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Collective vortex rectification effects in films with a periodic pinning array

Clecio C. de Souza Silva,1 J. Van de Vondel,1 B. Y. Zhu,2 M. Morelle,1 and V. V. Moshchalkov1, ∗

1Nanoscale Superconductivity and Magnetism Group,Laboratory for Solid State Physics and Magnetism,

Katholieke Universiteit Leuven, Celestijnenlaan 200 D, B-3001 Leuven, Belgium2Quantum Phenomena Observation Technology Laboratory,

The Institute of Physical and Chemical Research(RIKEN) and AdvancedResearch Laboratory Hitachi Ltd. Hatoyama, Saitama 350-0395, Japan

(Dated: February 2, 2008)

The vortex ratchet effect has been studied in Al films patterned with square arrays of submicronantidots. We have investigated the transport properties of two sets of samples: (i) asymmetricalantidots where vortices are driven by an unbiased ac current, and (ii) symmetrical antidots wherein addition to the ac drive a dc bias was used. For each sample, the rectified (dc) voltage ismeasured as a function of drive amplitude and frequency, magnetic field, and temperature. Asunambiguously shown by our data, the voltage rectification in the asymmetric antidots is inducedby the intrinsic asymmetry in the pinning potential created by the antidots, whereas the rectificationin the symmetric antidots is induced by the dc bias. In addition, the experiments reveal interestingcollective phenomena in the vortex ratchet effect. At fields below the first matching field (H1),the dc voltage–ac drive characteristics present two rectification peaks, which is interpreted as aninterplay between the one-dimensional motion of weakly pinned incommensurate vortex rows andthe two-dimensional motion of the whole vortex lattice. We also discuss the different dynamicalregimes controlling the motion of interstitial and trapped vortices at fields slightly higher than H1

and their implications for the vortex ratchet effect.

PACS numbers: 74.78.Na., 74.40.+k, 05.40.-a, 85.25.-j

I. INTRODUCTION

Ratchets are systems with an asymmetric periodicpotential capable of promoting unidirectional transportof classical or quantum objects subject to an unbi-ased fluctuating drive. For that purpose, the exter-nal forcing must be a non-equilibrium one, like an al-ternating force or a colored noise, since useful workcan not be extracted from equilibrium fluctuations only.The ratchet effect was first used to explain biochem-ical mechanisms such as the intracellular transport ofmacromolecules.1 Since then, parallel to advances in thetheoretical understanding,2 several devices have beenproposed and fabricated in order to test the ratchet ideaat the classical and quantum levels and to use the ratcheteffect for particle segregation and motion rectification ofelectrons and fluxons.3 Most of this work, however, is fo-cused on ratchet systems consisting of single objects oran assembly of weakly interacting objects. Much less isknown about the collective rectification of strongly in-teracting systems via the ratchet effect and work on thissubject carried out so far is mostly theoretical.

Vortices in superconductors with a periodic array ofpinning centers have proven to be an ideal system inwhich to study collective dynamical phenomena of par-ticles moving in a periodic potential. Several interestingdynamical effects have already been studied in these sys-tems. For instance, as demonstrated theoretically4 andexperimentally5,6,7, periodic pinning potentials are ableto guide vortex motion into high symmetry directions ofthe pinning lattice. Another remarkable dynamical ef-

fect in these systems is the presence of Shapiro steps inthe voltage-current characteristics when an rf current iscoupled to the dc drive.8 The steps appear due to phase-locking between the rf signal and the vortex motion overthe periodic pining structure.

Controllable motion of vortices by means of the ratcheteffect was recently demonstrated in experiments per-formed on superconducting samples with periodic arraysof asymmetric pinning centers9,10 and for asymmetricconfigurations of symmetrical pinning sites.7 These struc-tures are able to break the symmetry of the vortex pin-ning potential and promote vortex motion rectificationwhen an unbiased ac current is applied. In Ref. 9, forinstance, the vortex ratchet effect was demonstrated ina Nb film where the asymmetrical pinning potential wasgiven by an array of triangular magnetic dots. In Ref. 10,we observed pronounced rectification effects in Al filmswhere the symmetry of the pinning potential was bro-ken by engineering a composite configuration of antidotlattices with big and small antidots placed close to eachother.

Despite a great deal of theoretical efforts11,12,13,14,15

aimed at the description of the ratchet dynamics of vor-tices in asymmetrical pinning potentials, these systemsare far from being completely understood. The vortexmotion itself, in the presence of strong pinning sites, isstill a subject of intense study. Differently from manyother systems, vortices are soft objects; their core is ableto deform to better adjust to the available pinning po-tential. Due to this property, the usual approximation oftreating vortices as localized overdamped particles maybreak down and the resulting vortex ratchet effect can

2

be quite different from, for instance, the effects knownfor overdamped Brownian ratchets. Indeed, as we haverecently shown, the dynamics ruling the voltage rectifica-tion in a superconducting film with an array of asymmet-rical antidots can be described by underdamped equa-tions of motion.10 This means that this system can beregarded as an example of inertia ratchet systems, eventhough the true mass of a vortex is negligible.

In the present paper, we investigate in details vortexmotion rectification in Al films with nanoengineered an-tidot arrays as a function of ac excitation, magnetic fieldand temperature. Two systems are investigated: (a)square arrays of asymmetrical antidots excited by a sym-metrical ac current, an intrinsic ratchet system, and (b) asquare array of symmetrical antidots excited by an ac cur-rent coupled to a dc bias, i.e. a tilted-potential ratchet.For fields below the first matching field, H1 = Φ0/a2

p

(where Φ0 = h/2e is the flux quantum and ap is the an-tidot lattice constant), the vortex response in both sys-tems is characterized by hysteresis, which suggests under-damped motion, and a double rectification peak at someincommensurate fields. These results are interpreted interms of a minimal inertia ratchet model and moleculardynamics simulations of “inertial” vortices. The simu-lation results indicate that the double peak observed atincommensurate fields can be explained in terms of plas-tic deformation of the vortex lattice. For fields higherthan H1, our data reveal that the motion of interstitialvortices is essentially overdamped, in contrast with theunusual underdamped dynamics of the vortices trappedby the antidots. This result suggests that the underlyinginertial effect is connected to the strong interaction be-tween the vortices and the antidots and not to the realmass of the vortices.

The paper is organized as follows. In Sec. II, we de-scribe the sample preparation, the experimental proce-dure and the experimental results. In Sec. III, we presentthe details of our underdamped model and the moleculardynamics simulations. The experimental data, as wellas a comparison between modelling and experiment, arediscussed in Sec. IV. Finally, in Sec. V, we draw ourconclusions.

II. EXPERIMENTAL RESULTS

A. Sample preparation and experimental details

The experiments where performed on three differentAl thin films, thermally evaporated on a SiO2 substratepatterned by electron beam lithography. For all samplesthe pattern was designed in a cross shaped geometry toallow four point measurements in the two perpendicu-lar directions [see Fig. 1.(a)]. The cross consists of two300-µm-wide strips containing the nanoengineered array(period ap = 1.5 µm) of pinning sites. This gives a valueof 0.92 mT for the first matching field. The first twosamples (AAD1 and AAD2) have the same unit cell con-

TABLE I: Superconducting parameters for the three samplesstudied.

AAD1 AAD2 SAD

Thickness t (nm) 38 37 22

Critical temperature Tc (K) 1.469 1.438 1.532

Mean free path le (nm) 3.0 8.1 3.4

Coherence length ξ(0) (nm) 59 97 63

Penetration depth λ(0) (nm) 225 136 210

GL parameter κ 3.83 1.40 3.32

sisting of a small (300 × 300 nm2) and big (600 × 600nm2) antidot separated by a thin superconducting wall(90 nm) and approximate thicknesses (38 and 37 nm).The third sample (SAD) has a symmetric unit cell con-sisting of 600 x 600 nm2 square antidots and a thicknessof 22 nm. Figs. 1 (b) and (c) show atomic force micro-graphs (AFM) of samples AAD2 and SAD, respectively.The dashed lines depict the unit cell (1.5 × 1.5 µm2) ofthe respective sample. The white dots observed in theimages are particles of resist layer which did not come offduring the lift off procedure. These particles are innocu-ous to our experiment.

Table I shows the superconducting parameters for allthree samples. The superconducting critical tempera-ture Tc was obtained by using a resistance criterion of10% of the normal state resistance in a zero-field mea-surement. The resistive transition for all samples wassharp, with a width typically of ∆Tc ∼ 3mK. From theresidual resistivity at 4.2 K, the electron mean free pathle was found. From this we have calculated the coher-ence length ξ(0) and the penetration depth λ(0) usingthe dirty limit expressions. All samples are type-II su-perconductors, i.e., their Ginzburg-Landau parameter isκ = λ(0)/ξ(0) > 1/

√2. Note that, although samples

AAD1 and AAD2 share the same pattern and have simi-lar thicknesses, their mean free paths were quite different,which is a result of the different evaporating conditions(base pressure and growth rate) in which the sampleswere prepared.

The transport measurements were performed in a cryo-stat using ac and/or dc currents applied at the end of

FIG. 1: Layout of the Al samples. (a) Cross shaped geometryof the samples. AFM images of sample AAD2 (b) and SAD(c). The dashed lines indicate the corresponding unit cell.

3

the cross legs in the x-direction. The magnetic field wasapplied perpendicularly to the xy plane. In all ac mea-surements, the applied ac current was a sinusoidal wave,i.e., I(t) = Iac sin(2πft), where f is the excitation fre-quency and Iac is its amplitude. The voltage was ac-quired in four point mode by an oscilloscope, used as ananalogue-digital converter. The waveform was averagedover several cycles (typically 200-500 cycles) in order toenhance the signal-to-noise ratio. The dc voltage wasthen obtained by integrating the “clean” waveform.

B. Rectification by unbiased asymmetrical antidots

Fig. 2 (a) shows a contour plot of the dc voltage acrossthe sample (Vdc) as a function of reduced magnetic field(H/H1) and current amplitude (Iac) at a temperatureT = 0.98Tc and excitation frequency f = 1 kHz. AtH = 0, one can observe a clear sign inversion of the dcvoltage. Note that the voltage drop due to vortex motionis given by Vdc = L·(〈v〉×B), where B is the flux density,〈v〉 is the time averaged vortex velocity, and L is thedistance between the voltage contacts. Thus, the changeof Vdc sign as H (and, consequently, B) changes its signmeans that vortices and antivortices are rectified to thesame direction. This simple assumption demonstratesunambiguously the intrinsic rectifying properties of theasymmetrical antidots configuration. A similar behaviorof Vdc(H, Iac) was also observed in sample AAD1 and itsplot for the positive field side was presented in Ref. 10.Another important effect demonstrated by Fig. 2 (a) is aclear enhancement of the rectified voltage near the firstmatching field (see also Ref. 10, Fig. 2). This can beexplained by the high symmetry of the vortex lattice atthis field range which induces coherent vortex motion,that is, all vortices contribute to the rectified voltage ina constructive way.

In panels (b) and (c) of Fig. 2, we show detailed plots ofthe Vdc-Iac characteristics of samples AAD2 and AAD1,respectively. Similarly to single-object ratchet systems,these curves are characterized by a rectification window,in which the rectified voltage increases monotonically,and by the presence of a tail, in which Vdc drops smoothlytowards zero. The rectification window is defined by twocritical forces, F1, which marks the onset of rectification,and F2, for which rectification is maximum. These forcesalso determine the effective asymmetry of the pinningpotential, which we define here as α = 1 − F1/F2. Forsome field values (usually, for fields close to H1 or H1/2)a second rectification peak is observed, suggesting an in-terplay between two ratchets (c.f. Sec. IV), each havingtheir own critical forces [see Fig. 2(c)]. However, a clearsecond peak was not identified in the measurements onsample AAD2, which is probably due to a lack of res-olution in our measurements in the narrow rectificationwindow of this sample.

For fields higher than H1, a reversal in the net vortexmotion is observed at lower amplitudes [see the Vdc(Iac)

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ac amplitude (mA)- V

dc (m

V)

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Sample AAD2

Vdc (mV)

H/H1

ac a

mpl

itude

(mA

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(b) Sample AAD2

H/H1 0.95 0.97 1.03

(c) Sample AAD1

H/H1 0.49 0.82 0.96

FIG. 2: (a) Contour plot of the dc voltage measured acrosssample AAD2 as a function of reduced field and ac ampli-tude at T = 0.98Tc, and frequency of 1 kHz. (b) Detail ofthe amplitude dependence of the dc voltage shown in (a) forfields H/H1 = 0.95, 0.97 and 1.03. (c) Ac-drive amplitudedependence of the dc voltage measured across sample AAD1at T = 0.976Tc and f = 1 kHz for fields H/H1 = 0.49, 0.82and 0.96.

curve at H/H1 = 1.03 in Fig. 2(b)]. In Ref. 9, a similarvoltage reversal was observed at fields above 3H1, whichwas the saturation field of the sample studied. The au-thors interpreted this result by arguing that the vorticestrapped by the triangular magnetic dots arrange them-selves in such a way that the interstitial vortices expe-rience an effective asymmetric potential with oppositeasymmetry. In our experiment, the saturation field wasH1. This can be inferred from the strong drop of thecritical current just above this field, which indicates thatweakly pinned interstitial vortices come into play. Then,in fact, the voltage reversal observed at fields higher thanH1 can also be interpreted as a reversal in the recti-fication of interstitial vortices. However, in our case,the reverse asymmetry is not induced by an asymmet-ric arrangement of vortices inside the pins, but rather bythe current distribution (encircling the asymmetric an-tidot configuration) created by a single-quantum vortextrapped in the double-antidot pins.

By comparing the results obtained on samples AAD1and AAD2, one can notice the striking difference in therectified voltage of both samples. For the similar reducedtemperatures, the maximum dc voltage of AAD2 was sev-eral times smaller than that of AAD1. This can be ex-plained by the fact that the coherence length of AAD2

4

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ac Amplitude (mA)

T/Tc 0.976 0.978 0.980 0.982 0.984 0.986 0.988 0.990

Max. V

dc (mV

)

T/Tc

FIG. 3: Dc voltage measured across sample AAD2 versusac amplitude for H/H1 = 0.95 and f = 1 kHz at differenttemperatures, as indicated. Inset: Plot of the maximum dcvoltage at this field as a function of temperature.

at temperatures close to Tc may be bigger than the anti-dots (for instance ξAAD2(T = 0.97Tc) ≃ 560 nm). In thissituation, the vortices are not able to resolve the detailsof the antidots and the effective asymmetry is stronglysuppressed. This is very clear by calculating the asymme-try ratio of both samples at a relatively low temperatureT = 0.976Tc (where thermal escape can be neglected)and reduced field H/H1 = 0.96, which gives α ≈ 0.09 forAAD1 and α ≈ 0.02 for AAD2.

The coherence length, can also play an important rolein the temperature dependence of the rectified voltage.Fig. 3 shows Vdc-Iac characteristics of sample AAD2 atseveral temperatures for the same reduced field H/H1 =0.98. The maximum rectified voltage seems to increasemonotonically until a temperature T ≃ 0.982Tc, abovewhich it starts to drop, resulting in a bell-shaped T de-pendence (see inset). In Brownian ratchets, a decreasein the dc response as the temperature increases is in-deed expected.16 The reason is that, when the Brownianmotion of particles is enhanced by thermal noise, moreenergy is needed to move them in a given direction. How-ever, in the vortex case near the critical temperature,increasing temperature induces not only vortex fluctua-tions but also a strong increase of the vortex size, whichleads to a decrease in the effectiveness of the pinningpotential and its asymmetry. At low temperatures, thereduction of fluctuations and vortex size should lead to amonotonic increase in rectification. Nevertheless, whenvortices are small enough and their fluctuations are weak,they are more sensitive to the background pinning causedby the natural inhomogeneities of the sample. Due to thedisorder induced by the background pinning, the anti-dot pinning potential may become less effective at lowertemperatures, thus causing a reduction of the rectifiedvoltage.

In Fig. 4(a)–(d) we show the voltage output waveform

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221 µA292 µA

(g)

- V(m

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Time/Period

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Sample AAD1

320 µA413 µA479 µA

H/H1 = 1.50 (h)

H/H1 = 0.82

Time/Period

-12-606

12 (a)

(b)

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01020

Sample SAD

T = 0.973 TcH/H1 = 0.96

(c)

(d)

FIG. 4: (a)–(d) Time evolution of the voltage output of sam-ple AAD1 at a temperature T = 0.973Tc, field H = 0.96H1,and ac amplitudes Iac = 477 (a), 502 (b), 544 (c) and 704 µA(d). (e),(f) Voltage output of sample SAD at T = 0.966Tc,H = 0.82H1, a dc bias Idc = 64 µA, and ac amplitudes: Iac =553 and 569 µA (e); Iac = 624 and 680 µA (f). (g),(h) Volt-age output of sample SAD at T = 0.966Tc, H = 1.50H1, a dcbias Idc = 64 µA, and ac amplitudes: Iac = 102, 221 and 292µA (g); and Iac = 320, 413, and 479 µA (h).

in one cycle of the ac drive for different drive amplitudesof the Vdc-Iac characteristic presented in Fig. 2(c) (opencircles). Note that, as discussed above, here negativevoltage corresponds to positive vortex velocity, since theapplied field is positive. Panels (a) and (b) correspondto points in the rectification window, where the ratchetsystem behaves as a half-wave rectifier, whereas (c) and(d) correspond to points in the rectification tail. At eachhalf-period of the ac drive, the corresponding half-periodof the voltage output in all cases (a-d) is asymmetric; vor-tices seem to be depinned at a given force and repinnedat a lower force value, giving rise to hysteretic behavior.Such a hysteresis was also observed in sample AAD2, butsomewhat smoothed due to the strong effects of fluctua-tions in this sample. As we shall see below, this indicatesthat the system is ruled by underdamped dynamics.

C. Rectification by dc biased symmetrical antidots

Since the symmetrical antidot configuration of sam-ple SAD does not produce rectified vortex motion by it-self, we externally induce an asymmetry in the systemby “tilting” the pinning potential with a small dc cur-rent bias. In Fig. 5(a), we show the contour plot of the

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Vdc(mV)

T = 0.966 TcFreq. = 1 kHzdc offset = 64 µA

H/H1

ac A

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itude

(mA

)

8.06.75.34.02.71.30

(b)

H/H1 0.43 0.82 1.09

(c)

Idc (µA) 130 46 22

FIG. 5: (a) Contour plot of the dc voltage measured acrosssample SAD as a function of reduced field and ac amplitudeat T = 0.966Tc, frequency of 1 kHz, and a dc bias of 68µA. (b) Detail of the amplitude dependence of the dc voltageshown in (a) for fields H/H1 = 0.43, 0.82, and 1.09. Thedashed line indicates the saturation voltage Vsat = RnIdc =2.8 mV. (c) Ac-drive amplitude dependence of the dc voltageat T = 0.973Tc, f = 1 kHz, H/H1 = 0.93, and different tilts,Idc =130, 46 and 22 µA.

dc voltage across sample SAD in the H-Iac plane at atemperature T = 0.966Tc for a dc bias of Idc = 64 µAand ac excitations of 1 kHz. [Note that the steps observedin this plot are just artifacts induced by the low resolu-tion of the field sweep (0.05 mT).] In this experiment, noflip of the dc voltage sign is observed. This is indeed ex-pected because here the polarity of a vortex does matterin determining its net velocity. For instance, for a posi-tive current bias, i.e., in the positive x direction, a vortexwith positive polarity experiences an positive slope in thepinning potential (due to the Lorentz force pointing tothe negative y direction), whereas an antivortex senses anegative slope. Thus, as the magnetic field changes itssign, so does the asymmetry of the pinning potential, insuch a way that the cross product 〈v〉 ×B keeps its sign.

In Fig. 5(b), we present a detailed plot of the Vdc-Iac

characteristics at fields H/H1 = 0.43, 0.82, and 1.09,for the experiment of Fig. 5(a). Here, in comparisonto the Vdc-Iac characteristics of the asymmetric antidotsamples (Fig. 2), the important difference is that the rec-tification tail approaches asymptotically a finite voltagevalue, which depends on the applied dc tilt, whereas the

rectification tail of the intrinsically asymmetric samplesreaches zero voltage at high ac drives. In the rectificationwindow, however, samples SAD and AAD1 behave in asimilar way. For instance, at fields close to H1, the sec-ond rectification peak is also seen in the Vdc-Iac curvesof sample SAD.

Fig. 5(c) presents the rectification effect in sampleSAD for different dc bias, Idc = 22, 46, and 130 µA atH/H1 = 0.93 and T/Tc = 0.973. At this field, a doublerectification peak is observed for all dc biases. The dis-tance between peaks seems to be constant, whereas therectification window is wider for higher tilt values. Thewindow is roughly defined by the critical values Ic − Idc

and Ic + Idc, where Ic = 501 µA is the critical depinningcurrent obtained from conventional dc transport mea-surements. Thus, the system is given an asymmetry ofα = 2Idc/(Ic + Idc). For the tilt Idc = 22 µA, this givesα ≈ 0.086, which is similar to the intrinsic asymmetryobserved in sample AAD1 at T = 0.973Tc. Another note-worthy point is that the main rectification peak decreasesrelatively to the saturation voltage for higher tilts. Thisis consistent with the fact that, in the limit of the criticaltilt (Idc = Ic), no rectification peak should be observedbecause there the energy barrier for the easy flow direc-tion disappears completely. In this limiting case, Vdc(Iac)is expected to grow monotonically from zero to the sat-uration voltage.

The time evolution of the voltage output for differ-ent drive amplitudes of the Vdc-Iac characteristics shownin Fig. 5(b) is presented in Fig. 4(e)–(h). We considertwo different fields, H/H1 = 0.82 (e and f) and 1.50 (gand h), corresponding, respectively, to filled circles andcrossed circles of Fig. 5(b). In the H/H1 = 0.82 case, theasymmetrical shape of the half-periods of V (t) resemblesthat observed in sample AAD1 [shown in panels (a)–(d)of this figure], that is, here a hysteresis in the depinning-repinning process is also observed. The main differenceis that in the high drive limit [panel (f)], the amplitudeof the positive half-cycle is higher than the amplitudeof the negative one, due to the extrinsic nature of theasymmetry in sample SAD, whereas for sample AAD1the amplitude of both half-cycles are approximately thesame.

For fields higher than H1 [Fig. 4 (g) and (h)], there ex-ists an amplitude range for which the voltage half-cyclesare symmetrical, that is, no hysteresis is observed. Sincethis behavior is only present at low ac amplitudes, wecan assume that the symmetrical waveforms correspondto motion of interstitial vortices. At H/H1 = 1.5, theresponse is strongly enhanced in the positive half-cycleabove an amplitude Iac = 292 µA, suggesting that at thisvalue the vortices trapped by the antidots start moving.The voltage waveform corresponding to this amplitudeand above are asymmetrical [panel (h)], resembling thewaveforms observed at fields below H1. A similar evo-lution of the voltage waveforms as the ac amplitude isvaried was also observed in the samples with asymmetri-cal pinning sites for fields H > H1.

6

III. THEORETICAL MODELLING

A. Inertia ratchet model

Our experimental data clearly demonstrate a pro-nounced hysteresis in the voltage as a transport current iscycled. Typical systems exhibiting pronounced hystere-sis in their transport properties are driven underdampedparticles in periodic potentials. In these systems, thehysteresis in the depinning-repinning process is a conse-quence of the finite inertial mass of the particles, whichhas important effects on their transport properties. Forinstance, in microscopic boundary lubrication problems,the depinning of the (inertial) lubricant particles from theperiodic potential of a solid substrate is usually identifiedas the static friction force whereas the repinning of theparticles is identified as the dynamical friction force.17

Other “inertia” ratchet systems are asymmetrical under-damped Josephson junction arrays18 and dc SQUIDs19.In these two examples, the rectified object is the phaseof the superconducting order parameter across a junctionand the inertia is given by the shunt capacitance of thejunctions. This problem is formally the same as that ofthe mechanical inertia ratchet.

The rectification of inertial particles in a ratchetpotential has been studied theoretically by severalgroups.18,19,20,21 The dynamics of inertia ratchets is morecomplex then their overdamped counterpart due to theexistence of both regular (for moderate underdamped)and chaotic (for extremely underdamped) dynamics. Inthe regular regime, the amplitude dependent dc responsepresents a strong enhancement in the rectification win-dow and a faster decaying tail, as compared to the char-acteristics of overdamped particles. Inspired by the factthat, in our experiment, (i) the rectified voltage demon-strates hysteretic behavior and (ii) the Vdc-Iac character-istic generally has pronounced rectification effect in therectification window, we propose a description of vor-tex dynamics in antidot systems in terms of a simplifiedunderdamped model, in which vortices (still treated asparticles) are given an apparent mass to account for theobserved hysteresis.

We first consider the simpler case of a single vortexin a ratchet potential, here representing the asymmetricantidot array. The equation of motion is given by

Mx = −ηx − U ′p(x) + A sin(ωt), (1)

where η is the viscous drag coefficient, x is the particleposition. The pinning potential Up(x) is modelled in asimplified way as the double-sine function:

Up(x) = −U

[

sin

(

2πx

ap

)

+β

2sin

(

4πx

ap

)]

(2)

The parameter β defines the asymmetry of the pinningforce (α = 2β/(1 + β), for β < 1/4), the critical forcesbeing given by Fw = 2π(1 − β)U/ap, for drive along the(weak) positive x direction, and Fs = 2π(1 + β)U/ap,

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itude

00.090.170.26

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<v>

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FIG. 6: Time averaged particle velocity as a function ofac drive amplitude calculated in the single-particle inertiaratchet model [Eq. 1] for the viscosity-to-mass ratios η/M =0.1, 2.5, 4 and 10 (in units of fp). Inset: Contour plot of aver-age velocity as a function of amplitude and η/M ; the dashedlines define the rectification window in this case (α = 0.095).

for drive along the (strong) negative x direction. Notethat, since here we do not take into account thermalfluctuations, the critical forces which define the rectifi-cation window of this system are given deterministicallyby F1 = Fw and F2 = Fs.

The main characteristic frequencies setting the limitsof the dynamical regimes of Eq. (1) are the libration fre-quency in the potential minima, which for α <∼ 0.2 is

given approximately by fp =√

U/Ma2p, and the inverse

relaxation time γ = η/M . The quasi-adiabatic regime(i.e., close to the zero-frequency adiabatic limit), corre-sponds to frequencies much smaller than these two char-acteristic frequencies. In the numerical solution of Eq. (1)we have chosen excitation frequencies f < 10−4fp andγ ≥ 0.1fp, corresponding to the quasi-adiabatic regime.

In Fig. 6, we show the net dc velocity versus ac driveamplitude (〈v〉-Fac characteristics for different values ofγ. Forces are in units of 2πU/ap and frequencies (andγ) in units of fp. For these runs, we have chosen theasymmetry factor α = 0.095, which is close to the typicalvalues of I1/I2 observed in sample AAD1. The insetshows a detailed diagram of 〈v〉 as a function of A and M .Notice the crossover between the spread rectification inthe extreme overdamped limit and the sharp rectificationin the extreme underdamped regime. For η/M >∼ 4fp, thecurves are essentially overdamped, following the behaviorof the adiabatic solution of a deterministic overdampedratchet.10 For η/M <∼ 2fp, the curves are much morepronounced in the rectification window and have a fasterdecaying tail. We shall compare these results with theexperimental data in Sec. IV.

7

B. Molecular dynamics simulations

Now, we consider the 2D problem of a lattice of “iner-tial” vortices interacting with a 2D array of asymmetricpinning sites, which is modelled as two interpenetratingsquare sublattices of strong and weak attractive poten-tials. The sublattices have spacing ap (=1 here) and areshifted with respect to each other by a distance d = 0.2ap.All pinning centers are modelled by Gaussian potentialwells with a decay length Rp. Hence, the pinning forceexerted by the k-th pin of one sublattice on the i-th vor-tex reads

Fs,wp (ri) = −Fs,w

ri − Rs,wk

Rp

exp

(

−∣

∣

∣

∣

ri − Rs,wk

Rp

∣

∣

∣

∣

2)

,

(3)where ri represents the position of the i-th vortex andR

s,wk is the location of the k-th pin of the weak (w) and

the strong (s) sublattice, respectively. The intensity ofthe individual pinning force is denoted by Fs,w. We de-fine the ratio between the weak and strong pinning forcesas ǫ = Fw/Fs. In our simulation, forces (per unit length)are taken in units of F0 = Φ2

0/8π2Λ3, with Λ the ef-fective superconducting penetration depth. Vortices areconsidered to interact logarithmically, the correspondingrepulsive vortex-vortex force being given by

Fvv(ri) = Fvv0

Nv∑

j 6=i

Λrij/|ri − rj | ,

where rij = (ri − rj)/|ri − rj |. Fvv0 denotes the strengthof the vortex-vortex interaction. The underdamped equa-tion of motion for vortex i is given by

M ri = FL(t) + Fvv(ri) + Fp(ri) − ηri . (4)

The ac driving Lorentz force is FL(t) = A sin(ωt), whereω = 2π/P and P is the period of the ac drive. η is theviscosity coefficient and M is the effective mass of anindividual vortex. This equation is integrated by molec-ular dynamics (MD) simulations. Parameters used in thesimulations are Rp = 0.13a0, Λ = 0.26a0. Fvv0 = 0.06,F s

p0 = 0.5 and ǫ = 0.925, time step τ0 = 1/150. Allthe frequencies of the ac sine-wave drive are smaller than10−6/τ0, i.e., into the quasi-adiabatic regime. For themass and viscosity we have chosen, respectively, M = 1

and η = 1, which, in terms of fp =√

Up/Ma2p (here

Up = FsR2p/2 is the amplitude of the individual strong

pinning potential), leads to a viscosity-to-mass ratioη/M = 2.44fp. The MD simulation results will be dis-cussed in the next section.

IV. DISCUSSION

In Fig. 7, we present a fitting of our inertia ratchetmodel to the experimental data of samples AAD1 and

SAD. The data is plotted in normalized units. The nor-malized critical forces used in the model were the sameobserved in the respective measurement. In this way,the only adjustable parameter is the viscosity-to-massratio, γ = η/M , which is given in units of the libra-tion frequency fp. For comparison, we have shown inFig. 6 the Vdc-Iac calculated close to the overdampedlimit γ = 10. As it is clearly seen from this graph, theusual overdamped model fails to describe the ratchet ef-fect of vortices in an antidot lattice, more specifically inour Al samples AAD1 and SAD. On the other hand, theunderdamped model with γ values close to the under-damped limit provides very good fitting to our experi-mental data. For the measurement of sample AAD1 atT = 0.973Tc and H/H1 = 0.99, for example, we usedα = 0.095 (which gives approximately the same normal-ized critical forces observed in this experiment) and thebest fit was obtained with γ = 1.8 [see Fig. 7 (a)–inset],which is close to the underdamped limit. For the sampleSAD at H/H1 = 0.98 and T = 0.966Tc, we used α = 0(i.e. a symmetrical sinusoidal potential) and added tothe driving force a dc component Fdc = 0.066Fp (whereFp = 2πU/ap is the critical force of the pinning poten-tial), in agreement with the Idc/Ip ratio observed in theexperiment. In this case the best fit was obtained withγ = 2.1 [Fig. 7 (b)].

The success of an underdamped model to describe thevoltage induced by vortex motion is not a trivial re-sult. Vortex dynamics is usually treated in terms of over-damped particle models because it is a well establishedfact that in most samples the vortex mass is negligi-ble compared to the damping coefficient.23 Nevertheless,these models do not consider the effect of vortex deforma-tion that can lead to an irreversibility in the depinning-repinning process and, consequently, to an apparent in-ertial effect. In recent Ginzburg-Landau simulations ofa superconductor with an antidot lattice, it was shownthat the process of depinning of a vortex from an antidotcan be characterized by strong current-induced elonga-tion of the vortex core cross-section, which results in ahysteresis in the pinning mechanism.24 This effect seemsto mimic the behavior of an underdamped particle in aperiodic potential and can be responsible for the hystere-sis observed in our experiments.

A. Vortex rectification for H < H1

In our experiment, the Vdc-Iac curves for some fieldsbelow H1 are characterized by a double rectification peak.As we pointed out in our previous publication, the sec-ond rectification peak can be explained in terms of plas-tic deformation of the vortex lattice.10 In other words,the existence of a plastic dynamical phase with a lowercritical current gives rise to the appearance of a secondratchet system, which competes with the main ratchet,given by the motion of the whole vortex lattice. Theconsequences of the existence of two ratchets with dif-

8

1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.70.0

0.2

0.4

0.6

0.8

1.0

1.0 1.2 1.40.00.20.40.60.81.0

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Experiment Num. Fit

IVIIIIII H = 0.96 H1

(a)

Nor

mal

ized

dc

volta

geH = 0.99 H1

H = 0.98 H1

(b)

Normalized ac Amplitude

FIG. 7: (a) Main frame: normalized dc voltage versus cur-rent amplitude (circles) of sample AAD1 at T = 0.973Tc

and H = 0.96H1 and numerical fit obtained from the dou-ble ratchet model described in Sec IV(see also Sec. III) withγ = 1.9 and χ = 0.5. The dashed vertical lines define differentdynamical regimes of the system (see text). Inset: same asthe main frame but with H = 0.99H1. The fitting (line) wasobtained integrating Eq. 1 with γ = 1.8. (b) Normalized dcvoltage versus current amplitude (circles) of sample SAD atT = 0.966Tc and H = 0.98H1 using an excitation frequencyof 1 kHz and a dc bias of 46 µA (which gives a reduced tilt.The fitting (line) was obtained using γ = 2.1.

ferent critical forces can be heuristically illustrated byour one particle underdamped model. Let us considertwo independent inertia ratchets. The first one repre-sents the whole vortex lattice, with critical forces Fw andFs. The second ratchet represents the fraction χ of thevortices that deform plastically and has critical forcesF ′

w ≤ Fw and F ′s ≤ Fs. In this way, the total dynamics

is simply given by the solution v(t) of the first ratchetfor F (t) > Fw and F (t) < −Fs, and, otherwise, by thesolution v′(t) of the second ratchet, normalized by thefactor χ. This simple model provides an excellent fittingto the experimental data for H < H1. For instance, to fitthe data of the experiment performed in sample AAD1at H = 0.96H1 and T/Tc = 0.973, we solved Eq. (1) forboth ratchets independently, using for the critical forcesthe normalized values obtained experimentally, and ad-justed the parameters γ and χ. In this case, we obtainedγ = 1.9 and χ = 0.5 [see Fig. 7 (a)].

To get a better insight on how such plastic dynamicscan lead to a double peak in the Vdc-Iac characteristic,we performed MD simulations of “inertial” vortices forfields H ≤ H1. The parameters used in the simulationsare given in Sec. III-B. Fig. 8 (a)-(d), show the simu-lated net velocity of vortices 〈v〉 as a function of drive

amplitude Fac for reduced fields H/H1 = 1, 0.94, 0.92,and 0.88. The simulations reproduce quite well the dou-ble rectification peak observed experimentally. Panels(e) and (f) show the time dependence of the center-of-mass (c.m.) velocity of the vortex lattice at vortex den-sities H/H1 = 1 and 0.92, respectively. As in our ex-periments, here the waveforms of vortex motion duringeach half-period are asymmetrical, i.e., present hystereti-cal behavior, which is a result of the finite apparent massof the vortices. In particular, the simulated waveformsat H/H1 = 0.92 [panel (f)] evolve similarly to the ex-perimental data of sample AAD1 presented in Fig. 4 (a)-(d). In both, experiment and simulations, four differentratchet phases (I-IV) are clearly identified [these phasesare defined by vertical dashed lines in the Vdc-Iac charac-teristic of Fig. 7 (a) and in the 〈v〉-Fac characteristics ofFig. 8 (b) and (c)]. In phase I, only a fraction of vorticesmove in the easy direction of the pinning potential. Nomotion to the hard direction is observed. In phase II, thewhole vortex lattice is half-wave rectified, i.e., all vorticesmove to the easy direction and no vortices move to thehard one. In phase III, we have motion of all vortices inthe easy direction and motion of a fraction of them tothe hard direction. The averaged rectified voltage dropsabruptly at the onset of motion to the hard direction butincreases again at slightly higher amplitudes, since themain ratchet (the whole vortex lattice) is still in its rec-tification window, giving rise to the second rectificationpeak. Finally, in the phase IV, all vortices move backand forth, but in an asymmetric fashion, resulting in thelong tail observed in the experiments and simulations.

To illustrate the interplay between the plastic dynam-ics and motion of the whole vortex lattice, we plottedin Fig. 8 (g)–(i) the vortex trajectories for the simula-tion at H/H1 = 0.94 and a few drive amplitudes. AtFac = 0.2 (g), all vortices are still pinned (just oscillatinginside the pinning centers). Note that, since the numberof vortices does not match the number of pinning sites,there is a distribution of vacant pins, that is to say, ofdiscommensurations in the vortex lattice. In this case,and also for H/H1 = 0.92, there are some vortex rows,with a denser distribution of discommensurations, whichdepin more easily. Thus, at Fac = 0.205 (h), only theweakly pinned vortex rows are dragged away, giving riseto a one-dimensional channelling dynamics characterizedby a strong shear of the vortex lattice. Finally, at ahigher drive, Fac = 0.210 (i), the potential barriers forthe strongly pinned vortex rows are also overcome and allvortices move to the easy direction of the pinning poten-tial. At lower fields, such as H/H1 = 0.88 shown in panel(d) of this figure, the distribution of vacancies is approx-imately homogeneous in such a way that all vortex rowshas approximately the same depinning force. Thus, nochannelling dynamics is observed and the 〈v〉-Fac char-acteristic at this field presents a single rectification peak.

9

FIG. 8: Molecular dynamics simulationsresults for η = 1 and M = 1. Pan-els (a)-(d): net dc velocity as a functionof the ac driving force amplitude in nor-malized units (see text) at reduced fieldsH/H1 = 1 (a), 0.94 (b), 0.92 (c), and 0.88(d). The dashed lines in (c) and (d) definedifferent dynamical regimes of the system(see text). Panels (e) and (f): time evolu-tion of the c.m. velocity at H/H1 = 1 (e)and 0.92 (f). Panels (g)-(i): vortex tra-jectories (lines) for H/H1 = 0.94 at driveamplitudes 0.2 (g), where vortices are stillpinned, 0.205 (h), where part of the vor-tices move as 1D channels, and 0.21 (i),where all vortices move. Vortices are rep-resented by the small dots and the doublepinning sites are represented by the dou-ble discs.

B. Vortex rectification for H > H1

At fields higher than H1, the interpretation of vortexmotion in terms of underdamped dynamics may breakdown because of the presence of interstitial vortices.Since they do not interact directly with the antidots, theirdynamics is expected to be similar to that of vortices in aplain film, that is, essentially overdamped. This was in-deed suggested by the amplitude dependence of the volt-age waveforms V (t) [see Fig. 4 (g) and (h)]: whereas forhigh amplitudes the signal in a half-cycle is asymmetric,at lower amplitudes, the signal may be perfectly sym-metric, that is, completely reversible. Since apparentlyhysteresis only come out when the vortices trapped in theantidots are set into motion, one may then conclude thatthe apparent inertia comes from the interaction betweenvortices and antidots rather than from the real vortexmass.

V. CONCLUSION

In conclusion, we have studied the vortex ratchet effectin superconducting films with nanoengineered arrays ofasymmetric and symmetric antidots. In the asymmetricantidot sample, we observed rectification of an unbiased

ac current due to the intrinsic asymmetry of the system,thus demonstrating that this is an intrinsic vortex ratchetsystem. In the symmetric antidot sample, the rectifica-tion effect was due to the dc tilt of the periodic (symmet-ric) potential induced by the applied dc bias. The vor-tex rectification is strongly dependent on temperatureand magnetic field. The temperature dependence indi-cates that the coherence length plays an important rolein determining the effective asymmetry produced by theasymmetric antidot array; when the vortex size becomescomparable to the big antidot size, the rectification isstrongly suppressed.

The field dependence of the rectification effects foundin our experiments provides the following observations in-dicating collective behavior in the ratchet dynamics: (i)rectification is enhanced near the first matching field, ev-idencing the importance of vortex-vortex interactions inthe ratchet effect (at the first matching field, the vortex-vortex interactions cancel out by symmetry); (ii) at fieldsabove H1, we observed a sign reversal in the ratchet effectat low amplitudes, which indicates that weakly pinnedinterstitial vortices are rectified to the opposite directiondue to the inverse asymmetry produced by the currentdistribution of the vortices trapped in the antidots; and(iii) at some fields below H1, a double rectification peakwas observed, which can be explained by the competition

10

between rectification of (weakly pinned) incommensuratevortex rows and the rectification of the whole vortex lat-tice.

In addition, our data revealed that the ratchet effectfor fields lower than H1 is dominated by underdampeddynamics. These results were corroborated by moleculardynamics simulations of an underdamped vortex ratchetmodel. We argued that the inertial effect actually comesfrom vortex deformation due to the strong interactionwith antidots, rather than from a finite vortex mass. In-deed, at fields higher than H1, we observe that the dy-namics of interstitial vortices is essentially overdamped,since these vortices are not pinned directly by the anti-

dots.

Acknowledgement

We thank Alejandro Silhanek for useful discussions.This work was supported by the K.U.Leuven ResearchFund GOA/2004/02 and FWO programs. C.C.S.S. issupported by CNPq, an Agency of the Brazilian Govern-ment. M.M. acknowledges support from the Institute forthe Promotion of Innovation through Science and Tech-nology in Flanders (IWT-Vlaanderen).

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