Solid State Communications 151 (2011) 900–903

Contents lists available at ScienceDirect

Solid State Communications

journal homepage: www.elsevier.com/locate/ssc

Coherent motion of vortices driven by both dc and ac current in a Josephsonjunction ratchet arrayK.H. Lee ∗

Department of Applied Physics, Dankook University, Yongin 448-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 23 November 2010Received in revised form17 February 2011Accepted 29 March 2011by S. MiyashitaAvailable online 6 April 2011

Keywords:A. SuperconductorsC. Josephson junction arrayD. Ratchet effectD. Vortex dynamics

a b s t r a c t

We have studied the vortex dynamics in a ratchet array of Josephson junctions in the presence ofmagnetic field of 1/5 flux quantum per plaquette. The ratchet potential consists of both alternate criticalcurrents for all the vertical junctions and alternate shunt capacitances for all the horizontal junctions. Thevortices driven by an ac current in some parameters are found to show the directional motion as wellas the asymmetric current–voltage characteristics. We use the time-dependent vorticity and the time-dependent vorticity–vorticity correlation function to analyze the motion of vortices on a few fractionalShapiro steps. We have found that vortices on a fractional Shapiro n/5-step move coherently through nplaquettes during a single ac cycle. The asymmetric features of the ratchet array gradually disappear asfinite temperature increases.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Transport studies in ratchet potentials have received muchattention in recent years, particularly in view of their potentialapplications in nanotechnology. One of the main features inthe micro ratchet system is the directional motion of Brownianparticles driven by alternating bias or random fluctuations [1–7].The rectification effect can be greatly enhanced by temporallyturning on and off the ratchet potential depending on the spatialposition of the particles [8,9]. Such a ratchet effect has also beenstudied in various superconducting devices and networks [10–19].One recent experimental realization reports a pronounced voltagerectification effect of fluxons in a superconducting thin film withan array of asymmetric antidots [20]. A Josephson junction arraywith alternate junction parameters is considered to be one ofgood ratchet systems because of relative easiness in assigningthe symmetry breaking junction parameters. As for the vortexdynamics in a ratchet array, anunderdamped array ismore suitablethan an overdamped one because the vortex inertial mass can beadjusted simply by the junction capacitance [21–26].

In this paper, we investigate the motion of vortices driven byboth dc and ac current in a ratchet array of Josephson junctions. Theratchet potential consists of a set of alternate critical currents for

∗ Tel.: +82 31 8005 3213; fax: +82 31 8005 3208.E-mail address: khlee@dankook.ac.kr.

0038-1098/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2011.03.030

all the vertical junctions and a set of alternate shunt capacitancesfor all the horizontal junctions. Along the direction of vortex flow,the alternate critical currents serve as alternate potential barriersfor vortices while the alternate capacitances provide alternateplaquette areas. To see the time-dependent motion of vorticesmore clearly, we use the vorticity–vorticity correlation function,which points out the time lag between two vorticities separated bya certain distance of plaquettes along theprojectedpath of vortices.By comparing the correlation functions of positive time-lag and ofnegative time-lag, one may determine not only the average speedof vortices but also the direction of the motion. For certain rangesof the dc bias in which the external ac drive gives rise to coherentmotion of vortices, the vorticity–vorticity correlation functions are,indeed, useful for describing the time dependence of vortices.

Such a coherent motion of vortices originates from phaselocking of superconducting grains in an array. This transport phe-nomenon in a frustrated Josephson network, known as fractionalShapiro steps, has been studied well both experimentally andnumerically [26–29]. For a square array of N × N plaquettes witha transverse magnetic field of 1/q flux quantum per plaquette, thequantized voltage steps appear when the time-averaged voltagesatisfies ⟨V ⟩ = nNhν/2eq, where n is an integer and ν is thefrequency of external ac drive. Even for a ratchet array, Shapirosteps are also expected to appear in the same time-averagedvoltage although the transport property shows asymmetricbehavior. We focus on ratchet dynamics when vortices are drivenby external current that leads to a Shapiro step, and try to comparethe dynamics with vortex motion in a non-ratchet potential.

K.H. Lee / Solid State Communications 151 (2011) 900–903 901

2. Numerical calculation for a ratchet array

We use a square array of superconducting grains, each coupledto its neighbors by resistively- and capacitively-shunted junctions.The total current between grains i and j is

Iij = CijdVij

dt+

Vij

Rij+ Ic;ij sin(ϕi − ϕj − Aij), (1)

where ϕi is the phase of the superconducting order parameter.Cij, Rij, and Ic;ij are the shunt capacitance, the shunt resistance,and the critical current of the junction, respectively. The voltagedifference Vij and the magnetic gauge phase factor Aij are definedby

Vij = Vi − Vj =h̄2e

ddt

(ϕi − ϕj), (2)

Aij =2πΦ0

∫ rj

riA · dl, (3)

where Φ0 = hc/2e is the flux quantum, A is the vector potentialof the applied magnetic field and ri is the position of the centerof grain i. Aij satisfies

∑P Aij = 2π f , where the summation is

over a plaquette and f is the frustration. In the presence of finitetemperature T , a fluctuating Langevin noise current can be addedin Eq. (1) [30].

A ratchet array can be formed effectively by assigning bothalternate critical currents and alternate inter-capacitances alongthe horizontal direction of the vortex flow. The alternate criticalcurrents are set for all the vertical junctions: Ic;y = 1.2Ic followedby 0.8Ic . For the alternate capacitances, all the horizontal junctionsalong the same horizontal direction have the alternate values:Cx = 2C followed by C . To obtain a stable vortex flow, ahigher value of the critical current to all horizontal junctions isassigned. We choose Ic;x = 1.5Ic for such a guide bank for thevortices. A schematic diagram of 4 × 3 plaquettes in a ratchetarray is shown in the inset of Fig. 1, where Josephson junctionsare denoted by crosses (‘‘X ’’). The coupled equations of Eqs. (1)–(3), with Kirchhoff’s law at each grain, can be solved to obtainthe time-dependent phases ϕi(t) and voltages Vi(t) [24]. Weintroduce a uniform external current of I + Iac sin(2πνt) intothe bottom grains and extract from the top grains. In a directionperpendicular to the current injection, we use periodic boundaryconditions. For a typical cycle of current–voltage characteristics,we use the final phase and voltage configurations at the previousbias as the initial conditions for the new bias. The dimensionlessMcCumber–Stewart parameter, β = 2eR2IcC/h̄ = 10, is chosen inour calculation.

3. Vortex motion in a ratchet array for f = 1/5

We chose a magnetic field of 1/5 flux quantum per plaquette.Unlike most other frustration including f = 1/2, 1/3, 1/4, and2/5, this value of frustration forms a square vortex lattice andgives the fact that an individual vortex does not directly comeinto contact with its neighbors. For this reason, it is relatively easyto focus on an arbitrary vortex and trace its approximate path.The curves shown in Fig. 1 are the time-dependent vorticities at 7neighboring plaquettes around amid row of a ratchet array of 10×

10 plaquettes. The coordinates of the leftmost bottom plaquetteare assumed to be (0, 0). The array is rocking by an external accurrent only: Iac = 0.36Ic with frequency ν = 0.04ν0, where ν0 isthe characteristic frequency, ν0 = 1/t0 = 2eRIc/h̄. The vorticity isdefined at the center of each plaquette as the sumof supercurrents,the sum being taken in the counterclockwise direction:

vr(t) =

−P

Ic;ij sin(ϕi − ϕj − Aij). (4)

-1

0

1

2

3

1850 1900 1950 2000t / t0

Vr

V(3.5) V(7.5)

V(8.5)

V(9.5)

V(4.5)

V(5.5)

V(6.5)

Fig. 1. Time-dependent vorticities at 7 neighboring plaquettes in a mid row of aratchet array at zero dc current. The array is driven by an ac current of magnitudeIac = 0.36Ic with frequency ν = 0.04ν0 . The inset shows a schematic of 4 × 3plaquettes in a ratchet array. The Josephson junction is displayed by a cross (‘‘X ’’)mark, the size of which represents the magnitude of critical current.

Vortices at this zero dc current in Fig. 1 do not flow, but oscillatewith a period of 1/ν = 25t0 within their own plaquettes. Thetime-dependent vorticities of v(4,5) and v(9,5) have relatively highpeaks, indicating that vortices are located at these positions witha separation of 5a, where a is the lattice constant of the array.The nonidentical peaks of two vorticities are due to the ratchetpotential. One vortex sits on a smaller plaquette, while the otherone which is 5 plaquettes away from the first one sits on a largerplaquette. In other words, the non-identical peaks are caused bythe fact that the vortex lattice repeats every 5 plaquettes whilethe ratchet potential repeats every 2 plaquettes. The double-peakprofile for a period of 25t0 is also attributed to the ratchet potential.A regular array, however, shows only single-peak vorticities for atime period.

If the ac current is increased to Iac = 0.52Ic , the ratchetarray exhibits not only asymmetric but also off-centered I–Vcharacteristics, as shown in Fig. 2. The curve does not show asignificant hysteresis in our parameters used. The time-averagedvoltage at zero dc current is not zero, but has a finite value whichlies on a fractional Shapiro 1/5-step ranging from I = −0.07Icto 0. The fractional Shapiro steps appear in quantized voltageplateaus of ⟨V ⟩ = ±

n5

Nhν/2e for a frustration f = 1/5,

where n is an integer. Note that in our dimensionless quantitiesthe steps should occur at ±

n5

2πν/ν0. At this zero dc current,

the rocking ratchet array effectively produces a rectification effectfor vortices. The negative time-averaged voltage means that thevortices move to the left (i.e., -x axis). Such a moving directioncan also be confirmed from the time-dependent vorticities and thetime-dependent vorticity–vorticity correlation functions.

The time-dependent vorticity–vorticity correlation functionpoints out the time lag between two vorticities separated by asmall spatial distance along the projected path of vortices:

C(vrvr′)(t) = ⟨vr(τ )vr′(τ + t)⟩τ , (5)C(vrvr′)(−t) = ⟨vr(τ + t)vr′(τ )⟩τ , (6)

where ⟨· · ·⟩τ represents the average over time τ . Two sites,r and r′, are the plaquette coordinates separated by a certaindistance along the direction of the flux flow. The positive-timecorrelation function is mathematically identical to the negative-time correlation function with the reversed order of two spatially-separated positions,

C(vrvr′)(t) = C(vr′vr)(−t). (7)

902 K.H. Lee / Solid State Communications 151 (2011) 900–903

-0.5026

-0.2513

0.0000

0.2513

0.5026V

(t)

/ NR

I c

<V

> /

NR

I c

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6I / Ic

-0.2

0.0

0.2

1900 1950 2000 2050 2010

t / t0

Fig. 2. Asymmetric I–V characteristics of a ratchet array driven by an ac current ofIac = 0.52Ic and ν = 0.04ν0 . The ‘‘+’’ mark represents the origin of coordinates.The inset shows the time-dependent voltage at zero dc current. The time-averagedvoltage at zero dc current lies on the fractional Shapiro 1/5-step.

When there is a directional motion of vortices, the positive-timecorrelation should be different from the negative-time correlation.By comparing the sequential order of several neighboring peaks ofboth thepositive-time and thenegative-time correlation functions,one may determine the direction of the moving vortices.

We also define the space-averaged vorticity–vorticity correla-tion functions:

C1x(t) = ⟨C(vrvr+1xi)(t)⟩r, (8)

where ⟨· · ·⟩r represents the spatial average over all the sites withinthe entire array and i is the unit vector for the x-axis. The space-averaged vorticity–vorticity correlation functions in Eq. (8) mayhelp to enhance the correlation characteristics in the presence offluctuating Langevin thermal noise.

Now, we describe the coherent motion of vortices when theyare on fractional Shapiro steps in a ratchet array. The time-dependent vorticity–vorticity correlation functions at zero dccurrent are shown in Fig. 3, where the junction parameters arethe same as in Fig. 2. The positive-time correlation functionsof Fig. 3(a) are oscillating with the following peak sequence:C(v(3,5)v(7,5)), C(v(3,5)v(6,5)), C(v(3,5)v(5,5)), etc. The correlationfunction C(v(3,5)v(8,5)) is just the autocorrelation except the factthat one plaquette has a smaller area while the other one hasa larger area. The time interval between neighboring peaks is asingle period of the applied ac drive. The positive-time correlationfunctions describe the fact that just after a single ac cycle fromthe moment that a vorticity (e.g., v(4,5)) is activated somewhere,a second vorticity (e.g., v(3,5)) will be activated at a plaquettedistance to the left of the first one. The peaks of the negative-timecorrelation functions of Fig. 3(b) occur in the reverse sequence.Similarly, one may say that before a single ac cycle from themoment that a vorticity (e.g., v(3,5)) is activated somewhere, aprevious vorticity (e.g., v(4,5)) is activated at a plaquette distanceto the right of that vorticity (i.e., v(3,5)). These features ofboth correlation functions describe the fact that vortices movecoherently through one plaquette to the left during a single periodof the applied ac drive.

The motion of vortices at dc currents ranging from I = 0.04Icto 0.14Ic can be explained similarly. Since this range is in thepositive bias branch of the fractional Shapiro 1/5-step, vortices areexpected to move to the right. Fig. 4 shows the time-dependentvorticity–vorticity correlation functions for I = 0.1Ic . The positive-time correlation functions in Fig. 4(a) is almost the same as thenegative-time correlation functions in Figs. 3(b) and 4(b) is almost

0

0

2

2

0 100 200 300t / t0

C (

v r v

r,) (

+ t)

C (

v r v

r,) (

- t)

a

b

C (v(3, 5) v(6, 5))C (v(3, 5) v(5, 5))

C (v(3, 5) v(8, 5))C (v(3, 5) v(7, 5))

C (v(3, 5) v(4, 5))

I = 0

Fig. 3. Time-dependent vorticity–vorticity correlation functions at zero dc current.The ratchet array is driven by an ac current Iac = 0.52Ic with ν = 0.04ν0 . (a)Positive-time correlation functions of C(vrvr′ )(t), and (b) negative-time correlationfunctions of C(vrvr′ )(−t).

I = 0.1

2

0

2

0

0 100 200 300

C (

v r v

r,) (

+t)

C (

v r v

r,) (

-t)

t / t0

a

b

C (v(3,5) v(6,5)C (v(3,5) v(4,5) C (v(3,5) v(5,5)

C (v(3,5) v(8,5)C (v(3,5) v(7,5)

Fig. 4. Time-dependent vorticity–vorticity correlation functions at I = 0.1Icwhich are on a fractional Shapiro 1/5-step of the I–V characteristics in Fig. 2. (a)Positive-time correlation functions of C(vrvr′ )(t), and (b) negative-time correlationfunctions of C(vrvr′ )(−t).

the same as Fig. 3(a) also. The reverse sequence of the peaks isrelated to changing the polarity of the bias current. The polaritychange makes vortices move to the opposite direction. From thepositive-time correlation functions of Fig. 4(a), one may say thatafter a single ac cycle from the moment that a vorticity (e.g.,v(3,5)) is activated somewhere, a second vorticity (e.g., v(4,5)) willbe activated at a plaquette distance to the right of the first one.The dynamics from the negative-time correlation functions can beexplained in a similar manner. In short, vortices move coherentlyby one plaquette to the right during an ac cycle in spite of thepresence of the ratchet potential.

K.H. Lee / Solid State Communications 151 (2011) 900–903 903

T = 0.02

Δx = 5

Δx = 0 1

3

5

0.02

0.2

2

4

T = 00.10.5

0 100 200t / t0

< C

Δx (

+t)

><

CΔx

(+

t) >

-1

0

1

2

-1

0

1

2

a

b

Fig. 5. (a) Space-averaged positive-time correlation function from C1x=0(t) toC1x=5(t) at T = 0.02T0 . (b) C1x=5(t) with 5 temperatures. The other ratchetparameters are the same as in Fig. 2. The finite temperature eventually destroysthe vorticity–vorticity correlations.

For a fractional Shapiro 2/5-step ranging from I = 0.16Ic to0.26Ic in the I–V characteristics, vortices are expected to movetwice faster than for the 1/5-step. This expectation becomes quiteevident when dealing with the time-dependent vorticity–vorticitycorrelation functions. According to the sequential peaks ofboth positive-time and negative-time correlation functions (notshown), we have found that just after a time lag of a single ac cyclea second vorticity is activated at a distance of two plaquettes to theright from the position where the first one is activated. This meansthat vorticesmoveby twoplaquettes to the right during an ac cycle.

Let us turn to themotion of vortices in the presence of the finitetemperature. Now, we use the space-averaged vorticity–vorticitycorrelation functions, defined in Eq. (8), to reduce the fluctuatingnoises effectively. The rectification effect shown in Fig. 2 isexpected to disappear as temperature increases. The thermalfluctuation may also gradually destroy the correlations betweentwo vorticities separated by a certain distance. Fig. 5(a) shows thespace-averaged positive-time correlation functions from C1x=0 toC1x=5 at zero dc current at a fixed temperature of T = 0.02T0,where T0 = h̄Ic/2ekB. The ratchet array is driven by an ac currentof Iac = 0.52Ic with frequency ν = 0.04ν0 as before. The space-averaged correlations are still oscillating with a given ac cycle, butgradually decrease as timepasses. For a fixed distance of separationon the path of the flux flow, the correlations should also decreaseas temperature increases. An example of a separation of 1x = 5is shown in Fig. 5(b). Vortices are expected to gradually lose theirspatial correlations with increasing temperature. The correlationpeaks eventually disappear around T = 0.1T0. Below 0.1T0 (e.g.,T = 0.02T0), we can say that vortices move to the left coherentlyat zero dc current. If the temperature is larger than 0.1T0, it is notpossible to determine the direction of vortices due to the spatialuncorrelation.

4. Conclusions

We have investigated the motion of vortices driven by bothdc and ac current in a ratchet array of underdamped Josephsonjunctions. We choose a frustration f = 1/5 in which vorticesform a square vortex lattice of a lattice constant

√5a. For this

frustration, it is relatively easy to follow an arbitrary vortex

and trace its approximate path. In order to see the time-dependent motion of vortices more clearly, we use the time-dependent vorticity–vorticity correlation function. It indicates thetime lag between two vorticities separated by a certain distance ofplaquettes along the projected path of vortices. By comparing thesequential order of the correlation peaks for positive time-lag andnegative time-lag, one may obtain not only the flow direction butalso the average speed of vortices.

Depending on the bias polarity, the ratchet array showsasymmetric current–voltage characteristics in depinning currentsand in the step-widths of the time-averaged voltage plateaus.The off-center behavior, the so-called rectification effect, appearsat zero dc driving current for some ratchet parameters. Whenthe driving current is not big enough to make a finite time-averaged voltage, vortices oscillate within their own plaquettes.We have found that vortices on a fractional Shapiro n/5-stepmovecoherently through n plaquettes during a single ac cycle. It meansthat vortices on such a step move n times faster than vortices onthe first step. For vortices on the Shapiro steps, ratchet dynamics isnot different from non-ratchet dynamics.

In order to analyze the overall translational motion of vortices,the space-averaged vorticity–vorticity correlation function isbetter than the individual vorticity–vorticity correlation function.It is useful to discuss vortex dynamics especially in the presenceof finite temperature. Using the space-averaged correlationfunctions, we have found that the asymmetric features includingthe current-induced depinning and the step-widths graduallydecrease as temperature increases. The spatial correlation peaksalso decrease and eventually disappear at high temperatures.

Acknowledgement

The present research was supported by the research fund ofDankook University in 2010.

References

[1] P. Reimann, M. Grifoni, P. Hänggi, Phys. Rev. Lett. 79 (1997) 10.[2] S. Scheidl, V. Vinokur, Phys. Rev. B 65 (2002) 195305.[3] M. Grifoni, M.S. Ferreira, J. Peguiron, J.B. Majer, Phys. Rev. Lett. 89 (2002)

146801.[4] G. Costantini, U.M.B. Marconi, A. Puglisi, Europhys. Lett. 82 (2008) 50008.[5] M. Van den Broeck, C. Van den Broeck, Phys. Rev. Lett. 100 (2008) 130601.[6] R.M. da Silva, C.C. de Souza Silva, S. Coutinho, Phys. Rev. E 78 (2008) 061131.[7] D. Speer, R. Eichhornm, P. Reimann, Phys. Rev. Lett. 102 (2009) 124101.[8] F.J. Cao, L. Dinis, J.M.R. Parrondo, Phys. Rev. Lett. 93 (2004) 040603.[9] M. Feito, F.J. Cao, J. Stat. Mech. (2009) P01031.

[10] I. Zapata, R. Bartussek, F. Sols, P. Hänggi, Phys. Rev. Lett. 77 (1996) 2292.[11] R.D. Astumian, Science 276 (1997) 917.[12] F. Falo, P.J. Martínez, J.J. Mazo, S. Cilla, Europhys. Lett. 45 (1999) 700.[13] E. Trías, J.J. Mazo, F. Falo, T.P. Orlando, Phys. Rev. E 61 (2000) 2257.[14] C.-S. Lee, B. Jankó, I. Derényi, A.-L. Barabási, Nature 400 (1999) 337.[15] J. Rousselet, L. Salome, A. Ajdari, J. Prost, Nature 370 (1994) 446.[16] L.P. Faucheux, L.S. Bourdieu, P.D. Kaplan, A.J. Libchaber, Phys. Rev. Lett. 74

(1995) 1504.[17] D.E. Shalom, H. Pastoriza, Phys. Rev. Lett. 94 (2005) 177001.[18] V.I. Marconi, Phys. Rev. Lett. 98 (2007) 047006.[19] K.H. Lee, J. Korean Phys. Soc. 53 (2008) 722.[20] J. Van de Vondel, C.C. de Souza Silva, B.Y. Zhu, M. Morelle, V.V. Moshchalkov,

Phys. Rev. Lett. 94 (2005) 057003.[21] R.S. Newrock, C.J. Lobb, U. Geigenmüller,M.Octavio, Solid State Phys. 54 (2000)

263.[22] P. Martinoli, Ch. Leemann, J. Low Temp. Phys. 118 (2000) 699.[23] C. Reichhardt, A.B. Kolton, D. Domínguez, N. Grønbech-Jensen, Phys. Rev. B 64

(2001) 134508.[24] Wenbin Yu, K.H. Lee, D. Stroud, Phys. Rev. B 47 (1993) 5906.[25] Wenbin Yu, D. Stroud, Phys. Rev. B 46 (1992) 14005.[26] H.S.J. van der Zant, F.L. Fritschy, T.P. Orlando, J.E. Mooij, Phys. Rev. Lett. 66

(1991) 2531.[27] S.P. Benz, N. Rzchowski, M. Tinkham, C.J. Lobb, Phys. Rev. Lett. 64 (1990) 693.[28] K.H. Lee, D. Stroud, J.S. Chung, Phys. Rev. Lett. 64 (1990) 962.[29] D. Domínguez, J.V. José, Phys. Rev. Lett. 69 (1992) 514.[30] V. Ambegaokar, B.I. Halperin, Phys. Rev. Lett. 22 (1969) 1364.