RAPID COMMUNICATIONS
PHYSICAL REVIEW B
VOLUME 57, NUMBER 2
1 JANUARY 1998-II
Voltage jumps in current-voltage characteristics of Bi2Sr2CaCu2O81d superconducting films:
Evidence for flux-flow instability under the influence of self-heating
Z. L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian
Institut für Physik, Universität Mainz, Staudinger Weg 7, D-55099 Mainz, Germany
~Received 8 August 1997!
We present the observation of voltage jumps at high flux-flow velocities in current-voltage characteristics of
c-axis-oriented Bi2Sr2CaCu2O81d superconducting thin films. The experimental results agree quantitatively
with the prediction for viscous flux-flow instability including consideration of a finite heat-removal rate from
the samples. We obtained a characteristic magnetic field B T , which separates the regions where the instability
is dominated by nonthermal and heating mechanisms. The inelastic scattering rate 1/t in and the diffusion length
l e of the quasiparticles have also been extracted. @S0163-1829~98!50302-5#
Some 20 years ago Larkin and Ovchinnikov ~LO! predicted a vortex instability resulting in a voltage jump in the
current-voltage (I-V) characteristics at high flux-flow
velocities.1 The underlying idea is that in the flux-flow region the viscous friction, which a vortex experiences during
its movement, is a nonmonotonic function of the velocity
* . Above this critical
with a maximum at a critical value y LO
velocity, the viscous force decreases leading to an even
higher velocity accompanied by a still lower viscosity and so
on and so forth, giving rise to the above instability. Experimental investigations on this phenomenon have been done
both in low-temperature2–5 and high-temperature6–8 superconducting thin films. The LO theory allows to extract the
temperature dependence of the inelastic scattering times of
quasiparticles. The published experimental results could be
summarized as follows: ~1! The voltage jumps are intrinsic
to the vortex system and are not artifacts due to thermal
runaway; a reasonable agreement between experimental results and LO theory has been found.2–7 ~2! Both the critical
current at which the voltage jumps occur2–5,7 and the critical
vortex velocity corresponding to the instability3,5,7,8 exhibit
significant magnetic field dependence. This behavior is not
predicted by LO theory, and the magnetic-field dependence
of the critical velocity makes the extracted values for the
scattering rate uncertain. ~3! The voltage jumps could appear
in the vortex glass or collective creep state which correspond
to a negative curvature of the I-V characteristics in doublelogarithmic plots.5,7 Such voltage jumps could be caused by
other mechanisms given below, for example, as predicted by
the theory of self-organized criticality ~SOC!,9 voltage instabilities could appear near the pinning-depinning transition by
thermally activated jumps of vortices which trigger a chain
reaction of vortex movements leading to avalanches of diverging size.
Recently the magnetic-field dependence of the critical
current and the critical voltage has been explained theoretically by Bezuglyj and Shklovskij ~BS! considering the effect
of self-heating on the flux-flow instability predicted by Larkin and Ovchinnikov. They have successfully interpreted the
experimental data of classic superconducting indium films.10
In this paper we report experimental evidence for the BS
theory in high-T c superconductors. Because the instability
0163-1829/98/57~2!/736~4!/$15.00
57
caused by SOC and other mechanisms appears only before or
near the flux line depinning, the possible corresponding interpretation could be ruled out if flux-flow behavior V}I is
observed before the voltage jump. Due to its low magnetic
irreversibility field,11 the Bi2Sr2CaCu2O81d superconductor
should be an excellent candidate for the observation of the
flux-flow behavior in a large range of temperatures and magnetic fields. In our experiment we have investigated the I-V
characteristics of Bi2Sr2CaCu2O81d superconducting films.
Both the typical behavior of flux flow and the voltage jumps
were observed at lower and higher vortex velocities, respectively. Our experimental data agree quantitatively with the
prediction of BS theory for the viscous flux-flow instability
influenced by self-heating.
Bi2Sr2CaCu2O81d superconducting films of different
thicknesses ~typically 400 nm! were deposited by sputtering
on @100# oriented SrTiO3 substrates.12 As revealed by x-ray
analysis, all films were purely c-axis oriented. Silver contact
pads were evaporated and bridges ~length 100 mm, width 10
mm! were patterned by photolithography and wet chemical
etching, allowing the measurement of I-V curves using the
standard four point method. External magnetic fields up to 6
T, parallel to the c axis of the film could be applied. The
measurements were conducted with rectangular current
pulses of 1 s length and an interval time of 3 s between
pulses, with the direction of the current always perpendicular
to the field. For the present work two films were analyzed.
The experimental results for both samples agree, thus in the
following we concentrate on the sample with superconducting critical temperature ~midpoint! T c 586.0 K and a normalstate resistivity of 66 mV cm at 100 K.
Figure 1~a! shows the I-V characteristics at 77 K for different magnetic fields from 0.25 T up to 6 T. In order to
examine the vortex state at lower vortex velocities, the data
are plotted with a double-logarithmic scale. Within our experimental resolution the I-V characteristics in low magnetic
fields exhibit a clearly discontinuous voltage jump at a well
defined critical current I * . As an example for the data in a
magnetic field of 0.25 T, the definition of the experimentally
observed critical current I * and critical voltage V * are indicated by the dotted lines. The dashed line represents the
slope for typical flux-flow behavior V}I. Figure 1~a! shows
R736
© 1998 The American Physical Society
RAPID COMMUNICATIONS
57
VOLTAGE JUMPS IN CURRENT-VOLTAGE . . .
R737
FIG. 2. Magnetic-field dependence of ~a! the critical current
density J * and ~b! the critical vortex velocity y * at temperatures as
assigned to each curve.
FIG. 1. ~a! Current-voltage characteristics of a c-axis oriented
Bi2Sr2CaCu2O81d bridge at 77 K. The magnetic fields, from lower
right to upper left are B50.25, 0.50, 0.75, 1, 1.25, 1.5, 2, 3, 4, 5,
and 6 T. The dashed line represents the expected behavior V}I for
flux flow at lower vortex velocities. The definitions of the critical
current I * and the critical voltage V * are indicated by dotted lines.
~b! The magnetic-field dependence of the flux-flow resistivity in the
low current limit. The solid lines are guides to the eye.
that all curves at not too high vortex velocities are parallel to
the dashed line. In addition the flux-flow resistivity r f as
plotted in Fig. 1~b! shows the typical linear dependence on
B. This result positively demonstrates that the vortex system
is already in the flux-flow state before the voltage jumps
occur. Thus for the cause of the voltage jumps we can directly exclude the mechanisms of SOC, hot spot,13 and depinning of the flux lines,14 all of which should occur before
or at the flux line depinning, i.e., below the flux-flow regime.
Vortex lattice crystallization15 could lead to a jumplike voltage increase in the I-V curves after the flux line depinning
with the critical current shifted to a higher value with increasing temperature. However, as shown later, the critical
current I * observed in our experiments decreases with increasing temperature. Thus we conclude that the mechanism
based on flux-flow instability is most suitable to explain the
voltage jumps observed in our samples.
According to LO theory a voltage instability in I-V characteristics is expected to occur when the vortex velocity
reaches the critical value
* 51.02~ D/ t in! 1/2~ 12T/T c ! 1/4
y LO
~1!
where D is the quasiparticle diffusion coefficient; t in is the
inelastic scattering time of quasiparticles. The critical veloc-
* and the critical current I LO
* , at which the LO instaity y LO
bility occurs, are predicted to be field independent. As shown
in Figs. 1 and 2, however the experimentally obtained critical
current I * and hence the current density J * ~5I * /S, where
S is the area of the bridge’s cross section! depend strongly on
the magnetic field at all temperatures. The critical velocity
y * ~5V * /BL, where L is the length between the voltage
contacts! exhibits significant field dependence at higher temperatures ~e.g., 79 K!, while at lower temperatures ~e.g., 65
K! this dependence seems to disappear. These results are
consistent with observations both in low-T c superconducting
Al,2,3 In,3 Sn,2,3 and Ta/Ge ~Ref. 5! films, and high-T c superconducting YBa2Cu3O72d thin films.7,8 Recently it has been
shown in a theoretical work by Bezuglyj and Shklovskij ~BS!
that such a magnetic-field dependence could be caused by
the unavoidable heating of quasiparticles due to the finite
rate of removing the power dissipated in the sample.10 According to BS theory, the quasiparticles should have a higher
temperature T * than the experimentally measured temperature T ~bath temperature!. In an extension of the LO approach considering the finite heat removal rate, the magneticfield dependence of the critical current density J * and
critical electric field E * (E * 5V * /L5 y * B) are given by
J*
~ 3t21 ! 1/2
52&t 3/4
,
J0
~ 3t11 !
~2!
E * ~ 12t ! ~ 3t11 !
5
.
E 0 2&t 3/4 ~ 3t21 ! 1/2
~3!
with t5 @ 11b1(b 2 18b14) 1/2# / @ 3(112b) # and b
5B/B T . The introduced normalizing current density J 0 and
electric field E 0 are
RAPID COMMUNICATIONS
R738
Z. L. XIAO, P. VOSS-DE HAAN, G. JAKOB, AND H. ADRIAN
J 0 52.62s n /e 0 ~ D t in! 21/2k B T c ~ 12T/T c ! 3/4,
~4!
E 0 51.02B T ~ D/ t in! 1/2~ 12T/T c ! 1/4,
~5!
57
which are independent of the magnetic field. Here s n is the
conductivity in normal state, e 0 is the electron charge, and k B
is the Boltzmann constant.
The characteristic magnetic field B T depends on the heat
transfer coefficient h and the time t in of quasiparticle energy
relaxation
B T 50.374k 21
B e 0 R h h t in ,
~6!
where R h 5( s n d) 21 is the sheet resistance and d is the
thickness of the film. Furthermore, the ratio of the critical
electric fields with and without the influence of the quasipar* 5t 1/4(3t21) 1/2/&, where
ticle heating is found as E * /E LO
* 5 y LO
* B. It follows that, for B!B T , corresponding to t
E LO
'1, the heating effect is not important and the critical electric field E * is approximately equal to the predicted theoret* . In the other limit of B@B T , t reaches its
ical value E LO
minimum of 1/3; hence the heating effect dominates and the
* . Thereexperimentally observed E * is far smaller than E LO
fore the parameter B T separates the regions where the influence of self-heating is of minor and of major importance for
the flux-flow instability.
As shown above, we obtained experimentally the critical
current density J * and the critical electric field E * from the
instability point as defined in Fig. 1~a!. Equations ~2! and ~3!
are the parametric form of the corresponding E * and J *
taken at different magnetic fields at a fixed bath temperature
T. For suitably chosen J 0 and E 0 at the different temperatures, the experimental data should collapse on a common
curve. The corresponding results are given in Fig. 3~a!. The
symbols represent the experimental data shown in Fig. 2 and
the solid line is calculated from Eqs. ~2! and ~3!. Rather good
agreement between theory and our experimental data could
be found. The determined J 0 and E 0 , both of which decrease
with increasing temperature, are given in the inset of Fig.
3~a!. Following the approach of Bezuglyj and Shklovskij, the
inelastic quasiparticle scattering rate 1/t in could be extracted
from Eqs. ~4! and ~5! using the obtained J 0 , E 0 in combination with B T , which can be obtained by fitting the measured
J * (B) at fixed temperatures to the theoretical curve Eq. ~2!.
From Eqs. ~2! and ~3! the magnetic-field dependence of the
dissipated power could be expressed as J * E * /J 0 E 0 5(1
2t). Thus B T can be extracted through less complicated
fitting. The related results are shown in Fig. 3~b!. The magnetic field B T exhibits a strong temperature dependence varying from 5 T at 65 K to 0.4 T at 79 K. In our experiments the
values of b5B/B T fall in the range between 0.25 and 2. The
rate of the critical electric field with and without thermal
* varies from 0.82 ~0.1 T! to 0.47 ~0.75 T! at
influence E * /E LO
79 K, whereas at 65 K it changes from 0.71 ~2.5 T! to 0.55 ~6
T!. With these results one can also easily explain why the
magnetic-field dependence of the critical velocity y * at 79 K
is stronger than at 65 K as shown in Fig. 2~b!. As indicated
above, the measured E * , especially obtained in low magnetic fields, is not too far away from the expected critical
* of LO theory. In fact, the extracted inelastic quavalue E LO
siparticle scattering time t in of indium films differs by about
FIG. 3. Scaling of the data in Fig. 2 according to Eqs. ~2! and
~3!. The solid lines are the theoretical curves and the symbols ~identical to Fig. 2! are experimental data. The temperature dependence
of the related parameters E 0 , J 0 , and B T is given in the insets of ~a!
and ~b!, respectively.
a factor of 2 with and without consideration of the heating
effect;10 in other systems reasonable critical velocities and
inelastic quasiparticle scattering rates 1/t in were obtained
without considering heating effect.3–7
The corresponding inelastic quasiparticle scattering rate
1/t in for different temperatures is given in Fig. 4. Both the
inelastic quasiparticle scattering rate 1/t in itself and its temperature dependence are comparable to those reported for
YBa2Cu3O72d thin films.6,7 The strong temperature depen-
FIG. 4. Temperature dependence of inelastic quasiparticle scattering rate 1/t in and diffusion length l e 5(D t in) 1/2 extracted from
Eqs. ~4! and ~5! using the value of J 0 , E 0 , and B T obtained in Fig.
3.
RAPID COMMUNICATIONS
57
VOLTAGE JUMPS IN CURRENT-VOLTAGE . . .
dence of the inelastic quasiparticle scattering rate 1/t in is
most probably due to the spin-fluctuation phenomenon.16
The diffusion length l e 5(D t in) 1/2, which plays an important
role in this theory, must be larger than the core size, in order
for the excitations to leave the core,10 and larger than the
intervortex distance to get spatial homogeneity.8 The calculated values of the diffusion length from Eq. ~4! using the
obtained J 0 are presented also in Fig. 4. Clearly l e is larger
than not only the vortex core size ~typically 2 j 0 55 nm! in
Bi2Sr2CaCu2O81d superconducting films17 but also larger
than the intervortex distance ~about 50 nm in a magnetic
field of 1 T!. Thus we can conclude that, in the range of
magnetic fields and temperatures we have studied, the nonequilibrium distribution of the quasiparticles extends over
the whole volume of the superconductor.
The heat transfer coefficient obtained from Eq. ~6! varies
from 100 W/cm2 K at 79 K to 130 W/cm2 K at 65 K. This
value is lower than the typical heat transfer coefficient
103 W/cm2 K of the film-substrate boundary.18 As shown in
Ref. 18, the temperature rise in the film can also be caused
from the heat flow in the substrate. The temperature difference between the surface of the substrate and the bath is
given as:18 DT s 5 a Pt 0 / $ 2L(Dt 0 ) 1/2@ 4(Dt 0 ) 1/21W # c p % ,
where a '4/p 1/2, P is the dissipated power in the film, t 0 is
the duration of the current pulse, D is the heat diffusion
1
A. I. Larkin and Yu. N. Ovchinnikov, Sov. Phys. JETP 41, 960
~1976!.
2
L. E. Musienko, I. M. Dmitrenko, and V. G. Volotskaya, JETP
Lett. 31, 567 ~1980!.
3
W. Klein, R. P. Huebener, S. Gauss, and J. Parisi, J. Low Temp.
Phys. 61, 413 ~1985!.
4
A. V. Samoilov et al., Phys. Rev. Lett. 75, 4118 ~1995!.
5
B. J. Ruck et al., Phys. Rev. Lett. 78, 3378 ~1997!.
6
S. G. Doettinger et al., Phys. Rev. Lett. 73, 1691 ~1994!.
7
Z. L. Xiao and P. Ziemann, Phys. Rev. B 53, 15 265 ~1996!.
8
S. G. Doettinger, R. P. Huebener, and A. Kühle, Physica C 251,
285 ~1995!.
9
O. Pla and F. Nori, Phys. Rev. Lett. 67, 919 ~1991!; L. Legrand
et al., Europhys. Lett. 34, 287 ~1996!; Z. L. Xiao and P. Ziemann, Physica C 282-287, 2363 ~1997!.
R739
constant of the substrate, and L and W are the length and
width of the bridge, respectively. Using c p 51 J/cm3 K, D
50.18 cm2/s ~Ref. 18! and the values of W(510 m m), L
(5100 m m) and t 0 (51 s) we derived a heat transfer coefficient h s @ 5 P/(LWDT) # of the SrTiO3 substrate of about
500 W/cm2 K. Thus the effective heat transfer coefficient
from film to bath is about 300 W/cm2 K, which is reasonably
consistent with our experimental value. The disparity may be
due to an underestimated E 0 arising from inhomogeneities in
the film.10
In conclusion, we have attributed to the observation of
voltage instability in current-voltage characteristics at high
flux-flow velocities in high-T c superconducting films of
Bi2Sr2CaCu2O81d. A quantitative agreement between our
data and the theory for viscous flux flow under the influence
of power dissipation during the vortex motion was found.
Having integrated the effects of the unavoidable quasiparticle heating into the quantitative analysis of the experimental
data, we hope that our results will stimulate further investigation on the viscous flux-flow instability.
We would like to thank Th. Kluge and P. Haibach for
experimental advice and support. The project was financially
supported by the SFB 262.
A. I. Bezuglyj and V. A. Shklovskij, Physica C 202, 234 ~1992!.
K. Karada et al., Phys. Rev. Lett. 71, 3371 ~1993!.
12
P. Wagner et al., Physica C 215, 123 ~1993!.
13
W. J. Skocpol, M. R. Beasley, and M. Tinkham, J. Low Temp.
Phys. 16, 145 ~1974!; A. Vl. Gurevich and R. G. Mints, Rev.
Mod. Phys. 59, 941 ~1987!.
14
W. Henderson, E. Y. Andrei, M. J. Higgins, and S. Bhattacharya,
Phys. Rev. Lett. 77, 2077 ~1996!.
15
A. E. Koshelev and V. M. Vinokur, Phys. Rev. Lett. 73, 3580
~1994!.
16
S. M. Quinlan, D. J. Scalapino, and N. Bulut, Phys. Rev. B 49,
1470 ~1994!.
17
F. Gollnik et al., Physica C 235-240, 1933 ~1994!.
18
S. K. Gupta et al., Physica C 206, 335 ~1993! and references
therein.
10
11