RAPID COMMUNICATIONS
PHYSICAL REVIEW B
VOLUME 61, NUMBER 14
1 APRIL 2000-II
Periodic magnetization instabilities in a superconducting Nb film with a square lattice of Ni dots
A. Terentiev, D. B. Watkins, and L. E. De Long
Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055
L. D. Cooley
Applied Superconductivity Center, University of Wisconsin, Madison, Wisconsin 53706-1687
D. J. Morgan and J. B. Ketterson
Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208
共Received 23 September 1999兲
Isothermal magnetization curves of a superconducting Nb film perforated by a square lattice of Ni dots
exhibit quasiperiodic instabilities below ⬃4 K, with a field-dependent period equal to the first, second, or third
matching fields. The instabilities are found in a range of applied fields well above the saturation matching
value, and under conditions where a continuous Nb film with the same dimensions and magnetization remained
stable. The results suggest that terraces of matched flux density exist at the border of a flux-depleted zone
created by a geometric barrier near the film edge. Geometric effects may thus play an important role in
determining the presence or absence of matching effects.
Thin-film superconductors patterned with lattices of artificial pinning centers 共APC’s兲 exhibit a variety of interesting
mesoscopic and quantum phenomena.1–3 Lattices of antidots
共holes兲,3–5 or dots of normal6,7 or magnetic6–11 metals are
known to greatly enhance the low-field critical current density J c (H), as well as provide different opportunities to precisely control flux pinning and flow, as desired in practical
applications of superconducting films.12
The magnetization and electrical resistivity of patterned
films are of particular interest, since they exhibit strong
anomalies when the temperature is very close to the critical
temperature T c and the perpendicular applied field is very
near a ‘‘matching field’’ H n , at which the equilibrium number of Abrikosov flux lines 共FL’s兲 共expected to penetrate an
unpatterned film in the mixed state兲 is an integral multiple n
of the number of APC.1–3 However, the isolated Abrikosov
FL are energetically unstable to the formation of multiply
quantized (n⭓1) fluxoids at each APC.3,13–16 At intermediate applied fields H n ⬍H n⫹1 , the equilibrium state is a mixture of fluxoids 关 n⌽ 0 or (n⫹1)⌽ 0 ] that penetrate the
specimen,3–5 where ⌽ 0 is the flux quantum. The magnetization curve M (H) is then similar to the logarithmic reversible
flux penetration above the lower critical field H c1 of pristine
type-II superconductors;16 however, the transfer of flux 共e.g.,
via hopping of singly quantized FL兲 between APC is irreversible and increases the magnetic hysteresis well above
that of unpatterned films.3–5
From an applications viewpoint, more interesting behavior occurs when n is equal to a ‘‘saturation index’’ s
⫽D/4␰ (T), where D is the APC diameter and ␰ (T) is the
temperature-dependent coherence length.14 When H⬎H s , it
is not possible to add fluxoids with flux (s⫹1) ␾ 0 to the
saturated pins,13–15 so that additional FL are instead repelled
into the interstitial region between the APC.17–19 This results
in a sharp transition to high flux-flow rates15,18 and greatly
reduced magnetization hysteresis,4 because enhanced FL motion is permitted by relatively weak interstitial FL pinning
0163-1829/2000/61共14兲/9249共4兲/$15.00
PRB 61
very near T c . On the other hand, random pinning on interstitial defects, which grows in strength with decreasing temperature, could trap interstitial FL to produce quite disordered arrangements, even when H⬍H s . 12,17–21 It is therefore
not surprising that matching anomalies previously have been
observed in superconducting films with APC only when the
temperature is very near to T c . 3–12,18,19
The present work reports the unexpected observation of
matching anomalies in Nb films patterned with square Ni dot
lattices near and below 3 K, which is well below the zerofield T c ⬇8.8 K of our Nb film. The anomalies are ‘‘quasiperiodic’’ magnetization jumps whose period may abruptly
change from ⬃15 Oe, to ⬃28 Oe, to ⬃42 Oe, as the external
field is increased. These periods are remarkably close to the
first three matching fields (1⭐n⭐3) previously
detected7,10,11 very close to T c , strongly suggesting that the
quasiperiodicity is a result of field matching well below T c ,
in clear contrast to the general rule that matching anomalies
rapidly become ill-defined for T⭐0.9T c . 12
Sample films for this study were fabricated using
electron-beam lithography and electron-beam deposition.
Blocks of square Ni dot lattice of 90 ␮m⫻90 ␮m were
stitched together to produce an overall film area of 1 mm⫻1
mm. A Nb film of 95 nm thickness was then deposited
around Ni dots of 120 nm diameter and 110 nm height; thus
the Ni dots completely perforated the Nb layer. A control
film of Nb was fabricated with the same spatial dimensions
and zero-field T c ⫽8.8 K as the patterned film, which implies
there was no macroscopic reduction of T c due to magnetic
depairing or proximity effects. Detailed descriptions of the
fabrication and physical properties of sample films were
given elsewhere.7,10,11
dc magnetization and ac magnetic susceptibility measurements were done using a Quantum Design MPMS5 superconducting quantum interference device 共SQUID兲 magnetometer, using applied fields aligned perpendicular to the plane
of the films. Demagnetization of the dots was accomplished
R9249
©2000 The American Physical Society
RAPID COMMUNICATIONS
R9250
A. TERENTIEV et al.
PRB 61
FIG. 2. dc magnetic moment versus applied magnetic field for
an unpatterned Nb film held at a temperature of 2.5 K. Note that
narrow intervals of unstable magnetization as the last flux exits the
film on either positive or negative field sweeps.
FIG. 1. 共a兲 Magnetic moment versus applied magnetic field for a
Nb film perforated with Ni dots, and for an unperforated Nb film at
a temperature of 8.7 K (T c ⫽8.8 K兲. First and second matching
fields of ⬃13 Oe and 26 Oe are clearly seen, as well as a weak
matching anomaly near 39 Oe. 共b兲 Magnetic moment versus applied
magnetic field for the films of 共a兲 taken at a slightly lower temperature of 8.5 K, where matching anomalies are no longer well defined.
Note that there is a small field offset ⬇1 Oe in both plots due to a
residual field trapped in the magnetometer solenoid.
by oscillating the field of the superconducting magnet with a
gradual decrease of field amplitude, starting at 1 T, while the
sample was held at room temperature. After demagnetization, the sample was cooled in zero field and isothermal dc
magnetization was measured at several temperatures. Additional magnetization measurements were conducted using an
electromagnet fitted with a cryostat insert that accommodated a Hall-probe 共HP兲 sensor. The HP experiments were
conducted to eliminate possible effects of sample motion
through inhomogeneous magnetic field regions, and to
achieve better thermal stability by immersing the sample in a
coolant bath. The HP magnetometer data confirmed the general features of the SQUID experiments, and details of these
results are discussed elsewhere.22
dc magnetization data for the patterned and control films
at a temperature of 8.7 K⬃0.99 T c are shown in Fig. 1共a兲.
Matching anomalies appear at the first (H 1 ⬃13 Oe, second
(H 2 ⬃26 Oe兲, and third 共 H 3 ⬃39 Oe兲 matching fields for the
perforated film. These values are roughly consistent with the
calculated values H n ⫽n⫻(14.4 Oe兲 for the 1.2 ␮m dot
spacing 共recent data taken on another Quantum Design magnetometer indicate the low matching fields may be due to an
approximate 10% scaling error in the field/current calibration
of our solenoid兲. These matching anomalies previously were
found7,10,11 to be asymmetric, depending on the direction of
polarized Ni moments. The magnetization curve shown in
Fig. 1共a兲 is close to symmetric, consistent with a low remanent magnetization of Ni dots in demagnetized samples. The
data for the unpatterned film at 8.7 K are smooth, and the
hysteresis width is lower than that of the patterned film only
at low fields. This indicates that Ni dots act to enhance FL
pinning in the matching regime H⬍H 3 ⬇H S , but induce no
significant enhancement of magnetic hysteresis at higher
fields at temperatures very close to T c .
Lower temperature data11 indicate that there is little difference between the magnetization hysteresis for Nb films
with and without APC well below T c , suggesting that spatial
correlation between the FL and APC is lost at low temperatures. This is consistent with the smearing out of matching
anomalies by 8.5 K ⬃0.97T c , as shown in Fig. 1共b兲 and a
large body of experimental results3–12,18,19 on other patterned
films. Further decreases of temperature reduce thermal stability and result in flux jumps in unpatterned Nb films,11,22
similar to the behavior of commercial superconducting
wire.23 We observed this effect over a narrow range of fields
close to zero when flux was leaving an unpatterned film, as
shown in Fig. 2.
In contrast to Fig. 2, unexpected and remarkable lowtemperature behavior is exhibited by the film patterned with
Ni dots, as shown in Fig. 3. The collapse of the usual zerofield peak in magnetization of the patterned film yields a
‘‘fish tail’’ pattern recently attributed to random thermomagnetic instabilities in unpatterned Nb films.24 Nevertheless,
the anomalies shown in Fig. 3 are quite unique in that the
magnetization jumps are quasiperiodic over a large field
range at the lowest experimental temperature of ⬃2.5 K,
which is emphasized in Fig. 4. The saw-tooth oscillations
first exhibit a 14–15 Oe (⬇H 1 ) period at fields above 5H 1
⬇70 Oe, then abruptly switch to 27–29 Oe 共⬇H 2 兲 near
5H 2 ⬇140 Oe, followed by another switch to ⬃41 Oe 共⬇H 3 兲
near 5H 3 ⬇205 Oe. Moreover, our unpatterned control film
magnetization, shown in Fig. 2, is essentially the same as
that of the patterned film below 4 K, even though it exhibits
no quasiperiodic structure at high field. The presence of in-
RAPID COMMUNICATIONS
PRB 61
PERIODIC MAGNETIZATION INSTABILITIES IN A . . .
FIG. 3. Development of a saw-tooth structure in the magnetic
moment versus applied magnetic field of a Nb film perforated with
Ni dots at a temperature of 2.5 K.
stabilities in the patterned film data, and their absence at high
field in the control film data, suggest that the Ni dot lattice
exerts a crucial influence on the appearance and nature of
the instabilities.
Standard stability arguments23 predict stable flux fronts
for ␮ 0 J 2c a 2 / ␳ C(T c ⫺T)⬍3, assuming that J c has linear temperature dependence. The hysteresis width ⌬H⫽aJ c yields25
an estimated critical current density J c ⬇109 – 1010 A/m2 for
FIG. 4. Expanded view of the data of Fig. 3 共negative applied
field-top, positive applied field-bottom; arrows indicate field sweep
direction兲. Note the regions of pronounced field periodicity in the
field increasing data 共previously unmagnetized, virgin film兲 with
period switching from ⬃14 Oe at fields below 140 Oe, to ⬃28 Oe
at higher fields. The period ⬃42 Oe observed near 200 Oe corresponds to the third matching field.
R9251
both patterned and unpatterned samples. Taking into account
the high demagnetization factor (103 – 104 ), the film width
a⫽1 mm, the density ( ␳ ⫽8550 kg/m3兲 and specific heat
关 C(T c )⫽0.8 J/kg K兴 for Nb, and inserting J c ⫽109 A/m2,
yields the condition a⬍0.3 mm, which is close to the actual
sample sizes. This indicates that flux gradients should be
close to the instability threshold. Large instabilities abruptly
cease above a reproducible field H f , whose temperature dependence (H f ⬇200 Oe at T⫽4.0 K, 230 Oe at 3.5 K, and
800 Oe at 3.0 K兲 is consistent with a strong reduction of C
and increase in J c as temperature decreases.22
Explanation of the quasiperiodic instabilities clearly requires the introduction of additional mechanisms beyond
random thermomagnetic events. The fact that the initial lowfield slope 共i.e., the perfect diamagnetic susceptibility of the
virgin film兲 shown in Fig. 3 is very close to the slopes observed at fields just above quasiperiodic instabilities,
strongly suggests that these instabilities take place in the
screening currents located near the film edge region that is
traditionally expected to experience local magnetic fields
close to the applied field in the critical state.25 Therefore, the
strong quasiperiodic behavior that persists to at least 350 Oe
in Fig. 3 seems to contradict the general experimental trend
that matching anomalies are limited to very low-field
strengths 共e.g., n⭐8 for Nb films with comparably dimensioned antidot lattices19兲.
Alternatively, matching correlations could be retained
within interior domains whose sizes are comparable to the
magnetic penetration depth ␭(T). A ‘‘terraced critical
state,’’ or staircase flux profile was predicted20 in the case of
moderate interstitial pinning, as a compromise between uniform matching and the usual linear Bean critical state profile.
Molecular-dynamic simulations21 suggest that the FL lattice
is plastically deformed in domain walls between wellmatched zones. However, ␭(T) decreases with decreasing
temperature, and the well-ordered flux region within a given
domain should become small compared to the total flux
threading the sample well below T c . Therefore, complex
flux profiles are expected to form deep inside the film, and
no matching anomalies characteristics of large correlated domains in the film interior would be anticipated. A detectable
signature of matching then might only occur for supercurrents running through relatively homogeneous flux arrangements near the film edge, where their contribution to the
overall magnetization is maximal.
Taking these observations into account, we postulate that
the quasiperiodic magnetization anomalies observed for the
patterned film are the result of matching in flux-depleted
regions of the film that experience local magnetic fields on
the order of the lowest matching fields H n (n⫽1,2,3). In
particular, rarified regions in the FL density near sample
edges are expected at low fields, due to the ‘‘geometric barrier’’ and the strongly penetrating Meissner currents that act
to force flux toward the center of plate-like samples that have
strong demagnetizing effects.26–29 Indeed, effects due to the
‘‘geometric barrier’’ should be ubiquitous in patterned films,
but this scenario generally has not been considered in the
literature to date.
The field range where quasiperiodic anomalies are most
clearly observed, from ⬃70 to ⬃250 Oe, is consistent with
the field range within which the geometric barrier should be
RAPID COMMUNICATIONS
R9252
A. TERENTIEV et al.
influential. Current lithography techniques limit the APC diameter D⬎0.1 ␮m and the APC separation to d⬃1 ␮m,
which restricts H 1 to ⬃10–30 G. At high temperatures,
where H c1 is low, we expect the enforcement of uniform
matching across the entire pin array. However, far below T c ,
this ‘‘single-terrace’’ matching will be lost because of the
inhomogeneities generated in FL density due to the combined effects of the geometrical barrier and interstitial pinning. Using the low critical field H c1 of bulk Nb⬃2000 G,
and the given width W⬇1 mm and thickness t⬇0.1 ␮m of
our films 共in the absence of pinning兲, we estimate26 that FL
first penetrate the edge of the film when the applied field is
above H p ⫽H c1 (t/2W) 1/2⬇14 Oe 共as observed in Fig. 2兲. As
H is increased above H p , a domelike distribution of FL is
initially formed by the inward force of Meissner currents.
This ‘‘compression region’’ spreads outward from the center
of the sample into a flux-depleted region,30 and reaches the
film edge at H e ⬇5H p ⬇70 Oe. When H⬎H e , the geometric
barrier is overcome and FL are redistributed more uniformly
throughout the film. Note that pattern in Fig. 4 共bottom plot
for a virgin magnetization sweep兲 suggests this process may
‘‘reset’’ or repeat itself for 2H e or 3H e , implying that each
matching field functions as a ‘‘effective lower critical field
scale,’’ since the geometrical barrier might reestablish itself
at these threshold fields where complete filling of the depletion region is just achieved at successively higher-field
1
A. T. Hebard, A. T. Fiory, and S. Somekh, IEEE Trans. Magn.
13, 589 共1977兲.
2
P. Martinoli, Phys. Rev. B 17, 1175 共1978兲.
3
V. V. Metlushko et al., Solid State Commun. 91, 331 共1994兲.
4
M. Baert et al., Phys. Rev. Lett. 74, 3269 共1995兲.
5
Y. Bruynseraede et al., Proc. SPIE 2697, 328 共1996兲.
6
A. F. Hoffmann, Ph.D. thesis, University of California, San Diego, 1999.
7
Y. Jaccard et al., Phys. Rev. B 58, 8232 共1998兲.
8
D. J. Morgan and J. B. Ketterson, Phys. Rev. Lett. 80, 3614
共1998兲.
9
J. I. Martin, M. Velez, J. Nogues, and I. K. Schuller, Phys. Rev.
Lett. 79, 1929 共1997兲.
10
David J. Morgan, Ph.D. thesis, Northwestern University, 1998.
11
A. Terentiev et al., Physica C 324, 1 共1999兲
12
V. V. Moshchalkov et al., Phys. Scr. T55, 168 共1994兲.
13
A. Buzdin, Phys. Rev. B 47, 11 416 共1993兲.
14
G. S. Mkrtchyan and V. V. Schmidt, Zh. Eksp. Teor. Fiz. 61, 367
共1971兲 关Sov. Phys. JETP 34, 195 共1972兲兴.
15
I. V. Khalfin and B. Ya. Shapiro, Physica C 207, 359 共1993兲.
16
V. V. Moshchalkov et al., Phys. Rev. B 54, 7385 共1996兲.
17
K. Harada et al., Science 274, 1167 共1996兲.
PRB 61
matching densities. It is then plausible that at low enough
applied fields the flux profile near the film edge has terraces
that consist solely of strongly pinned fluxoids 共i.e., no interstitial FL兲. Flux jumps then occur when the terrace edges
become unstable, due to the increasing gradients associated
with the overall FL profile.
We conclude the quasiperiodic instabilities, which we observe for applied fields approaching 1 kOe, are related to a
‘‘geometric barrier’’26–29 that produces steep flux gradients.
We identify these instabilities as unexpected lowtemperature matching anomalies, most probably initiated
near the edge of the sample where the flux density can be
much lower than in the film interior. This interpretation suggests that the geometrical barrier plays an important role in
determining the FL distribution in films with APC lattices,
and that sample geometry should be a crucial consideration
in developing patterned films for applications.
Research at the University of Kentucky was supported by
the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Science, Grant No. DE-FG0297ER45653. Research at Northwestern University was supported by National Science Foundation Grant No. DMR9309061. Research at the University of Wisconsin was
supported by the U.S. Department of Energy, Division of
High-Energy Physics, Grant No. DE-FG02-96ER40961 and
the NSE MRSEC for Nanostructured Materials.
V. V. Metlushko et al., Europhys. Lett. 41, 333 共1998兲.
V. Metlushko et al., Phys. Rev. B 60, R12 585 共1999兲.
20
L. D. Cooley and A. M. Grishin, Phys. Rev. Lett. 74, 2788
共1995兲.
21
C. Reichhardt et al., Phys. Rev. B 54, 16 108 共1996兲.
22
A. Terentiev, B. Watkins, L. E. De Long, L. D. Cooley, D. J.
Morgan, and J. B. Ketterson, Physica C 共to be published兲.
23
M. N. Wilson, Superconducting Magnets 共Oxford University
Press, New York, 1983兲, Chap. 7, pp. 131–158.
24
Y. Kopelevich and P. Esquinazi, J. Low Temp. Phys. 113, 1
共1998兲.
25
C. P. Bean, Phys. Rev. Lett. 8, 250 共1962兲.
26
E. Zeldov et al., Phys. Rev. Lett. 73, 1428 共1994兲.
27
M. Benkraouda and J. R. Clem, Phys. Rev. B 53, 5716 共1996兲.
28
Th. Schuster et al., Phys. Rev. Lett. 73, 1424 共1994兲.
29
E. Zeldov, J. R. Clem, M. McElfresh, and M. Darwin, Phys. Rev.
B 49, 9802 共1994兲.
30
If pinning is strong, the FL distribution also contains a central
region of reduced density 共‘‘doughnut hole’’兲 共Refs. 26–28兲. In
any event, the critical state region is predicted to have a highly
nonlinear profile until the applied field is well above H e
共Ref. 27兲.
18
19