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 1Departement Fysica, Universiteit Antwerpen, B-2020 Antwerpen, Belgium 2Advanced Science Institute, RIKEN, Wako-shi, Saitama 351-0198, Japan 3Physikalisches Institut-Experimentalphysik II and Center for Collective Quantum Phenomena, Universität Tübingen, D-72076 Tübingen, Germany 4Physics 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.
B1= 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 = fi = f
model 共see Ref. , the maximum value of the critical de- i is the total force per unit length acting on vortex i, fi
pinning current in quasiperiodic pinning arrays of antidots and fvp are the forces due to vortex-vortex and vortex-pin
reached the values of about 0.5 of the Ginzburg-Landau de- interactions, respectively, fT is the thermal stochastic force,
pairing current, for the used parameters.
and fd is the driving force; v
i is the velocity and ␩ is the viscosity. All the forces are expressed in units of f0= ⌽2/ 8␲2␭3, where ⌽ IV. CRITICAL CURRENT IN QP TILING APS
0 = hc / 2e, 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 fvp = 兺 p 兩r
共p兲兩⌰冉rp− 兩ri− rk rˆ共p兲, 共2兲 a function of applied magnetic flux, for different QP tilings.
i − rk
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 rˆ共p兲
son, Fig. also shows the J 共⌽兲 dependence for a fivefold i − r
i − rk
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 j0, 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 j0 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dAPS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 dAPS/ ␭. 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 dAPS/ ␭ 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.
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