PHYSICAL REVIEW B 84, 214529 (2011)

Visualizing the ac magnetic susceptibility of superconducting films via magneto-optical imaging

M. Motta,1 F. Colauto,1 R. Zadorosny,1,* T. H. Johansen,2,3 R. B. Dinner,4 M. G. Blamire,4 G. W. Ataklti,5

V. V. Moshchalkov,5 A. V. Silhanek,5,6 and W. A. Ortiz1,3

1Departamento de Fı́sica, Universidade Federal de São Carlos, 13565-905 São Carlos, SP, Brazil
2Department of Physics, University of Oslo, POB 1048, Blindern, NO-0316 Oslo, Norway

3Centre for Advanced Study, Norwegian Academy of Science and Letters, NO-0271 Oslo, Norway
4Department of Materials Science, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, UK

5INPAC - Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and Magnetism Group,
Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium

6Département de Physique, Université de Liège, B-4000 Sart Tilman, Belgium
(Received 8 February 2011; revised manuscript received 4 October 2011; published 30 December 2011)

We have established a link between the global ac response and the local flux distribution of superconducting
films by combining magnetic ac susceptibility, dc magnetization, and magneto-optical measurements. The
investigated samples are three Nb films: a plain specimen, used as a reference sample, and other two films
patterned with square arrays of antidots. At low temperatures and small ac amplitudes of the excitation field,
the Meissner screening prevents penetration of flux into the sample. Above a certain ac drive threshold, flux
avalanches are triggered during the first cycle of the ac excitation. The subsequent periodic removal, inversion,
and rise of flux occurs essentially through the already-created dendrites, giving rise to an ac susceptibility signal
weakly dependent on the applied field. The intradendrite flux oscillation is followed, at higher values of the
excitation field, by a more drastic process consisting of creation of new dendrites and antidendrites. In this more
invasive regime, the ac susceptibility shows a clear field dependence. At higher temperatures a smooth penetration
occurs, and the flux profile is characteristic of a critical state. We have also shown that the regime dominated by
vortex avalanches can be reliably identified by ac susceptibility measurements.

DOI: 10.1103/PhysRevB.84.214529 PACS number(s): 74.78.Na, 74.78.Fk, 74.25.Dw, 74.25.Op

I. INTRODUCTION

Superconductivity is a state of matter characterized by a
diamagnetic response caused by the exclusion of magnetic
flux from its interior. In type II superconductors, the resulting
field-dependent magnetic moment can be linked to the internal
distribution of the quantum units of flux, so-called vortices. In-
deed, a picture of the flux distribution at microscopic scales can
be indirectly achieved from integrated response techniques, via
the assumption of certain predetermined models. Arguably, the
most successful and popular mapping between the microscopic
and macroscopic superconducting worlds is the Bean critical
state model.1,2 This model allows one to estimate, from
magnetization and susceptibility measurements, the maximum
current a superconductor can bear before dissipating, i.e., its
critical current. The basic assumptions validating the Bean
model are the existence of a continuous flux distribution
throughout the specimen and a field-independent critical
current. Clearly, such restrictive conditions do not account
for all possible scenarios of flux distribution. Attempts for
adjusting to more realistic situations included the decrement
of critical current with magnetic field,3 the presence of edge
barriers,4,5 and the existence of a finite lower critical field.6,7 In
most cases, these improvements result in marginal corrections
to the simpler Bean approximation.

A situation in which the Bean critical state model fails
dramatically occurs when sudden flux bursts develop as a
consequence of a thermomagnetic instability.8 Under these
circumstances, any estimation of the critical current based on
global response techniques is futile and local magnetic probes
become imperative. An unambiguous identification of the field
range dominated by avalanches can be obtained through the

field dependence of the magnetization, which exhibits clear
jumps within the instability region.9–11 However, signatures
of the avalanche regime in the ac susceptibility are not easy
to recognize, most specially for values of the ac drive h

sufficiently small to avoid disturbance on the flux distribution
which one intends to probe. For larger values of h, however,
both components of the ac susceptibility, χ = χ ′ + jχ ′′, are
reentrant in temperature and applied field; i.e., shielding
becomes stronger and dissipation decreases upon increase
of either of those two variables. This reentrant behavior has
also been observed earlier in Pb films12,13 and ascribed to the
occurrence of flux avalanches. The present work reveals the
cause for such change of the diamagnetic response through
investigations of the early stages of flux penetration in Nb
films. In our approach we performed magnetic measurements
employing global techniques—namely, susceptibility and
magnetization—as well as magneto-optical imaging (MOI).
Our experiments revealed that the first avalanches, while
serving as a track for flux entrance, do also guide flux exit, thus
suggesting why the initial ac response of the superconducting
film is nearly constant for variations of the applied field H or
temperature T . Upon increase on H or T , the ac response reen-
ters, i.e., becomes more diamagnetic as the avalanches cease.

II. SAMPLES AND EXPERIMENTAL TECHNIQUES

The samples investigated consist of Nb films of approx-
imately rectangular shape, with thickness d = 50 nm. Their
lateral dimensions appear in Table I. Two of them have a
square array of antidots (ADs) fabricated by electron-beam
lithography. The Nb was deposited via dc sputtering on top

214529-11098-0121/2011/84(21)/214529(7) ©2011 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.84.214529


M. MOTTA et al. PHYSICAL REVIEW B 84, 214529 (2011)

TABLE I. Superconducting critical temperature Tc, coherence
length ξGL(0), penetration depth λGL(0), and lateral dimensions l and
w for the three samples investigated.

Sample Tc (K) ξGL(0) (nm) λGL(0) (nm) l (mm) w (mm)

Pristine 7.91 9.4 168.0 3.0 2.5
AD04 7.40 9.1 170.0 2.7 2.6
AD08 6.42 9.1 173.0 2.7 2.5

of a SiO2 insulating substrate. Our preliminary investigations
included a variety of similar samples, all of them leading to
comparable results. The present report, however, is restricted
to results on only three samples, one of each type. Thus,
the sample investigated through magnetic measurements
is the very same which was imaged using the MO technique.
The patterned samples—named AD04 and AD08—have
antidots with sizes 0.4 μm and 0.8 μm, respectively. The
third specimen—the reference sample—is a plain film of Nb.
The period of the patterns is a = 1.5 μm, which corresponds
to a commensurability field H1 = �0

a2 = 9.2 Oe at which
the densities of vortices and antidots match each other.14

Using the expression for the upper critical field and for the
dirty limit,15 the zero-temperature superconducting coherence
length, ξGL(0), and the penetration depth, λGL(0), were
estimated. The obtained values are shown in Table I, as well
as the superconducting transition temperature at zero field, Tc.
Noticeably, a drop in Tc results from the insertion of antidots.

Measurements were made for temperatures ranging from
2 K up to Tc, and the ac sinusoidal drive was simulated, in MOI
experiments, by cycling the dc field in complete loops made
up of 0.1 Oe steps. Ac susceptibility and dc magnetization
measurements were carried out with commercial Quantum
Design equipment (MPMS and PPMS). The susceptibility
measurements were recorded using a fixed frequency of 1 kHz,
for values of the excitation field within the ±15 Oe interval.
Prior to fixing the value of the excitation frequency, we verified
that the susceptibility response is independent of frequency
within 5 orders of magnitude, up to values above 1 kHz.
In all cases, the excitation field h and the external magnetic
field H were applied perpendicular to the film plane. The
magneto-optical imaging technique employed relies on the
occurrence of the Faraday effect16 on a magnetic indicator
film placed on top of the superconducting specimen. The
indicators used in the present work are Bi-substituted yttrium
iron garnet films (Bi:YIG) with in-plane magnetization. Data

for the so-called Cole-Cole plots of the susceptibility, i.e., its
imaginary part χ ′′ as a function of the real part χ ′, were taken
isothermally, varying the ac drive. Data normalization of the
susceptibility was performed using the reference value χ0 of
its real component, which represents the Meissner plateau,
measured in the PPMS at T = 2 K with a drive amplitude h =
20 mOe, in the absence of magnetic field.17

III. RESULTS AND DISCUSSION

It is well known that thin films of most superconducting
materials exhibit flux avalanches within specific intervals
of temperature and applied field.9–11,18–25 Such behavior is
intimately correlated to thermomagnetic instabilities expe-
rienced by the material when heat generated by a sudden
displacement of vortices cannot be dissipated, creating thus
an increase in the local temperature. This warmer region, in
turn, has its pinning capability reduced, being thus likely to
host even more vortex motion, reinforcing the process.26,27

There is a threshold temperature Tth above which no avalanches
occur and a rather smooth penetration of the magnetic flux is
observed. It has been shown that Tth can be substantially in-
creased if antidots are introduced into the specimen.9,12,13,28,29

In this case, the most relevant difference, compared to plain
films, is that vortices are guided preferentially along the rows
of ADs.13,29 This aspect will be further discussed later in this
paper. Above Tth, flux invades the sample gradually, and the
system appears to obey a critical-state regime. Assuming the
simplest of such critical-state regimes, i.e., the Bean model,
one can obtain the superconducting critical current density
Jc by measuring the field h at which the magnetic flux
front reaches a characteristic geometry-dependent penetration
distance.30,31 If one considers a thin disk of thickness d and
diameter 2R � d, the two-dimensional effective penetration
depth in the perpendicular geometry is � = 2λ2

d
. For � � 2R,

the field amplitude at which χ ′′ peaks, hp, corresponds to
72% of the full penetration field and can be related to the
critical current density by the expression Jc = 1.03hp

d
.32 On the

other hand, an approximate expression for Jc has been recently
developed by Chen and co-authors33 for films of rectangular
shape. We have employed Eq. (20) of Ref. 33 to calculate Jc

of our rectangular samples:

Jc = 4.85ϒ0(γ )

(3 − γ −1)w
hp, (1)

where γ = l/w, τ = d/w, and

ϒ0(γ ) = arctan(1 − 0.7223γ −0.954 + 0.3522γ −2.57 − 0.141γ −3.66)

τ
. (2)

Figure 1 compiles results obtained from isothermal ac
susceptibility measurements on the three films, taken with the
PPMS in zero applied field,17 using the excitation amplitude
h � 15 Oe as the external variable. Panel (a) shows Jc( T

Tc
)

obtained as discussed in the previous paragraph, with hp

determined from the peak in χ ′′(h) curves. We have used
the reduced variable t = T

Tc
to unify the superconducting

domain of all studied samples. Curves for the plain film
and sample AD04 have two distinct regions: At higher
temperatures, Jc(T ) is smooth, with upward curvature; below

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VISUALIZING THE AC MAGNETIC SUSCEPTIBILITY OF . . . PHYSICAL REVIEW B 84, 214529 (2011)

(a)

(b) (c)

(d)

FIG. 1. (Color online) Panel (a): Temperature dependence of the
critical current in the absence of a dc field (see main text and Ref. 17),
as determined from the peak on the imaginary part of the susceptibility
measured as a function of the excitation amplitude. Panels (b)–
(d): Cole-Cole plots, at different temperatures, for the studied
samples: pristine (b), AD04 (c), and AD08 (d). The theoretical plot
for a thin circular disk is also shown. Smoothness or noisiness of
the Cole-Cole plots emphasizes different regimes (critical state or
avalanches) of flux penetration.

a certain limiting temperature, however, it deviates from this
canonical behavior, a change that is intimately connected to the
appearance of avalanches in both specimens, as explained later
in this paper. The equivalent curve for sample AD08 is quite
different: A downward curvature is maintained throughout
the whole interval, as a consequence of avalanches being
present within the entire temperature domain. As a matter

of fact, values of Jc shown for temperatures below the
avalanche threshold should not be interpreted as representative
of the critical current density, since they were obtained under
the erroneous assumption that the system is in a critical-state
regime. In the avalanche regime, the actual critical current
is a local, extremely inhomogeneous variable, which cannot
be properly described by a unique global value. Nonetheless,
displaying all points in a single picture stresses the idea that
a frontier between both regimes can be reliably obtained
from ac susceptibility measurements. It should be noticed
that the curves in Fig. 1(a) confirm that the inclusion of ADs
of moderate size promotes an increase in Jc

34 and Tth.9,13

Comparing the high-T portions of Jc(T ) for the plain film and
sample AD04, one clearly sees this improvement, as well as
the relative enlargement of the area where avalanches occur.
For sample AD08, however, this comparison does not apply,
since avalanches take place in the whole temperature window.

A second important signature of the occurrence of
avalanches that can be drawn from susceptibility measure-
ments is depicted in panels (b) through (d) of Fig. 1, which
show Cole-Cole plots of the susceptibility. Displaying data
in such a manner can be quite advantageous to emphasize
the occurrence of different regimes of vortex matter. For
example, all Cole-Cole plots for a system obeying a critical
state dynamics should collapse into one single universal
plot, irrespective of the values of applied field, temperature,
frequency, and amplitude of the ac drive. Panel (b) shows
results around the peak for the pristine film, taken at three
different values of T : For temperatures below the threshold
limit for avalanches (t = 0.38 and 0.57), χ ′′(χ ′) is very noisy,
contrasting with the behavior observed at higher temperatures
(t = 0.69), for which the curves are noticeably smooth. A
similar set of curves is shown in panel (c) for sample AD04,
which exhibits avalanches for t = 0.54 and t = 0.77, but not
for t = 0.85. Complete Cole-Cole plots for sample AD08 are
shown in panel (d) for t = 0.39, t = 0.62, and t = 0.78. The
inset features the central portion of the curves, emphasizing
the noisy behavior which, for this sample, is present over the
entire range of temperatures. It should be noticed that, at the
left lower part of Fig. 1(d), χ ′′ goes to zero while χ ′ is not −1
for larger values of T. The reason for this behavior is that the
initial screening on curves χ ′(h) remains flat but imperfect,
while there is essentially no dissipation [χ ′′(h) ∼ 0]. This
means that some flux enters but remains pinned. Since this
behavior is also exhibited by sample AD08 and the pristine
film, one is led to associate it with the intrinsic pinning of Nb.
We will return to this point further ahead.

It is also worth comparing the experimental results with
those expected theoretically. Chen et al.33 have shown that the
ac susceptibility of rectangular films can be adequately treated
using the same expression derived by Clem and Sanchez for an
infinitely thin circular disk.32 This result, also included in all
Cole-Cole plots of Fig. 1, indicates that, at all temperatures, the
plain-film response [panel (b)] is different from that obtained
for the Bean approximation, a result that is possibly due to the
fact that Jc is not field independent, as assumed in that simple
model. As a matter of fact, the Bean model predicts a peak at
χ ′

max = 0.38, whereas the assumption of a field dependence on
Jc shifts the peak to the right and up,35 just as observed for the
pristine film. On the other hand, the Cole-Cole plots for the

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M. MOTTA et al. PHYSICAL REVIEW B 84, 214529 (2011)

FIG. 2. (Color online) MO images taken at 3 K. White indicates
positive flux, dark negative, medium gray stands for zero flux
(screened). Panels (a) to (e) are for sample AD04: After zero-field
cooling, field increased to 2 Oe (a), decreased to zero (b) and −2
Oe (c), and increased back to zero (d) and 2 Oe (e). Same protocol
followed for sample AD08, panels (f) to (j), with field extremes of
±1 Oe. Bottom panel: Sketch of time evolution of the applied field
during data collection.

patterned samples deviate from the one predicted by Bean’s
model in a nonsystematic manner, so that the difference cannot
be ascribed to a specific mechanism—e.g., the introduction of
creep or of a field-dependent Jc.35–37

We turn now the discussion to the images of profiles
of flux penetration and exit at relatively small fields. Such
images where taken after cyclic field excursions, as in the case

FIG. 3. MO images of the pristine film, taken at 3 K, following
the same protocol depicted in Fig. 2: After zero-field cooling, field
was cycled between ±4 Oe. Panel (a) was taken at 0 Oe after the first
half cycle of the field; panel (b) is for −1.5 Oe.

of those performed during susceptibility measurements. To
emulate the magnetic history imprinted by the ac excitation,
the external field was ramped up and down in steps of 0.1
Oe. However, given the time needed to acquire images, on the
order of 100 ms, the cycles were not sinusoidal, but steplike, as
roughly indicated at the bottom of Fig. 2. The similarity among
this procedure and that taking place during a sinusoidal cycle
in a susceptibility measurement is ensured by the frequency
independence of the ac measurements. Figures 2 and 3 present
some selected MO images for the three samples studied here.
All pictures were taken for T = 3 K. White regions correspond
to positive penetrated flux, dark areas represent negative flux
(antiflux), while medium gray stands for unpenetrated portions
of the sample (zero flux). Panels (a) to (e) in Fig. 2 are for
sample AD04: After a zero-field cooling procedure, the field
was increased to 2 Oe (a), decreased back to zero (b) and to
−2 Oe (c), and then increased again to zero (d) and finally
to 2 Oe (e). All steps were performed with the ramp rate
of 20 Oe/s. We find that near 1.4 Oe, abrupt avalanches
suddenly invade the sample from the edge, in the form of
finger-type dendrites, clearly guided by the rows of ADs (a).
Upon decrease of the field, reversed flux penetrates the sample
through the same tracks [(b) and (c)], transforming them
progressively into antidendrites. When the field is increased
again [(d) and (e)], the tracks are penetrated once more by
positive flux, so that the original dendrites are gradually
restored. The same behavior is seen in panels (f) to (j) for
sample AD08, for which the field protocol is just the same as
for AD04, with field extremes of ±1 Oe. This feature is also
presented by the pristine sample, as exemplified in panels (a)
and (b) of Fig. 3, taken respectively at fields zero and −1.5
Oe, after a field excursion to 4 Oe.

It is worth mentioning that all samples studied here
exhibit dendritic penetration at certain values of field and
temperature. However, for experiments repeated under the
same conditions, flux dendrites follow different tracks during
each run. This further emphasizes that the onset of flux
avalanches is governed by an instability condition. The lattice
of antidots is nevertheless causing some degree of guidance
for the full avalanche, in contrast to the dendritic penetration
in unpatterned samples.

We have also used magnetic measurements and MO
imaging to investigate the reentrant behavior of suscepti-
bility curves. The inset of Fig. 4(a) shows the temperature

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FIG. 4. (Color online) (a): Real and imaginary parts of
χ as a function of the ac amplitude, taken at T = 3, 5,
and 6.5 K. The inset shows the temperature dependence
of the same quantities, measured with h = 1, 2.5, and 3.8
Oe. Stars represent values of both components, at 3, 5,
and 6.5 K, calculated from magnetization loops (see text).
(b): Similar to inset in (a) for the pristine film.

dependence of χ ′ and χ ′′, measured with ac amplitudes of
1.0, 2.5, and 3.8 Oe, for sample AD04. For the lower value
of h, one sees the ordinary ac response of a superconductor;
at larger amplitudes, however, the reentrant behavior appears,
as already seen earlier in Pb films:12 Upon increase of the
temperature, the real part first decreases, to a more diamagnetic
level, and then increases toward zero at the transition. The
imaginary part is also odd, starting at a relevant dissipation
level and then decreasing before peaking as the transition
is approached. One can also follow the evolution of both
susceptibility components with the excitation field at fixed
temperatures. The main graph of Fig. 4(a) shows isothermal
measurements at 3, 5, and 6.5 K. It is rather intriguing that
the loss of diamagnetism starts earlier for 3 K than for 5 K,
a feature that is also matched by the peaks on χ ′′. Further
increasing T , however, restores the ordinary behavior; i.e.,
the transition is broadened and starts at lower amplitudes
h. The vertical dashed lines on the main panel are guiding
lines to connect both experiments: For h = 1 Oe, the real
part χ ′ increases monotonically as the temperature changes
from 3 to 5, and then to 6.5 K. The imaginary part is also

monotonous: negligible for 3 and 5 K, and nonzero at 6.5 K.
On the other hand, for h = 2.5 Oe both components have an
initial decrease as T is switched from 3 to 5 K, followed by
an increase, for T growing from 5 to 6.5 K. This feature is
even more pronounced at large h. One can also notice that,
for small values of h, χ ′(h) ∼ −1 at low temperatures (e.g.,
3 K), but is less negative for larger values of T , whereas the
corresponding χ ′′(h) ∼ 0 in all temperatures. As discussed
earlier, this is due to efficient pinning: Flux enters the sample
but is prevented from moving. For this reason, the left-lower
parts of the Cole-Cole plots for all samples studied here do not
collapse, as seen, for example, in Fig. 1(d) for sample AD08.
This reentrant behavior, discussed here for sample AD04, is
also presented by the pristine film, as illustrated in Fig. 4(b)
for the temperature dependence of χ ′ and χ ′′ for a variety of
values of the excitation amplitude. However, as sample AD08
exhibits avalanches in the whole temperature interval, there is
no transition between different dynamic regimes, and therefore
no reentrance could occur, as is actually the case.

Invoking frequency independence of the susceptibility, one
can emulate an ac measurement by cycling the dc field and
capturing images of the penetration profile of the sample
at adequate values of H . Panels (a)–(c) on Fig. 5 show
hysteresis loops for sample AD04 at temperatures 3 K, 5 K, and
6.5 K, respectively. At 3 K one sees a hysteretic loop whose
nonvanishing area is due to the viscous motion of entering and
exiting vortices within the AD-guided dendrites [panel (d)]. As
shown on panel (c), at 6.5 K the loop is wide open, as could be
expected for temperatures approaching Tc. The corresponding
image is shown on panel (f), with a critical-state-like envelope
and a certain “microtexture.” Flux entrance is not abrupt,
as in an avalanche, exhibiting a filamentary though smooth
penetration. Mostly interesting, however, is the fact that the
magnetic response at 5 K is nonhysteretic,38 which means

FIG. 5. (Color online) Panels (a)–(c): Hysteresis loops for sample
AD04 at temperatures 3, 5, and 6.5 K. The area of the loop, which
is nonzero at 3 K, closes down at 5 K and reopens at 6.5 K. Panels
(d)–(f) are the corresponding images at the maximum field (2.5 Oe),
taken immediately after the virgin curve.

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M. MOTTA et al. PHYSICAL REVIEW B 84, 214529 (2011)

that, upon increase of the temperature, the loop first closes
down and then opens up again. The MO image, represented
on panel (e), shows quite clearly that the heart of the sample
is not penetrated by magnetic flux at 5 K and 2.5 Oe. We
have also compared hysteresis loops and MO images for the
pristine sample, obtaining similar results. The collapse of the
loop, however, does not occur for sample AD08 which, as
already discussed, always exhibits avalanches. We take this as
an additional evidence that the closing down of the loop is a
feature intimately related to the suppression of avalanches in
samples AD04 and pristine.

One further evidence that ac measurements can be emulated
using dc magnetization loops arises from the argument that χ ′
is the average slope of the magnetization loop (throughout
one full cycle), whereas χ ′′ is related to the energy losses per
sample volume per cycle by

χ ′′ =
∮
M dH

πH 2
m

= A

πH 2
m

, (3)

where A is the area of the magnetization hysteresis loop, which
extends from −Hm to +Hm.32,39 Employing this reasoning, we
calculated both components of χ using data from the dc loops.
As an illustration we have included, in the inset of Fig. 4(a),
stars representing such results as obtained from the ±2.5 Oe
loops measured at 3, 5, and 6.5 K. A similar procedure for
the pristine film, taken with Hm = 5 Oe, leads to the set of
stars on Fig. 4(b). One can thus conclude that the reentrant
behavior of χ ′—i.e., a reinforced diamagnetism—is related to
the temperature limit for the occurrence of avalanches.10,11

As a final remark, we comment on the filamentary structure
of the penetrated flux on sample AD04 at high temperatures.
Panel (f) in Fig. 5 shows this clearly: While the penetration
front has the typical format of a critical-state regime, it is in fact
an envelope for a filamentary finger-type structure. Noticeably,
however, no avalanches take place at that temperature, and the
filamentary inner structure develops smoothly. From time to
time, the smooth penetration is perturbed by small amounts
of flux entering the film. Minor fluctuations seen on M(H )
at 6.5 K [Fig. 5(c)] are the corresponding signatures of those
tiny perturbations. This interesting feature, which is in straight

connection with the existence of the array of ADs, can be
compared with another occurrence of a filamentary structure,
observed by Welling et al. in a YBCO film with an array
of ADs.40 Flux penetration in the form of thin filaments
was also observed in YBCO films deposited on vicinal cut
substrates.41–44

IV. CONCLUSIONS

Combining magneto-optical visualization of penetrated
flux with magnetic ac susceptibility and dc magnetization
measurements, we have investigated the early stages of flux
penetration on Nb films with and without arrays of antidots.
Our results show that ac susceptibility measurements can be
used to detect vortex avalanches, either constructing Jc(T )
curves or monitoring the occurrence of a noisy behavior in
Cole-Cole plots. We have also shown that, in the low-field
regime, the roots of most dendrites are reused during the
process of entrance and exit of flux, although some new
dendrites and antidendrites might also be created at different
points along the sample edges. From hysteresis loops measured
at different temperatures, we were able to calculate both
components of the ac susceptibility. MO images taken at
several points of those loops enabled us to establish a reliable
link among those three experimental techniques and, through
this correspondence, visualize the flux distribution throughout
the sample after an ac field cycle.

ACKNOWLEDGMENTS

This work was partially supported by the Methusalem
Funding of the Flemish Government, the NES-ESF program,
the Belgian IAP, the Fund for Scientific Research-Flanders
(FWO-Vlaanderen), the UK Engineering and Physical Sci-
ences Research Council, and the Brazilian funding agencies
FAPESP and CNPq. A.V.S. is grateful for the support from the
FWO-Vlaanderen. T.H.J. acknowledges the financial support
of the Norwegian Research Council. W.A.O. thanks the Centre
for Advanced Study (Norway) for the hospitality during the
last stage of this work.

*Present address: Faculdade de Engenharia, UNESP - Universidade
Estadual Paulista, Departamento de Fı́sica e Quı́mica, 15385-000
Ilha Solteira, SP, Brazil.
1C. P. Bean, Phys. Rev. Lett. 8, 250 (1962).
2C. P. Bean, Rev. Mod. Phys. 36, 31 (1964).
3Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev. 129, 528
(1963).

4H. A. Ullmaier, Phys. Status Solidi 17, 631 (1966).
5J. R. Clem, J. Appl. Phys. 50, 3518 (1979).
6M. V. Indenbom, Th. Schuster, H. Kuhn, H. Kronmüller, T. W. Li,
and A. A. Menovsky, Phys. Rev. B 51, 15484 (1995).

7A. V. Silhanek, J. Gutierrez, R. B. G. Kramer, G. W. Ataklti, J. Van
de Vondel, V. V. Moshchalkov, and A. Sanchez, Phys. Rev. B 83,
024509 (2011).

8D. V. Denisov, D. V. Shantsev, Y. M. Galperin, Eun-Mi Choi, Hyun-
Sook Lee, Sung-Ik Lee, A. V. Bobyl, P. E. Goa, A. A. F. Olsen, and
T. H. Johansen, Phys. Rev. Lett. 97, 077002 (2006).

9S. Hebert, L. Van Look, L. Weckhuysen, and V. V. Moshchalkov,
Phys. Rev. B 67, 224510 (2003).

10F. Colauto, E. M. Choi, J. Y. Lee, S. I. Lee, V. V. Yurchenko,
T. H. Johansen, and W. A. Ortiz, Supercond. Sci. Technol. 20, L48
(2007).

11F. Colauto, E. J. Patino, M. G. Blamire, and W. A. Ortiz, Supercond.
Sci. Technol. 21, 045018 (2008).

12A. V. Silhanek, S. Raedts, and V. V. Moshchalkov, Phys. Rev. B 70,
144504 (2004).

13M. Menghini, R. J. Wijngaarden, A. V. Silhanek, S. Raedts, and
V. V. Moshchalkov, Phys. Rev. B 71, 104506 (2005).

214529-6

http://dx.doi.org/10.1103/PhysRevLett.8.250
http://dx.doi.org/10.1103/RevModPhys.36.31
http://dx.doi.org/10.1103/PhysRev.129.528
http://dx.doi.org/10.1103/PhysRev.129.528
http://dx.doi.org/10.1002/pssb.19660170220
http://dx.doi.org/10.1063/1.326349
http://dx.doi.org/10.1103/PhysRevB.51.15484
http://dx.doi.org/10.1103/PhysRevB.83.024509
http://dx.doi.org/10.1103/PhysRevB.83.024509
http://dx.doi.org/10.1103/PhysRevLett.97.077002
http://dx.doi.org/10.1103/PhysRevB.67.224510
http://dx.doi.org/10.1088/0953-2048/20/8/L02
http://dx.doi.org/10.1088/0953-2048/20/8/L02
http://dx.doi.org/10.1088/0953-2048/21/4/045018
http://dx.doi.org/10.1088/0953-2048/21/4/045018
http://dx.doi.org/10.1103/PhysRevB.70.144504
http://dx.doi.org/10.1103/PhysRevB.70.144504
http://dx.doi.org/10.1103/PhysRevB.71.104506


VISUALIZING THE AC MAGNETIC SUSCEPTIBILITY OF . . . PHYSICAL REVIEW B 84, 214529 (2011)

14Matching effects are less pronounced at the temperature interval
reported here (i.e., not very close to Tc). The commensurability
field, characteristic of the lattice of ADs, is a reference for the
window of magnetic fields at which the events treated here take
place.

15V. V. Schmidt, The Physics of Superconductors, edited by P. Muller
and A. V. Ustinov (Springer, Berlin, 1997).

16L. E. Helseth, R. W. Hansen, E. I. Il’yashenko, M. Baziljevich, and
T. H. Johansen, Phys. Rev. B 64, 174406 (2001).

17The level of control of the zero-field condition was different for
the 3 experimental setups employed: below 30 mOe at the MPMS,
below 250 mOe at the MOI, and around 1 Oe at the PPMS.

18P. Leiderer, J. Boneberg, P. Brüll, V. Bujok, and S. Herminghaus,
Phys. Rev. Lett. 71, 2646 (1993).

19C. A. Duran, P. L. Gammel, R. E. Miller, and D. J. Bishop, Phys.
Rev. B 52, 75 (1995).

20T. H. Johansen, M. Baziljevich, D. V. Shantsev, P. E. Goa, Y. M.
Galperin, W. N. Kang, H. J. Kim, E. M. Choi, M.-S. Kim, and S. I.
Lee, Europhys. Lett. 59, 599 (2002).

21A. V. Bobyl, D. V. Shantsev, T. H. Johansen, W. N. Kang, H. J. Kim,
E. M. Choi, and S. I. Lee, Appl. Phys. Lett. 80, 4588 (2002).

22I. A. Rudnev, S. V. Antonenko, D. V. Shantsev, T. H. Johansen, and
A. E. Primenko, Cryogenics 43, 663 (2003).

23S. C. Wimbush, B. Holzapfel, and C. Jooss, J. Appl. Phys. 96, 3589
(2004).

24I. A. Rudnev, D. V. Shantsev, T. H. Johansen, and A. E. Primenko,
Appl. Phys. Lett. 87, 042502 (2005).

25D. V. Denisov, A. L. Rakhmanov, D. V. Shantsev, Y. M. Galperin,
and T. H. Johansen, Phys. Rev. B 73, 014512 (2006).

26M. Baziljevich, A. V. Bobyl, D. V. Shantsev, E. Altshuler, T. H.
Johansen, and S. I. Lee, Physica C 369, 93 (2002).

27F. Colauto, E. Choi, J. Y. Lee, S. I. Lee, E. J. Patino, M. G. Blamire,
T. H. Johansen, and W. A. Ortiz, Appl. Phys. Lett. 96, 092512
(2010).

28S. Kolesnik, V. Vlasko-Vlasov, U. Welp, G. W. Crabtree,
T. Piotrowski, J. Wrobel, A. Klimov, P. Przystupski, T. Skoskiewicz,
and B. Dabrowski, Physica C 341-348, 1093 (2000).

29V. Vlasko-Vlasov, U. Welp, V. Metlushko, and G. W. Crabtree,
Physica C 341-348, 1281 (2000).

30E. H. Brandt and M. Indenbom, Phys. Rev. B 48, 12893 (1993).
31E. Zeldov, J. R. Clem, M. McElfresh, and M. Darwin, Phys. Rev.

B 49, 9802 (1994).
32J. R. Clem and A. Sanchez, Phys. Rev. B 50, 9355 (1994).
33D.-X. Chen, C. Navau, N. Del-Valle, and A. Sanchez, Physica C

470, 89 (2010).
34V. V. Moshchalkov, M. Baert, V. V. Metlushko, E. Rosseel, M. J.

Van Bael, K. Temst, Y. Bruynseraede, and R. Jonckheere, Phys.
Rev. B 57, 3615 (1998).

35D. V. Shantsev, Y. M. Galperin, and T. H. Johansen, Phys. Rev. B
61, 9699 (2000).

36E. H. Brandt, Phys. Rev. B 55, 14513 (1997).
37E. H. Brandt, Phys. Rev. B 58, 6523 (1998).
38Notice that the same scale is used for the 3 magnetic loops shown.

Further amplification of the 5 K loop would evidence some amount
of hysteresis, due to penetration at the borders.

39R. B. Goldfarb and A. F. Clark, IEEE Trans. Magnetics, MAG-21,
332 (1985).

40M. S. Welling, R. J. Wijngaarden, C. M. Aegerter, R. Wördenweber,
and P. Lahl, Physica C 404, 410 (2004).

41Ch. Jooss, R. Warthmann, and H. Kronmüller, Phys. Rev. B 61,
12433 (2000).

42A. Polyanskii, R. L. S. Emergo, J. Z. Wu, T. Aytug, D. K. Christen,
G. K. Perkins, and D. Larbalestier, Phys. Rev. B 72, 174509 (2005).

43M. Djupmyr, G. Cristiani, H.-U. Habermeier, and J. Albrecht, Phys.
Rev. B 72, 220507(R) (2005).

44V. V. Yurchenko, A. J. Qviller, P. B. Mozhaev, J. E. Mozhaeva, J. B.
Hansen, C. S. Jacobsen, I. M. Kotelyanskii, A. V. Pan, and T. H.
Johansen, Physica C 470, 799 (2010).

214529-7

http://dx.doi.org/10.1103/PhysRevB.64.174406
http://dx.doi.org/10.1103/PhysRevLett.71.2646
http://dx.doi.org/10.1103/PhysRevB.52.75
http://dx.doi.org/10.1103/PhysRevB.52.75
http://dx.doi.org/10.1209/epl/i2002-00146-1
http://dx.doi.org/10.1063/1.1485304
http://dx.doi.org/10.1016/S0011-2275(03)00157-7
http://dx.doi.org/10.1063/1.1778816
http://dx.doi.org/10.1063/1.1778816
http://dx.doi.org/10.1063/1.1992673
http://dx.doi.org/10.1103/PhysRevB.73.014512
http://dx.doi.org/10.1016/S0921-4534(01)01226-6
http://dx.doi.org/10.1063/1.3350681
http://dx.doi.org/10.1063/1.3350681
http://dx.doi.org/10.1016/S0921-4534(00)00799-1
http://dx.doi.org/10.1016/S0921-4534(00)00895-9
http://dx.doi.org/10.1103/PhysRevB.48.12893
http://dx.doi.org/10.1103/PhysRevB.49.9802
http://dx.doi.org/10.1103/PhysRevB.49.9802
http://dx.doi.org/10.1103/PhysRevB.50.9355
http://dx.doi.org/10.1016/j.physc.2009.10.012
http://dx.doi.org/10.1016/j.physc.2009.10.012
http://dx.doi.org/10.1103/PhysRevB.57.3615
http://dx.doi.org/10.1103/PhysRevB.57.3615
http://dx.doi.org/10.1103/PhysRevB.61.9699
http://dx.doi.org/10.1103/PhysRevB.61.9699
http://dx.doi.org/10.1103/PhysRevB.55.14513
http://dx.doi.org/10.1103/PhysRevB.58.6523
http://dx.doi.org/10.1016/j.physc.2003.11.026
http://dx.doi.org/10.1103/PhysRevB.61.12433
http://dx.doi.org/10.1103/PhysRevB.61.12433
http://dx.doi.org/10.1103/PhysRevB.72.174509
http://dx.doi.org/10.1103/PhysRevB.72.220507
http://dx.doi.org/10.1103/PhysRevB.72.220507
http://dx.doi.org/10.1016/j.physc.2010.02.085