Physica C 291 Ž1997. 143–148
A new method for study the mixed state of high-Tc
superconductors by using the relaxation of resistivity
L.P. Ma
a,b,)
, H.C. Li b, R.L. Wang b, L. Li
b
a
b
Beijing Laboratory of Vacuum Physics, Center for Condensed Matter Physics, Chinese Academy of Sciences, P.O. Box 2724,
Beijing 100080, China
National Laboratory for SuperconductiÕity, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, China
Received 2 June 1997; revised 5 July 1997; accepted 11 August 1997
Abstract
A new method to study the mixed state of high-Tc superconductors by the relaxation of resistivity R ff Ž t . experiments is
theoretically and experimentally discussed. The theoretical method about how to calculate the decayed magnetization current
JiŽ R ff Ž t .. and the effective pinning energy of flux lines UŽ R ff Ž t ., JiŽ t .. from R ff Ž t . is discussed. Two equations about
JiŽ R ff Ž t .. and UŽ R ff Ž t ., JiŽ t .. are obtained by the two-kind-of-flux-creep model. R ff Ž t . experiments are performed at
various temperatures and magnetic fields. U–J data are calculated. Reasonable physical parameters are derived by fitting the
U–J data with two U–J relation models. Experiments and the U–J relation models support the new method and the
two-kind-of-flux-creep model. q 1997 Elsevier Science B.V.
Keywords: Effective pinning energy; Relaxation; Resistivity; Flux creep; High-Tc superconductor
1. Introduction
When the temperature is near the superconducting
transition temperature ŽTc . the decayed magnetic signal becomes weak, and measuring the relaxation of
magnetization M Ž t . is nearly impossible w1x. The
method of the relaxation of resistivity R ff Ž t . was
proposed previously for describing the relaxation
state of high-Tc superconductors ŽHTSCs. near Tc
w1x. When a DC current flows through a superconductor that is in a magnetic relaxation, a damping
resistivity can be measured, this phenomenon is the
relaxation of resistivity. Experiments proved that the
)
Corresponding author.
relaxation of resistivity can give a good description
of the relaxation state of HTSCs. It is possible to
study the mixed state of HTSCs near Tc by R ff Ž t .
experiments. In a previous paper w1x we proposed a
two-kind-of-flux-creep model to describe the relaxation of resistivity. A general value Ji Žto some
extent it stands for the average value of the amplitude of magnetization current density. was defined
for describing the magnetization current density, and
a general value Jt was used to stand for the transport
current density. In R ff Ž t . state, The flux formed by
Ji is defined as relaxation flux, which decays with
time. The flux that does not decay with time is the
conventional flux. The interactions among the relaxation flux, conventional flux, and transport current
determine the nature of R ff Ž t .. Since in R ff Ž t . state,
0921-4534r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved.
PII S 0 9 2 1 - 4 5 3 4 Ž 9 7 . 0 1 6 5 8 - 4
L.P. Ma et al.r Physica C 291 (1997) 143–148
144
Ji is decayed with time, the creep of relaxation flux
under the Lorentz force of Jt causes a time-decayed
resistivity. The creep of the conventional flux under
the Lorentz force of Jt causes a time-independent
resistivity. We found that the two-kind-of-flux-creep
model can give a good description for R ff Ž t . experiments at various temperatures. The relation between
the effective pinning energy ŽU . of flux lines and the
current density Ž J . is very important in the study of
the mixed state of HTSCs. In this article, we report
our further study of the relaxation of resistivity. A
new method for deriving the U–J data from R ff Ž t .
experiments is theoretically and experimentally discussed.
duced by the creep of conventional flux, m s 1 y
UorkT, and t o s JcorŽ mA..
Combining Eqs. Ž1. and Ž2., we get a general
relation between R ti Ž t . and Ji Ž t .:
R ti Ž t . s
R ti Ž 0 . Ji Ž t .
AJco
d Jird t.
Ž 4.
Let:
GŽ t . s
t
H0 R
ti
Ž t. dt,
Ž 4a .
`
G Ž `. s
H0 R
ti
Ž t . d t.
Ž 4b .
Considering the conditions Ji Ž0. s Jco , Ji Ž`. s 0,
and GŽ0. s 0, the solution of Eq. Ž4. is
2. The equations of Ji( R ff ( t )) and U( R ff ( t ), Ji( t ))
We use the two-kind-of-flux-creep model to discuss how to obtain U–J data from R ff Ž t . experimental data. In slab geometry, the conventional-flux
creep under the influence of magnetization current
density Ji causes the decrease of Ji , which is governed by the rate equation w2x:
d Jird t s A exp yU Ž JirJco . rkT .
Ž 1.
Here, A is a constant, and U is the effective pinning
energy. We found w1x that the resistivity produced by
the relaxation-flux creep is:
R ti Ž t . s
R ti Ž 0 . Ji Ž t .
Jco
exp yU Ž JirJco . rkT .
Ž 2.
Here, R ti Ž0. sm o Mi Ž0. LforJt , which represents the
amplitude of the relaxation of resistivity; L is the
average jumping distance of flux lines; f o is the
attempt jumping frequency; Mi Ž0. is the irreversible
part of magnetization at t s 0. The physical meaning
of R ti Ž0. is the resistivity produced by the creep of
relaxation flux when t s 0. It is proportional to the
attempt creep velocity Vo Ž Vo s Lfo ..
For logarithmic U–J relation, we found w1x that
the resistivity produced by the creep of relaxation
flux and conventional flux is
s
R ff Ž t . s R o q R ti Ž 0 . Ž 1 q trto . ,
with s s Ž 2 y m . rm.
Ž 3.
Here, R o is the time-independence resistivity pro-
Ji Ž t . rJco s 1 y G Ž t . rG Ž ` .
1r2
.
Ž 5.
From Eq. Ž2. we get UŽ JirJco ., that is
U Ž JirJco . rkT s ln R ti Ž 0 . rR ti Ž t .
q ln Ji Ž t . rJco .
Ž 6.
Substituting Eq. Ž5. into Eq. Ž6., we obtain
U Ž JirJco . rkT s ln R ti Ž 0 . rR ti Ž t .
q 1r2 ln 1 y G Ž t . rG Ž ` . . Ž 7 .
From Eqs. Ž5. and Ž7., Ji Ž t . and UŽ JirJco . can be
fully determined by R ff Ž t . experimental data. Using
the two equations, one can obtain the U–J data from
R ff Ž t . experiments to study the mixed state of
HTSCs.
In the calculation of GŽ t . and GŽ`. in Eqs. Ž4a.
and Ž4b., a fit formula is used,
a
b
R ff Ž t . s R 1 q R 2 Ž 1 q trt1 . q R 3 Ž 1 q trt1 . . Ž 8 .
Here, R 1 , R 2 , R 3 , t 1 , a and b are the fit parameters,
which are determined by using the equation to fit the
R ff Ž t . experimental data.
In some U–J relation models w3–9x for conventional flux, if the current density J is substituted by
Ji , they can be applied to the case of the relaxation
flux. The vortex-glassrcollective-pinning ŽVGrCP.
model gives a power law U–J relation w4–8x
U Ž JirJco . s Ž Uorm . Ž JcorJi .
my1
.
Ž 9.
The Griessen model is based on Griessen’s argument
L.P. Ma et al.r Physica C 291 (1997) 143–148
145
w9x that for any continuously different potential, the
effective pinning energy can be written as
n
U Ž JirJco . s Uo Ž 1 y JirJco . ,
with n s 3r2.
Ž 10 .
Eqs. Ž9. and Ž10. will be applied to fit the obtained
U–J data from R ff Ž t . experiments.
3. Experimental
An epitaxial Gd 0.8Y0.2 Ba 2 Cu 3 O 7 superconducting
thin film with the c-axis perpendicular to the film
surface, was fabricated on a Ž100. YSZ substrate
using the in situ DC magnetron sputtering method.
Detailed information about the fabrication can be
found elsewhere w10x. The film’s thickness is about
˚ Photolithography and chemical etching tech3000 A.
niques were used to define a four-probe pattern with
a center strip that is nominally 0.3 = 30 = 300 mm3
in size. The resistivity of the sample at 95 K and
zero field is about 225 mV P cm. The zero resistance
transition temperature measured by the four-probe
method is 91.5 K, and the transition width Ž10–90%.
measured by AC susceptibility measurements is 0.5
K. X-ray diffraction shows that it is a single phase.
R ff Ž t . experiments are performed at the state of the
tail part of magnetization versus external applied
magnetic field Ž M–H . curve. R ff Ž t . experiments are
performed after establishing fully critical state in the
sample. The applied transport current was varied
according to different applied magnetic field. Because we use the four-probe method to measure
R ff Ž t ., the voltage is measured directly. Voltage can
be measured when the applied transport current is
large enough. We select a proper current to let the
initial-measured voltage be over 60 mV, so a good
time-dependence voltage can be measured. The
R ff Ž t . measurement was described elsewhere w1x.
4. Results and discussion
R ff Ž t . experiments at various temperatures and
magnetic fields have been performed. Fig. 1 shows
the relative change of R ff Ž t . at 88 K and four
magnetic fields. The circles and triangles represent
Fig. 1. The relative change of R ff Ž t . at T s88 K, and m o H s1, 2,
3 and 4 T for a Gd 0.8Y0.2 Ba 2 Cu 3 O 7 superconducting thin film.
The circles and triangles represent experimental data, the solid
lines are the fits by Eq. Ž3.. The amplitude R ti Ž0. is 8.62, 43.5,
56.8 and 16.4 mVPcm for m o H s1, 2, 3 and 4 T, respectively.
the experimental data. The curves are the fits by Eq.
Ž3.. The relaxation amplitude R ti Ž0. are 8.62, 43.5,
56.8 and 16.4 mV P cm at magnetic fields of 1, 2, 3
and 4 T, respectively. It can be seen from Fig. 1 that
Eq. Ž3. derived from the two-kind-of-flux creep
model describes the experimental data at various
magnetic fields well.
Eqs. Ž5. and Ž7. that are derived from the twokind-of-flux-creep model can be used to calculate
J–U data from R ff Ž t . experimental data. The integration terms GŽ t . and GŽ`. in Eqs. Ž4a. and Ž4b.
can be easily obtained by using the fit formula, Eq.
Ž8.. The inserts of Fig. 2a–d show the R ff Ž t . data
fitted by the fit formula. The open squares are the
experimental R ff Ž t . data, the curves are the fits by
Eq. Ž8.. It is confirmed from the inserts of Fig. 2 that
fit formula, Eq. Ž8., is good enough to calculate the
integration terms GŽ t . and GŽ`..
The U–J data are calculated from R ff Ž t . data at
T s 88 K and four magnetic fields. The U–J relation
models of VGrCP and Griessen are used to fit the
calculated U–J data. R ff Ž t .. The parameters Uo , m ,
and n in the models are determined by fits. Fig.
2a–d show the current density dependence of the
effective pinning energy at T s 88 K, for m o H s 1,
2, 3 and 4 T, respectively. The open circles stand for
L.P. Ma et al.r Physica C 291 (1997) 143–148
146
Fig. 2. The current density dependence of effective pinning energy at T s 88 K and four magnetic fields for a Gd 0.8Y0.2 Ba 2 Cu 3 O 7
superconducting thin film. Ža. m o H s 1 T, Žb. m o H s 2 T, Žc. m o H s 3 T and Žd. m o H s 4 T. The open circles stand for the calculated U–J
data from R ff Ž t . data. The solid lines are the fits by the VGrCP model. The dashed lines are the fits by the Griessen model. The inserts of
Ža., Žb., Žc. and Žd. show the fits of R ff Ž t . data by fit formula, Eq. Ž8., for calculating GŽ t . and GŽ`..
the U–J data derived from R ff Ž t . experimental data
by using Eqs. Ž5. and Ž7.. The solid curves are the
fits by Eq. Ž9.. The dashed curves are the fits by Eq.
Ž10.. The physical parameters in the two U–J relation models obtained by fits are shown in Table 1.
We have given the R ff Ž t . experimental data at 3
Table 1
Fit parameters by two models at T s 88 K and four magnetic fields
Model
Parameter
Hs1 T
Hs2 T
Hs3 T
Hs4 T
VGrCP
VGrCP
Griessen
Griessen
UorkT
m
UorkT
n
1.09 " 0.02
0.09 " 0.02
2.34 " 0.06
1.53 " 0.05
1.17 " 0.02
0.16 " 0.04
2.61 " 0.05
1.56 " 0.03
1.08 " 0.01
0.30 " 0.02
2.58 " 0.05
1.57 " 0.03
1.87 " 0.01
0.22 " 0.01
4.1 " 0.1
1.51 " 0.02
L.P. Ma et al.r Physica C 291 (1997) 143–148
147
Fig. 3. The calculated U–J data at m o H s 3 T, and T s 80, 84, 86 and 88 K for a Gd 0.8Y0.2 Ba 2 Cu 3 O 7 superconducting thin film. Ža.
T s 80 K, Žb. T s 84 K, Žc. T s 86 K and Žd. T s 88 K. The open circles stand for calculated derived U–J data. The solid lines are the fits
by the VGrCP model. The dashed lines are the fits by the Griessen model.
T, and four temperatures in Ref. w1x. The U–J data at
the conditions are also calculated from R ff Ž t .. Fig.
3a–d show the current density dependence of effective pinning energy at 3 T and T s 80, 84, 86 and 88
K, respectively. The open circles stand for experi-
mentally derived U–J data by using Eqs. Ž5. and Ž7..
The solid curves are the fits by Eq. Ž9.. The dashed
curves are the fits by Eq. Ž10.. The physical parameters in the two U–J relation models that are obtained
by fits are shown in Table 2.
Table 2
Fit parameters by two models at H s 3 T and four temperatures
Model
Parameter
T s 80 K
T s 84 K
T s 86 K
T s 88 K
VGrCP
VGrCP
Griessen
Griessen
Uo rkT
m
Uo rkT
n
2.552 " 0.005
0.174 " 0.005
5.5 " 0.1
1.49 " 0.03
1.822 " 0.006
0.354 " 0.006
4.7 " 0.2
1.6 " 0.1
1.09 " 0.01
0.30 " 0.01
2.6 " 0.1
1.58 " 0.05
1.08 " 0.01
0.30 " 0.02
2.58 " 0.05
1.57 " 0.03
148
L.P. Ma et al.r Physica C 291 (1997) 143–148
that the VGrCP model describes the U–J relation in
R ff Ž t . state of HTSCs better, which is in agreement
with the results of magnetic measurements w11x.
5. Conclusion
Fig. 4. The temperature dependence of the amplitude of the
relaxation of resistivity R ti Ž0. for a Gd 0.8Y0.2 Ba 2 Cu 3 O 7 superconducting thin film.
From R ff Ž t . experiments we find an interesting
phenomenon. The relaxation amplitude R ti Ž0. shows
a maximum value at a temperature and magnetic
field as shown in Fig. 4. In Fig. 4 the open circles
represent R ti Ž0. data. When the applied magnetic
field of HTSCs samples is at a high enough temperature, R ff Ž t . is occurs, when the magnetic field is
over Hc2 , R ff Ž t . becomes zero. So it can be understood that R ff , or the amplitude of relaxation resistivity R ti Ž0., must have a maximum at a certain state
ŽTm , Hm .. From the two-kind-of-flux-creep model
R ti Ž0. is proportional to the maximum flux-creep
velocity. The peak in R ti Ž0. versus magnetic field at
a temperature reflects the flux motion. The physical
mechanism will be studied in our next step.
If reasonable physical parameters are derived by
fitting the U–J data with U–J relation models, it
will be proved that the calculated U–J data and
the-two-kind-of-flux-creep model are correct. We
have applied VGrCP and Griessen models to fit the
U–J data calculated at various temperatures and
magnetic fields. It is found that the value of the
parameter n in the Griessen model is about 1.5, and
the value of the parameter m in VGrCP model
ranges from 0.1 to 0.4. These values are quite reasonable w9,11x. It can be derived from Figs. 2 and 3
A new method to study the mixed state of HTSCs
by R ff Ž t . experiments has been theoretically and
experimentally discussed. Equations about the decayed magnetization current Ji Ž R ff Ž t .. and the effective pinning energy UŽ R ff Ž t ., Ji Ž t .. have been obtained. The J–U data have been calculated from
R ff Ž t . experimental data at various temperatures and
magnetic fields by using the two-kind-of-flux-creep
theory. Reasonable physical parameters have been
derived by fitting the U–J data with VGrCP and
Griessen models. R ff Ž t . experiments and U–J relation models support the proposed method to study
the mixed state of HTSCs.
Acknowledgements
We would like to thank Prof. Z.X. Zhao, B. Yin
and J.W. Li, for their support and discussion. This
work is supported by the National Center for R & D
on Superconductivity of China.
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