414 368
[20] Gorkov L P, Kopnin N B 1988 Usp. Fiz. Nauk 156 117 (Sov. Phys. Usp. 31 850)
http://arxiv.org/abs/cond-mat/0410112
http://arxiv.org/abs/cond-mat/0609541
http://arxiv.org/abs/cond-mat/0605427
	Introduction
	Experiment
	Model
	I-V curve of network
	I-V curve of a typical weak link
	Current - voltage characteristics
	Conclusion

  The current-voltage ($I$-$V$) characteristics of break junctions of
polycrystalline La$_{1.85}$Sr$_{0.15}$CuO$_4$,
Y$_{0.75}$Lu$_{0.25}$Ba$_2$Cu$_3$O$_{7-\delta}$,
Bi$_{1.8}$Pb$_{0.3}$Sr$_{1.9}$Ca$_2$Cu$_3$O$_x$ and composite
YBa$_2$Cu$_3$O$_{7-\delta}$ + Ag are investigated. The experimental $I$-$V$
curves exhibit the specific peculiarities of
superconductor/normal-metal/superconductor junctions. The relation between an
$I$-$V$ characteristic of network of weak links and $I$-$V$ dependencies of
typical weak links is suggested to describe the experimental data. The $I$-$V$
curves of typical weak links are calculated by the K\"{u}mmel - Gunsenheimer -
Nicolsky model considering the multiple Andreev reflections.

Introduction
Measurements of current - voltage (I-V ) characteristics are accompanied with the heat
emission and the selfheating. The selfheating can modify dramatically the resulting I-V
curve. A heat hysteresis of I-V curve and a dependence of I-V curve on the velocity of
scanning of current are signs of selfheating.
A removal of the selfheating is very important for transport measurements of high-Tc
superconductors because their heightened temperature sensibility. By reducing the cross
section S of a bulk sample one can measure I-V curve at the fixed range of the current
density for the smaller values of the measuring current. The selfheating decreases as
well. In the case of non-tunneling break junction (BJ) technique, a significant reducing
of S is achieved by the formation of a microcrack in a bulk sample. The non-tunneling
BJ of high-Tc superconductors represents two massive polycrystalline banks connected
by a narrow bottleneck (Figure 1a). The bottleneck is constituted by granules and
intergranular boundaries which are weak links (Figure 1b). The current density in
the bottleneck is much larger than that in the banks. If the bias current I is less
than the critical current Ic of the bulk sample then the weak links in the banks have
zero resistance. Provided small transport currents, (i) Ic and the I-V curve of the BJ
http://arxiv.org/abs/0704.0694v1
Current - voltage characteristics of break junctions of high-Tc superconductors 2
are determined by the weak links in the bottleneck only, (ii) the selfheating effect is
negligible.
Figure 1. a) Break junction of polycrystalline sample. The crack 1 and the bottleneck
2 are displayed. b) Granules in the bottleneck. Filled circles mark weak links that are
intergranular boundaries. Dotted lines are the main paths for transport current. c)
Simplified circuit for the network (Sec. 3.1).
The experimental I-V curves of BJs of high-Tc superconductors have rich
peculiarities reflecting physical mechanisms of a charge transport through weak links.
It was a topic of many investigations [1, 2, 3, 4, 5, 6, 7]. Here we analyze the earlier
works of our group [4, 5, 6, 7] and the new experimental data (Section 2). The model for
description of the I-V curves is suggested in Section 3. The peculiarities observed on the
experimental I-V curves of BJs have been explained in Section 4. Also the parameters
of weak links in the investigated samples are estimated in Section 4.
2. Experiment
La1.85Sr0.15CuO4 (LSCO), Y0.75Lu0.25Ba2Cu3O7−δ (YBCO) and Bi1.8Pb0.3Sr1.9Ca2Cu3Ox
(BSCCO) were synthesized by the standard ceramic technology. The composite 67 vol.%
YBa2Cu3O7−δ + 33 vol.% Ag (YBCO+Ag) was prepared from YBa2Cu3O7−δ powder
and ultra-dispersed Ag [8]. The initial components were mixed and pressed. Then the
composite was synthesized at 925◦C for 8 h. The critical temperatures Tc are 38 K for
LSCO, 112 K for BSCCO, 93.5 K for YBCO and YBCO+Ag.
Samples with a typical size of 2 mm x 2 mm x 10 mm were sawed out from
synthesized pellets. Then the samples were glued to a sapphire substrates. The sapphire
was chosen due to its high thermal conductivity at low temperatures. The central part
of the samples was polished down to obtain a cross-sectional area S ≈ 0.2 x 1 mm2.
For such a value of S, the critical current Ic of YBCO and BSCCO has a typical value
about 2 A at 4.2 K (current density ≈ 1000 A/cm2). Further controllable decrease in S
is very difficult due to an inevitable mechanical stresses breaking the sample. In order
Current - voltage characteristics of break junctions of high-Tc superconductors 3
to obtain a contact of the break junction type, the sample with the above value of S
was bent together with the substrate with the help of screws of spring-loaded current
contacts. It led to the emergence of a microcrack in the part of sample between the
potential contacts. As a result, either a tunnelling contact (no bottleneck, the resistance
R > 10 Ohm at the room temperature) or a metal contact (R < 10 Ohm) was formed.
Only the metal contacts were selected for investigation.
The drop of Ic(4.2 K) when the sample was cracked shows that the values of
S decreased by ≈30 times for LSCO and ≈100 times for YBCO and BSCCO. For
YBCO+Ag Ic(77.4 K) decreases by ∼500 times.
The I-V curves were measured by the standard four probe technique under bias
current. A typical V (I) dependence of BJ has the hysteretic peculiarity which decreases
as temperature increases. Also there is the excess current on the I-V curves. The I-V
curves of LSCO and BSCCO BJs exhibit an arch-like structure at low temperature.
The I-V curves of BJs investigated are independent of scanning velocity of bias current.
Thus, the experimental conditions provides that the hysteretic peculiarity on these I-V
curves is not caused by the selfheating.
3. Model
3.1. I-V curve of network
A polycrystalline high-Tc superconductor is considered to be the network of weak links.
The I-V curve of a network is determined by the I-V curves of individual weak links
and their mutual disposition.
Let us consider firstly an influence of mutual disposition of weak links on the I-V
curve. For a bulk high-Tc superconductor the I-V curve resembles the one of typical
single weak link [9]. However the I-V curves of BJs are distorted usually in comparison
with the one of a single weak link. It is because the combination of finite number
of weak links remains in the bottleneck of BJ (Figure 1 a, b). So the contribution
of different weak links to the resulting I-V curve is more stronger in a BJ than in a
large network. The characteristics of a chaotic network is difficult to calculate [9]. To
simplify the calculation of resulting I-V curve of BJ we consider an equivalent network:
the simple parallel connection of a few chains of series-connected weak links (Figure 1
c). Indeed there are percolation clusters [10] in a network that are paths for current
(Figure 1 b). The each percolating cluster in the considered network is considered to be
the series-connected weak links.
The V (I) dependence of the series-connected weak links is determined as V (I) =
Vi(I), where the sum is over all weak links in the chain, Vi(I) is the I-V curve of each
weak link. The weak links and their I-V curves may be different. It is conveniently to
replace here the sum over all weak links with the sum of a few more typical weak links
multiplied by a weighting coefficient Pi. The relation for the series-connected weak links
Current - voltage characteristics of break junctions of high-Tc superconductors 4
is resulted:
V (I) = NV
PiVi (I) , (1)
where NV is the number of typical weak links, Pi shows the share of ith weak link in
the resulting I-V curve of the chain,
Pi = 1.
The parallel connection of chains is considered further. If the current I flows
through the network then the current Ij through jth chain equals IP‖j/N‖ and
Ij = I.
Here N‖ is the number of parallel chains in the network, P‖j is the weighting coefficient
determined by the resistance of jth chain,
P‖j = 1.
An addition (a subtraction) of chains in parallel connection smears (draws down)
the I-V curve of network to higher (lower) currents. It is like to the modification of
I-V curve due to the increase (the decrease) of cross section of sample. For the sake of
simplicity the difference of parallel chains can be neglected and the typical chain may
be considered only. Then the expression for I-V curve of network of weak links follows:
V (I) = NV
, (2)
where the sum is over the typical weak links with weighting coefficients Pi, NV is the
number of series-connected weak links in the typical chain in the network, I/N‖ = Ii is
the current through the ith weak link of the typical chain.
3.2. I-V curve of a typical weak link
The metal intergranular boundaries were revealed in the polycrystalline YBCO
synthesized by the standard ceramic technology [11]. The excess current and
other peculiarities on the I-V curves of the studied samples are characteristic for
superconductor/normal-metal/superconductor (SNS) junctions [12]. These facts verify
that the intergranular boundaries in the high-Tc superconductors investigated are
metallic. Therefore the networks of SNS junctions are realized in the samples.
The Kümmel - Gunsenheimer - Nicolsky (KGN) theory [13] only among theories
developed for SNS structures predicts the hysteretic peculiarity on the I-V curve of
weak link. The KGN theory considers the multiple Andreev reflections of quasiparticles.
According to the KGN model, the hysteretic peculiarity reflects a part of I-V curve with
a negative differential resistance which can be observed under bias voltage [12, 13]. The
KGN approach was used earlier to the description of experimental I-V curves of low-Tc
[14, 15] and high-Tc weak links [16, 14].
The approach based on consideration of the phase slip in nanowires [17] may
alternatively be employed to compute the hysteretic I-V curve. The model [17] is
valid at T ≈ Tc while the KGN model is appropriate at temperature range T < Tc.
We use the simplified version [14] of KGN to describe the I-V curves of individual
weak links. According to [14] the expression for the current density of SNS junction is
given by:
Current - voltage characteristics of break junctions of high-Tc superconductors 5
Table 1. Parameters of superconductors
Sample ∆0 [meV] m
∗/me kF [Å
BSCCO 25 6.5 0.61
LSCO 9 5 0.35
YBCO 17.5 5 0.65
YBCO+Ag 17.5 5 0.65
j(V ) =
em∗2d2
2π3~5
−∆+neV
∆2 − E2
1− C 2|E|
E2 −∆2
with C = π/2(1 − dm∗∆/2~2kF ) for C > 1 and C = 1 otherwise, E1 = −∆ + neV for
−∆ + neV ≥ ∆ and E1 = ∆ otherwise. Here A is the cross section area and d is the
thickness of normal layer with the inelastic mean free path l and resistance RN , e is the
charge and m∗ is the effective mass of electron, ∆ is the energy gap of superconductor,
n is the number of Andreev reflections which a quasiparticle with energy E undergoes
before it moves out of the normal layer.
One should calculate a few I(V ) dependencies by Eq.(3) for different parameters
to simulate the I-V curve of network by Eq.(2). Almost all parameters in Eq.(3) can be
dispersing for different weak links. Indeed there are some distribution functions of the
parameters of intergranular boundaries (d, A, RN) or the parameters of superconducting
crystallites (∆, the angle of orientation) in the SNS network.
4. Current - voltage characteristics
Figures 2-5 show the experimental I-V curves of BJs (circles) and the calculated I-V
curves of SNS networks (solid lines). The right scale of V -axis of all graphs is given in
the units eV/∆ to correlate the position of peculiarities on I-V curve with the value of
energy gap.
The parameters of superconductors are presented in Table 1. The mean values of
energy gap ∆0 at T = 0 known to be for high-Tc superconductors were used. Parameters
kF , m
∗ were estimated by the Kresin-Wolf model [18].
For a fitting we have calculated the I(V ) dependencies of different SNS junctions
by Eq.(3) to describe different parts of the experimental I-V curve. The parameters
varied were d and RN . Then we have substituted the arrays of I-V values to Eq.(2).
The most experimental I-V curves are satisfactory described when the sum in Eq.(2)
contains at least two members. The first member describes the hysteretic peculiarity,
Current - voltage characteristics of break junctions of high-Tc superconductors 6
the second one describes the initial part of I-V curve. The fitting is illustrated in detail
on Figure 2 were curve 1 is calculated for d = 78 Å, curve 2 is calculated for d = 400 Å
Figure 2. I-V curve of YBCO+Ag break junction at T = 77.4 K. Experiment (circles)
and computed curves (solid lines). Arrows display the jumps of voltage drop. Curve 1
that is N‖I1(V1) fits the hysteretic peculiarity. Curve 2 that is N‖I2(V2) fits the initial
part of I-V curve. Curve 3 is the dependence V (I) = NV
P1V1(I/N‖) + P2V2(I/N‖)
Figure 3. I-V curve of BSCCO break junction at T = 4.2 K. Experiment (circles)
and computed curve (solid line). Arrows display the jumps of voltage drop.
The main fitting parameters are d/l, NV , N‖A, RN/N‖, P1,2. The parameter P1
is the weighting coefficient for the stronger (with the thinner d) typical weak link,
P2 = 1− P1. Some parameters used are presented in Table 2. Values of l are estimated
from the experimental data of resistivity (2, 3, 1.6, 3.6 mOhm cm at 150 K for bulk
BSCCO, LSCO, YBCO, YBCO+Ag correspondingly) and data of works [19, 20]. The
value of l for Ag at 77 K is known to be ∼ 0.1 cm. But it is more realistic to use much
Current - voltage characteristics of break junctions of high-Tc superconductors 7
Figure 4. Temperature evolution of I-V curve of LSCO break junction. Experiment
(circles) and computed curves (solid lines). The I-V curves at 11.05 K, 23.6 K, 32.85
K are shifted up by 0.2 V, 0.4 V, 0.6 V correspondingly.
Figure 5. Temperature evolution of I-V curve of YBCO break junction. Experiment
(circles) and computed curves (solid lines). The I-V curves at 21.65 K, 41.1 K, 61.95
K are shifted up by 10 mV, 20 mV, 30 mV correspondingly.
smaller value for composite. Table 2 shows the possible different values of l and the
corresponding values of d for YBCO+Ag.
The number of the parallel paths N‖ is estimated by assuming A ≃ 10−11 cm2
for the weak links in polycrystalline high-Tc superconductors. Such choice of A is
reasonable because the cross section area of weak link should be more smaller than
D2 (Figure 1b), where D ∼ 10−4 cm is the grain size of high-Tc superconductors. This
rough estimation of N‖ is influenced by a form of the percolation clusters in the sample
[10] and imperfections of weak links.
Figures 2-5 demonstrate that the hysteretic peculiarity on the experimental I-V
curves is resulted from the region of negative differential resistance. This region is due
to the number of the Andreev reflections decreases when the voltage increases.
Current - voltage characteristics of break junctions of high-Tc superconductors 8
Table 2. Parameters of SNS junctions in the networks
Sample l [Å] d1 [Å] d2 [Å] P1 NV N‖
BSCCO 72∗ 3.5 - 1 1 1
LSCO 50∗ 4.8 20 0.905 15 20
YBCO 90∗ 2 20 0.333 3 1
YBCO+Ag 1000∗∗ 78 400 0.75 4 5
100∗∗ 7.8 40 0.75 4 5
∗ value at T = 4.2 K
∗∗ value at T = 77.4 K
The experimental I-V curves for LSCO and YBCO at different temperatures and
the corresponding curves computed are presented in figures 4 and 5. We account a
decreasing of l and ∆ to compute I-V curves at higher temperatures (for LSCO l(11.05
K) = 50 Å, ∆(11.05 K) = 0.93 meV, l(23.6 K) = 50 Å, ∆(23.6 K) = 0.69 meV, l(32.85
K) = 47 Å, ∆(32.85 K) = 0.46 meV; for YBCO l(21.65 K) = 81 Å, ∆(21.65 K) = 17.3
meV, l(41.1 K) = 70 Å, ∆(41.1 K) = 16.6 meV, l(61.95 K) = 60 Å, ∆(61.95 K) = 13.3
meV). The coincidence of computed curves and experimental I-V curves becomes less
satisfactory then T approaches to Tc. As possible, this discrepancy is due to an influence
of other thermoactivated mechanisms.
As the simulation curves demonstrate (Figs. 3 and 4), the arch-like peculiarity
on the experimental I-V curves of LSCO and BSCCO is one of the arches of the
subharmonic gap structure [13]. By using Eq.(2) we account for the arch-like peculiarity
at voltages ≫ ∆/e for LSCO (Figure 4) that should seem to contradict the KGN model
prediction for the subharmonic gap structure at V ≤ 2∆/e [13].
Also we have used Eq.(2) to estimate the number of resistive weak links in the
sample of composite 92.5 vol. % YBCO + 7.5 vol. % BaPbO3 [16]. The I-V curve of
this composite was described earlier by the KGN based approaches [16, 14]. We obtained
NV = 13, N‖ ≈ 4000 and the full number of resistive weak links is 52000. Small number
NV is the evidence that the shot narrowest part of bulk sample is resistive only.
5. Conclusion
We have measured the I-V characteristics of break junctions of polycrystalline high-Tc
superconductors. The peculiarities that are typical for SNS junctions are revealed on
the I-V curves.
The expression for I-V curve of network of weak links (Eq.(2)) was suggested to
describe the experimental data. Eq.(2) determines the relation between the I-V curve
of network and the I-V characteristics of typical weak links.
The I-V curves of SNS junctions forming the network in the polycrystalline high-
Tc superconductors are described by the Kümmel - Gunsenheimer - Nicolsky approach
[13, 14]. The multiple Andreev reflections are found to be responsible for the hysteretic
Current - voltage characteristics of break junctions of high-Tc superconductors 9
and arch-like peculiarities on the I-V curves. The shift of subharmonic gap structure
to higher voltages is explained by the connection of a few SNS junctions in series.
We believe that the expression suggested (Eq.(2)) allows to estimate the number
of junctions with nonlinear I-V curves and R > 0 in various simulated networks.
Acknowledgements
We are thankful to R. Kümmel and Yu.S. Gokhfeld for fruitful discussions. This
work is supported by program of President of Russian Federation for support of young
scientists (grant MK 7414.2006.2), program of presidium of Russian Academy of Sciences
”Quantum macrophysics” 3.4, program of Siberian Division of Russian Academy of
Sciences 3.4, Lavrent’ev competition of young scientist projects (project 52).
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http://arxiv.org/abs/cond-mat/0410112
http://arxiv.org/abs/cond-mat/0609541
http://arxiv.org/abs/cond-mat/0605427
	Introduction
	Experiment
	Model
	I-V curve of network
	I-V curve of a typical weak link
	Current - voltage characteristics
	Conclusion
