## Physik.uni-stuttgart.de

PHYSICAL REVIEW B

**82**, 184512 共2010兲

**Enhancing the critical current in quasiperiodic pinning arrays below and above**
**the matching magnetic flux**
V. R. Misko,1,2 D. Bothner,3 M. Kemmler,3 R. Kleiner,3 D. Koelle,3 F. M. Peeters,1 and Franco Nori2,4
1

*Departement Fysica, Universiteit Antwerpen, B-2020 Antwerpen, Belgium*
2

*Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan*
3

*Physikalisches Institut-Experimentalphysik II and Center for Collective Quantum Phenomena, Universität Tübingen,*
*D-72076 Tübingen, Germany*
4

*Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA*
共Received 2 July 2010; published 8 November 2010兲
Quasiperiodic pinning arrays, as recently demonstrated theoretically and experimentally using a fivefold
Penrose tiling, can lead to a significant enhancement of the critical current

*Ic *as compared to "traditional"regular pinning arrays. However, while regular arrays showed only a sharp peak in

*I *共⌽兲
at the matching flux
⌽1 and quasiperiodic arrays provided a much broader maximum at ⌽⬍⌽1, both types of pinning arrays turnedout to be inefficient for fluxes larger than ⌽1. We demonstrate theoretically and experimentally the enhance-ment of

*I *共⌽兲
for ⌽⬎⌽1 by using non-Penrose quasiperiodic pinning arrays. This result is based on a
qualitatively different mechanism of flux pinning by quasiperiodic pinning arrays and could be potentiallyuseful for applications in superconducting microelectronic devices operating in a broad range of magneticfields.

PACS number共s兲: 74.25.Sv, 74.25.Wx, 74.78.⫺w
but very

*narrow *peaks in the critical current versus magneticflux,

*I *共⌽兲
. This shortcoming can be overcome by using
Pinning of Abrikosov vortices in superconductors is a
more sophisticated pinning topologies, e.g., quasiperiodic
topic that has attracted considerable interest, both with re-
共QP兲 In particular, using a fivefold Penrose tiling
spect to the fundamental physical properties of so-called
共see, e.g., Refs. as an APS 共i.e., placing pins on the
"vortex-matter" and with respect to device applications. In
vertices of a Penrose tiling兲 results in a considerable en-
the latter case, for bulk or thick film superconductors, the
hancement of

*I *共⌽兲, i.e., significant broadening of the peak
inclusion of nanodefects, acting as pinning sites, typically
in

*I *共⌽兲, as was recently demonstrated and
randomly distributed within the superconductors, has been
experimentallyThe underlying idea of using QP APS is
shown to significantly enhance the critical current density,
that, contrary to regular APS, they include

*many *periodicities
which is important for increasing the current-carrying capac-ity of wires and tapes, e.g., for applications in superconduct-
and thus can trap vortices for magnetic fluxes other than
ing magnets 共see, e.g., Ref. and references therein兲. On the
⌽

*i*. As shown for Penrose-tiling
other hand, recent progress in the fabrication of nanostruc-
in addition to the matching-flux peak at ⌽
tures has provided a wide variety of well-controlled litho-
acquires a remarkably

*broad *maximum for ⌽
graphically defined artificial pinning sites, and experiments
⬍⌽1. This maximum turned out to be even more robust with
can control both the location and the strength of each pinning
respect to variations in parameters of APS than the sharp
center in vortex-confining arrays of pinning sites 共APS兲 共see,
peak at ⌽ = ⌽1However, in theory and experiments,
e.g., Refs. In this context it has been shown that, e.g.,

*I *共⌽兲
decreases rapidly for fluxes ⌽⬎⌽1 for QP Penrose-
microholes 共antidots兲 or magnetic dots may improve the per-
tiling APS as well as for regular APS. Here, we discuss the
formance of microelectronic thin-film devices. Examples are
essential difference in the flux pinning for magnetic fluxes
the suppression of low-frequency flux noise in superconduct-
below and above the matching value ⌽ = ⌽1 and we demon-
ing quantum interference devices by strategically positioned
strate, theoretically and experimentally, how the critical cur-
or the enhancement of the quality factor of super-
rent

*I *共⌽兲
can be improved in the important regime where
conducting microwave resonators operated in magnetic
⌽⬎⌽1, by using novel artificial APS.

fields, which can be achieved by incorporating antidot arrays
The paper is organized as follows: the model and experi-
into the thin-film From a basic point of view, it is
ment are described in Secs. and respectively. In Sec.

interesting to explore the optimum pinning landscape pro-
we analyze the enhancement of the critical current below
vided by APS. One convenient way to do this is to investi-
the first matching field in different aperiodic pinning arrays
gate the critical current

*Ic *versus magnetic flux ⌽ for differ-
and compare the results to those for a fivefold Penrose-tiling
ent arrangements of pinning sites.

APS. The mechanisms of vortex "conductivity" leading to
Periodic artificial APS were proposed as devices able of
the enhancement of the critical current below and above the
trapping magnetic flux and thus enhancing

*Ic*. However, the
matching flux are discussed in Sec. In Sec. we also
efficiency of periodic APS is strongly limited to so-called
present our experimental and numerical results for QP APS
matching fields, i.e., when the number of vortices is com-
demonstrating the enhancement of the critical current for ⌽
mensurate with the number of pins, thus resulting in strong
⬎⌽1. Finally, the conclusions are given in Sec.
2010 The American Physical Society
MISKO

*et al.*
PHYSICAL REVIEW B

**82**, 184512 共2010兲

pattern

*Aqp *is less than

*A*, the critical current Eq. acquiresa prefactor

*A */

*A *⬇ 0.58 共Refs. and in the simula-
We model a three-dimensional slab, infinitely long in the

*z*
tions below. We use narrow potential wells as pinning sites,
direction, by a two-dimensional 共2D兲 共in the

*xy *plane兲 simu-
with the maximum pinning force

*f*
lation cell with periodic boundary conditions. Note that fol-
and the radius

*rp*
= 0.04 to 0.1.

lowing Refs. and we apply periodic boundary condi-tions at the boundaries of the square simulation cell while thequasiperiodic array is taken smaller than the cell. This is
done in order to prevent imposing

*ad hoc *periodicity to thevortex motion in a QP APS: the free-of-pinning region be-
Experiments were performed on

*d *= 60-nm-thick Nb films
tween the QP APS and the boundary of the simulation cell
containing circular antidots 共diameter

*D *= 400 nm兲 at the
serves as a reservoir of vortices that mimics the external
vertices of various types of QP APS 共see insets of Figs. and
applied magnetic field. This reservoir 共of thickness on the
in an area of 200⫻ 200 m2. The average antidot density
order of few intervortex distances兲 erases the memory, i.e.,
is

*np *0.5 m−2, corresponding to a first matching field
the information on the coordinate of a vortex leaving the cell.

*B*1=

*np *0 1 mT. Electric transport measurements for the
Each additional vortex enters the QP sample at the current
determination of the critical current

*Ic*
point of "least resistance to entry," similarly to the way vor-
bridges were performed at well-controlled temperatures

*T*
tices enter any other sample when placed in an external mag-
and magnetic fields; for details of sample fabrication and
netic field. This approach has been successfully used in nu-
experimental setup see Refs. and The bridges had
merous simulations in the past 共see, e.g., Refs. and
transition temperatures

*Tc *8.4 K and transition widths
varying from 15 to 25 mK. The temperature dependence of
To study the dynamics of vortex motion, we perform
the critical current at

*B *= 0 differed from bridge to bridge
simulated annealing simulations by numerically integrating
leading to different absolute values of

*Ic B *= 0兲 at the same
the overdamped equations of motion 共see, e.g., Refs. and
reduced temperatures

*T */

*Tc*. The experimental data shown be-
for a description of the method兲,
low, were recorded at

*T */

*Tc*= 0.9995, where the absolute val-ues of

*Ic *for different samples were very similar. Note that
according to our estimates based on a simple core pinning

*i *=

**f***i *=

**f**
model 共see Ref. , the maximum value of the critical de-

*i *is the total force per unit length acting on vortex

*i*,

**f***i*
pinning current in quasiperiodic pinning arrays of antidots
and

**f***vp *are the forces due to vortex-vortex and vortex-pin

reached the values of about 0.5 of the Ginzburg-Landau de-
interactions, respectively,

**f***T *is the thermal stochastic force,

pairing current, for the used parameters.

and

**f***d *is the driving force;

**v**
*i *is the velocity and is the
viscosity. All the forces are expressed in units of

*f*0= ⌽2/ 823, where ⌽

**IV. CRITICAL CURRENT IN QP TILING APS**
0 =

*hc */ 2

*e*, and lengths 共fields兲 are in
units of 共⌽ / 2兲
For a fivefold Penrose-tiling pinning arrays
Following Refs. and we model vortex pinning by
with many can enhance the critical
short-range parabolic potential wells located at positions

**r**共

*p*兲.

for a broader range of magnetic fluxes as compared to peri-
The pinning force is
odic APS. In order to examine and possibly further optimize

*J *共⌽兲 in APS, in this section we study the critical current as

**f***vp *= 兺

*p *兩

**r**
共

*p*兲兩⌰冉

*rp*− 兩

**r***i*−

**r***k ***r***ˆ*共

*p*兲, 共2兲 a function of applied magnetic flux, for different QP tilings.

*i *−

**r***k*
First, we consider a tiling consisting of the simplest
shapes, squares and triangles 共both compatible with the
where

*Np *is the number of pinning sites,

*fp *is the maximum
lowest-energy vortex lattices兲. The result for a quasiperiodic
pinning force of each potential well,

*rp *is the range of the
square-triangle is shown in Fig. For compari-
pinning potential, ⌰ is the Heaviside step function, and

**r***ˆ*共

*p*兲

son, Fig. also shows the

*J *共⌽兲 dependence for a fivefold

*i *−

**r**
*i *−

**r***k*
Penrose-tiling APS.

*J *共⌽兲 for the QP square-triangle tiling
The ground state of a system of moving vortices is ob-
has a broad maximum at ⌽ ⬍⌽
tained by simulating the field-cooled experiments 共e.g., Ref.

1, which is formed due to
the rearrangement of vortices between the square and trian-
. For deep short-range 共␦-like兲 potential wells, the energy
gular tiles when their density changes. Note the absence of a
required to depin vortices trapped by pinning sites is propor-
tional to the number of pinned vortices,

*N*共

*p*兲. Therefore, in
1, indicating that the local matching conditions be-
tween the APS and the vortex lattice are not fulfilled at the
this approximation, we can define the normalized critical
given parameters of the APS 共i.e., the average distance be-
tween the pinning sites, their maximum strength and radius兲.

However, a peak in

*J *共⌽兲
at ⌽1 is observed for the Penrose-

*I *共⌽兲
tiling APS. Another tiling consisting of the same simple

*N *共⌽兲
shapes 共i.e., squares and triangles兲 is the nonpisot square-
and study the dimensionless value

*J*
*c *=

*Ic j*0, where

*Nv *is the
triangular shown in the inset of Fig. Although
total number of vortices within the simulation cell of the area
these two types of tilings are characterized by different in-

*A*, and

*j*0 is a constant 共i.e., the current needed to depin

*Nv*
flation rules 共i.e., different arrangements of the tiles兲, they
noninteracting vortices,

*j*
contain the same elements, i.e., hexagons formed by six tri-
. Since the area of the QP

ENHANCING THE CRITICAL CURRENT IN…
PHYSICAL REVIEW B **82**, 184512 共2010兲

Square-triangle nonpisot
Magnetic flux F/F1
Magnetic flux F/F1
FIG. 1. 共Color online兲 Normalized critical current, *J*
*c *versus normalized magnetic flux, ⌽ / ⌽1
number of vortices *Nv *, for different QP
APS 关shown by solid blue 共dark gray兲 lines兴: 共a兲 square-triangle quasiperiodic tiling, 共b兲 square-triangle nonpisot tiling, 共c兲 Goodman-Strausstiling, and 共d兲 nonpisot Penrose tiling. Insets show the corresponding tiling patterns. For comparison, the function *J *共⌽ / ⌽ 兲
Penrose-tiling APS is shown by a red 共gray兲 line. The *J *共⌽ / ⌽ 兲
1 curves are shown for *f p f *0 = 2 and *rp*
= 0.1. These values of *f p *and *rp *are
also used in Figs.
angles surrounded by six squares. This similarity in the struc-
the maximum at the matching flux ⌽ 关
ture determines similar local matching effects between vor-
These examples indicate that the considered tilings pro-
tices and the pinning centers for these two tilings. As a result,
vide, in general, a weaker enhancement of the critical current
the *J *共⌽兲
curve for the nonpisot square-triangular tiling 关Fig.

as compared to the fivefold Penrose-tiling APS which seems
turns out to be practically identical to that for the qua-
to be the best among these competitors. Let us then consider
siperiodic square-triangular tiling 关compare to Fig. An
a different modification of a Penrose tiling, namely, a nonp-
example of a tiling consisting of three different tiles 共three
isot Penrose This tiling also consists of two types of
types of rhombuses兲 is a Goodman-Strauss shown in
rhombuses. It is shown in the inset of Fig. As shown in
the inset of Fig. Surprisingly, the behavior of *J *共⌽兲
the main panel of Fig. this APS provides an even better
this APS is very similar to that of the Penrose-tiling APS,
enhancement of *J *共⌽兲
within nearly the same, as for the
although it has an obvious shortcoming: it does not contain
fivefold Penrose tiling, range of magnetic flux: it is charac-
MISKO *et al.*
PHYSICAL REVIEW B **82**, 184512 共2010兲

FIG. 2. 共Color online兲 Flow pattern for vortices driven by an external current illustrating the holelike vortex conductivity in the
Shield-tiling APS for ⌽ⱗ⌽1: ⌽ ⬇ 0.95⌽1. The pinning sites are shown by filled large red 共gray兲 circles. Vortex positions are shown for tensuccessive time intervals *ti*, where *i *= 1 – 10: by open small circles, for *i *= 1 – 9, and by filled small green 共gray兲 circles, for *i *= 10. The Lorentzforce drives vortices from the left to the right. Most of the vortices do not move: their positions just slightly change 共polarized兲 with respectto the centers of the pinning sites. The trajectories of moving vortices 共from the left to the right兲 are shown by thick blue 共gray兲 arrows. Themotion of vortices results in effective propagation of vacancies 共holes兲 from the right to the left 共shown by thin dashed black arrows兲,including holes entering the sample from outside, as in the region marked by the dotted rectangle. Newly created holes are shown by thinsolid black circles while annihilated holes 共i.e., filled by vortices兲 are shown by thin dashed black circles.

terized by an almost-flat plateau within the broad maximum
**V. "HOLELIKE" AND "ELECTRONLIKE" VORTEX**

in *J *共⌽兲
. This improvement of the pinning properties is ex-
plained by the specific arrangement of the rhombuses in this
As can be seen in Fig. all the above results obtained
kind of tiling, which turns out to be closer to the energeti-
using different APS, while providing different profiles of
cally favorable arrangement of vortices in the lattice, thus
*J *共⌽兲
, still have a common feature: *Jc *drops drastically for
providing better vortex-pin matching conditions for various
⌽⬎⌽1 as it does in periodic APS 共for both cases, ⌽⬎⌽1
densities. For the fivefold Penrose-tiling the
and ⌽⬍⌽ 兲
maximum of the critical current at ⌽ = ⌽1 is separated from
Can we improve this? One might think that in practice we
the main broad maximum and is not as stable as the broad
can increase the density of the pinning sites in the array and
maximum with respect to changes in the parameters of the
thus trap flux for any ⌽. Indeed this might work to some
APS 共i.e., the maximum pinning strength兲The advantage
extent but the efficiency of an APS, especially QP APS, is
of the APS based on the nonpisot Penrose tiling is that it
determined not only by the ratio of the average intervortex
provides a smooth transition from the matching-field con-
distance to the average period of the APS 共*d*APS兲 but also by
figuration 共i.e., at ⌽ = ⌽ 兲
1 to lower-density pinned states 共i.e.,
the ratio of those distances to the magnetic field penetration
when some of the vertices of the small tiles become
depth , which is specific for a given superconductor. To
. This provides a flattening of the *J *共⌽兲
understand this, let us now briefly recall the essential differ-
关Fig. and thus increases the overall value of the maxi-
ence between vortex lattice pinning by periodic and quasip-
mum *Jc *within the range of the plateaulike maximum. This
eriodic APS. In the former case, if the symmetry of the vor-
important improvement could be useful for applications of
tex lattice coincides with that of the pinning array 共e.g., for a
triangular pinning array; in case of a square APS, transitions

ENHANCING THE CRITICAL CURRENT IN…
PHYSICAL REVIEW B **82**, 184512 共2010兲

Magnetic flux F/F1
Magnetic flux F/F1
Magnetic flux F/F1
FIG. 3. 共Color online兲 Normalized critical current 共experiment and theory兲, *J*
*c *versus normalized magnetic flux, ⌽ / ⌽1
vortices *N *兲
*v *, for different QP APS 关solid blue 共dark gray兲 line兴: 共a兲 Watanabe-Ito-Soma tiling, 共b兲 Tuebingen tiling, and 共c兲 Shield tiling.

Insets show the corresponding tiling patterns. The function *J *共⌽ / ⌽ 兲
for a fivefold Penrose-tiling APS is shown by red 共gray兲 line, for
between the triangular, half-pinned, and square lattices
but the "vortex conductivity" is still provided by the motion
occur—see, e.g., Ref. the vortex lattice and the pinning
of vortices 共Fig. However, in contrast to periodic arrays
array are commensurate for any *d*APS/ . In contrast to peri-
where this motion is possible due to the channeling of vorti-
odic APS, the pinning efficiency of a QP APS is sensitive to
ces 关leading to the drop of *J *共⌽兲
for ⌽⬍⌽1 , the process of
this ratio *d*APS/ since the elastic deformations of the vortex
channeling is strongly suppressed in QP systems 共as in me-
lattice are involved in its collective pinning by QP APS.

tallic quasicrystals where the usual periodic Bloch solution
This is explained by the fact that a vortex lattice is incom-
for electrons does not . Thus this can explain the pres-
mensurate with any QP tiling 共see Refs. and . Thus, the
ence of the maxima and the fast drop of *J *共⌽兲
efficiency of a QP APS does *not *scale with the density of
for ⌽⬍⌽1 and for ⌽⬎⌽1, correspondingly.

pinning sites. On the other hand, shifting the broad maxi-
Therefore, the problem is how to immobilize the addi-
mum in *J *共⌽兲
toward larger values of ⌽, i.e., ⌽⬎⌽1, results
tional 共i.e., electronlike兲 vortices for ⌽⬎⌽1. Although in a
in the loss of pinning efficiency for ⌽⬍⌽1. Moreover, the
QP APS they appear to be less mobile than in periodic APS
problem of the effective pinning for magnetic flux for ⌽
共no Bloch-type solutions兲, they can jump to adjacent tiles
1 is of fundamental interest and, as we show below, is
hopping conductivity兲 or "push" pinned vortices and thus
related to a qualitatively different type of flux pinning in
unpin them producing secondary interstitial vortices. To pre-
aperiodic APS.

vent this scenario, we propose to immobilize extra vortices
The fast drop of *J *共⌽兲
for ⌽⬎⌽1 is explained by the
in a special way, namely, to create well-separated "reser-
enhanced mobility of interstitial vortices, which is rather
voirs" for them which themselves are arranged in a *quasip-*
high even in QP APS. Note that a very low concentration of
*eriodic order*. This will prevent the extra vortices from 共i兲
interstitial vortices 共as compared to pinned vortices兲 is suffi-
hopping-type motion 共due to the isolation of the reservoirs兲
cient to drastically reduce the critical current. Using the lan-
and 共ii兲 channeling 共due to the QP order兲.

guage of conductivity in metals and semiconductors, we can
Among 2D tilings, the Watanabe-Ito-Soma
call the motion of interstitial vortices "electronlike vortex
共consisting of squares and rhombuses兲 and Tuebingen
conductivity." Correspondingly, the situation ⌽⬍⌽
consisting of two types of triangles兲 shown in the insets of
there are vacancies, i.e., unoccupied pinning sites, is similar
Figs. and respectively, are possible candidates.

to hole-type conductivity, since in this case the motion of
These tilings contain isolated "islands" formed by triangles
vacancies is similar to the motion of holes in semiconduc-
and rhombuses and the density of pinning sites on these is-
tors. For fluxes ⌽ⱗ⌽
1, the concentration of "holes" is low
lands is rather high. The plots of *Jc*
MISKO *et al.*
PHYSICAL REVIEW B **82**, 184512 共2010兲

traps for interstitial vortices 共see Fig. These tiles are well
separated 共i.e., do not have common sides兲 and are arrangedin a QP order 关see inset of Fig. Thus, *J *共⌽兲
pronounced maximum of the same magnitude as the maxi-mum at the first matching flux ⌽1. 共However the broad maxi-mum for ⌽⬍⌽1 is suppressed, compared to the fivefoldPenrose-tiling APS.兲
The critical depinning current *J *共⌽兲
was analyzed theo-
retically and experimentally for different quasiperiodic-tilingarrays of pinning sites. We showed that a fivefold Penrose-tiling APS provides stronger enhancement of *J *共⌽兲
other 2D QP tilings for a broad range of the applied magneticflux. It was demonstrated that the pinning properties of the
Penrose-tiling APS can be improved by using a differentmodification of the tiling, i.e., nonpisot Penrose tiling. Thisleads to an overall higher value of *Jc *for a broader range ofmagnetic fields. We proposed a mechanism for the flux pin-ning in QP APS which implies an effective trap and spaceseparation of interstitial vortices. This prevents hopping-typemotion and channeling of vortices in the APS and thus re-sults in an increase in the critical current for flux valueslarger than the matching flux ⌽1. These results might beuseful for future applications in microelectronic devices re-quiring a high *Jc *over a broad range of the applied flux. Ourproposal can be easily extended, mutatis mutandi, to otherrelated systems, including colloidal suspensions interactingwith pinning traps provided by arrays of optical or vortices in rotating Bose-Einstein condensates pinned by
FIG. 4. 共Color online兲 Distribution of vortices in a Shield-tiling
APS calculated for:
b . The vortices are
potential landscapes created by corotating
shown by green 共light gray兲 solid dots while the pinning sites by red
共gray兲 open circles.

show that indeed there is an appreciable increase in the criti-
This work was supported by the "Odysseus" Program of
cal current for ⌽⬎⌽
the Flemish Government and the Flemish Science Founda-
1, although the overall value of *Jc *for
tion 共FWO-Vl兲, the Interuniversity Attraction Poles 共IAP兲
1 for both tilings is rather low. These features of *Jc*
can be seen in both experimental and theoretical *J *共⌽兲
Programme—Belgian State—Belgian Science Policy, the
curves. Note that the experimental curves in Figs.
FWO-Vl, and by the DFG via SFB/TRR21. V.R.M. is grate-
qualitatively 关and to a certain extent quantitatively—see
ful to the FWO-Vl for the support of the research stay at the
Figs. and reproduce the features under discussion
DML 共ASI, RIKEN兲, and to F.N. for hospitality. M.K. grate-
of the calculated *J *共⌽兲
curves, i.e., the enhancement of
fully acknowledges support from the Carl-Zeiss-Stiftung,
*J *共⌽兲
for ⌽⬎⌽1. A more detailed quantitative comparison
and D.B. from the Evangelisches Studienwerk e.V. Villigst.

will require additional analysis of *J *共⌽兲
for different param-
F.N. acknowledges partial support from the Laboratory of
eters of pinning arrays 共e.g., radii of the antidots兲 and various
Physical Sciences, National Security Agency, Army Research
temperatures. Such analysis is beyond the scope of the
Office, DARPA, AFOSR, National Science Foundation under
present work and will be reported
Grant No. 0726909, JSPS-RFBR under Contract No. 09-02-
A significant enhancement of *J *共⌽兲
for ⌽⬎⌽1 is ob-
92114, Grant-in-Aid for Scientific Research 共S兲, MEXT Kak-
tained with the Shield 共with squares, triangles, and
enhi on Quantum Cybernetics, and Funding Program for In-
threefold hexagons兲 关Fig. The large tiles are excellent
novative R&D on S&T 共FIRST兲.

1 J. Gutiérrez, A. Llordés, J. Gázquez, M. Gibert, N. Romà, S.

Van Bael, K. Temst, R. Jonckheere, and Y. Bruynseraede,
Ricart, A. Pomar, F. Sandiumenge, N. Mestres, T. Puig, and X.

3 J. E. Villegas, S. Savel'ev, F. Nori, E. M. Gonzalez, J. V. An-
2 V. V. Moshchalkov, M. Baert, V. V. Metlushko, E. Rosseel, M. J.

guita, R. García, and J. L. Vicent,
ENHANCING THE CRITICAL CURRENT IN…
PHYSICAL REVIEW B **82**, 184512 共2010兲

4 R. Wördenweber, P. Dymashevski, and V. R. Misko,
25 K. Harada, O. Kamimura, H. Kasai, T. Matsuda, A. Tonomura,
and V. V. Moshchalkov,
5 V. R. Misko, S. Savel'ev, A. L. Rakhmanov, and F. Nori,
26 Small variation in this factor is related to slightly different num-
bers of pinning sites which fit within area *A*
*qp *for different QP
P. Selders and R. Wördenweber,
27 M. Kemmler, D. Bothner, K. Ilin, M. Siegel, R. Kleiner, and D.

7 P. Lahl and R. Wördenweber,
8 P. Bushev, D. Bothner, J. Nagel, M. Kemmler, K. Konovalenko,
28 Note that while QP APS contain an infinite number of periodici-
A. Loerincz, K. Ilin, M. Siegel, D. Koelle, R. Kleiner, and F.

ties, the corresponding diffraction patterns in the reciprocal *k*
space form a QP hierarchical self-similar set of ␦ peaks of de-
V. Misko, S. Savel'ev, and F. Nori,
creasing intensity. As a result, only few periodicities turn out to
10 V. R. Misko, S. Savel'ev, and F. Nori,
be significant in QP APS.

We would like to emphasize that using various quasiperiodic
11 J. E. Villegas, M. I. Montero, C.-P. Li, and I. K. Schuller,
pinning arrays we increase the critical current *J *共⌽兲
below 共⌽⬍⌽ 兲
and above the matching flux 共⌽⬎⌽1 where
12 M. Kemmler, C. Gürlich, A. Sterck, H. Pöhler, M. Neuhaus, M.

*J *共⌽兲
decreases very rapidly in periodic APS. Thus by broaden-
Siegel, R. Kleiner, and D. Koelle,
ing the maximum in *J *共⌽兲
we significantly *enhance *the critical
current as compared to the case of periodic APS. However, we
13 A. V. Silhanek, W. Gillijns, V. V. Moshchalkov, B. Y. Zhu, J.

do *not *increase the magnitude of the *J *共⌽兲
peak at ⌽ = ⌽1.

Moonens, and L. H. A. Leunissen,
30 R. Paredes, J. L. Aragón, and R. A. Barrio,
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31 W. V. Pogosov, V. R. Misko, H. J. Zhao, and F. M. Peeters,
V. V. Moshchalkov,
32 B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N.

16 D. Levine and P. J. Steinhardt,
Christodoulides, and J. W. Fleischer,
17 H. C. Jeong and P. J. Steinhardt,
Shahar Even-Dar Mandel and R. Lifshitz,
18 P. J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe, and
33 Y. Watanabe, M. Ito, and T. Soma,
19 *Quasicrystals: The State of the Art*, 2nd ed., edited by P. J. Stein-
hardt and D. P. Divincenzo 共World Scientific, Singapore, 1999兲.

34 M. Baake, P. Kramer, M. Schlottmann, and D. Zeidler,
20 *Quasicrystals*, edited by J.-B. Suck, M. Schreiber, and P. Häus-
sler 共Springer, Berlin, 2002兲.

35 D. Bothner, M. Kemmler, V. R. Misko, F. M. Peeters, F. Nori, R.

21 B. Grünbaum and G. C. Shephard, *Tilings and Patterns *共W.H.

Kleiner, and D. Koelle, 共unpublished兲.

Freeman, New York, 1987兲.

36 F. Gähler, in *Quasicrystalline Materials*, edited by Ch. Janot and
22 R. A. Richardson, O. Pla, and F. Nori,
J. M. Dubois 共World Scientific, Singapore, 1988兲.

37 J. C. Crocker and D. G. Grier,
24 C. J. Olson, C. Reichhardt, and F. Nori,
38 S. Tung, V. Schweikhard, and E. A. Cornell,

Source: https://www.physik.uni-stuttgart.de/TR21/common/show_file.php/publications/423/publication.pdf

Cult Med Psychiatry (2010) 34:132–168DOI 10.1007/s11013-009-9163-1 Post-Soviet Placebos: Epistemology and Authorityin Russian Treatments for Alcoholism Published online: 5 December 2009 ! Springer Science+Business Media, LLC 2009 The dominant modalities of treatment for alcoholism in Russia are suggestion-based methods developed by narcology—the subspecialty of Russianpsychiatry which deals with addiction. A particularly popular method is the use ofdisulfiram—an alcohol antagonist—for which narcologists commonly substituteneutral substances. Drawing on 14 months of fieldwork at narcological clinics inSt. Petersburg, this article examines the epistemological and institutional conditionswhich facilitate this practice of ‘‘placebo therapy.'' I argue that narcologists'embrace of such treatments has been shaped by a clinical style of reasoning specificto a Soviet and post-Soviet psychiatry, itself the product of contested Soviet politicsover the knowledge of the mind and brain. This style of reasoning has facilitatednarcologists' understanding of disulfiram as a behavioral, rather than a pharmaco-logical, treatment and has disposed them to amplify patients' responses throughattention to the performative aspects of the clinical encounter and through man-agement of the treatment's broader reputation as an effective therapy. Moreover,such therapies have generally depended upon, and helped to reinforce, clinicalencounters premised on a steeply hierarchical physician–patient relationship.

A21045 schizophrenia doc 2 28/5/04 12:46 pm Page 1 Support and Guidance forEveryone Affected by Schizophrenia A21045 schizophrenia doc 2 28/5/04 12:46 pm Page 2 Inside front cover Copyright © Bristol-Myers Squibb Company and Otsuka Pharmaceutical Co., Ltd. 2004 All rights reserved, including the right of reproduction in whole or in part in any form.