PHYSICAL REVIEW B
VOLUME 60, NUMBER 21
1 DECEMBER 1999-I
Frequency and ac field scaling of the nonlinear ac susceptibility of a HgBa2CaCu2O6ⴙ ␦ thin film
B. J. Jönsson-Åkerman* and K. V. Rao
Department of Condensed Matter Physics, Royal Institute of Technology, S-100 44 Stockholm, Sweden
E. H. Brandt
Max-Planck-Institut für Metallforschung, Institut für Physik, D-70506 Stuttgart, Germany
共Received 16 February 1999兲
We confirm a recently proposed scaling relation for the nonlinear ac susceptibility response of type-II
superconductors by high-precision measurements on a c-axis oriented HgBa2CaCu2O6⫹ ␦ thin film. From
measurements in different ac field and frequency combinations, given by the scaling relation, the flux creep
exponent n(T,H) of the superconducting film can be determined. At T⫽100 K and in the field range studied,
n(T,H) is found to vary with the dc bias field as H ⫺0.18
. 关S0163-1829共99兲00245-3兴
dc
I. INTRODUCTION
Scaling procedures have played a significant role in gaining insight into the basic understanding of the length scale of
various fundamental properties of high-T c superconductors.
One of the most widely reported scaling procedures1 uses
nonlinear dc conductivity data ␴ (J) to extract the glass temperature T g and the static and dynamic critical exponents v
and z. Similar information can also be extracted from a scaling analysis of the linear ac susceptibility response of thin
superconducting disks.2 Recently another scaling relation has
been predicted for the nonlinear ac susceptibility response of
type-II superconductors.3 For any nonlinear conductor with
power-law E-J characteristics E(J)⫽E c (J/J c ) n the ac susceptibility response should be invariant to a simultaneous
change in ac field amplitude and frequency following
␻
n⫺1
H ac
⫽const.
共1兲
This scaling relation does not originate from a phase transition, as is the case for glass scaling around T g , but is rather
a consequence of Maxwell’s equations and the material law
relating the current density and the electric field.
In this work we experimentally confirm that the ac susceptibility response of a c-axis oriented HgBa2CaCu2O6⫹ ␦
共Hg-1212兲 thin film is invariant under the transformation Eq.
共1兲 and show how this invariance can be used to determine
the flux creep exponent n(T,H). By varying the applied dc
bias field we also determine the dc field dependence of the
flux creep exponent at 100 K.
The explicit form of the kernel Q(r,r⬘ ) for a cylinder has
been given in Ref. 4. The momentary screening current can
be found from numerical time integration of Eq. 共2兲 and the
corresponding magnetic moment of the disk is at each time
given by
m 共 t 兲 ⫽2 ␲
R
d
dr
0
0
冋
dy ⬘ Q 共 r,r⬘ 兲 ⫺1 E 共 J 兲 ⫺
0163-1829/99/60共21兲/14913共4兲/$15.00
drr 2
0
冕
d
0
␹⫽
1
␲ h 0V
冕
2␲
0
册
r⬘
␮ Ḣ 共 t 兲 .
2 0 a
共2兲
PRB 60
dyJ 共 r,y,t 兲 .
m 共 t 兲 e ⫺i ␻ t d 共 ␻ t 兲 ,
共3兲
共4兲
where V is the sample volume. Once the material law E(J)
is known, the ac susceptibility response of a thin circular
disk can be readily calculated using Eqs. 共2兲–共4兲.
Both the collective creep6 and the vortex glass7 theories
predict a current-density-dependent activation energy for
vortex depinning,
U 共 J 兲 ⫽U 0
共 J c /J 兲 ␣ ⫺1
␣
,
共5兲
where ␣ is a small positive exponent. When inserted into the
Arrhenius law,
冉
E 共 J 兲 ⫽E c exp ⫺
The problem of how an external axial magnetic field
␮ 0 H a (t) penetrates into a superconducting cylinder of radius
R and thickness 2d can be formulated as an equation of
motion for the screening current J(r,t)⫽J(r,y,t),
冕 ⬘冕
R
In a typical ac susceptibility measurement the external magnetic field varies sinusoidally in time as H a (t)⫽h 0 cos ␻t.
The fundamental ac susceptibility ␹ ⫽ ␹ ⬘ ⫺i ␹ ⬙ can then be
defined5 as
II. SCALING RELATION
J̇ 共 r,t 兲 ⫽ ␮ ⫺1
0
冕
冊
U共 J 兲
,
kT
共6兲
Eq. 共5兲 yields a highly nonlinear E-J relation. In the limit
␣ →0, one has the so-called logarithmic approximation8
U(J)⫽U 0 ln(Jc /J) that is generally considered to describe
single-vortex creep.9 Within this approximation Eq. 共6兲 becomes
E 共 J 兲 ⫽E c 共 J/J c 兲 n with n⫽U 0 /kT,
14 913
共7兲
©1999 The American Physical Society
B. J. JÖNSSON-ÅKERMAN, K. V. RAO, AND E. H. BRANDT
14 914
PRB 60
which is often experimentally observed.10,11 Even in an experimental situation when the full expression for the activation energy 关Eq. 共5兲兴 should be used, an effective flux creep
exponent can be defined as4
n̄⬅
冉冊
⳵ ln E U 0 J c
⫽
⳵ ln J kT J
␣
⭓
U0
.
kT
共8兲
Such an effective flux creep exponent will depend on the
ratio J c /J, i.e., on how much the current density has relaxed
from its initial value J c , and hence on the driving frequency
␻. For small enough ␣ this frequency dependence may, however, be neglected.
It has recently been shown3 that when E(J) is given by a
power-law relation as in Eq. 共7兲, the equation of motion for
the screening current density 关Eq. 共2兲兴 is invariant to a simultaneous change in the time unit, by a factor of c, and in the
current and field units, by a factor of c 1/(n⫺1) . This means
that if Eq. 共2兲 is expressed in a new time unit t̃ ⫽t/c, then
the new functions J̃(r, t̃ )⫽J(r,t) * c 1/(n⫺1) and H̃ a ( t̃ )
⫽H a (t) * c 1/(n⫺1) will satisfy the same equation. The resulting magnetic moment will consequently scale by the same
factor, m̃( t̃ )⫽m(t) * c 1/(n⫺1) . In the case of a periodic applied field H a (t)⫽h 0 cos ␻t, the ac susceptibility ␹ (h 0 , ␻ )
will hence remain unchanged if the frequency is increased
by, say, a factor of 10 and the amplitude by a factor of
101/(n⫺1) , which is the condition given by Eq. 共1兲. The same
scaling relation will hold approximately for the effective flux
creep exponent n̄, provided n̄ does not change considerably
in the frequency range studied.
III. EXPERIMENT
Hg-1212 films with nominal thickness of 400 nm were
prepared by a conventional two-step method involving deposition of Hg-free precursor films on SrTiO3 substrates followed by annealing at 820 °C for 30 min in a controlled
Hg-vapor atmosphere. The x-ray-diffraction pattern collected
within 5°⬍2 ␪ ⬍70° shows predominantly lines corresponding to the c-axis oriented Hg-1212 phase with minor traces
of c-axis oriented Hg-1223. A rectangular sample with lateral dimensions 2.8⫻3.6 mm2 was chosen for this study.
A home-built high-sensitivity susceptometer,12,13 based
on a three-coil mutual inductance bridge and a two-position
background subtraction scheme, was used for all measurements. ac fields of H ac⫽1 – 2 Oe 共root-mean-square, H ac
⫽2 ⫺1/2h 0 ) and dc fields in the range H dc⫽20– 100 Oe were
applied by the same primary coil. Driving frequencies ranged
from f ⫽89.1– 891 Hz. The software allows for consecutive
measurements at different field and frequency settings specified by a list that is run through repeatedly. To minimize flux
trapping, a decrease in ac field is always done gradually with
the minimum step given by the resolution of the digital
lock-in output 共⬃0.3%兲. The temperature may be ramped
slowly or held constant during the measurement. The ac susceptometer is calibrated in emu/Oe and hence measures ␹ ⬘ V
and ␹ ⬙ V. For thin films measured in the perpendicular geometry, it is not practical to divide by the film volume. In the
following we will therefore always express the susceptibility
in emu/Oe, which for brevity will be denoted by ␹ ⬘ and ␹ ⬙ .
FIG. 1. ␹⬘ and ␹⬙ vs T for H ac⫽1 mOe, f ⫽181 Hz. In the inset
is shown ␹⬘ vs T for H ac⫽15 Oe, f ⫽8.91, 27.3, 89.1, 273, and 891
Hz.
IV. RESULTS AND DISCUSSION
From a typical low-field 共1 mOe兲 ac susceptibility temperature scan, shown in Fig. 1, we find T c ⫽120 K with
⌬T c ⫽4.5 K given by the temperatures where 10% and 90%
of complete screening is reached, respectively. The fullscreening susceptibility of a thin circular disk with radius a
is ␹ 0 ⫽2a 3 /3␲ in cgs units. Numerical simulations of thin
rectangular samples with sides 2a and 2b show that the thincircular-disk expression can be used provided it is multiplied
with a prefactor that depends on the aspect ratio b/a. 14 The
experimentally determined susceptibility at 77 K in Fig. 1,
␹ 0,exp⫽⫺9.4⫻10⫺4 emu/Oe, is only about 3% smaller than
␹ 0,th⫽⫺1.67⫻2a 3 /3␲ ⫽⫺9.7
the
theoretical
value
⫺4
⫻10 emu/Oe. In the inset is shown a temperature scan in
an ac field of H ac⫽15 Oe and f ⫽8.91– 891 Hz. Due to the
higher ac field the transition broadens considerably and also
there is a clear increase in the transition width as the frequency is decreased. The observation of these two different
causes for a broadened transition gives additional insight into
the physical meaning of Eq. 共1兲: broadening due to an increase in H ac can be annulled by a corresponding increase in
excitation frequency; the ratio between the factors describing
the respective increases is given by the flux creep exponent
through Eq. 共1兲.
It is experimentally known that n(T,H) in Hg-1212 thin
films depends strongly on both field and temperature.10 Since
the scaling relation assumes that the flux creep exponent
does not vary during the ac field period, we perform all measurements in a dc bias field to ensure that n(T,H) is kept
constant. It has recently been shown that for H dcⱿ2H ac ,
n(T,H) in Hg-1212 thin films does not show any detectable
ac field dependence.15 In this study we use H ac⬇2 Oe in dc
bias fields of H dc⫽20– 100 Oe.
In Table I we give the applied ac fields and frequencies
used in the experiments, which satisfy Eq. 共1兲 for trial flux
creep exponents n ⬘ ⫽10– 17. If n ⬘ equals the actual flux
creep exponent n(T,H) we expect a collapse of all the data;
if n ⬘ is too large, the increased broadening due to a lower
frequency will not be fully compensated for by the corre-
FREQUENCY AND ac FIELD SCALING OF THE . . .
PRB 60
14 915
TABLE I. Listing of ac field 共in Oe rms兲 and frequency combinations used in experiments to test the trial
exponents n ⬘ ⫽10– 17.
f 关Hz兴
891
413
273
181
127
89.1
n ⬘ ⫽10
n ⬘ ⫽11
n ⬘ ⫽12
n ⬘ ⫽13
n ⬘ ⫽14
n ⬘ ⫽15
n ⬘ ⫽16
n ⬘ ⫽17
2.000
1.836
1.754
1.675
1.611
1.549
2.000
1.852
1.777
1.795
1.646
1.589
2.000
1.865
1.796
1.730
1.675
1.622
2.000
1.876
1.812
1.751
1.700
1.651
2.000
1.885
1.826
1.769
1.722
1.675
2.000
1.893
1.838
1.785
1.740
1.697
2.000
1.900
1.848
1.798
1.756
1.715
2.000
1.906
1.857
1.810
1.770
1.732
sponding weaker ac field and consequently 兩 ␹ ⬘ ( f ) 兩 will be
smaller than 兩 ␹ ⬘ (891 Hz) 兩 for all frequencies f ⬍891 Hz; if
n ⬘ is too small, on the other hand, 兩 ␹ ⬘ ( f ) 兩 will be larger than
兩 ␹ ⬘ (891 Hz) 兩 .
A typical measurement run at T⫽100 K and H dc⫽20 Oe
with ac fields corresponding to n ⬘ ⫽10 is shown in Fig. 2.
Data are taken over a period of approximately 2 h and averaged to improve the accuracy. For this particular choice of
n ⬘ ⫽10 the data clearly do not collapse. On the contrary, the
magnitude of the in-phase component increases with decreasing frequency, which indicates that the applied ac field
is decreased too rapidly with frequency for a collapse to
occur, i.e., n ⬘ ⫽10 is smaller than the actual n(T,H).
In Fig. 3 we show the same data together with data for all
trial flux creep exponents n ⬘ ⫽10– 17 in dc fields ranging
from 20 to 100 Oe. For all dc fields we see that there is a
monotonic change in how well the data collapse for different
values of the trial value n ⬘ . While for n ⬘ ⫽10 in a field of
H dc⫽20 Oe no overlap was observed in Fig. 2, the overlap is
almost perfect for n ⬘ ⫽16. For too small n ⬘ values all data lie
below ␹ ⬘ (891 Hz) and for larger n ⬘ values data points consequently lie above ␹ ⬘ (891 Hz). When data points taken at
the same frequency are connected with straight lines, as
shown for H dc⫽45 Oe, the lines intersect almost perfectly in
a single point. For each dc field we thus determine the actual
flux creep exponent n(T⫽100 K, H dc) from the intersection
point of these lines. We also take the extension of the inter-
FIG. 2. ␹ ⬘ ( f )/ 兩 ␹ ⬘ (891 Hz) 兩 for n ⬘ ⫽10, H dc⫽20 Oe at T
⫽100 K measured during 2 h. ac fields and frequencies taken
from the first column of Table I. To the right is plotted the time
average of the same data.
section region along the trial exponent axis as a measure of
how well the data overlap.
The flux creep exponent is plotted vs dc field in Fig. 4. n
clearly decreases with increasing field and a log-log plot of
the same data, shown in the inset, suggests a field depen⫺0.18
dence as H dc
in the field range studied. The error bars
indicate the size of the intersection regions extracted from
Fig. 3 and confirm a close overlap of the experimental data.
In a dc bias field of 100 Oe there is, however, an increased
uncertainty in n that might indicate the scaling as given by
Eq. 共1兲 is not adequate at this field.
FIG. 3. Time-averaged data ␹ ⬘ ( f )/ 兩 ␹ ⬘ (891 Hz) 兩 for all combinations n ⬘ ⫽10– 17 and H dc⫽20– 100 Oe. The lines for H dc⫽45 Oe
show how the value for the actual flux creep exponent n(T,H) is
determined from their intersection. The dashed line is a guide to the
eye to indicate how this intersection point moves with H dc .
14 916
B. J. JÖNSSON-ÅKERMAN, K. V. RAO, AND E. H. BRANDT
PRB 60
n̄ is a useful concept that well describes the vortex dynamics
in this field and temperature range. It is, however, quite
likely that the increased uncertainty observed for H dc⫽100
Oe is due to a breakdown of this approximation. Since both
␣ and the ratio J c /J in Eq. 共8兲 are expected to increase with
field, H dc⫽100 Oe might be a threshold field where n̄ depends too strongly on frequency for the scaling relation to be
meaningful.
V. CONCLUSION
FIG. 4. n(T,H) vs H dc at T⫽100 K. The error bars correspond
to the width of the intersection region in Fig. 3. In the inset is
shown a log-log plot of the same data together with a power-law fit
n(T,H)⬀H ⫺0.18
.
dc
In recent studies on the vortex dynamics of Hg-1212 thin
films it has been shown that the logarithmic approximation
关Eq. 共7兲兴 can accurately describe the frequency dependence
of the nonlinear ac susceptibility response.11 Below a certain
line T cm(H) in the H-T plane, n(T,H) shows a monotonic
decrease with temperature and is inversely proportional to
the square root of the applied field, n(T,H)⬀H ⫺0.5. 12 Above
this line, n(T,H) increases with increasing temperature and
shows a weaker field dependence. Since T cm(H) is ascribed
to a crossover from single-vortex creep to collective creep of
flux bundles, a finite and increasing value for ␣ in Eq. 共5兲 is
expected. At T⫽100 K we are above T cm for the fields used
in this work, which is confirmed by the rather weak field
dependence of the experimentally determined flux creep exponent. From the close overlap of the data in Fig. 共3兲 we
conclude that, although the logarithmic approximation may
no longer be strictly valid, the effective flux creep exponent
*Current address: Physics Department, University of California–
San Diego, 9500 Gilman Drive, La Jolla CA 92093-0319. Electronic address: jjonsson@ucsd.edu
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In conclusion, we have shown that high-precision ac susceptibility measurements on a c-axis oriented Hg-1212 thin
film experimentally confirm a recently suggested scaling relation for the nonlinear ac susceptibility response of type-II
superconductors with flux creep. By varying the applied dc
bias field we demonstrate how the field dependence of the
flux creep exponent n(T,H) can be determined. At T
⫽100 K and in a dc field range of H dc⫽20– 100 Oe we find
that the field dependence of the flux creep exponent is well
⫺0.18
. From the close overlap of the
described by n(T,H)⬀H dc
data and the rather weak field dependence we conclude that
our measurements are made in a regime where an effective
flux creep exponent n̄ describes the vortex dynamics rather
well, although, in a strict sense, the logarithmic approximation might not be valid anymore. At H dc⫽100 Oe the scaling
is not adequate, which we interpret as an increase in the
frequency dependence of n̄.
ACKNOWLEDGMENTS
We are grateful to S. H. Yun and U. O. Karlsson for
providing us with their Hg-1212 thin-film sample. We thank
A. M. Grishin for many discussions and comments during
this work. B.J.J-Å. is most obliged to the Ericsson Research
Foundation for a partial travel grant. This research was supported by the Swedish Natural Science Research Council
共NFR兲.
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