Accepted Manuscript

Vortex Transport and Pinning in Conformal Pinning Arrays

D. Ray, C. Reichhardt, C.J. Olson Reichhardt, B. Jankó

PII: S0921-4534(14)00146-4DOI: http://dx.doi.org/10.1016/j.physc.2014.04.038Reference: PHYSC 1252647

To appear in: Physica C

Received Date: 30 December 2013Revised Date: 17 April 2014Accepted Date: 22 April 2014

Please cite this article as: D. Ray, C. Reichhardt, C.J. Olson Reichhardt, B. Jankó, Vortex Transport and Pinning inConformal Pinning Arrays, Physica C (2014), doi: http://dx.doi.org/10.1016/j.physc.2014.04.038

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Vortex Transport and Pinning in Conformal Pinning Arrays

D. Ray1,2, C. Reichhardt2, C. J. Olson Reichhardt2, and B.Janko1

1Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Abstract

We examine the current driven dynamics for vortices interacting with conformal crystal pinning arrays and compareto the dynamics of vortices driven over random pinning arrays. We find that the pinning is enhanced in the conformalarrays for field densities less than 2.5 times the matching field. At higher fields, the effectiveness of the pinning in themoving vortex state is enhanced in the random arrays compared to the conformal arrays, leading to crossing of thevelocity force curves.

Key words: vortex, conformal pinning, dynamics

1. Introduction

Many of the applications of type-II superconduc-tors require that the system maintain a large criticalcurrent or the pinning of vortices in the presence ofa magnetic field [1]. One of the key questions is, fora fixed number of pinning sites, what is the arrange-ment of the sites that maximizes the pinning. Oneapproach to this question is the use of lithographicmeans to create various types of periodic arrays ofartificial pinning sites [2–6] out of nanohole lattices[2–5] or arrays of magnetic dots [6]. In these periodicpinning systems, strong commensuration or match-ing effects can occur when the number of vorticesequals an integer multiple n of the number of pin-ning sites [2,3,7,8]. At the matching conditions, therecan be a peak in the critical current when the vor-tices form an ordered state [3,7,8]. For fields close tomatching fields, interstitials or vacancies can appearin the ordered vortex structure and act as effectiveparticles that are weakly pinned [9]. The enhance-ment of pinning at commensurate fields has alsobeen observed in colloidal experiments [10] on peri-odic optical trap arrays. The colloids are repulsivelyinteracting particles that have behavior similar to

that of vortices, showing that understanding vortexdynamics on periodic or semi-periodic substrates isalso useful for the general understanding of dynam-ics near commensurate-incommensurate transitions[11].

In superconducting samples at matching fieldswith two or more vortices per pinning site, thereare several possibilities for how the vortices canbe pinned. Multi-quanta vortices can be stabilizedby each pin at the matching fields [8,12], multiplesingly-quantized vortices can be stabilized at ornear each pinning site [13,14], or there can be onlya single vortex located at each pinning site withthe remaining vortices trapped more weakly in theinterstitial regions between pins [2,3,7,8]. It is alsopossible for combinations of these scenarios to oc-cur, or for there to be a crossover from one regimeto another as the field increases [2,8,15–17].

Another approach to pinning enhancement is theuse of quasiperiodic pinning arrays such as Penrosetilings [18–20]. Commensuration effects still occurfor such arrays; however, the strength of the pin-ning at nonmatching fields is generally greater thanthat found for nonmatching fields in periodic or ran-dom pinning arrays [18,19]. The random dilution of

Preprint submitted to Elsevier 28 April 2014

0 12 24 36

x

0

12

24

36

y

Fig. 1. The conformal pinning geometry used for simulatedtransport measurements. The drive is applied in the positivex-direction.

periodic pinning arrays produces peaks in the criti-cal current not only at the matching fields but alsoat fields where the number of vortices matches thenumber of pinning sites in the original undiluted ar-ray [21]. Strong non-integer matching peaks in thecritical current also appear in honeycomb pinningarrays, which are an example of an ordered dilutedtriangular pinning array [22]. Enhancement of thepinning at fractional fields can be achieved using ar-tificial ice pinning array geometries [23], and thesefractional matching peaks can be as strong as oreven stronger than the integer matching peaks [24].

Recently, a new type of pinning geometry calleda conformal crystal pinning array (illustrated inFig. 1) was proposed [25]. This array is constructedby performing a conformal transformation [26] on atriangular lattice to create a gradient in the pinningdensity while preserving the local sixfold orderingof the original triangular array. Flux gradient simu-lations show that the overall critical current in theconformal pinning array (CPA) is enhanced overthat of a uniform random pinning array [25] and isalso higher than that of uniform periodic pinningarrays except for fields very close to integer match-ing, where the periodic pinning gives a marginallyhigher critical current. In the CPA, commensurationeffects are absent and there are no peaks in the criti-cal current. The gradient in pinning density presentin the CPA enhances the pinning since it can match

the gradient in the vortex density. Random pinningarrays with a density gradient equivalent to that ofthe CPA produce a small critical current enhance-ment compared to uniform random arrays; however,a CPA with the same number of pinning sites gives asubstantially larger critical current, indicating thatthe preservation of the sixfold ordering of the pin-ning array is important for enhancing the pinning[25]. The simulation predictions were subsequentlyconfirmed in experiments which compared CPAs torandom and periodic arrays [27,28]. Other work onnon-conformal pinning arrays containing gradientsincludes numerical studies of hyperbolic tessela-tions [29], as well as experiments on non-conformalpinning arrays with gradients in which the pinningwas enhanced compared to uniform arrays [30].

The first numerical work on CPAs focused on flux-gradient driven simulations where the critical cur-rent is proportional to the width of the magnetiza-tion loop [25]. In such simulations, there is a gra-dient in the vortex density across the sample. Onequestion is whether the CPA still produces enhancedpinning in systems driven with an applied current.Previous work indicated that the CPA produces apinning enhancement compared to random arraysin this case as well [25]. In this work we further ex-plore the current-driven system by varying the ap-plied magnetic field and analyzing the vortex veloc-ity as a function of external drive to produce a mea-surement that is proportional to an experimentallymeasurable voltage-current curve. We find that atvery low vortex densities, the difference in criticalcurrent between random arrays and CPAs is small,and that as the field increases, the conformal ar-rays have stronger effective pinning, producing botha larger depinning force and a lower average vortexvelocity in the moving state compared to randomarrays. At higher fields, the CPA still has a high de-pinning threshold; however, once the vortices are inthe moving state, the average vortex velocity for therandom arrays can be lower than that for the CPA,indicating that the effectiveness of the pinning in thedynamic regime is suppressed for the CPA comparedto the random pinning. We show that this arises dueto the earlier onset of dynamical ordering [31,32] inCPAs compared to random pinning arrays at thesehigher magnetic fields.

2

2. Simulations

We consider an effective 2D model of vorticeswhere a single vortex i obeys the following equationof motion:

ηdRi

dt= Fvv

i + FPi + FD

i . (1)

Here η = φ20d/2πξ2ρN is the damping constant, d is

the sample thickness, φ0 = h/2e is the elementaryflux quantum, and ρN is the normal-state resistivityof the material. The vortex vortex interaction forceis Fvv

i =∑Nv

j 6=i F0K1(Rij/λ)Rij , where K1 is themodified Bessel function, λ is the London penetra-tion depth, F0 = φ2

0/(2πµ0λ3), Rij = |Ri −Rj | is

the distance between vortex i and vortex j, and theunit vector Rij = (R−Rj)/Rij . The force from thepinning sites is given by FP . Various models for thepinning can be considered; here, we use parabolicattractive sites with

FPi = −

Np∑

k=1

(Fp/rp)(Ri −R(p)k )Θ[(rp − |Ri −R(p)

k |)/λ],(2)

where R(p)k is the location of pinning site k, Fp is

the maximum pinning force, and Θ is the Heavi-side step function. Other short-range pinning po-tentials should give essentially the same results asthose reported here. An externally applied currentproduces a Lorentz force FD

i = J × B that is per-pendicular to the applied current. Previous studiesof this system predominantly focused on flux gradi-ent driven geometries [25,33] in which vortices areadded to or subtracted from a pin-free region out-side of the sample. Here, we apply the driving forceFD = Fdx in the x-direction and measure the vor-tex velocity in the x-direction. 〈Vx〉 =

∑Nv

i vi · x,where vi = dRi/dt. Using a spatially heterogeneousdrive of the type that would arise in an experimentalsample would not significantly alter the results; weassume that heating effects do not occur. The sys-tem geometry is illustrated in Fig. 1. There are Np

pinning sites of radius rp = 0.12λ with a maximumforce of Fp. The system size is 36λ× 36λ, with peri-odic boundary conditions in the x and y-directions.

3. Transport

In Fig. 2 we plot 〈Vx〉 versus Fd for samples withnp = 1.0/λ2, rp = 0.12λ, and Fp = 0.55F0 for uni-form random arrays and CPAs at fields ranging from

0 0.1 0.2 0.30

0.1

0.2

0.3

0 0.1 0.20

0.1

0.2

0 0.1 0.20

0.1

0.2

0 0.1 0.20

0.1

0.2

0 0.1 0.20

0.1

0.2

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

H/Hφ=0.5 H/Hφ=0.8

H/Hφ=1.4 H/Hφ=1.7

H/Hφ=2.0 H/Hφ= 0.2, 2.2

Fig. 2. 〈Vx〉 versus Fd curves for the CPA (lower darklines) and random arrays (upper light lines) in samples withnp = 1/λ2, rp = 0.12λ, and Fp = 0.55F0 for B/Bφ = (a)0.5, (b) 0.8, (c) 1.4, (d) 1.7, (e) 2.0, and (f) 0.2 (right lines)and 2.2 (left lines). In panels (a-e) the effectiveness of thepinning for the CPA is higher than for the random array.

0 0.1 0.2 0.3 0.4F

d

0

0.02

0.04

0.06∆V

Fig. 3. The difference between the velocity response in therandom array and the CPA, ∆V = 〈V rand

x 〉 − 〈V CPAx 〉, vs

Fd for samples with B/Bφ = 0.5, 1.4, and 2.2 (lower leftto upper left). In this field range, ∆V is positive, indicatingthat the pinning is more effective in the CPA at low andintermediate values of Fd. At the highest drives, ∆V goesto zero.

B/Bφ = 0.2 to 2.2. In Figs. 3 and 4 we plot the dif-ference between the velocity response in the randomarray and the CPA, ∆V = 〈V rand

x 〉 − 〈V CPAx 〉, as a

function of Fd. At lower fields, shown in Fig. 3, thepinning in the CPA is more effective than in the ran-dom array, giving a positive ∆V , for all but the high-est values of Fd. At B/Bφ = 1.4, ∆V reaches a max-imum value of 0.045 in Fig. 3, while for B/Bφ = 2.2the maximum value of ∆V is only 0.015, indicatingthat as B/Bφ increases, the difference in the pinning

3

0 0.05 0.1 0.15 0.2F

d

-0.015

-0.01

-0.005

0

0.005

∆V

Fig. 4. ∆V vs Fd for samples with B/Bφ = 2.5, 2.8, and3.0 (upper right to lower right). For low Fd, the pinning ismore effective in the CPA, as indicated by the positive valueof ∆V ; however, at intermediate Fd the pinning is moreeffective in the random arrays. At the highest drives, ∆Vgoes to zero.

effectiveness of the CPA compared to the randompinning array is reduced. At higher fields, shown inFig. 4, for low Fd above depinning, the pinning inthe CPA is more effective than in the random array,but at higher Fd, the drag on the moving vorticescreated by the CPA is smaller than that of the ran-dom array. In each case, for large Fd, ∆V goes to 0as the system enters the Ohmic regime where all thevortices are moving and the effects of the pinningare minimal. The transition to the Ohmic responseregime occurs near Fd = 0.26 for B/Bφ = 1.4 inFig. 3, while for B/Bφ = 2.2, the transition drops toa lower value of Fd = 0.07. Our results are not sen-sitive to a minor misalignment of the driving forcewith the x axis. Unlike a periodic pinning array, theCPA does not have easy flow channels, and we findalmost no dependence of the 〈Vx〉 versus Fd curveson driving direction until quite large misalignmentsof order 40 degrees are applied.

In Fig. 5 we plot 〈Vx〉 versus Fd for the same sys-tem in Fig. 2 at higher values of B/Bφ = 1.7, 2.2,and 2.8. The critical depinning force Fc is higherin the CPA than in the random array for B/Bφ =1.7 and 2.2, but at B/Bφ = 2.8 〈Vx〉 for intermedi-ate drives is lower in the random array than in theCPA. This can be seen more clearly in the ∆V plotin Fig. 4, where for each of the fields ∆V is initiallypositive, but drops below zero as Fd increases, in-dicating that CPA pinning becomes less effective atintermediate values of Fd and that the average vor-tex velocity is higher in the CPA than in the ran-dom array. The reversal of the effectiveness of the

0

0.1

0.2

<v x

>

0

0.1

0.2

<v x

>

0 0.05 0.1 0.15 0.2F

d

0

0.1

0.2

<v x

>

0

0.4

0.8

P6

0

0.4

0.8

P6

0

0.4

0.8

P6

H/Hφ=1.7

H/Hφ=2.2

H/Hφ=2.8

Fig. 5. Lower solid curves: 〈Vx〉 vs Fd for a random array(light lines) and a CPA (dark lines). Upper symbols: P6,the fraction of sixfold coordinated vortices, vs Fd for a ran-dom array (light symbols) and a CPA (dark symbols). (a)B/Bφ = 1.7. (b) B/Bφ = 2.2. (c) B/Bφ = 2.8. In (c), thevortices dynamically order at a lower drive for the CPA thanfor the random array, giving a lower value of 〈Vx〉 for therandom array at the intermediate drives 0.1 < Fd < 0.2.

pinning in the moving state produces as a crossingin the velocity vs force curves as shown in Fig. 5(c).

The reversal of the effectiveness in the pinning atintermediate drives occurs because the vortices dy-namically order or partially crystallize at a lowerdrives in the CPA than in the random pinning ar-ray. It is known from current-driven simulation stud-ies of random pinning arrays that a dynamical re-ordering transition can occur into a moving statethat is partially crystalline or smectic-like [31,32].In the dynamically ordered state, the vortex veloci-ties are generally higher than in moving states withmore random ordering, since the shear modulus of arandom structure is much lower. A disordered vor-tex configuration has a higher probability of somevortices being temporarily pinned by the substrate,while in a moving crystal state, the vortices all movetogether and can not be individually trapped bypinning sites. For the random array, as the field in-creases, the drive FOr

d at which the vortices begin todynamically order decreases. FOr

d is also a function

4

0 0.5 1 1.5 2 2.5 3B/Bφ

0

0.01

0.02

0.03

∆ F d

Fig. 6. The difference in the external drive Fd at whichVx = 0.05 between the random and the conformal arrays,∆F = F rand

d (Vx = 0.05) − FCPAd (Vx = 0.05), vs B/Bφ. At

low fields, ∆Fd is small, at intermediate fields the CPA hasstronger pinning (∆Fd > 0), and for B/Bφ > 2.5 the randomarray has stronger pinning (∆Fd < 0).

of the pinning density np, and as np decreases, FOrd

also decreases. In the CPA, the pinning density hasa gradient, and as a result, there is a gradient in thevalue of FOr

d across the sample. At the higher mag-netic fields, the vortices can start to locally dynam-ically order in the lower pin density portions of theCPA sample. The partially ordered state forms inthe low pin density regions during a transient timeτo, and this state becomes disordered while passingthrough the high pin density regions during a tran-sient time τd. As the field increases, these transienttimes change, and the vortices remain disordered ifτd > τo, while for τo < τd the vortices can order.This means that in a random pinning array, the vor-tices are disordered when Fd < FOr; however, for aCPA at the same value of Fd, if τo < τd, an orderedmoving vortex state will form and hence 〈Vx〉 forthe CPA will be higher than for the random array.As B/Bφ increases, τo decreases. This is consistentwith the behavior in Fig. 2, where the extent of therange of Fd over which 〈Vx〉CPA/〈Vx〉rand > 1 growsas B/Bφ increases. It may be possible that at highenough B/Bφ, the vortices in the CPA would imme-diately dynamically order as soon as they depin; inthis case, the critical current for the random arraywould be higher than that of the CPA.

In Fig. 5 we plot simultaneously 〈Vx〉 and the frac-tion of six-fold coordinated vortices P6 versus Fd forthe random pinning and the CPA. In the dynami-cally ordered moving crystal state, P6 is close to 1[31,32]. In Fig. 5(a) at B/Bφ = 1.7, the pinning ismore effective in the CPA over the entire window of

Fd shown in the figure. At depinning, P6 drops forboth types of pinning as the system enters a plas-tic flow regime. At higher Fd, P6 increases when thevortices begin to reorder, and in Fig. 5(a), P6 for therandom array is higher than that for the CPA forFd > 0.15. In Fig. 5(b) for B/Bφ = 2.2, we find asimilar trend; however, in Fig. 5(c) for B/Bφ = 2.8,P6 is higher for the CPA than for the random ar-ray for Fd > 0.1. This also corresponds to the rangeof Fd over which 〈Vx〉 in the CPA is higher than inthe random array. At Fd = 0.2, P6 reaches nearlythe same value for both arrays, and the differencein 〈Vx〉 between the two arrays also vanishes. Thisresult confirms that at high magnetic fields, the vor-tices dynamically order at a lower drive for the CPAthan for a random pinning array.

We can roughly estimate the relative effective-ness of the pinning in the CPA and random pinningarrays by plotting the difference in the value Fd

at which 〈Vx〉 = 0.05 for the two arrays, ∆Fd =F rand

d (Vx = 0.05) − FCPA(Vx = 0.05). Figure 6shows that at low B/Bφ, ∆Fd is small and the dif-ference between the random and conformal arrayis minimal due to the weak vortex-vortex interac-tions. In the limit of single vortex pinning, there isno difference between the two arrays and ∆Fd = 0.Near B/Bφ = 1.4, ∆Fd reaches a maximum value;it then decreases and becomes negative for B/Bφ >2.5. The value of B/Bφ at which ∆Fd drops be-low zero depends on the velocity value chosen forthe measurement; however, Fig. 6 indicates thatthe enhanced effectiveness of the CPA is limitedto B/Bφ < 2.5. The CPA effectiveness may alsodepend on the size of the pinning sites and on Fp,both of which can be sample dependent.

4. Summary

We investigated the current driven dynamics ofvortices interacting with conformal pinning arraysand compared the effectiveness of the pinning to thatof random pinning arrays with the same total num-ber of pinning sites. The conformal pinning array isconstructed by performing a conformal transforma-tion of a triangular pinning array to create a new pin-ning array that has a density gradient but still con-serves the local sixfold ordering of the original trian-gular array. We find that the for vortex densities lessthan twice the matching field, the critical depinningforce for the conformal array is higher than that ofthe random array, and that in the moving vortex

5

state that the velocity of the vortices in the confor-mal array is also lower than in the random array. Athigher drives the difference between the two types ofarrays is washed out due to dynamical reordering ofthe vortices. At higher fields, the critical depinningforce for the conformal array remains higher thanthat of the random array, but at intermediate drivesthe average vortex velocity in the random arrays islower than that in the conformal array, leading to acrossing of the velocity-force curves. This reversal ofthe pinning effectiveness arises because the vorticesdynamically order at a lower drive in the conformalarray than in the random array. There are still is-sues to consider in the conformal pinning array, suchas performing conformal transformations on latticestructures other than a triangular array. It wouldalso be interesting to investigate vortex ratchet ef-fects of the type previously found in samples withrandom or periodic pinning arrays [34,35], as theseeffects may be enhanced in the conformal pinningarrays.

This work was carried out under the auspices ofthe NNSA of the U.S. DoE at LANL under ContractNo. DE-AC52-06NA25396.

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7

We perform current-driven simulations comparing conformal and random pinning.

Pinning is enhanced by the conformal array over a wide range of fields.

Dynamic vortex reordering occurs at a lower drive for the conformal array.