PHYSICAL REVIEW B
VOLUME 58, NUMBER 10
1 SEPTEMBER 1998-II
Dynamics of the second peak in the magnetization of Bi2Sr2CaCu2O8 crystals
S. Anders, R. Parthasarathy, and H. M. Jaeger
The James Franck Institute and Department of Physics, The University of Chicago, Chicago, Illinois 60637
P. Guptasarma and D. G. Hinks
Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439
R. van Veen
Delft Institute for Microelectronics and Submicrontechnology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
~Received 2 April 1998!
We use a combination of relaxation measurements and magnetic hysteresis loops at different field ramp rates
to explore the dynamical behavior of the second peak in the magnetization of Bi2Sr2CaCu2O8 crystals. We find
that the second peak is absent in the short-time limit. It evolves at intermediate time scales due to a different
decay rate of the Bean profile at fields above and below the second peak. At long time scales, when the Bean
profile for fields below the second peak has decayed, the size of the second peak saturates. Finally, while the
Bean profile above the second peak field slowly decays, the second peak decreases and vanishes.
@S0163-1829~98!00634-1#
INTRODUCTION
An intriguing feature in the phase diagram of
Bi2Sr2CaCu2O8 is an anomalous second maximum in the
magnetization hysteresis loops.1–11 This so-called ‘‘second
peak’’ is located at fields of a few hundred Gauss and appears at temperatures between 20 and 40 K. It has attracted
much attention, in part because it may be a signature of a
transition from three- to two-dimensional fluctuation behavior of vortices in the superconductor. However, there have
been several explanations for the second peak, and roughly
they can be divided into two classes. The first is based on a
‘‘static’’ picture, in which defects in a given crystal provide
a fixed distribution of pinning sites.1–3 A change in the relaxation behavior could then occur, for example, at magnetic
fields where there is a matching of the vortex spacing with
the dislocation network in the crystal. In this case, the second
peak would be associated with this matching and should be
visible at all ramp rates of the external field.12 However,
there has been much recent evidence that the second peak
does change significantly in size as the ramp rates are
varied.1,4–6 These observations have led to a second class of
explanations that are based on the idea that the vortex dynamics changes at the second peak, resulting in a fielddependent magnetic relaxation rate. The precise nature of
this change so far has not been established unambiguously.
In addition to the above-mentioned dimensional crossover in
the vortex response, factors such as the increasing importance of surface barriers at elevated temperatures at which
bulk pinning is weak come into play. In general, as for any
dynamical mechanism, there should be an associated set of
characteristic time scales that compete with the ramp rate.
There has been some controversy,1,4–6 however, as to
whether the second peak should be most pronounced at short
or at long time scales, corresponding to fast or slow ramp
rates of the external field.
Here we report on the evolution of the second peak and its
behavior on different time scales by extracting data from two
complimentary types of measurements: ~a! direct measure0163-1829/98/58~10!/6639~6!/$15.00
PRB 58
ments of the magnetization relaxation at fixed external fields,
which is the standard method for studying magnetic decay,
and ~b! magnetization loops taken at different external-field
ramp rates, which allow us to determine a dynamic
relaxation rate. Our results on several high-quality
Bi2Sr2CaCu2O8 crystals with varying oxygen content show
clearly that the second peak evolves with a temperature- and
doping-dependent time scale. We find that the second peak
does not appear on time scales shorter ~or ramp rates faster!
than this characteristic time. As the second peak builds up,
the relaxation rate has a marked field dependence, but becomes field independent at long time scales, when the second
peak is fully established. This shows that the mechanism
behind the second peak works on short time scales and dies
out at long time scales.
EXPERIMENT
High-quality Bi2Sr2CaCu2O8 crystals were grown at stoichiometry using a variation of the traveling-solvent floating
zone method. Growth was performed in a double-mirror image NEC SC-M15HD furnace modified with an external
home-built mechanism for very slow controlled growth, less
than 0.1 mm/h. Oxygen content was changed by annealing
optimally doped crystals in high-purity flowing gases adjusted for differential partial pressures of oxygen. Control of
Cr/Ca stoichiometry yielded T c 595 K (DT c ,1 K) at ‘‘optimal’’ oxygen doping, the highest observed in this
material.13 Single crystals from the same batch were also
investigated by a variety of probes including Raman spectroscopy, NMR spectroscopy, microwave, infrared and fourprobe conductivity, Laue diffraction, and four-circle x-ray
scattering, indicating crystals of very high quality. For example, tunneling spectra14,15 with scanning tunneling microscopy and point-contact tunneling yield repeatable superconducting gaps with a 2–4 % scatter in measured gap value
over micrometer distances, indicating very high chemical
and electronic homogeneity.
We performed measurements on four samples, two near
optimally oxygen doped ~T c 594 and 92 K! and two oxygen
6639
© 1998 The American Physical Society
S. ANDERS et al.
6640
FIG. 1. Magnetization vs local field B for a near optimally
doped crystal at several ramp rates of the external field. Ramp rates
are 9.6, 6.4, 4.8, 3.2, 1.6, 0.8, 0.4, 0.2, 0.1, and 0.05 G/s. The slower
ramp rates correspond to the narrower loops. The distance between
the Hall probe and the edge of the crystal was 120 mm.
overdoped ~T c 576 and 72 K!. Typical crystal dimensions
were 200–400 mm on the side at 20–30 mm thickness. For
magnetization measurements each crystal was attached to an
array of 11, in-line microfabricated Hall probes that measured the local field, along the c axis, at the surface of the
crystal. The Hall probes were fabricated in a n-type Si/SiGe
heterostructure grown by molecular-beam epitaxy as described elsewhere.16 Each of the 11 Hall probes had an active
area of ~2 mm!2 and was separated from the neighboring
probes by 7 mm. The external magnetic field H was always
applied parallel to the c axis of the crystal and could be
ramped at rates 50 mG/s,Ḣ,80 G/s. Measurements were
performed over the temperature range 2,T,31 K.
RESULTS
Figure 1 shows magnetization loops for one of the near
optimally doped crystals at T526.25 K for several ramp
rates. The magnetization, M 5B2H, is taken as the difference between the applied field H and the local field B as
measured by the Hall probes. The data in Fig. 1 correspond
to a probe 120 mm from the edge of the sample that was near
the center of the crystal. Because the local-field profile inside
the crystal continually relaxes towards the equilibrium configuration, the overall width of the magnetization loops decreases with decreasing ramp rate. The second peak can be
seen at B 2pk'200 G ~arrows!. Its position depends on the
local field and is almost ramp-rate and temperature independent ~at a ramp rate Ḣ50.2 G/s it could be observed between
22 and 30 K!. The size of the second peak, by contrast,
shows a remarkable ramp-rate dependence. It is not observed
at ramp rates above 10 G/s, reaches a maximum at about 0.2
G/s, and saturates for even lower ramp rates.
To explore this more explicitly, we measured the magnetic relaxation after the field ramp had been stopped. In Fig.
2, these relaxation data are plotted together with the magnetization loops ~only the branches with negative magnetization are shown!. For this plot, we use the applied field H as
~horizontal! field axis, because H remains fixed during relaxation. Each of the relaxation measurements spans 1000 s
with data points equally spaced in time at 10-s intervals. We
PRB 58
FIG. 2. Magnetization vs applied field H ~lines; this is the same
data set as in Fig. 1! and relaxation measurements at fixed H ~data
points!. The relaxation runs start at the magnetization loops with 9.6
~filled circles! and 3.2 G/s ~open circles!, respectively. The data
points are separated by 10 s and extend to 1000 s.
performed two sets of relaxation measurements, with ramp
rates of 9.6 and 3.2 G/s before stopping. The resulting magnetization profiles track the magnetization loops well. This is
clarified in Fig. 3, where relaxation data points corresponding to the same times are connected. Note again that the
second peak becomes most pronounced at long time scales.
For the overdoped crystal we show magnetization and relaxation data in Fig. 4. The data correspond to a Hall probe
located 110 mm from the edge of the crystal and about 60
mm from its center. Here the second peak is much wider than
for the optimally doped crystal, and is most pronounced at
about 14 K. It remains discernable for ramp rates less than 30
G/s. For the relaxation data in Fig. 4, the field had been
ramped with 50 G/s to the target field prior to taking relaxation measurements ~the hysteresis loop taken at 50 G/s lies
almost on top of the one with 30 G/s and is not shown for
clarity!. Relaxation data are shown at 1-s intervals for times
FIG. 3. Relaxation-time profile. Relaxation data points corresponding to the same time ~0, 10, 20, 30, 40, 100, and 1000 s, going
from the bottom of each plot to the top! for the relaxation data in
Fig. 2.
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DYNAMICS OF THE SECOND PEAK IN THE . . .
FIG. 4. Magnetization during ramping ~lines! and during relaxation ~data points! for an overdoped crystal. The ramp rates were
30, 10, 5, 1, 0.2, and 0.1 G/s, going from the bottom to the top of
the plot. For the relaxation data, the ramp rate prior to relaxation
was 50 G/s. The distance between the Hall probe and the edge of
the crystal was 110 mm. Inset: second peak field for several oxygenoverdoped crystals vs their T c . Circles, our data; squares, Ref. 8;
diamonds, Ref. 9; triangle, Ref. 15.
up to 100 s. As in the optimally doped crystal, the relaxation
profile tracks the ramp-rate profile.
The inset to Fig. 4 compares B 2pk for several crystals of
varying oxygen doping, based on this work and data published in the literature.7,9,17 For this plot we restrict ourselves
to data taken by local Hall probes since the exact position of
the second peak in terms of the local field B cannot, in general, be determined with global measurements. One exception is the datum at T c 564 K, which was obtained from
vibrating sample magnetometer measurements and confirmed by muon spin rotation. From this plot we find that
B 2pk(T c ) is well approximated by a linear relationship
B 2pk5a2bT c with best fitting parameters a54536 G and
B548 G/K.
In order to further elucidate the dynamic vortex response
above and below B 2pk , we calculate the unnormalized dynamical relaxation rate R(B)5dM (Ḣ)/d ln Ḣ introduced by
Pust et al.18 We can obtain R(B)5(M f 2M s )/(ln Ḣ f /Ḣs)
from magnetization loop data taken at two different ramp
rates ~the indices f and s indicate fast and slow ramp rate,
respectively!. We note that we cannot use the normalized
dynamical relaxation rate19 Q(B)5d ln M(Ḣ)/d ln Ḣ since
the reversible magnetization and therefore the absolute value
of M was not accessible to our experiment. The main difference between dynamical relaxation rates such as Q and R and
the conventional relaxation rate S5dM /d ln t is that the
former are obtained for constant local field B, whereas the
latter is measured at fixed applied field H ~and, consequently,
varying local field!. Therefore, whenever the relaxation rate
changes considerably with field, it is desirable to use the
dynamical relaxation rate. Figure 5 shows R(B) for the field
exit case ~corresponding to the upper right quadrant in Fig.
1! in an optimally doped crystal. Each trace corresponds to
two particular ramp rates, decreasing from 9.6 and 6.4 G/s
~top! to 0.2 and 0.05 G/s ~bottom!. R appears to consist of
two components: a B-independent background level that in-
6641
FIG. 5. Dynamical relaxation rates R for the optimally doped
crystal vs local field B. R is obtained from magnetization loops at
ramp rates of 9.6/6.4, 4.8/1.6, 3.2/0.8, 1.6/0.4, and 0.2/0.05 G/s,
going from the top of the plot to the bottom.
creases with faster ramp rates ~and is due to the overall increase of the magnetization loop width that has not been
normalized for!, and a bulge at fields just below B 2pk . The
maximum of this bulge occurs at B'60 G, which corresponds to the development of the dip in the field exit branch
of the magnetization in Fig. 1. The overall size of this bulge
is strongly ramp-rate dependent and is most pronounced at
fast ramp rates, i.e., over short time scales. It decreases below our experimental resolution for Ḣ,0.2 G/s, 0.05 G/s.
Thus, while the second peak in the magnetization loops in
Fig. 1 is most apparent at slow ramp rates or long time
scales, the dynamic relaxation rate indicates that most of the
underlying vortex dynamics actually occurs at short time
scales.
DISCUSSION
Our data show a clear ramp-rate dependence and thus
provide strong support for a dynamical origin of the second
peak. A dynamical mechanism was also discussed in Refs.
4–6. However, an unresolved issue has been whether the
second peak should appear most pronounced for slow or fast
ramp rates. Several authors have attempted to shift the experimentally observable time window to shorter time scales
to gain information about the early part of the magnetization
relaxation out of the initial, unrelaxed state where one might
expect the critical current to show a strong field dependence
in the vicinity of the second peak. Yeshurun et al.6 moved
their time window by decreasing the temperature and thus
lowering the relaxation rate. They found that the second peak
vanishes in the limit of very fast time scales. Similarly,
Cohen5 found the second peak is absent at fast ramp rates at
20 K. This is corroborated by Tamegai et al.4 who performed
direct relaxation measurements down to very short times
(1022 s) after a steplike field change. By connecting data
points corresponding to the same times for all measured
fields, similar to our Fig. 3, they reconstructed hysteresis
loops and found that the second peak is diminished at short
times and builds up for larger times.
Our findings are consistent with these observations. They
6642
S. ANDERS et al.
contrast, however, with the conclusions of Cai et al.,1 who
reconstructed M outside of the experimentally accessible
time window by extrapolating relaxation data towards
shorter times. From fitting their relaxation data to a power
law M a t 2s(T,B) they inferred that the second peak should be
most pronounced in the initial, unrelaxed state. This argument is based on the assumption that the same power law
holds over the whole range of time scales. However, our
relaxation data, some of which extend over almost five decades in time, cannot be described by a simple power law.
Figures 1, 2, and 4 demonstrate that the second peak vanishes from the magnetization loops for fast ramp rates in
both the near optimally doped and the overdoped crystals
~we ramped as fast as 80 G/s and found no sign that the
second peak might reappear at ramp rates faster than those
shown in the figures!.
We next discuss how the shape of the magnetic flux profile inside the crystal relates to the second peak. The magnetic profile can essentially have two shapes. If bulk pinning
dominates, there is a roughly constant field gradient throughout the crystal, the so-called Bean profile. If bulk pinning is
not dominant, as is the case for high temperatures or large
time scales when the vortex system approaches equilibrium,
the magnetic profile is governed by edge effects: a surface
barrier due to vortex image forces at the crystal surface20 and
a geometrical barrier due to the thin flat shape of the
crystal.21 Edge effects lead to a dome-shaped magnetic profile as has been shown by Berry8 and Zeldov.10 Our field
resolution is not high enough to allow us to map out the
details of the dome shape. ~It appears as an essentially flat
region.!
Berry et al.8 found that, below B 2pk , the magnetic profile
is dome shaped at all times, whereas above B 2pk the profile is
Bean-like in the short-time limit and then decays to the
dome-shaped profile. Similarly, Cohen et al.5 inferred from
the shape of global magnetization loops that a Bean-type
profile penetrates through the sample for B.B 2pk . With our
array of Hall probes we find, by contrast, that for fields just
above B 2pk the profile is Bean-like ~and therefore governed
by bulk pinning! at all ramp rates. For B,B 2pk , however,
we observe a transition from a Bean-like profile at high ramp
rates to a flat profile at low ramp rates ~as stated before, the
profile appears flat within our field and spatial resolution, but
may be compatible with a slight dome shape!. For the optimally doped crystals, this transition takes place at ramp rates
between 0.1 to 0.4 G/s ~the ramp rate at which the transition
occurs depends on the position of the Hall probe!. We note
that for T526.25 K, the temperature corresponding to the
data shown in the figures, the second peak reaches its maximum for a ramp rate of about 0.2 G/s.
We thus find that the growth of the second peak is accompanied by the decay of the Bean profile and the establishment of a flat, or possibly dome-shaped profile, in the regime
B,B 2pk . This scenario is further supported by the dynamic
relaxation rate data in Fig. 5. Specifically, we can associate
the ramp rate-dependent ‘‘bulge’’ in R for B,B 2pk with the
time decay of the Bean profile: for shorter time scales, the
second peak evolves due to a difference in the relaxation
rates of the Bean profile for fields above and below B 2pk .
Once the Bean profile for fields below B 2pk has decayed, the
second peak stops growing. The magnetic profile in this field
PRB 58
range is now governed by edge barriers that give rise to a
flat-/dome-shaped profile. At fields above B 2pk , the magnetic
profile is still Bean-like, resulting in a large magnetization
for Hall probes sufficiently far away from the edge of the
crystal, where the effect of the Bean-like field gradient is
biggest. Eventually, this Bean profile will decay as well,
leaving a flat, possibly dome-shaped profile at all fields. At
this time scale, the second peak will disappear. At 26.25 K,
this should be observable at very large time scales, beyond
our experimental time window.22 An alternative way to access the large time scales is to speed up the relaxation process by going to higher temperatures, since temperature and
time rescale each other for the process of relaxation.6 We
observed that the second peak disappears gradually if we
increase the temperature at fixed ramp rate ~0.2 G/s! and has
completely decayed at 31 K. A more detailed study of the
decay of the second peak at higher temperatures is given by
Yeshurun6 and Cohen.5
So far, we have not talked about a possible origin for the
second peak, but merely showed that it evolves due to a
field-dependent relaxation rate of the Bean profile. Mechanisms that lead to such a field-dependent relaxation rate include a crossover from a single vortex to a collective vortex
regime, melting from a vortex solid at high fields to a liquid
at low fields. However, neutron-diffraction studies23 and
muon spin-rotation studies15,24 found a sudden decrease in
the intensity of the signal as the field is increased, indicative
of a transition from vortex lines to two-dimensional vortex
‘‘pancakes.’’ @The crossover from single to collective creep
would not show up in those data, and the melting scenario
would result in a larger signal at higher fields ~in the solid!.#
We can explore the three-dimensional–two-dimensional
~3D-2D! crossover scenario and extract the anisotropy parameter G. Vinokur et al.25 found that vortex fluctuations become quasi two dimensional when the magnetic field exceeds the characteristic value B 2D'F 0 /Gs 2 , where F 0 is
the flux quantum and G 1/2s is the effective Josephson length
in layered superconductors with interlayer spacing s and anisotropy factor G. If we use s51.5 nm for the near optimally
doped crystals and equate B 2D with the second peak field
B 2pk , we obtain G 1/25210 and 190. This is compatible with
magnetic torque measurements by Martinez et al.26 who
found G 1/2.150. For our overdoped crystals we estimate that
the change in s is less than 1%.27 B 2pk for these samples is
difficult to determine because the feature is very wide. Nevertheless, with B 2D5B 2pk51200~6150! and 1000~6100! G,
we find reasonable anisotropy parameters of G 1/258665
and 9464. A consistency check of this picture comes from
comparing a wide range of crystals with varying oxygen content. In general, overdoped crystals are less anisotropic than
optimally doped ones,28 and B 2pk increases as the anisotropy
decreases.7,9 The linear relationship between B 2pk and T c ~inset to Fig. 4!, together with B 2D5B 2pk , can be used for a
quantitative estimate of G 1/2 if either T c or B 2pk is known.
Remarkably, Fig. 5 shows that in the long-time limit ~bottom traces!, R is independent of B. This suggests that, although the bulk profile at our longest times is dome shaped
for B,B 2pk and still bean shaped for B.B 2pk , the relaxation for both field ranges is governed by the decay of the
surface barrier that shifts the dome and the Bean profiles
towards the equilibrium magnetization by the same amount.
DYNAMICS OF THE SECOND PEAK IN THE . . .
PRB 58
For this scenario, surface relaxation needs to be faster than
bulk relaxation, which is believed to be the case at low
temperatures.29
Finally, we note that although the time development of
the relaxation data in Figs. 2 and 4 generally follows the time
development of the magnetization loops, there is one slight
difference. In the field entry case, the relaxation for fixed H
proceeds further than expected from the magnetization loops,
and, conversely, the relaxation in the field exit case is slower
than expected from the magnetization loop data. Thus, the
magnetic decay proceeds slightly differently for relaxation
during ramping and for relaxation at fixed field. In other
words, despite many similarities, there remain subtle differences in the nonequilibrium dynamic behavior between the
continuously driven steady state and the state decaying towards equilibrium.
CONCLUSIONS
We have shown that the second peak in both optimally
doped and overdoped Bi2Sr2CaCu2O8 crystals is a purely dynamical phenomenon. We stress that both relaxation mea-
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6643
surements and magnetization loops at different ramp rates
yield important complimentary information about the time
behavior of the magnetization. We find that the second peak
evolves due to a field-dependent magnetic relaxation rate of
the Bean profile and is absent in the short-time limit. Once
the Bean profile at fields below B 2pk has decayed, the second
peak is fully established. The different shapes of the magnetic profile for fields below and above B 2pk are not the
origin, but rather the result of the crossover in flux creep
behavior.
ACKNOWLEDGMENTS
We thank A. W. Smith and H. Claus for stimulating discussions, and A. Verbruggen, E. van der Drift, and S. Radelaar from the Delft Institute for Microelectronics and Submicron Technology ~DIMES! for help with the Hall probe
fabrication. This work was supported by the NSF through the
Science and Technology Center for Superconductivity
~Grant No. DMR 91-20000! and by the DOE, Basic Energy
Science–Materials Science ~Contract No. W/31/109/ENG/
38!.
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For ramp rates fast enough to drive the system into the flux-flow
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