PHYSICAL REVIEW B
VOLUME 57, NUMBER 17
1 MAY 1998-I
Flux noise in Bi2Sr2CaCu2O8 displaying the paramagnetic Meissner effect:
Evidence of spontaneous magnetic moments
J. Magnusson, P. Nordblad, and P. Svedlindh
Department of Materials Science, Uppsala University, Box 534, S-751 21 Uppsala, Sweden
~Received 28 April 1997; revised manuscript received 5 November 1997!
Flux noise measurements of a ceramic Bi2Sr2CaCu2O8 sample displaying the paramagnetic Meissner effect
~PME! have been performed in the temperature range 0.5,T/T c ,1. The frequency dependence of the zerofield noise power scales as S F ;1/f a with the exponent a monotonically decreasing from a '1.2 at the lowest
temperature to a '1.0 close to the superconducting transition temperature. The superposition of a weak
magnetic field suppresses the flux noise and the frequency dependence of the noise power changes in such a
way that the exponent a decreases in magnitude with increasing field. The spectral noise density is found to
relate to the low-field out-of-phase component of the ac susceptibility according to the fluctuation-dissipation
theorem. The results favor a description of the PME in terms of fluctuating spontaneous magnetic moments
with a typical magnitude of 108 m B . @S0163-1829~98!05917-7#
A few years after the discovery of high-T c superconductors, it was found that some sintered Bi-based hightemperature superconductors displayed a positive fieldcooled ~FC! magnetization when cooled in a sufficiently low
magnetic field.1 This astonishing phenomenon, which is now
known as the paramagnetic Meissner effect ~PME!, was later
confirmed by a number of groups.2–5 Detailed work presented in Ref. 2, showed that the PME exists in Bi-2212 and
Bi-2223 ceramics processed by different techniques. PME
samples have also been shown to display other anomalous
behaviors than a positive FC magnetization. In microwave
absorption measurements2 one finds a maximum in the absorption at zero external field for PME samples, while for a
sample not displaying the PME the microwave absorption
shows a minimum at zero field. In ac susceptibility experiments, the modulus of the second harmonic can be used to
monitor internal magnetic fields.4 In PME samples internal
fields were detected even after cooling through the superconducting transition temperature in zero field. This is in contrast to the behavior observed for non-PME samples, where
internal fields in zero external field are only observed when
the sample is in a remanent magnetic state. In zero-fieldcooled ~ZFC! magnetic relaxation measurements,5 the relaxation rate of PME samples at low fields displays a nearly
linear field dependence, while the field dependence of the
relaxation rate of a non-PME sample follows an H a dependence with a>2. Also, in a temperature interval close to T c ,
the low-field zero-field-cooled susceptibility, M ZFC /H, displays a diamagnetic susceptibility which increases with increasing external field,1,5 quite the opposite to what is observed for ordinary type-II superconductors.
To explain the PME, different theoretical models have
been proposed.6–8 Some models are based on the idea of
spontaneous magnetic moments developing below T c .
Kusmartsev6 introduced the concept of an orbital glass, a
state characterized by the existence of randomly oriented
spontaneous orbital currents. The spontaneous currents are
due to superconducting grains forming closed loops with so
called p junctions9 between different grains. In such a junc0163-1829/98/57~17!/10929~7!/$15.00
57
tion, the wave function of the Cooper pairs acquires an extra
phase shift of p in the tunneling process. If there is an odd
number of p junctions in a closed superconducting loop,
spontaneous currents may be generated, giving rise to an
orbital magnetic moment. One possible origin of p junctions
is that the superconducting order parameter belongs to the
pairing state of d-wave symmetry.7 This conjecture is supported by recent phase sensitive tests10,11 of the orderparameter symmetry in high-T c superconductors. In particular, in Ref. 10 it was shown using a scanning
superconducting quantum interference device ~SQUID! microscope that spontaneous orbital currents appear in
YBa2Cu3O7 rings containing three grain-boundary junctions,
generating half a magnetic-flux quantum, while no spontaneous currents are generated in rings containing two junctions.
In addition, new tetracrystal experiments indicate pure
d x 2 2y 2 pairing in tetragonal Tl2Ba2CuO6 superconductors.12
These results on other high-T c materials indirectly support
an explanation of the PME in granular Bi-Sr-Ca-Cu-O
samples in terms of localized magnetic moments having their
origin in a superconducting order parameter of d-wave symmetry.
Recently a positive magnetization has been observed in
Nb discs when cooled below T c in a small magnetic
field.13,14 This led Koshelev and Larkin8 to propose a model
where the positive magnetization is due to a Bean state with
compressed flux, which for instance can be stabilized as a
result of inhomogeneous cooling of the sample. A necessary
ingredient in this model is the formation of an essentially
flux-free region close to the circumference of the disc where
circular currents will flow. If this flux-free region grows, the
trapped flux is compressed, and the total magnetization of
the sample can become positive. Based on the fact that a
positive magnetization has been observed in Nb discs and
also in YBa2Cu3O7 single crystals,15 the authors of Ref. 14
concluded that the paramagnetism in any superconductor, including that observed in ceramic Bi-Sr-Ca-Cu-O samples, is
not likely to be related to d-wave superconductivity. This
conclusion has been criticized by Sigrist and Rice,16 who
10 929
© 1998 The American Physical Society
10 930
J. MAGNUSSON, P. NORDBLAD, AND P. SVEDLINDH
FIG. 1. Field-cooled susceptibility, M FC /H dc , versus temperature for applied fields in the range 0.01–0.3 Oe. T c 587 K. PME
sample.
pointed out that although Nb and Bi-Sr-Ca-Cu-O samples
have a positive low-field FC magnetization in common, there
are also clear differences in the behavior of the two systems,
suggesting that different mechanisms are needed in order to
explain the behavior of the respective materials. The same
conclusion is arrived at in Ref. 17, where a comparative
study of the PME in Bi-based high-T c and conventional Nb
superconductors is presented.
In this paper, we present results from flux noise measurements on a Bi-2212 sintered sample displaying the PME.
Comparing these with results from ac susceptibility measurements, it is found that the noise spectral density is related to
the out-of-phase component of the ac susceptibility according to the fluctuation-dissipation theorem. The suppression
of the noise power when superimposing a weak magnetic
field strongly favors a description of the observed noise in
terms of fluctuating spontaneous magnetic moments. Moreover, it allows one to estimate the average size of the magnetic moments to 10215 A m2. The observed behavior is in
accordance with the magnetic moments originating from p
loops carrying a magnetic flux of half a flux quantum.
A batch of granular Bi2Sr2CaCu2O8 was prepared by reaction sintering of Bi2O3, SrCO3, CaCO3, and CuO. The
material was ground and resintered several times at temperatures 790–860 °C followed by annealing, first at 850 °C in
an argon atmosphere mixed with 8% oxygen and then at
670 °C in 100% Ar. X-ray powder analysis indicated a single
phase material. The size of the sample was 2.532.1
31.7 mm3. Scanning electron microscopy studies of the
sample revealed a rather porous, flaky structure with typical
grain areas and thicknesses of 1 – 4 m m2 and 0.2–0.4 mm,
respectively. The critical temperature ~onset! determined
from ac susceptibility and dc magnetization was T c 587 K.
Low-field magnetization5 as well as ac susceptibility results
show that for the temperature range studied in the noise measurements, the granular sample can be described as a decoupled superconducting network. Moreover, from dc magnetization measurements it was confirmed that the sample
showed a positive field-cooled magnetization for low applied
magnetic fields ~cf. Fig. 1!.
A sintered sample of Bi2Sr2CaCu2O8 not displaying a
positive field-cooled magnetization was used as a reference
sample in the noise measurements. Apart from the final an-
57
nealing at 670 °C in 100% Ar, which was excluded for this
sample, the sample preparation followed the same steps as
described above for the PME sample. The critical temperature ~onset! determined from ac susceptibility was T c
566 K. Dc magnetization as well as ac susceptibility results
show that the coupling between superconducting grains is
stronger for this sample than for the PME sample. Henceforth, this sample will be referred to as the non-PME sample.
The flux-noise measurements were performed at different
temperatures in the interval 40–95 K ~55–70 K! for the PME
~non-PME! sample using a noncommercial SQUID
magnetometer.18 The sample space was magnetically
screened using m metal and superconducting shields. The
longitudinal background field at the sample was of the order
0.1 mOe. A small superconducting solenoid, working in persistent mode, was used to superimpose magnetic fields in the
range 10 mOe–30 Oe. When studying the flux noise in superimposed magnetic fields, the sample was first warmed to
a temperature above T c where the field was applied. The
sample was subsequently field cooled to the measurement
temperature where it was given ample time for the temperature to stabilize. The output from the SQUID electronics was
connected to an HP35670A dynamic signal analyzer. The
spectral noise density was measured in the frequency range
0.1–400 Hz. At each temperature, an averaged noise spectrum was obtained by averaging over 250–1000 independent
noise spectra. The background noise power in the SQUID
setup ~approximately 1029 F 20 /Hz at a frequency of 1 Hz!
was obtained above T c and all noise spectra presented below
have been corrected for this contribution to the measured
noise. In addition, the noise spectra were smoothed by averaging over logarithmically spaced frequency segments.
The FC susceptibility, M FC /H dc , of the PME sample is
plotted versus temperature for fields in the interval 10 mOe–
0.3 Oe in Fig. 1. The most striking feature of a PME sample
is clearly seen in this figure; at low enough applied fields ~for
this particular sample for fields H dc<0.1 Oe!, M FC /H dc becomes positive. At the lowest temperature and for the smallest field shown in this figure, the field-cooled susceptibility
takes a value of about 0.40 in SI units, which is considerably
larger than the maximum paramagnetic signal predicted by
the flux compression model.8 For fields larger than 0.1 Oe,
M FC /H dc becomes negative and the absolute value of the
diamagnetic susceptibility increases up to fields of the order
10 Oe. For even larger fields M FC /H dc decreases with increasing field, implying that the grains become penetrated by
vortices.
In Fig. 2~a!, the zero-field noise-power spectrum, S F ( f ),
of the PME sample is shown for different temperatures. The
noise increases when cooling below T c , attains its largest
magnitude at about 76 K and then decreases in magnitude on
further cooling of the sample. It is seen that the noise-power
spectra approximately follow a S F ;1/f a dependence at all
temperatures. The value of the exponent a gradually increases with decreasing temperature, from a '1.0 close to T c
to a '1.2 at the lowest temperature studied. In Fig. 2~b!, the
noise-power spectrum at T580 K is shown for different superimposed dc fields (H dc). It can be seen that the magnitude
of the noise, except for the lowest fields (H dc<0.01 Oe)
where no significant change can be resolved, gradually de-
57
FLUX NOISE IN Bi2Sr2CaCu2O8 DISPLAYING THE . . .
FIG. 2. Magnetic-flux noise power, S F ( f ), versus frequency for
the PME sample. The noise power given corresponds to the sample
magnetic-flux noise coupled to the SQUID loop. In ~a! the different
curves correspond to different temperatures at zero superimposed
dc field and in ~b! the dependence on dc field at a fixed temperature,
T580 K, is shown.
creases with increasing field. At a field of 0.1 Oe the noise
has been reduced by one order of magnitude.
The corresponding noise results for the non-PME sample
are shown in Fig. 3. The noise increases when cooling below
T c , attains its largest magnitude at about 62 K and then
decreases in magnitude on further cooling of the sample
@Fig. 3~a!#. Also for this sample, the noise-power spectra
follow a S F ;1/f a dependence, with a '0.9 close to T c and
increasing to a '1.1 at the lowest temperatures investigated.
In Fig. 3~b!, the noise-power spectrum at T562 K is shown
for different superimposed dc fields (H dc). It can be seen
that the magnitude of the noise at fields H dc.0.1 Oe increases with increasing field, while at fields lower than this
no significant change can be resolved. Similar field dependence of the flux noise was also observed at T559 K.
The behavior visualized in Fig. 2~b! is distinctly different
from the behavior of the non-PME sample. The behavior of
the latter sample is however similar to the behavior observed
for Y-Ba-Cu-O and Bi-Sr-Ca-Cu-O thin films and crystals.19
In those cases when the magnetic-flux noise originates from
a random motion of vortices, the noise increases with increasing magnitude of the superimposed dc field. The reason
for this is obvious, a higher field yields a higher density of
vortices and therefore a larger number of elementary fluctuators. Moreover, the field dependence of the noise is often
weakened below a sample-dependent effective field H eff .
Reported values for H eff typically fall in the range 0.1–10
Oe.19 One explanation of the weakened field dependence of
10 931
FIG. 3. Magnetic-flux noise power, S F ( f ), versus frequency for
the non-PME sample. The noise power given corresponds to the
sample magnetic-flux noise coupled to the SQUID loop. In ~a! the
different curves correspond to different temperatures at zero superimposed dc field and in ~b! the dependence on dc field at a fixed
temperature, T562 K, is shown.
the flux noise is that there exists an intrinsic density of vortices in the sample not generated by the applied field. Such a
vortex density can for instance be created by thermally generated vortex-antivortex pairs.19
The behavior visualized in Fig. 2~b! suggests a different
origin of the noise: thermal fluctuation of localized spontaneous magnetic moments that become polarized by the superimposed dc field. The implications of such a polarization
will be discussed below. Figure 2~b! also shows that the
S F ;1/f a dependence remains in superimposed dc fields.
The exponent a however decreases with increasing magnitude of the field, at a field of H dc50.3 Oe and at T580 K
a '0.8, which can be compared to the zero-field value at the
same temperature of a '1.0. Lastly, it is worth noting that
increasing the magnitude of the superimposed field above
H dc53 Oe, the flux noise is increased. It seems plausible
that at such large superimposed dc fields, the measured noise
becomes dominated by conventional random vortex motion.
To further investigate the zero-field and low-field dynamics of the PME sample, we have also performed ac susceptibility measurements. Figure 4 shows the temperature dependence of the out-of-phase component of the ac
susceptibility, x 9 , for zero superimposed field @Fig. 4~a!# and
for a superimposed field of H dc50.3 Oe @Fig. 4~b!#. The different curves correspond to different frequencies of the ac
field. The ac susceptibility was measured using an ac field
amplitude of H ac50.12 Oe. It should be noted that the ac
susceptibility is virtually independent of the ac field amplitude up to H ac50.12 Oe. A nonlinear response is only re-
10 932
J. MAGNUSSON, P. NORDBLAD, AND P. SVEDLINDH
FIG. 4. The out-of-phase component of the ac susceptibility, x 9 ,
versus temperature for ac field frequencies in the range 0.017–1700
Hz. H ac50.12 Oe. ~a! H dc50 and ~b! H dc50.3 Oe. PME sample.
solved at even larger amplitudes of H ac . The behavior of the
ac susceptibility is anomalous and does not reconcile with
any critical state model, nor can it be understood in terms of
viscous flux motion since the observed frequency dependence differs from that predicted in a flux-flow model. Comparing the results shown in Figs. 4~a! and 4~b!, it is seen that
x 9 is suppressed by the application of a weak dc field, and
that also the frequency dependence changes. In the zero-field
case, the out-of-phase component in the larger part of the
investigated temperature window increases with decreasing
frequency. The only exceptions to this behavior are at temperatures close to T c , where an almost frequency independent behavior is observed, and at the lowest temperatures
where the superconducting order parameter in different
grains starts to phase lock, thereby creating an intergranular
critical state. In a superimposed field of H dc50.3 Oe, the
out-of-phase component close to T c increases with increasing frequency, while at lower temperatures the reversed frequency dependence is observed.
As a comparison, ac susceptibility results for the nonPME sample are shown in Fig. 5. Figure 5~a! shows the
temperature dependence of the out-of-phase component of
x 9 for zero superimposed field. The different curves correspond to different ac field amplitudes. In contrast to the PME
sample, x 9 depends on H ac for all fields investigated,
2 mOe<H ac<1.2 Oe. Figure 5~b! shows the temperature dependence of the out-of-phase component of x 9 for different
superimposed fields. For H dc,0.3 Oe, x 9 is independent of
the superimposed field, while for larger superimposed fields,
the effect of the field is to shift the x 9 curve to lower temperatures. This is in sharp contrast to the behavior of the
PME sample, where the x 9 curves are strongly suppressed by
57
FIG. 5. The out-of-phase component of the ac susceptibility, x 9 ,
versus temperature for the non-PME sample. f 517 Hz. ~a! H dc
50, the different curves correspond to different ac field amplitudes
H ac ; 4 mOe, 12 mOe, 40 mOe, 0.12 Oe, 0.4 Oe, and 1.2 Oe. ~b!
H ac50.12 Oe, the different curves correspond to different superimposed fields H dc ; 10 mOe, 0.1 Oe, 1 Oe, and 5 Oe.
superimposed fields H dc<1 Oe.
The frequency and field dependence of the out-of-phase
component of the ac susceptibility for the PME sample resembles that found for the magnetic-flux noise. This observation suggests that the fluctuations responsible for the measured flux noise are equilibrium fluctuations which may be
linked to the measured dissipation through the fluctuationdissipation theorem ~FDT!. The FDT can for a magnetic system be written as
S F ~ f ,T,H dc! 5k B T
x 9 ~ f ,T,H dc!
,
pf
~1!
where x 9 ( f ,T,H dc) should be independent of the ac field
amplitude, i.e., the response to a small ac field should be
linear in field. To test if the FDT holds for the PME sample,
one can, for given values of f and H dc , plot x 9 and f S F /T
versus temperature. Such an analysis is visualized in Fig.
6~a!, where the results for three different frequencies and two
different superimposed dc fields are presented. The figure
clearly shows that the noise power and the out-of-phase component of the ac susceptibility results are linked through the
FDT. The agreement is less good at the lowest temperatures,
where intergranular vortices, created by the ac field, affect
the ac susceptibility. It is thus to be expected that x 9 should
be larger than f S F /T in the low-temperature region where
the superconducting order parameter in nearby grains starts
to phase lock.
57
FLUX NOISE IN Bi2Sr2CaCu2O8 DISPLAYING THE . . .
FIG. 6. ~a! The magnetic-flux noise f S F /T ~left-hand scale! at
0.2 Hz ~diamonds!, 2 Hz ~squares!, and 20 Hz ~circles! and out-ofphase component of the ac susceptibility x 9 for the frequencies,
from left to right, 0.17, 1.7, and 17 Hz ~right-hand scale, solid lines!
versus temperature. Filled symbols correspond to H dc50 Oe and
open symbols to H dc50.3 Oe. PME sample. ~b! The magnetic-flux
noise S F (H)/S F (0) ~left-hand scale, solid squares! and out-ofphase component of the ac susceptibility x 9 ~right-hand scale, open
circles! at f 517 Hz and T580 K versus the superimposed dc field
for the PME sample. For the ac susceptibility H ac50.12 Oe was
used. S F (H)/S F (0) at f 517 Hz and T562 K versus the superimposed dc field for the non-PME sample ~solid triangles! is also
plotted.
In Fig. 6~b!, the x 9 and S F (H)/S F (0) data for the PME
sample corresponding to T580 K and f 517 Hz are plotted
versus superimposed dc field. Again, it is seen that the two
types of data are correlated according to the prediction of the
FDT. Both x 9 and S F (H)/S F (0) are suppressed by the applied field in the same way. The decrease levels off at fields
H dc>0.3 Oe, and at a field of H dc'0.5 Oe both types of data
have decreased approximately by a factor of 30 as compared
to the magnitude at H dc50. The field dependence of the flux
noise of the non-PME sample at T562 K and f 517 Hz has
been included for a comparison. For this sample, it is
not possible to find a good correlation between the
S F (H)/S F (0) and x 9 data.
The remarkable dc field dependence of the noise and the
out-of-phase component of the ac susceptibility for the PME
sample can be understood in a model where noninteracting
spontaneous magnetic moments exist and where each magnetic moment m can point in one of two possible directions
corresponding to two energy minima of the potential energy
of the magnetic moment. The potential energy of a magnetic
10 933
moment consists of two terms; one is related to the energy
barrier U 0 separating the two energy minima and the other
corresponds to the polarization energy due to the applied
field U H 52 m 0 mH dccos(u), where u is the angle between
the applied field and the direction of the magnetic moment.
Since the sample we are trying to model is a granular material, one should expect that there exists a distribution of energy barriers as well as a random orientation of the magnetic
moments with respect to the applied field. The described
model resembles that used to describe a system consisting of
noninteracting uniaxial magnetic nanoparticles.20
There are two effects that influence the magnitude of the
flux noise and the out-of-phase component of the ac susceptibility. The first effect always suppresses x 9 and relates to
the equilibrium population of the two energy minima which
can be expressed by Boltzmann factors normalized by the
partition function. At low fields, the expressions for the equilibrium populations of the two energy minima give the magnetization as M ;tanh@m0mHdc^ cos(u)&/kBT#, with ^ cos(u)&
51/3 for a random orientation of the magnetic moments with
respect to the applied field. Assuming, for simplicity, that the
ac field amplitude is much smaller than the dc field, the
equilibrium ac susceptibility can be calculated as x 0
5( ] M / ] H) H dc;cosh22(m0mHdc/3k B T). To obtain the dynamic susceptibility, or more specifically the out-of-phase
component of the ac susceptibility, it is necessary to introduce the distribution of relaxation times, g( t ), which will
appear as a natural consequence of the above mentioned distribution of energy barriers. Provided that g( t ) is a slowly
varying function of ln(t), it is possible to show that x 9 ( v )
} x 0 g( t ) with t '1/v . 20
The second effect is related to the change of the distribution of relaxation times due to the superimposed dc field. The
transition rate from one energy minimum to the other depends on the energy barrier seen by the magnetic moment
21
according to t 21
1,2 ' t 0 exp(2U0 /kBT), where U 0 is the zerofield energy barrier and t 0 is a microscopic attempt time.
Applying a dc field adds the energy term U H to the potential
energy of the magnetic moment. As a result of this extra
contribution to the potential energy, the energy barriers seen
by the magnetic moment change. For small fields, the main
effect of the extra term is to change the transition
rates between the two energy minima according to
21
21
t 21
and t 21
1 ' t 0 exp@2(U02uUHu)/kBT#
2 ' t 0 exp@2(U0
1uUHu)/kBT#, where t 1 ( t 2 ) corresponds to the field pointing
in a direction being in the same ~opposite! hemisphere as the
direction of the magnetic moment. The relaxation time, that
is the time it will take the magnetic moments to reach the
new equilibrium state defined by the applied dc field, is defined by 1/t 51/t 1 11/t 2 . The effect of the field will thus be
to decrease the relaxation times, implying that the distribution of relaxation times, at a given temperature, will be
shifted towards shorter time scales. This effect can suppress
or increase x 9 depending on the shape of g( t ).
Both effects described above will act as to change the
magnitude of the flux noise and the out-of-phase component
of the ac susceptibility. At the lowest fields, however, a plausible assumption is that g(1/v ) will not change significantly
due to the applied dc field. This does not imply that the
relaxation times are unaffected by the dc field, but merely
10 934
J. MAGNUSSON, P. NORDBLAD, AND P. SVEDLINDH
that the spectral weight at the relaxation time defined by t
'1/v does not change significantly due to the applied dc
field. Thus by fitting the expression for x 0 to the experimental results shown in Fig. 6~b!, with m as the only fitting
parameter, it is possible to estimate the typical magnitude of
the magnetic moments being responsible for the measured
dissipation and magnetic-flux noise. This analysis yields
a magnetic moment of magnitude m'6310216 A m2
~'10 8mB!.
It is interesting to compare our estimate of the magnitude
of the fluctuating magnetic moments with the prediction
from the p loop model. We will begin with identifying the
typical size for a superconducting loop resulting in spontaneous magnetic moments. Recent high-resolution transmission electron microscopy ~TEM! studies on a large number
of Bi-2212 samples, some of which do show and some of
which do not show the PME, indicate an important difference in the microstructure of the two types of samples.21 The
samples which do not display the PME are monodomain
along the a and b axes on length scales below 1 mm. In
contrast, in all PME samples many sharp boundaries between
grains with different orientation of the a and b axes were
found. These boundaries may provide the p junctions necessary for creating spontaneous orbital currents. The observed
differences on a length scale below 1 mm are also consistent
with the observation of the PME in Bi-2212 powders consisting of grains with a typical size of 1 mm.21 These results
indicate that it is possible to associate the p loops, if they
exist in granular Bi-2212 samples, with an intragranular
length scale of the order 1 mm.
Having established the relevant length scale of p loops, it
is possible to estimate the magnitude of the magnetic moments expected in a p loop model. Since each p loop carrying a spontaneous orbital current is associated with a magnetic flux approximately equal to F 0 /2, one may connect this
with a magnetic moment F 0 t/2m 0 , where t is the average
thickness of the platelets. This expression gives in our case a
magnetic moment of size 2 – 4310216 A m2. However, as
1
P. Svedlindh, K. Niskanen, P. Norling, P. Nordblad, L. Lundgren,
B. Lönnberg, and T. Lundström, Physica C 162-164, 1365
~1989!.
2
W. Braunisch, N. Knauf, V. Kataev, S. Neuhausen, A. Grütz, B.
Roden, D. Khomskii, and D. Wohlleben, Phys. Rev. Lett. 68,
1908 ~1992!; W. Braunisch, N. Knauf, G. Bauer, A. Kock, A.
Becker, B. Freitag, A. Grütz, V. Kataev, S. Neuhausen, B.
Roden, D. Khomskii, D. Wohlleben, J. Bock, and J. Preisler,
Phys. Rev. B 48, 4030 ~1993!.
3
B. Schliepe, M. Stindtmann, I. Nikolic, and K. Baberschke, Phys.
Rev. B 47, 8331 ~1993!.
4
Ch. Heinzel, Th. Theilig, and P. Ziemann, Phys. Rev. B 48, 3445
~1993!.
5
J. Magnusson, J.-O. Andersson, M. Björnander, P. Nordblad, and
P. Svedlindh, Phys. Rev. B 51, 12 776 ~1995!; J. Magnusson, M.
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6
F. V. Kusmartsev, Phys. Rev. Lett. 69, 2268 ~1992!.
7
M. Sigrist and T. M. Rice, J. Phys. Soc. Jpn. 61, 4283 ~1992!; J.
57
argued in Ref. 23, such a calculation ignores the current that
will flow close to the edge of the platelet due to the return
flux. Even though a general expression for the magnetic moment of a superconducting platelet of radius R containing
one flux line close to its center is not available, one can
estimate, using what is known for the case of an infinite
platelet of thickness t, the extra contribution to the magnetic
moment arising from the induced current to be of the order
F 0 R/ m 0 . 23 Using this expression and R'1 m m, the magnetic moment is calculated to be 2310215 A m2, which is in
fair agreement with the magnetic moment estimated from the
ac susceptibility and magnetic-flux noise measurements.
Moreover, to be able to observe spontaneously generated orbital currents ~magnetic moments!, the condition
2 p LI c /F 0 .1, where L is the self-inductance of the p loop
and I c is the junction critical current, need to be fulfilled.
This implies a junction critical current density as high as
109 – 1010 A/m2, which will require a highly transparent
grain
boundary
of
the
superconductor–metal–
superconductor type junction.22 It seems plausible that the
sharp boundaries between grains with different orientation of
the a and b axes seen by TEM ~Ref. 21! can provide such
junctions.
In conclusion, we have shown that the magnetic-flux
noise and the out-of-phase component of the ac susceptibility
for a PME sample are linked via the fluctuation-dissipation
theorem. This implies that the measured noise is due to equilibrium magnetic fluctuations. The suppression of the
magnetic-flux noise with the application of a weak magnetic
field strongly favors a description in terms of large localized
magnetic moments, which become polarized by the superimposed field. The estimated magnitude of the magnetic moments is in accordance with the expectations of a model
based on p loops, where each loop can be associated with a
magnetic flux approximately equal to F 0 /2.
Financial support from the Swedish Natural Science Research Council ~NFR! and the Carl Trygger foundation is
acknowledged.
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~1995!.
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