Dynamics of the second peak in the magnetization of Bi2Sr2CaCu2O8 crystals

Dynamics of the second peak in the magnetization of Bi

Sr

CaCu

O

crystals

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 Bi2Sr2CaCu2O8crystals. 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 ap-pears 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 behav-ior 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 re-laxation 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.1,4–6These observations have led to a second class of explanations that are based on the idea that the vortex

dy-namics changes at the second peak, resulting in a

field-dependent 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 impor-tance 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

measure-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 be-comes 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

sto-ichiometry using a variation of the traveling-solvent floating zone method. Growth was performed in a double-mirror im-age 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 ad-justed for differential partial pressures of oxygen. Control of Cr/Ca stoichiometry yielded Tc595 K (DTc,1 K) at ‘‘op-timal’’ 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 spec-troscopy, NMR specspec-troscopy, microwave, infrared and four-probe conductivity, Laue diffraction, and four-circle x-ray scattering, indicating crystals of very high quality. For ex-ample, tunneling spectra14,15with scanning tunneling micros-copy and point-contact tunneling yield repeatable supercon-ducting 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~Tc594 and 92 K! and two oxygen

PRB 58

overdoped ~Tc576 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 mea-sured 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 de-scribed elsewhere.16Each of the 11 Hall probes had an active area of ~2 mm!2 and was separated from the neighboring probes by 7mm. 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,H˙,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, M5B2H, is taken as the differ-ence 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 120mm 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 con-figuration, the overall width of the magnetization loops de-creases with decreasing ramp rate. The second peak can be seen at B2pk'200 G ~arrows!. Its position depends on the

local field and is almost ramp-rate and temperature

indepen-dent~at a ramp rate H˙50.2 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 mag-netic relaxation after the field ramp had been stopped. In Fig. 2, these relaxation data are plotted together with the magne-tization loops ~only the branches with negative magnetiza-tion are shown!. For this plot, we use the applied field H as

~horizontal! field axis, because H remains fixed during

relax-ation. Each of the relaxation measurements spans 1000 s with data points equally spaced in time at 10-s intervals. We

performed two sets of relaxation measurements, with ramp rates of 9.6 and 3.2 G/s before stopping. The resulting mag-netization profiles track the magmag-netization loops well. This is clarified in Fig. 3, where relaxation data points correspond-ing 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 re-laxation 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 relax-ation 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. 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 120mm.

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.

FIG. 3. Relaxation-time profile. Relaxation data points corre-sponding 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.

up to 100 s. As in the optimally doped crystal, the relaxation profile tracks the ramp-rate profile.

The inset to Fig. 4 compares B2pkfor several crystals of

varying oxygen doping, based on this work and data pub-lished in the literature.7,9,17For 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 gen-eral, be determined with global measurements. One excep-tion is the datum at Tc564 K, which was obtained from vibrating sample magnetometer measurements and con-firmed by muon spin rotation. From this plot we find that

B2pk(Tc) is well approximated by a linear relationship B2pk5a2bTc with best fitting parameters a54536 G and

B548 G/K.

In order to further elucidate the dynamic vortex response above and below B2pk, we calculate the unnormalized

dy-namical relaxation rate R(B)5dM(H˙)/d ln H˙ introduced by Pust et al.18 We can obtain R(B)5(Mf2Ms)/(ln H˙f/H˙s)

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(H˙)/d ln H˙ since

the reversible magnetization and therefore the absolute value of M was not accessible to our experiment. The main differ-ence 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-creases with faster ramp rates~and is due to the overall in-crease of the magnetization loop width that has not been normalized for!, and a bulge at fields just below B2pk. The

maximum of this bulge occurs at B'60 G, which corre-sponds 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 be-low our experimental resolution for H˙ ,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 ex-perimentally 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, Cohen5found the second peak is absent at fast ramp rates at 20 K. This is corroborated by Tamegai et al.4who performed direct relaxation measurements down to very short times (1022s) 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

FIG. 4. Magnetization during ramping~lines! and during relax-ation ~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 110mm. Inset: second peak field for several oxygen-overdoped crystals vs their Tc. Circles, our data; squares, Ref. 8; diamonds, Ref. 9; triangle, Ref. 15.

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.

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 Mat2s(T,B)they inferred that the second peak should be most pronounced in the initial, unrelaxed state. This argu-ment 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 de-cades in time, cannot be described by a simple power law. Figures 1, 2, and 4 demonstrate that the second peak van-ishes 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 pro-file inside the crystal relates to the second peak. The mag-netic profile can essentially have two shapes. If bulk pinning dominates, there is a roughly constant field gradient through-out 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 surface20and a geometrical barrier due to the thin flat shape of the crystal.21 Edge effects lead to a dome-shaped magnetic pro-file 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.8found that, below B2pk, the magnetic profile

is dome shaped at all times, whereas above B2pkthe 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.B2pk. With our

array of Hall probes we find, by contrast, that for fields just above B2pkthe profile is Bean-like~and therefore governed

by bulk pinning! at all ramp rates. For B,B2pk, 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 opti-mally 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 maxi-mum for a ramp rate of about 0.2 G/s.

We thus find that the growth of the second peak is accom-panied by the decay of the Bean profile and the establish-ment of a flat, or possibly dome-shaped profile, in the regime

B,B2pk. 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_2pkwith 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 B2pk.

Once the Bean profile for fields below B2pkhas decayed, the

second peak stops growing. The magnetic profile in this field

range is now governed by edge barriers that give rise to a flat-/dome-shaped profile. At fields above B2pk, 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 ac-cess the large time scales is to speed up the relaxation pro-cess 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 Yeshurun6and 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. Mecha-nisms that lead to such a field-dependent relaxation rate in-clude 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

pa-rameterG. Vinokur et al.25found that vortex fluctuations be-come quasi two dimensional when the magnetic field ex-ceeds the characteristic value B2D'F0/Gs2, where F0 is

the flux quantum andG1/2s is the effective Josephson length

in layered superconductors with interlayer spacing s and an-isotropy factorG. If we use s51.5 nm for the near optimally doped crystals and equate B2D with the second peak field

B_2pk, we obtainG1/25210 and 190. This is compatible with magnetic torque measurements by Martinez et al.26 who foundG1/2.150. For our overdoped crystals we estimate that the change in s is less than 1%.27 B_2pkfor these samples is difficult to determine because the feature is very wide. Nev-ertheless, with B2D5B2pk51200~6150! and 1000~6100! G,

we find reasonable anisotropy parameters of G1/258665 and 9464. A consistency check of this picture comes from comparing a wide range of crystals with varying oxygen con-tent. In general, overdoped crystals are less anisotropic than optimally doped ones,28and B2pkincreases as the anisotropy

decreases.7,9The linear relationship between B_2pkand T_c ~in-set to Fig. 4!, together with B2D5B2pk, can be used for a

quantitative estimate ofG1/2if either Tcor B2pkis known.

Remarkably, Fig. 5 shows that in the long-time limit ~bot-tom traces!, R is independent of B. This suggests that, al-though the bulk profile at our longest times is dome shaped for B,B_2pk and still bean shaped for B.B_2pk, the relax-ation 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.

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 differ-ences in the nonequilibrium dynamic behavior between the continuously driven steady state and the state decaying to-wards equilibrium.

CONCLUSIONS

We have shown that the second peak in both optimally doped and overdoped Bi2Sr2CaCu2O8crystals is a purely

dy-namical phenomenon. We stress that both relaxation

mea-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 B2pkhas decayed, the second

peak is fully established. The different shapes of the mag-netic 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 dis-cussions, and A. Verbruggen, E. van der Drift, and S. Rade-laar from the Delft Institute for Microelectronics and Submi-cron 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!.

1_{X. Y. Cai, A. Gurevich, D. C. Larbalestier, R. J. Kelley, M.}

Onellion, H. Berger, and G. Margaritondo, Phys. Rev. B 50, 16 774~1994!.

2_{G. Yang, P. Shang, S. D. Sutton, I. P. Jones, J. S. Abell, and C. E.}

Gough, Phys. Rev. B 48, 4054~1993!.

3_{A. K. Pradhan, S. B. Roy, P. Chaddah, D. Kanjilal, C. Chen, and}

B. M. Wanklyn, Physica C 264, 109~1996!.

4

T. Tamegai, Y. Iye, I. Oguro, and K. Kishio, Physica C 213, 33

~1993!.

5_{L. F. Cohen, J. T. Trotty, G. K. Perkins, R. A. Doyle, and K.}

Kadowaki, Supercond. Sci. Technol. 10, 195~1997!.

6_{Y. Yeshurun, N. Bontemps, L. Burlachkov, and A. Kapitulnik,}

Phys. Rev. B 49, 1548~1994!.

7_{B. Khaykovich, E. Zeldov, D. Majer, T. W. Li, P. H. Kes, and M.}

Konczykowski, Phys. Rev. Lett. 76, 2555~1996!.

8_{S. Berry, M. Konczykowski, P. H. Kes, and E. Zeldov, Physica C} 282-287, 2259~1997!.

9_{S. Ooi, T. Tamegai, and T. Shibauchi, J. Low Temp. Phys. 105,}

1011~1996!.

10_{E. Zeldov, D. Majer, M. Konczykowski, A. I. Larkin, V. M.}

Vinokur, V. B. Geshkenbein, N. Chikumoto, and H. Shtrikman, Europhys. Lett. 30, 367~1995!.

11_{K. Kadowaki and T. Mochiku, Physica C 195, 127~1992!; N.}

Chikumoto, M. Konczykowski, N. Motohira, and A. P. Maloz-emoff, Phys. Rev. Lett. 69, 1260~1992!; S. Ooi, T. Tamegai, and T. Shibauchi, Physica C 259, 280~1996!; V. N. Kopylov, A. E. Koshelov, I. F. Schegolev, and T. G. Togonidze, ibid. 170, 291~1990!; A. K. Pradhan, B. K. Roul, S. B. Roy, P. Chaddah, D. Kanjilal, C. Chen, and B. M. Wanklyn, Solid State Commun.

101, 367 ~1997!; A. K. Pradhan, S. B. Roy, P. Chaddah, D.

Kanjilal, C. Chen, and B. M. Wanklyn, Supercond. Sci. Technol.

9, 743~1996!; N. Chikumoto, M. Konczykowski, N. Motohira,

K. Kishio, and K. Kitazawa, Physica C 185, 2201~1991!.

12_{For ramp rates fast enough to drive the system into the flux-flow}

state, any pinning effects should be gone. To be in the flux-flow regime, we would have to ramp about four orders of magnitude faster than our fastest ramp rate, which was 80 G/s.

13_{P. Guptasarma and D. G. Hinks~unpublished!.}

14_{N. Miyakawa, P. Guptasarma, J. F. Zasadzinski, D. G. Hinks, and}

K. E. Gray, Phys. Rev. Lett. 80, 157 ~1998!; Y. deWilde, N. Miyakawa, P. Guptasarma, M. Lavarone, L. Ozyuzer, J. F. Zasadzinski, P. Romano, D. G. Hinks, C. Kendziora, G. W. Crabtree, and K. E. Gray, ibid. 80, 153~1998!.

15_{N. Miyakawa and J. F. Zasadzinski~private communication!.} 16_{R. van Veen, A. H. Verbruggen, E. W. I. M. van der Drift, S.}

Radelaar, S. Anders, and H. M. Jaeger~unpublished!.

17_{C. Bernhard, C. Wenger, Ch. Niedermayer, D. M. Pooke, J. L.}

Tallon, Y. Kotaka, J. Shimoyama, K. Kishio, D. R. Noakes, C. E. Stronach, T. Sembiring, and E. J. Ansaldo, Phys. Rev. B 52, R7050~1995!.

18_{L. Pust, J. Kadlecova, M. Jirsa, and S. Durcok, J. Low Temp.}

Phys. 78, 179~1990!.

19_{M. Jirsa, L. Pust, H. G. Schnack, and R. Griessen, Physica C 207,}

85~1993!.

20_{C. P. Bean and J. D. Livingston, Phys. Rev. Lett. 12, 14~1964!.} 21_{E. Zeldov, A. I. Larkin, V. B. Geshkenbein, M. Konczykowski,}

D. Majer, B. Khaykovich, V. M. Vinokur, and H. Shtrikman, Phys. Rev. Lett. 73, 1428~1994!.

22_{We speculate that this is what happened in the data of Ref. 8: the}

field was ramped so slowly that the experimental time window captured the decay of the Bean profile for B.B2pk.

23_{R. Cubitt, E. M. Forgan, G. Yang, S. L. Lee, D. McK. Paul, H. A.}

Mook, M. Yethiraj, P. H. Kes, T. W. Li, A. A. Menovsky, Z. Tarnawski, and K. Mortensen, Nature ~London! 365, 407

~1993!.

Schauwecker, D. Zech, R. Cubitt, E. M. Forgan, P. H. Kes, T. W. Li, A. A. Menovsky, and Z. Tarnawski, Phys. Rev. Lett. 71, 3862~1993!.

25

V. M. Vinokur, P. H. Kes, and A. E. Koshelev, Physica C 168, 29

~1990!.

26_{J. C. Martinez, S. H. Brongersma, A. Koshelev, B. Ivlev, P. H.}

Kes, R. P. Griessen, D. G. de Groot, Z. Tarnavski, and A. A.

Menovsky, Phys. Rev. Lett. 69, 2276~1992!.

27_{T. W. Li, P. H. Kes, W. T. Fu, A. A. Menovsky, and J. J. M.}

Franse, Physica C 224, 110~1994!.

28

Y. Kotaka, T. Kimura, H. Ikuta, J. Shimoyama, K. Kitazawa, K. Yamafuji, K. Kishio, and D. Pooke, Physica C 235-240, 1529

~1994!.