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Configuration-induced vortex motion in type-II superconducting films with periodic magnetic

dot arrays

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2014 Supercond. Sci. Technol. 27 065004

(http://iopscience.iop.org/0953-2048/27/6/065004)

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Superconductor Science and Technology

Supercond. Sci. Technol. 27 (2014) 065004 (7pp) doi:10.1088/0953-2048/27/6/065004

Configuration-induced vortex motion intype-II superconducting films with periodicmagnetic dot arraysQ H Chen1,2, D Q Shi1, W X Li1,3, B Y Zhu4, V V Moshchalkov2 andS X Dou1

1 Institute for Superconducting and Electronic Materials, University of Wollongong, Innovation Campus,Squires Way, North Wollongong, NSW 2500, Australia2 INPAC-Institute for Nanoscale Physics and Chemistry, Nanoscale Superconductivity and MagnetismGroup, K U Leuven, Celestijnenlaan 200D, B-3001 Heverlee, Belgium3 Solar Research Technologies, School of Computing, Engineering and Mathematics, University ofWestern Sydney, Penrith, NSW 2751, Australia4 National Laboratory for Superconductivity, Institute of Physics, Beijing National Laboratory forCondensed Matter Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

E-mail: shi@uow.edu.au

Received 23 January 2014, revised 14 February 2014Accepted for publication 14 February 2014Published 3 April 2014

AbstractUsing the molecular dynamic method we investigate numerically the current driven vortexmotion in a superconducting film with periodic arrays of both ferromagnetic (FM) andanti-parallel ferromagnetic (AFM) dots. The simulation results show that in the absence ofthermal fluctuation the vortex motion is dominated by the configurations of the magnetic dotarray. This guided vortex motion is only observed at the onset of the depinning of theinterstitial vortices. Two guided vortex motion mechanisms are discussed in this work. For theAFM configuration the vortex motion is mainly originated by the existence of magnetic dotswith opposite magnetized moments, while for the FM case it is mainly dependent on thedegree of the condensed packing of the magnetic dot lattice constant.

Keywords: guided vortex motion, configuration induced, ferromagnetic dot array, anti-parallelmagnetic dot array

(Some figures may appear in colour only in the online journal)

1. Introduction

Type-II superconducting films with artificial periodic pinningarrays have drawn a lot of interest in the past decades [1–10].Such arrays are usually composed of nanoscale antidots ormagnetic dots with a diameter significantly smaller than thespacing between them. They can trap magnetic flux and henceincrease critical current density Jc, which is advantageous forhigh current applications, where the development of resistanceand thus of power consumption must be minimized. Recentexperimental [1–6] and theoretical [7–10] studies have shownthat the maximum Jc occurs at the integer or fractionalmatching fields, where the number of vortices Nv is equal

to an integer or some fractional multiple, respectively, of thenumber of pinning centers Np, in which all the vortices arepinned in an orderly way in the pinning centers. This matchingeffect can also be obtained analytically in the framework of theGinzburg–Landau theory [11, 12]. When the external currentsurpasses a critical value, Jc, all the pinned vortices will bedepinned and moved.

In general, the motion of vortices results in a loss ofpractical superconductivity, and eventually changes the super-conductor into a normal state. Thus, the study of vorticesmoving in superconducting systems is also very importantand popular. Previous research has shown that vortex motion

0953-2048/14/065004+07$33.00 1 c© 2014 IOP Publishing Ltd Printed in the UK

Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

is dependent on the applied driving force and pinning strengthof the pinning centers as well as the densities of the vorticesand pinning centers [13–19]. When the pinning strength is notweak and the vortex number Nv > Np (or H > H1, where His the external magnetic field and H1 is the first matching fieldat which the density of the vortices is equal to the densityof pinning centers), with increasing driving force vorticesin the system go through five stages: pinned, linear Ohmic,disordered, incommensurate 1D (dimensional), and movingphases [13, 14]. If the applied force is along a symmetric axisthe vortex movement trajectories will be linear (except for thethird stage, which is too messy to be defined). Most recently,Reichhardt et al [20] have reported that the transport ofinterstitial vortices in superconducting systems with periodicpinning arrays, such as square, triangular, honeycomb, andkagome, can be anisotropic in two perpendicular directions.The degree of the anisotropy varies as a function of field due tothe fact that the interstitial vortex lattice has distinct orderingsat different matching fields. The anisotropy is most pronouncedat the matching fields but persists at incommensurate fields,and it is most prominent for triangular, honeycomb, andkagome pinning arrays. Square pinning arrays can also showanisotropic transport at certain fields in spite of the fact thatthe perpendicular directions of the square pinning array areidentical [20].

Based on the field polarity-dependent effect of magneticpinning centers [21], we constructed an innovative periodicmagnetic pinning array named an anti-parallel ferromagneticarray (AFM) of magnetic dots in our previous work [10, 17]and reported that besides the same benefits as a traditionalferromagnetic dot array (FM) configuration [10], for instancethe enhancement of the critical current density Jc and therich variety of dynamical plastic flowing phases, such anAFM system also brings us an extra enhancement of Jc [10]and causes hysteresis with the first order transitions in itscurrent–voltage (I –V ) curve due to the stress overshoot [17].Most recently, Verellen et al [22] have experimentally foundevidence of magnetically controlled vortex motion in anAl film on top of a periodic array of permalloy squarerings with in-plane magnetization, equivalent to a systemwith an AFM configuration of the magnetic dot array. Theyattributed this phenomenon to the strongly anisotropic pinningpotential landscape for vortices in the superconducting layer.Their transport measurements showed that this anisotropy isable to confine the flux motion along the high symmetryaxes of the square lattice of dipoles. This guided vortexmotion can be rerouted by 90◦ simply by changing the dipoleorientation. The research is so interesting and valuable forfuture superconductor-based devices to modulate locally themagnetic field that deserves to be studied further. Therefore, inthis paper we use the molecular dynamic method to investigatethe origins of the guided vortex motion. The work is organizedas follows. In section 2, we set up the samples used in thiswork. In section 3 we describe the numerical algorithm andthe procedure during simulations. In section 4, we show oursimulation results in detail with discussion. Finally, a summaryis given in section 5.

Figure 1. The black open and gray filled circles are the up- anddown-magnetized dots (pinning centers), respectively. (a) Theanti-parallel ferromagnetic dot array (AFM), where half of themagnetic dots are up-magnetized while the other half of the dots aredown-magnetized. (b) The parallel ferromagnetic dot array (FM),where all the magnetic dots are up-magnetized. The black solid lineis the unit cell with lattice constant a. For generality, we choose arandom lattice angle, such as 52◦, in contrast to the special angleused in [22]. The dashed lines are the directions (AB (orx-direction), AC, BC, DE and DB) that will apply driving force EFd

during simulations.

2. Sample configurations

In this work we model a two-dimensional superconductingsystem with periodic boundary conditions in the x and ydirections. Figure 1 is the two ordered configurations of themagnetic dot arrays used in this work. The black open and grayfilled circles are the up- and down-magnetized dots (pinningcenters), respectively. In figure 1(a) half of the magnetic dotsare up-magnetized, which always attracts vortices towards tothe dots, and the other half of the dots are down-magnetized,which repels vortices from them, while in figure 1(b) allthe magnetic dots are up-magnetized, which always attractsvortices towards to the dots. For convenience, we call theformer an anti-parallel ferromagnetic dot array (AFM) and thelatter a parallel ferromagnetic dot array (FM). As can be seenin figure 1, the black solid line is the unit cell with latticeconstant a. For generality, we choose the lattice angle as arandom value, such as 52◦, in contrast to a special angle asused in [22]. The dashed lines are the directions (AB (orx-direction), AC, BC, DE, and DB) that will apply drivingforce EFd during simulations.

During simulation, the array we selected is of 12 × 12magnetic dots and the lattice constant a = 1, 1.5, and 23,where 3 is the effective penetration Depth, which is equalto 2λ2/t , and λ and t are the London penetration depth andthe thickness of the superconducting film, respectively. Theparameters of each magnetic dot are the same as in [10], exceptthe maximum pinning force, that is taken to be fp = 0.558 f0,whereby f0 =8

20/λ

3 is the force per unit length. Here 80 =

2.07 mTµm2 is the flux quantum. Therefore, we calculate thatthe sample size is 123× 9.456 133.

It is obvious that the pinning potential landscape forvortices in the superconducting layer is strongly anisotropicfor a system with AFM or FM configurations of the magneticdot array. The symmetric degree of the potential varies withthe movement directions. For an AFM configuration with 52◦

angle BC and DE are the two easy directions for vortex motionsince they are more symmetric than any other directions, whilefor the FM case the AB direction will be the easiest.

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Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

3. Model and simulations

We assume that the motion of vortices is overdamped. Thetotal force acting on vortex i is then governed by [23]

EFi = EFd+ EFvv

i +EFvpi +

EFT− ηEvi = 0, (1)

where EFd is the applied driving force EFL acting equally onall vortices, EFvv

i is the vortex–vortex interaction force from allother vortices, EFvp

i is the vortex-pinning interaction force fromall pinning sites, EFT is a stochastic noise term to model thetemperature effect, η is the viscosity coefficient, taken to beunity, and Evi is the velocity of vortex i .

In our simulation EFvvi is modeled by the interaction force

between the Pearl vortices [24]

EFvvi =−

Nv∑j 6=i

∇

(∫d2Ek4π2 E0

2π tk2+ k3−1 eiEk·Eri j

), (2)

where Eri j is the displacement vector between vortex i andvortex j , E0 is the energy constant (energy per length,82

0/(2πµ0λ2)), and µ0 is the vacuum permeability. The

repulsive forces from all other vortices are included becauseof their long-range character. A smoothed method [25, 26]is introduced here to deal with such interaction, which is alook-up table up to distance 1003 with a 0.043 step and aninterpolation item for distance longer than 1003.

Finally, in the case of magnetic dots with out-of-planemagnetization, EFvp

i is modeled as [27]

EFvpi = −

Np∑k

M R820

λ2

∫∞

0dq

1Q

J1(q R)

J1(qρik)E(q, l, D), (3)

where Q = p(p + q coth(pt/2)) with p =√

1+ q2,

E(q, l, D) = e−ql(e−q D− 1), J1(x) is the Bessel function,

and l is the distance between the bottom of the SC film andthe upper surface of the dot (in experiments, SC film is usuallydeposited on top of the magnetic dots with an insulating layerin between to avoid proximity effects [21, 22, 28, 29]). R andD are the radius and thickness of the dots respectively, ρikis the distance between vortex i and the mapping center ofthe magnetic dot on the SC film, and M is the magnetizationwith M = m/(πR2 D) (in units of M0 = 80/λ

2), where mis the total magnetic moment of a dot. When the magneticmoment and the flux of the vortex have the same polarity ofFM configuration, the interaction force is attractive; however,if they have the opposite polarity of AFM configuration, theinteraction force is repulsive. The pinning center–vortex forceEFvpi is a short-range interaction. It decreases asρ−4

ik at distanceslarger than the magnetic dot lattice constant a, and we use thecutoff assuming that the force is negligible for distances greaterthan a. The forces in equations (1)–(3) are all in units of f0.

Since the up-magnetized magnetic dots for the AFM andFM cases are N↑p = 72 and 144, respectively, in order to keepthe same number of vortices, 12, at the interstitial positionswe fix the vortex number to Nv = 84 for the AFM and to

Nv = 156 for the FM. During calculations, the vortices arerandomly introduced first, then we anneal the sample from aninitial temperature (e.g. critical temperature Tc) to zero in 2500steps, and it remains constant at each step for 1000 moleculardynamic (MD) steps. Once the vortices are stable at zerotemperature, we slowly increase the driving force Fd alongone of the five given directions (AB, AC, BC, DE, and DB),respectively, for both AFM and FM configurations from 0 to1.0 f0 by 0.002 f0 every 1000 MD steps. We also calculate thevelocity and position of each vortex and compute the averagevelocity in the x and y directions, vs = N−1

v∑Nv

i=1 vi · s withs = (x, y) at every MD step, and write out this average velocityevery ten MD steps.

4. Results and discussion

4.1. Guided motion in a system with AFM configuration

Figure 2 shows the I –V curves (the first column) and vortexmovement trajectories at relatively small (second column) andlarge (third column) driving force for a system with AFMconfiguration of the magnetic dot array. Each row correspondsto the case of applying driving force along AB, AC, BC, DE,and DB directions, respectively. The black and red curves arethe average vortex velocities in the x-direction (AFM x) andy-direction (AFM y) with respect to the norm value of thedriving force Fd. The inset in (b1) is an expanded view ofthe dotted region. In the second and third columns the blackopen and gray filled circles are the up- and down-magnetizeddots, and the trajectory figures are obtained at driving forceFd= 0.26 f0 for (a2), (b2), (d2), and (e2), Fd

= 0.4 f0 for(c2), and Fd

= 0.80 f0 for (a3)–(e3), which are marked byvertical arrows in the first column.

As we mentioned in section 3, before the driving force Fd

is applied, we first anneal the system to zero temperature. Atthis moment 72 vortices are strongly pinned at the magneticpinning centers while the other 12 vortices are weakly pinned atthe interstitial positions. With increasing Fd, at the beginning,since the driving force Fd is too small to depin any vortex,all vortices in the samples remain pinned and thus the averagevelocities in all directions are zero. Further increasing theforce to a value around 0.1 f0, the 12 interstitial vortices willbe depinned first and start to move. Consequently, the averagevelocities will not be zero now. This Fd value corresponds tothe critical current density Jc of the system as Jc = Fd/80.From figure 2 one can find that the critical current densities arealmost equal to each other in AB, AC, DE, and DB directionsbut not the BC direction (further explanation can be seen laterin this subsection). This result implies that the critical currentdensity is mainly dependent on the density of the interstitialvortices when H > H1, and slightly changes with the drivingforce direction if the dot lattice is ordered but anisotropic.

When Fd is continuously increased along the AB direction(see figure 2(a1)), we observe non-zero velocities in they-direction, which means that the direction of vortex motiondoes not coincide with the driving force. Guided vortex motionhappens. For more detail we also compute the trajectories ofmoving vortices at 0.26 f0 lasting 200 MD steps, as shown

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Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

Figure 2. I –V curves (first column) and vortex movement trajectories at relatively small (second column) and large (third column) drivingforce for a system with an AFM configuration of magnetic dot array. Each row corresponds to the case applying a driving force along AB,AC, BC, DE, and DB directions, respectively. In the first column the black and red curves are the average vortex velocities in the x-direction(AFM x) and y-direction (AFM y) with respect to the norm value of the driving force Fd. The inset in (b1) is an expanded view of thedotted region. In the second and third columns the black open and gray filled circles are the up- and down-magnetized dots, and thetrajectory figures are obtained at driving force Fd

= 0.26 f0 for (a2), (b2), (d2), and (e2), Fd= 0.4 f0 for (c2), and Fd

= 0.80 f0 for(a3)–(e3), which are marked by vertical arrows in the first column.

figure 2(a2). From the figure one can easily find that theinterstitial vortices are moving not along the AB directionbut along the DE direction. We attribute this guided vortex

motion phenomenon to the existence of the down-magnetizedpinning centers in the system, i.e. the anisotropic pinningpotential landscape. When a vortex tends to move toward

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Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

the AB direction, a repulsive magnetic dot confronts it. Thevortex therefore deviates to its original direction AB and ismuch more easily attracted by the up-magnetized dot in DEdirections, giving rise to a unusual movement trajectory shownin figure 2(a2). When Fd > 0.40 f0, the pinned vortices in themagnetic dots start to be depinned and the guided motionbehavior is degenerated. The vortices in the sample are alldepinned at a value around Fd

= 0.50 f0. At this momentthe driving force is strong enough to overcome the repulsiveforces from the down-magnetized dots and hence the guidedmotion behavior disappears. Figure 2(a3) is the trajectoriesof the moving vortex at Fd

= 0.80 f0 with 100 MD steps.From the figure one can find that vortices completely moveback along the AB direction, which means the guided motionphenomenon disappears for relatively large driving force. Thatis, the guided vortex motion effect only occurs before stage Vby the interstitial vortices. Nevertheless, this value of thedriving force is still not strong enough to make the movingchannel a washboard [18], and thus the trajectories of theAFM configuration are not 1D lines but wavy in nature [10].From figure 2(a1) one can find that in the stage from whenthe first pinned vortex starts to be depinned to when allthe pinned vortices are depinned (Fd

= 0.40–0.50 f0) theaverage velocity is nonlinear. However, in other movementstages (Fd

= 0.10–0.40 f0 and Fd > 0.50 f0) the I –V curveis almost linear. Since I –V curves in other directions, AC, DE,DB, and BC, have the same characteristics as that in the ABdirection, we will not specifically mention this in the followingdiscussions.

The case along the AC direction, as can be seen infigure 2(b1), also has guided vortex motion behavior in theregime Fd

= 0.10–0.30 f0. The inset is an expanded viewof the dotted region. The red curve (or the average vortexvelocity in the y-direction) is not negative as one intuitivelyimagines but positive, which means that vortex motion isalong the DE direction, not the other easy direction BC. Thetrajectory in figure 2(b2) confirms this surprising behavior.This is probably because, at the onset of the depinning, theinterstitial vortices are quickly dragged to positions belowthe repulsive dots since the x-component of the driving forceis larger than the y-component. Consequently, vortices resultin fairly strong repulsive force in the DE direction. If weincrease the y-component of the driving force, i.e. Fd along adirection with an angle larger than 39◦, vortices will move indirection BC as by intuition. Figure 2(b3) is the trajectory atFd= 0.80 f0. Vortices move along the direction of the driving

force but are wavy, which mainly results from the repulsivemagnetic dots in the system.

The driving force in figure 2(c) is along the easy directionBC. In this case the vortices are in crystal lattice motionas expected. Figures 2(c2) and (c3) are the trajectories atcorresponding values of the driving force. No guided motionhappens in these two configurations. It should be noted thatthe critical current density for the AFM configuration alongthis direction is higher than any others. It is at Fd

= 0.30 f0.This is because along the BC direction the pinned positionsof the 12 interstitial vortices are far away from the magneticdots and, thus, we need to applied a larger driving force toovercome the interaction between vortices.

Figure 2(d1) shows the other easy direction for vortexmotion. It is along direction DE. In this situation there is noguided motion for the AFM configuration. Since vortices mov-ing in this direction are perfectly symmetric, the interactionsfrom the attractive and repulsive magnetic dots and the pinnedvortices in the pinning centers cancel each other. Therefore,the motion is only subject to the driving force and thus thetrajectories are perfectly linear, seen in figures 2(d2) and (d3)at 0.26 f0 and 0.80 f0, respectively.

Finally, for a driving force along the DB direction, asshown in figures 2(e1) and (e2), guided vortex motion appearsfor the AFM configuration. The reason is similar to the caseswe have discussed above.

4.2. Guided motion in a system with FM configuration

Unlike the AFM configuration, in a system with the FMconfiguration of the magnetic dot array, all the magnetic dotsare up-magnetized. Thus, a vortex always results in attractiveforces by the pinning centers. If the lattice parameter of themagnetic dot array is not large enough, and if the system is notisotropic, the guided vortex motion may happen inevitably.

In figures 3(b2) and (d2) one can clearly find that thevortex motions are not following the external driving forcealong directions AC and DE, respectively. They move along theAB direction. We conclude that another mechanism of guidedvortex motion happens in this situation. This guided motion isprobably due to the condensed packing of the up-magnetizeddots. If a vortex movement channel is close to attractivemagnetic dots on either side along the movement direction, thedots will exert a strong attractive force on the moving vortex,even pulling it into another movement direction as shown inthe figures. On the other hand, the trajectories in figure 3 arenot linear but somewhat wavy because of the competition ofthe forces. Figure 3(e2) demonstrates that the vortex motiontrajectories are in ann ‘S’ shape along the DB direction. Weconsider this is just an intermediate state of the guide motionby condensed packing and the normal motion along the DBdirection. We believe the trajectories will change to guidedmotion or normal motion with increasing or decreasing degreeof condensed packing, respectively. In the I –V characteristicsin the first column appear five standard stages, which havebeen intensively discussed in [14].

To further confirm the condensed packing effect we carryout similar simulations by changing the lattice parameter afrom 1.03 to 1.53 and 2.03, respectively.

Figures 4 and 3(d2) clearly shows that, with the increaseof the lattice parameter, vortex trajectories vary from a strongguided motion (a = 1.03) to a weak one (a = 1.53, whichis much wavier than the former one), then to an intermediateone (a = 2.03). We believe it will finally turn to one that hasno guided motion at all when we further increase the latticeconstant.

5. Summary

In conclusion, in the absence of thermal fluctuations vortexmotion in a superconducting film with periodic magnetic

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Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

Figure 3. I –V curves (first column) and vortex movement trajectories at relatively small (second column) and large (third column) drivingforces for a system with an FM configuration of the magnetic dot array. Each row corresponds to the case of applying the driving force alongthe AB, AC, BC, DE, and DB directions, respectively. In the first column the black and red curves are the average vortex velocities in thex-direction (FM x) and y-direction (FM y) with respect to the norm value of the driving force Fd. The inset in (a1) is an expanded view ofthe dotted region. In the second and third columns the trajectory figures are obtained at driving force Fd

= 0.26 f0 for (a2), (b2), (d2), and(e2), Fd

= 0.4 f0 for (c2), and Fd= 0.80 f0 for (a3)–(e3), which are marked by vertical arrows in the first column.

dot arrays is greatly dependent on the configuration of themagnetic dot array and the degree of condensed packing. Itis obvious that for a system with an AFM configuration of

the magnetic dot array the pinning potential landscape forvortices in the superconducting layer is strongly anisotropic.Two easy directions exist for vortex motion in such a system.

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Supercond. Sci. Technol. 27 (2014) 065004 Q H Chen et al

Figure 4. (a) I –V curves with different lattice parameters. (b), (c) Vortex motion trajectories at Fd= 0.26 f0 along DE direction for FM

configuration with lattice constants 1.53 and 2.03, respectively.

If the driving force is along these two directions, the depinninginterstitial vortices will exactly follow the directions giving riseto perfectly linear trajectories, indicating that there is no guidemotion in this situation. However, if we change the direction ofthe driving force, due to the repulsive magnetic dots the vortexmotion is guidable. Vortices always move along one of the twoeasy directions if the driving force is not strong enough. Thisguided vortex motion will disappear if only the driving forcesurpasses a critical value, and then all the vortices move backto the direction of the driving force.

Our simulations also show that even in the traditional FMconfiguration system we can also control vortex motion alongan axis of the system. This only happens in close packing cases.If the system is in condensed packing, the moving channel ofa vortex is much closer to the neighboring magnetic dots, andthe dots will attract the moving vortex and thus change themovement direction of the vortex.

Acknowledgments

This work has been supported by the Australian ResearchCouncil (ARC) Fund DP0770205, the FWO-Vlaanderenprojects and Methusalem funding by the Flemish govern-ment, and BYZ acknowledges the support by the MOST973 projects Nos 2011CBA00110, 2009CB930803, and theNational Natural Science Foundation.

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