Rectification and Flux Reversals for Vortices Interacting with Triangular Traps

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by  C. J. Olson Reichhardt

Rectification and Flux Reversals for Vortices Interacting with Triangular Traps
C.J. Olson Reichhardt and C. Reichhardt
Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
(February 2, 2008)
Wesimulatevorticesinsuperconductorsinteractingwithtwo-dimensionalarraysoftriangulartraps.
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Wefindthat,uponapplicationofanacdrive,anetdcflowcanoccurwhichshowscurrentreversals
0
withincreasingacdriveamplitudeforcertainvortexdensities,inagreementwithrecentexperiments
0
2 andtheoreticalpredictions. Weidentifythevortexdynamicsresponsibleforthedifferentrectification
regimes. We also predict theoccurrence of a novel transverse rectification effect in which a dcflow
n
appears that is transverse to thedirection of theapplied ac drive.
a
J PACS numbers: 74.25.Qt

2
Vortices interacting with periodic pinning structures This change in the rectification direction is explained in

] havebeenattractingconsiderableattentionsincetheyare Ref.[15]asoriginatingfromtheseparatemotionofinter-
n
anidealsysteminwhichtostudythestaticanddynami- stitial vortices at low drives,giving −y rectification, and
o
cal behaviors of collectively interacting particles coupled the motion of the pinned vortices at higher drives, pro-
c
- toperiodicsubstrates. Inthesesystems,avarietyofcom- ducing +y rectification. It is not clear how the motions
r
p mensurability effects and novelvortexcrystalscanoccur of these two vortex species can be fully separated, espe-
u asafunctionofthe ratioofthenumberofvorticestothe cially in the presence of thermal fluctuations, and thus
s number of pinning sites [1–3]. The vortex crystals are a clearer picture of the vortex dynamics producing the
.
t composed of one or more vortices trapped at each pin rectification is needed.
a
m withanyremainingvorticeslocatedinthe interstitialre- In this work we present simulations of vortices inter-
gions between the pinning sites. A variety of dynamical acting with 2D square arrays of triangular pinning sites
-
d flow phases occur when a drive is applied, including the for parameters corresponding to the recent experiments
n flowofinterstitialvorticesbetweenthepinningsites[4,5] of Ref. [15]. We find a +y rectification at low ratios n
o
as well as channeling along rows of pinning [4]. Much of ofthe number of vorticesto the number ofpinning sites,
c
the physics of vortices interacting with periodic pinning withamaximum+ydcfluxasafunctionofacamplitude.
[
is also observedfor repulsively interacting colloids in pe- Wheninterstitialvorticesarepresent,theinitialrectifica-

1
riodic optical trap arrays [6–8]. In addition to the basic tionis inthe−y direction,andis followedbya crossover
v
science issues, vortices interacting with periodic pinning to +y rectificationat higher ac amplitude, as seen in ex-
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1 arraysare relevant to possible technologicalapplications periments. We find that for large ac amplitudes, when
0 ofsuperconductors,suchascriticalcurrentenhancement the motion of the vortex lattice becomes elastic, the rec-
1 or controlled motion of flux for new types of devices. tificationswitchestothe−ydirection,whichisexplained
0
Severalmethodshavebeenproposedforusingperiodic in terms of channeling effects. We also predict a novel
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pinning or controlleddisorder in superconductorsto cre- transverse ratchet effect where a dc motion is generated
0
/ ate vortex ratchets, rectifiers, and logic devices [9–13]. inthe directiontransverseto the appliedac drive. All of
t
a As in general ratchet systems, which can be determin- the effects we describe here should also occur for repul-
m istic or stochastic in nature [14], a vortex ratchet trans- sively interacting colloidal particles interacting with 2D
- formsanacinputintoadcresponse. Vortexratchetscan triangular traps.
d be created via asymmetric periodic pinning lines [9,10], We consider a thin film superconductor containing a
n
asymmetric channels, or the asymmetry introduced by square array of N = 64 triangular pinning sites with
o p
multiple ac drives in a system with a symmetric sub- periodic boundary conditions in the x and y-directions.
c
: strate [11,12]. Ratchets constructed of periodic two- We add N =nN vortices to the sample, each of which
v v p
dimensional(2D)arraysofasymmetricpinningsiteshave obeys the overdamped equation of motion
i
X alsobeenproposed[13]. Recently,bothpositiveandneg-
dr
r ative vortex rectification have been experimentally real- η i =fvv +f +f +f . (1)
a dt i p AC T
izedinperiodicarraysoftriangularpinning sites[15]. In
Ref. [15], the triangles are arranged in a square lattice Here the vortex-vortex interaction force, appropri-
with the tips of the triangles oriented in the +y direc- ate for a thin film superconductor, is fvv =
i
tion. Whenanacforceisappliedintheydirectionatlow PNj6=vi(Φ20/µ0πΛ)rˆ/rij, where Φ0 is the elementary flux
matching fields, rectification of the vortex motion in the quantumandΛisthethinfilmscreeninglength[16]. We
+y directionoccursoverarangeofacamplitudes,witha useafastsummationmethodtoevaluatethis long-range
peak rectification at a particular amplitude. For higher interaction[17]. Thepinningforcef comesfromasquare
p
matching fields, there is a −y rectification at lower ac arrayof equilateral triangles with one vertex pointing in
drives,followedby a +y rectificationathigher ac drives. the +y direction, as shown in Fig. 1. Every triangular
1
0 1 (a) 0.08 (b)
>-0.04 >
Vy 0.5 Vy
<-0.08 <0.04
fdp
-0.12 0
0n 2 4 6 0
0.06 (c) 0.03 (d)
V>y0.03 V>y 0
< <
0 -0.03
y y -0.03
(e) 0.01 (f)
x (a) x (b) 0.01
V>y 0 V>y 0
<-0.01 <-0.01
-0.02 -0.02
0 0.5 1 1.5 0 0.5 1 1.5
A A
FIG.2. Netdcvelocity<Vy >inunitsofm/sasafunction
ofapplied accurrentAinunitsoff0 forn=(a)1, (b)2,(c)
3,(d)4,(e)5,and(f)6. Insetto(a): dcdepinningforce|f |
dp
for +y (circles) and −y (x’s) directions.
y y
lattice. All of the vortices sit in the pins for n ≤ 3, but
x (c) x (d)
FIG.1. Pinningsites(triangles)andvortexpositions(dark for n ≥ 4 the vortex-vortex interaction is strong enough
circles) at thematchingfieldsn= (a) 1, (b)2, (c)3, and (d) that some vortices move to the interstitial regions.
4. Wefirstconsiderthedcdepinningforcef inthepos-
dp
itive and negative y directions by applying a dc driving
pin is modeled as three half-parabolas of equal strength, forceofincreasingmagnitudeandmeasuringthe average
eachofwhichattractsthevortextoalinepassingthrough vortex velocity <V >. In the inset to Fig. 2(a) we plot
y
the centerofthe triangleandparallelto oneofthe sides. |f | as a function of n. At n = 1 the depinning is sym-
dp
The pinning force is cut off at the edge of the triangle. metricinthe ±y directions,asexpectedgivenourmodel
The applied ac force f = Asin(ωt)ˆr acts in the di- for the pinning. For n = 2, |f−y| is twice as large as
AC dp
rection perpendicular to an applied ac current. Here we |f+y|,duetothefactthatthevorticesalignverticallyfor
consider either fAyC = Asin(ωt)yˆ or fAxC = Asin(ωt)xˆ, +dypdriving, and horizontally for −y driving. For n = 3
with no mixtures of ac drives. Temperature is modeled and 4, |f+y| is 25% larger than |f−y|. Here, the vortex
dp dp
asrandomthermalkickswiththeproperty<fT(t)>=0 at the top of each pin acts to assist in the depinning of
and <fT(t)fT(t′)>=2ηkBTδ(t−t′). the vortices along the bottom of the pin for −y driving.
We match our parameters to those of the experiment For n > 4, the depinning is dominated by the intersti-
in Ref. [15], performed in Nb films with η =1.4×10−12 tial vortices and the asymmetry in the depinning is lost.
Ns/m. We take T/Tc = 0.98, giving fT = 0.46f0, where The depinning asymmetry at n=2 to 4 should be large
f0 = φ20/2πµ0λ3 = 1.09 × 10−5 N/m. At this tem- enough to be observable experimentally. Depinning in
perature, the London penetration depth λ = λ(0)/(1− the ±x directions is symmetric for all n. Asymmetric
(T/Tc)2)1/2 =368nm, so our pin spacing is 2.09λin the depinning at noninteger n was observed in Ref. [18].
x direction and 2.03λ in the y direction, and the pins Wenextconsidertheeffectofanappliedacdrivefy .
AC
are 1.68λ on a side. In the experiment, each pin held a We monitor the net dc velocity < V > at fixed ω and
y
maximum of three vortices, so to match this we set the sweepthe ac amplitude A, averaging<V > for 20 peri-
y
pinning strength to fp = 1.05f0. We fix ω = 78 kHz, ods at each increment of A. In Fig. 2(a) we plot <Vy >
higher than experiment due to the limitation of simula- vsAforn=1. ForA<0.9,thereisnonetflowinxory,
tion time; however, since the behavior of the system is indicating that the vortices are still pinned. For A=0.9
controlled by the ratio A/ω, we compensate by taking afinite +y dc velocityappearswithno netx dc velocity,
A larger than in experiment. We note that we find the corresponding to a +y rectification. As A increases fur-
samegenericbehaviorsforother parameters,including a ther,<V >increases,reachingamaximumatA=0.99
y
pinning array that is triangular rather than square. andthendroppingbacktozeroatA=1.05. Thisbehav-
InFig.1weshowthevortexpositionsandpinlocations iormatchesthatseenintheexperimentsofRef.[15]. We
at A = 0 for a system with a square array of triangular findnostepsinthe+y rectificationregime,inagreement
pins at fillings n = 1, 2, 3, and 4. A global symmetry with the experiments, but in disagreement with recent
breaking occurs at n = 2 between the two possible ar- T =0simulations ofa verysmallsystemfor asymmetric
rangementsofthevorticesallowedbythesquarepinning pinning sites that predicted steps in the velocity vs
2
from thermal noise.
n=2, -y n=2, +y
Forn=1inFig. 2(a),athigherA>1.05,<V >be-
y
comesnegative,indicating−yrectificationathighdrives.
Thisreversaloccursforallfillingsbutismostpronounced
for n=1 and n=4, when the onset of the reversalfalls
at lower values of A. We predict that this effect should
be easily observable in experiment, although it was not
reported in Ref. [15]. The high-drive −y rectification
originateswhenthe−y vortexmotionoccursthroughin-
y y
terstitial channels while the +y vortex motion becomes
x (a) x (b)
localized on top of a column of pinning, giving a larger
n=4, -y n=4, +y
drag during the +y portion of the cycle.
To understand the role of interstitial and pinned vor-
tices inthe rectificationprocess,we examinethe dynam-
ics of the vortices. In Fig. 3(a,b) we illustrate a typical
example of vortex motion leading to +y rectification at
n=2 and A=1.0. Fig. 3(a) shows the vortex trajecto-
riesduring1/4ofthe drivecyclewhenthe driveisinthe
−ydirection. Atthesetemperatures,noneofthevortices
y y remainpinned. The vortexmotion is uncoordinatedand
x (c) x (d) the vortices wander stochastically as they pass through
n=4, -y n=4, +y the pinning. In contrast, during the +y portion of the
cycleshowninFig. 3(b),thevorticesarechanneledboth
within and between the columns of pinning sites, and
there is little wandering in the x direction. This more
focused motion leads to a greater overall displacement
in the +y direction, and a net +y rectification. Similar
motionoccursforn=1andn=3,exceptthatforn=1
all of the motion is confined to the pinning channel and
there is no interstitial motion, so the net rectification is
y y
correspondingly reduced as seen in Fig. 2(a).
x (e) x (f)
At n = 4 and above, when interstitial vortices are
FIG.3. Pins(triangles),vortices(dots),andvortextrajec- present, −y rectification occurs at low drives, as shown
tories during1/4of thedrivingperiod (lines) for: (a) N =2,
in Fig. 2(d-f). In Fig. 3(c-d) we illustrate an example
A = 1.0, −y portion of drive cycle, (b) +y portion of same
ofvortexmotionin the −y rectificationregimefor n=4
cycle; (c) N =4, A=0.35, −y portion of drivecycle, (d)+y
andA=0.35. Fig. 3(c)showsthatthevorticesmeander
portion of same cycle; (e) N = 4, A = 0.75, −y portion of
moderately in the x direction during the −y portion of
drivecycle, (f) +y portion of same cycle.
the drive cycle. Note that none of the vortices remain
pinned; instead, all of the vortices are participating in
f curve[19]. Thetransportdoesnotoccurinanorga-
AC
the motion, with pinned and interstitial vortices switch-
nized step like flow, but is instead a stochastic process,
ingplacesfrequently. Duringthe+y portionofthe drive
aswe willshowinmoredetailin Fig.3. InFig.2(b) and
cycleshowninFig. 3(d),thedriveistoolowtoovercome
Fig. 2(c) we plot < V > for n = 2 and 3, respectively.
y
the pinning force and allow the formation of channels of
These curves are similar to the n = 1 case, with an ini-
the type seen in Fig. 3(b). Instead, the +y vortex flow
tial pinned phase followed by +y rectification that goes
is strongly diverted into the x direction by the pins, re-
through a maximum with increasing A.
ducing the net motion in the +y direction, and resulting
InFig.2(d)weshow<V >forthecaseofn=4when
y
in an overall −y rectification. As A is further increased,
interstitialvorticesarepresent,asseeninFig.1(d). Here
the vortices begin to overcomethe pinning in the +y di-
there is an initial negative rectification regime in the −y
rection and form channels during the +y portion of the
direction, followed by a crossover to a +y rectification
cycle, as illustrated in Fig. 3(f) for n=4 and A=0.75.
as seen in experiments [15]. In Fig. 2(e,f) we plot the
Thisgives+yrectificationjustasinFig. 3(a-b). Asnin-
cases for n = 5 and n = 6, respectively, which show a
creasesabove4,thechannelsbecomemorecloselypacked
similar rectification to that in Fig. 2(d). In all of these
and the amount of vortex wandering in the x direction
cases, a combination of interstitial and pinned vortices
during the −y portion of the cycle decreases, progres-
are present. As n is increased further, the ratcheting ef-
sivelyloweringthe net+y rectification. The onsetofthe
fectsaregraduallyreducedandbecomeindistinguishable
−y and +y rectification phases also drops to
3
0.05 (a) 0 (b) ment with experiments. We also predict that a novel
0.04 -0.01 transverse ratchet effect should occur when the ac drive
0.03 -0.02 is applied in the x direction. Here, the rectification is
> >
<Vx0.02 <Vy-0.03 in the y direction. Our results should be generic to any
0.01 -0.04
type of repulsively interacting particles moving through
0 -0.05
triangular traps and may have potential applications for
-0.01 -0.06
0 0.5 1 1.5 0 0.5 1 1.5 species fractionalization in colloidal systems.
A A
0.02 (c) 0.025 (d) This workwassupportedby the U.S. DoEunder Con-
0.015 0.02 tract No. W-7405-ENG-36.
> 0.01 >0.015
Vx Vy
<0.005 < 0.01
0 0.005
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