Supercond. Sci. Technol. 11 (1998) 1177–1180. Printed in the UK
PII: S0953-2048(98)94396-2
Observation of thermally activated
flux creep in YBa2Cu3O7−y
microbridges
S K H Lam
CSIRO, Telecommunications and Industrial Physics, PO Box 218, NSW 2070, and
University of Sydney, NSW 2006, Australia
Received 2 February 1998
Abstract. Current voltage characteristics of YBa2 Cu3 O7−y microbridges with
different thicknesses 20—110 nm were measured. The resistivity–current density
characteristics within the temperature range of 0.06 < T /Tc < 0.17, can be fitted by
the thermally activated flux creep model with a linear current density dependence
of the activation energy U (J ). The vortex activation energy within this temperature
range has a value of 33–59 meV whereas the hopping frequency was found to be
in the range of 109 –1010 Hz.
1. Introduction
3. Theory
The study of vortex dynamics in high-temperature
superconducting (HTSC) materials has both fundamental
and technological importance. For example, the materials
must provide strong vortex pinning in order to transport
large current without power dissipation. In HTSC materials,
thermally activated depinning of the flux line would
introduce severe problems for their application [1]. One
of the common methods used to investigate the vortex
dynamics is to study the measured voltage–current or
resistivity–current density characteristics (ρJC). The aim
of this work is to investigate the strength of the pinning
potential in YBCO thin films and the possible nature of the
pinning sites by following the framework of the thermally
activated flux creep model suggested by Anderson and Kim
[2].
Flux bundles can be thermally activated over a energy
barrier U with a hopping rate ν where
U
.
(1)
ν = ν0 exp −
kB T
2. Experiment
Here, ν0 is the attempt frequency of hopping and kB is the
Boltzmann constant. The flux creep model, proposed by
Anderson and Kim [2], takes a bias current into account
and assumes a current dependence of U as U (B, T , J ) =
U0 (B, T )(1−J /Jc,0 ) where B is the flux density and Jc,0 is
the current density for which the barrier U is zero, i.e. the
critical current density at zero temperature. Therefore, the
effective hopping rate νn which is the difference between
the hopping rates along and against the Lorentz force can
be written as
J U0
U0
νn = 2ν0 exp −
sinh
.
(2)
kB T
Jc,0 kB T
In order to study the intrinsic pinning properties of YBCO,
the damage to the material during patterning must be
minimized. In our experiment, the selective expitaxy
growth (SEG) technique was used to fabricate microbridges
which do not require any film patterning after the HTSC
thin film is deposited. SEG with Nb as an inhibitor has
been found to be a simple technique [3] to obtain stable
and high quality YBCO submicron structures. The details
of microbridge fabrication using the SEG technique has
been reported elsewhere [3]. Au contact pads were DC
sputtered on top of the YBCO through a shadow mask
and a four point contact method was used in all electrical
measurements.
νn can be expressed in terms of the average flux bundle
velocity v and the average hopping distance L by νn =
v/L. The time averaged electric field E observed in the
electric field–current density measurement is E = vB. In
the absence of an external applied field, the self-field of
the bias current is the only field present in the system.
An estimate of the maximum self-field near the edges,
where the vortices are expected to move first, is given by
Bse = µ0 wdJ /2(w + d) [4] where w and d are width
and thickness of the bridge respectively. In the case of
large current density J ≈ Jc,0 and low temperature U0 kB T , the hopping against the Lorentz force is energetically
unfavourable. As sinh(x) = exp(x)/2 for x 1, (2) can
c 1998 IOP Publishing Ltd
0953-2048/98/101177+04$19.50 1177
S K H Lam
Table 1. The measured, fitted and calculated parameters
for the three microbridges. Data in the parentheses denote
the range of values obtained within the temperature range
0.06 < t < 0.17.
d (nm)
Tc (K)
Jc (MA cm−2 )
77.4 K
5.05 K
Jc ,0 (MA cm−2 )
U0 (meV)
S69
S70
S71
110
85.5
60
85.3
20
81.8
2.63
48.2
49.3
(33–52)
1.86
31.9
33.2
(38–59)
0.46
9.20
9.28
(35–59)
be written as
ρ=
µ0 wdLν0
U0
J U0
E
=
exp −
exp
. (3)
J
2(w + d)
kB T
Jc,0 kB T
Therefore, a plot of ρ versus J in a log-linear scale will
give a straight line with slope m = U0 /Jc,0 kB T and yintercept ρ0 . In the next section, (3) will be used to fit
our data and the values of U0 and ν0 will be extracted and
discussed. There are reports ([5], [6], [7] p 120) which
also suggested that the self-field is proportional to current
density. Bse = µ0 wdJ /2(w+d) was used in the derivation
of ρJC (3) which gave a reasonable estimate of the hopping
frequency from our experimental data (section 4.3).
Figure 1. The resistivity–current density characteristics of
S69 at two different temperatures. The solid lines are fits to
the thermally activated flux creep model (3).
4. Results and discussion
4.1. Transport properties
Three microbridges of the same length l = 20 µm and
width w = 4 µm but with different film thicknesses were
fabricated and studied. Table 1 lists their thickness, the
measured critical temperature Tc and critical current density
Jc (using an electric field criterion Ec = 2 µV cm−1 ) at
77.4 and 5.05 K. The thinnest bridge S71 (d ≈ 20 nm)
clearly shows much lower Tc and Jc values compared with
the two thicker bridges.
4.2. Pinning potential
In figure 1, the ρJC of S69 at two different temperatures
is shown; the lines through the data are fits using (3) with
fitting parameters m and ρ0 . Similar fits were also done on
S70 and S71 in the temperature range 0.06 < t < 0.15 (t =
T /Tc ). To calculate U0 (mJc,0 kB T ), a Jc,0 value for each
sample is needed which was estimated by extrapolating
the experimental values of Jc (T ) to zero temperature. By
choosing an appropriate value of Jc,0 (table 1) as a fitting
parameter for each sample, the normalized values Jc (t)/Jc,0
of all three samples can be fitted well by a straight line with
a slope of 0.365, i.e. Jc (t)/Jc,0 = 1 − 0.365t (figure 2).
Figure 3 shows the calculated U0 values as a function
of reduced temperature t. All three samples have similar
values of U0 within the range of ≈ 33–59 meV.
The average distance between vortices R is given by
R ≈ (80 /B)0.5 [8] where 80 is the flux quantum. In
our experiment, the self-field was ≈1–30 mT and R ≈
1178
Figure 2. Normalized critical current density Jc (t )/Jc ,0
versus reduced temperature t . The line
Jc (t )/Jc ,0 = 1 − 0.365t is the best fit to data. The fitting
parameters Jc ,0 of each sample are shown in table 1.
0.3–2 µm which is larger than the penetration depth λ.
Hence, the interaction between vortices is excluded and the
measured U0 would correspond to the value of the individual vortex pinning energy. It would be instructive to compare U0 with the pinning energy Us of an atomic vacancy
(e.g. oxygen), which is given by Us ≈ 820 ξ/8π 2 µ0 λ2 [1]
where µ0 is the vacuum permeability and ξ is the coherence
length. Using λ0 ≈ 140 nm and ξ ≈ 1.5 nm [9] at zero temperature, Us ∼
= 35 meV (cf U0 ≈ 33–59 meV). The similar
values of U0 and Us suggest that the size of the pinning site
Thermally activated flux creep in YBa2 Cu3 O7−y microbridges
oxygen vacancies) as suggested by the calculated U0 values,
L would depend on the density of the atomic defects. At
present, the value of L is not known in detail. From the
measurement of YBCO nanobridges, de Nivelle ([7] p 115)
reported L ≈ 3 and 1 nm for d ≈ 85 and 25 nm. We
assume the densities of atomic defects are similar for similar
film thickness and use L = 3 nm for S69 (d = 110 nm) and
1 nm for S71 (d = 20 nm) to estimate the attempt hopping
frequency ν0 from the y-intercept value of figure 1. ν0 of
S69 and S71 was calculated in the range of 2.5–8 × 109
and 1.5–5.8 × 1010 Hz in our measurement temperature
range. These values have the same order of magnitude as
the values reported by the magnetic relaxation measurement
[13].
4.4. Correlation length
Figure 3. A plot of the calculated activation energy U0
versus the reduced temperature t . Lines are guides to the
eye.
in our samples is of the order of an atomic vacancy. This
kind of intrinsic pinning centre may also explain why U0
has similar values for all three samples (figure 3) although
there is a spread of Tc and Jc (table 1) between them.
In our measurement temperature range t 1, Us is almost constant as ξ ∝ (1 − t)−1/2 and λ ∝ (1 − t 4 )−1/2 [10].
The monotonic increase of U0 (figure 3) with temperature
contradicts the temperature independence of Us at t 1. It
is suggested that the variation of U0 with temperature does
not indicate a single value of the pinning potential which
varies with temperature: instead there is a distribution of
the pinning potentials rather than a single value. As the
length of the microbridges is much longer than the distance
between the vortices, the dissipation by vortex movement
would involve multiple vortices hopping at different positions on the bridge with different pinning potential e.g. due
to film inhomogeneity. Therefore, our observed U0 could
represent an average value of the distribution [11]. Assuming the hopping of vortices with pinning energy in the range
of U1 –U2 at temperature t1 leads to an observed average
value of U0 (t1 ). When temperature increases to t2 , thermally activated hopping becomes energetically favourable
for the vortices sitting at higher values, say U3 where
U3 > U2 . Hence, the observed average value U0 (t2 ) is
expected to be larger than U0 (t1 ) which may explain the
monotonic increase of U0 (t) in figure 3. The 1/f flux noise
data of YBCO films obtained by Ferrari et al [12] also indicate that the pinning energy has a distribution with a minimum value of ≈40 meV which agrees well with our data.
4.3. Attempt frequency
As the vortices are far apart and uncorrelated (section 4.2),
the hopping distance of each vortex would probably be
equal to the distance between the nearest pinning centres
L. If the pinning centres are due to atomic defects (e.g.
It is also interesting to study the magnitude of the
correlation length Lv of the vortices along the flux line.
The energy gain of a vortex jump UL driven by the Lorentz
force is UL = J BV L where V ≈ λ2 Lv is the volume of
the uncorrelated flux line. At UL ≈ U0 , J ≈ Jc,0 and
U0 = Jc,0 Bλ2 Lv L. In the case of very thin film d λ
(S71), λ must be replaced by an effective penetration depth
δ where δ ≈ 2λ2 /d [14]. The extracted U0 (33–59 meV)
from the measured ρJC give Lv equal to 75–100 nm for
S69 and 15–25 nm for S71. These values are approximately
equal to the thickness of the samples. Although Lv of S71
is much smaller than the value of S69, their U0 values are
similar which is due to S71 having a much larger value of
effective penetration depth.
5. Conclusion
Microbridges with thickness d in the range of 20–110 nm
were fabricated by the selective epitaxy growth method.
The ρJC of these microbridges in the temperature range of
0.06 < t < 0.17 was found to agree well with the thermally
activated flux creep model in the regime of J ≈ Jc and low
temperature i.e. U0 kB T [2]. The pinning potential U0
was found to be within the range of 33–59 meV on all
three samples with different film thicknesses. The hopping
frequency of the vortices was calculated to be ≈1010 Hz
and the correlation length of the flux line was found to
have values close to the thickness of the microbridges.
Acknowledgments
The author would like to thank Bagula Sankrithyan for
his work on YBCO film deposition and Keith Leslie for
technical support. The discussions with C P Foley and
K-H Müller are helpful during the preparation of this
manuscript.
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