Table Of ContentReversible Vortex Ratchet Effects and Ordering in Superconductors with Simple
Asymmetric Potential Arrays
Qiming Lu1,2, C.J. Olson Reichhardt1, and C. Reichhardt1
1Center for Nonlinear Studies and Theoretical Division,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
7
2Department of Physics, Applied Physics, & Astronomy,
0
Rensselaer Polytechnic Institute, Troy, New York, 12180-3590
0
2 (Dated: February 6, 2008)
n We demonstrate using computer simulations that the simplest vortex ratchet system for type-II
a superconductors with artificial pinning arrays, a simple asymmetric potential array, exhibits the
J samefeatures asmore complicated two-dimensional (2D) vortexratchetsthathavebeenstudied in
0 recent experiments. We show that the simple geometry, originally proposed by Lee et al. [Nature
1 400, 337 (1999)], undergoes multiple reversals in the sign of the ratchet effect as a function of
vortex density, substrate strength, and ac drive amplitude, and that the sign of the ratchet effect
] is related to the type of vortex lattice structure present. Thus, although the ratchet geometry has
n
the appearance of being effectively 1D, the behavior of the ratchet is affected by the 2D structure
o
of the vortex configuration. When the vortex lattice is highly ordered, an ordinary vortex ratchet
c
effect occurs which is similar to the response of an isolated particle in the same ratchet geometry.
-
r Inregimeswherethevorticesformasmecticordisorderedphase,thevortex-vortexinteractionsare
p
relevant and weshow with force balance arguments that theratchet effect can reversein sign. The
u
dcresponse of thissystem features areversible diode effect and a varietyof vortexstates including
s
triangular, smectic, disordered and square.
.
t
a
m PACSnumbers: 74.25.Qt
-
d
I. INTRODUCTION originalwork on vortex ratchets was followed by a series
n
of proposals for alternative vortex ratchets based on 2D
o
c asymmetricpinning arraygeometries,eachofwhichpro-
Whenanoverdampedparticleisplacedinanasymmet-
[ duces similar types of vortex ratcheteffects with the net
ric potential and an additional ac drive is applied, a net
vortexflowintheeasydirectionoftheunderlyingpoten-
2 dcdriftvelocityorrectificationcanoccurwhichisknown
v tial[6,7,8,9,10]. Theseratchetsoperateeitherthrough
as the ratchet effect. Stochastic ratchets can be con-
0 abottleneck effect[6,9]orbymeansofaneffectively1D
structed with Brownian particles, while ordinary ratch-
6 mechanism [7, 8, 10]. Similar ratchet effects have also
5 etscanbecreatedindeterministicsystems[1]. Typically, beenfoundforsuperconductorswithasymmetricsurface
9 an applied ac drive or periodic flashing of the potential barriers [11].
0 couples with some form of asymmetry in the substrate,
Recently,new types of2Dperiodic pinning geometries
6 breaking the symmetry of the particle motion. Ratchet
have been proposed that exhibit not only the regular
0
effectshavebeenstudiedinthecontextofmolecularmo-
/ ratchet effect but also a reverse ratchet effect, in which
t tors [1] as well as in the motion of colloidal particles [2],
a thenetdcvortexmotionisinthehardsubstratedirection
granular materials [3] and cold atoms in optical trap ar-
m [12, 13, 14, 15, 16, 17]. In some cases, ratchet reversals
rays [4].
occur when the vortex-vortex interactions at higher fill-
-
d Vorticesintype-IIsuperconductorsactasoverdamped ingsbecomerelevant,resultinginaneffectiveasymmetric
n particlesmovinginnanostructuredpotentiallandscapes. potentialorientedinthedirectionoppositetothepinning
o
The possibility of realizing a ratchet effect to transport structure. As a result, some of the vortices move in the
c
superconducting vortices was originally proposed by Lee hard direction of the pinning substrate, and the net mo-
:
v et al. [5], who studied vortices interacting with a sim- tionisdescribedbyeitheraneffectively1D[14,15,17]or
i
X ple asymmetric or sawtooth substrate potential and an 2D [16] mechanism. Other vortex ratchet systems have
additional external ac drive. This type of asymmetric multiple sign reversals due to symmetry breaking from
r
a potentialcouldbefabricatedbycreatingasuperconduct- multiple circularac drives[12] orthree state pinning po-
ingsamplethathasanasymmetricthicknessmodulation. tentials [13]. The reverse vortex ratchet effect has been
The externalac drive is providedby either an oscillating observedinexperimentswith2Darraysoftriangularpin-
externalmagneticfieldorbyanappliedaccurrentwhich ning sites [15] and other types of 2D arrays [17]. In the
producesanoscillatingLorentzforceonthevortices. The work of Silva et al. [17], the periodic pinning array in
numerical simulations of Ref. [5] indicated that a net dc the experimental sample has a two component unit cell
motion of vortices could arise upon application of an ac containing a large pinning site with a small pinning site
drive, and that the net vortex drift is in the easy direc- placed to one side. This system exhibits a remarkable
tion of the asymmetric substrate, as would be expected field dependent multiple reverse ratchet effect. For the
forasingleparticleinteractingwithsuchapotential. The first matching field and higher odd matching fields, a
2
3.82
3000
1500
v
N 1.04
>
vx 0 n
< v
1.04
-1500
-3000
3.82
0.0 0.4 0.8 1.2 1.6
F
dc
FIG. 2: The dc non-normalized average velocity hVxiNv
vs Fdc for varied nv and both directions of dc drive. The
FIG. 1: Schematic of the simulation geometry. Vortices are curves for Feixt = xˆFdc have hVxiNv ≥ 0, while the curves
subjected to an applied current J = Jyˆ which produces a for Feixt = −xˆFdc have hVxiNv ≤ 0. In order of increas-
dtiroinvinogf tfohrecefigFuerxet.=TFhdecxˆbootrtFomacxˆp,oirlltuiosntraotfedthienfithguerteopshpoowrs- i1n.5g6|/hλV2x,iN1.v7|4,/tλh2e,c1u.9r1v/esλ2h,a2v.e08n/vλ=2,12..0443//λλ22,,12..2728//λλ22,,13..3193//λλ22,,
theperiodicpotentialU(x)throughwhichthevorticesmove. 3.47/λ2, and 3.82/λ2. For nv < 2.08/λ2, the negative drive
Thewidth oftheeasydirection ofthepinningpotentialisl+ critical depinning force is larger than the positive drive crit-
and thewidth of thehard direction is l−, where l+ >l− and ical depinning force, fc− > fc+, but for nv ≥ 2.08/λ2, this
l++l− =a, thepinning lattice constant. reverses and fc− <fc+.
vortexlatticehasamoreorderedorcrystallinestructure,
regularratchet effect occurs, but at even matching fields
the effective vortexinteractions cancelor are reducedby
theratcheteffectchangessign. Simulationsindicatethat
the symmetry of the lattice, and the response is the or-
thepinningsitesarecapturingmultiplevortices,andthat
dinary ratchet effect as in the case of a single vortex.
this can change the direction of the asymmetric effec-
At fields where the vortex lattice is disordered or has an
tive potential at every matching field, as described by a
intrinsic asymmetry, the interactions between individual
1D model. A ratchet effect has also been observed in
vortices become relevant and a portion of the vortices
a Josephson-junction array structure which has spatial
move in the easy direction of the pinning potential, re-
asymmetry [18]. Here, a diode effect occurs where the
sultinginareversalofthesignoftheratcheteffectwhich
depinning field is higher in the harddirection. A ratchet
can be predicted from force balance arguments. In addi-
and reversed ratchet effect are also seen for matching
tiontothe ratchetresponseofthis system, weshowthat
fields less than one.
a richvarietyofdistinct vortexphasescanbe realizedas
Inthiswork,weperformadetailedinvestigationofthe
a function of density, including crystalline, smectic, dis-
simplestvortexratchetsystemcreatedbyanasymmetric
orderedandsquarephases. Commensurationeffects also
periodic modulation, which was originally shown to ex-
create oscillations in the critical depinning force which
hibit only an ordinary ratchet effect [5]. By considering
are distinct from those observedfor two dimensional pe-
magnetic fields much higher than those used in Ref. [5],
riodic pinning arrays.
where the effectively single vortex regime was studied,
we show that this system can also exhibit ratchet sign
reversals as a function of vortex density when the vor-
tex interactions become relevant. Unlike the majority of
II. SIMULATION
previously studied vortex ratchets, the behavior of this
systemis determined by the detailed 2Dstructure ofthe
vortexconfiguration,andcannotbeexplainedintermsof Weconsidera2DsystemofsizeL =L =24λ,where
x y
asimple1Dmodel. Wefindthatthedcresponseexhibits λistheLondonpenetrationdepth, withperiodicbound-
areversiblediodeeffectcorrelatedwiththeregionswhere ary conditions in the x and y directions. The sample
the ratchet effect reverses. The sign of the ratchet effect containsN vorticesinteractingwithanasymmetricsub-
v
is shownto be relatedto the nature of the overallvortex strate U(x) of period a = λ, illustrated schematically in
lattice structure. At magnetic field densities where the Fig. 1. The vortex density n = N /L L . A given
v v x y
3
vortex i obeys the overdamped equation of motion
dR
η dti =Fviv +Fsi +Feixt+FTi . (1) 1.0
The damping constant η = φ20d/2πξ2ρN, where d is the 0.8 Neg. Depinning Force
Pos. Depinning Force
sample thickness, ξ is the coherence length, ρ is the
N
+ c
normal-stateresistivity,andφ =h/2eistheelementary
0 f 0.6
flux quantum. The vortex-vortex interaction force is ,
- c
f
Nv r 0.4
Fvv = f K ij ˆr (2)
i 0 1 λ ij
Xj6=i (cid:16) (cid:17)
0.2
Here r = |r − r |, ˆr = (r − r )/r , and r is
ij i j ij i j ij i(j)
the position of vortex i (j). Force is measured in units 0.0
of f = φ2/2πµ λ3, and time in units of τ = η/f . 0 2 4 6 8 10 12
0 0 0 0
K1(rij/λ) is the modified Bessel function which falls off nv
exponentially for r > λ. For computational efficiency,
ij
the vortex-vortex interaction force is cut off at 6λ. We
have previously found that using longer cutoff lengths FIG. 3: The depinning force curves vs nv for the system in
−
Fig.2fordrivinginthenegativedirection,f (blackcircles),
produces negligible effects [20]. The pinning potential is c
and in the positive direction, f+ (open circles). The diode
modeled as a sawtooth as shown in Fig. 1, c
effect reverses near nv = 1.91/λ2 and nv = 9.90/λ2. Several
oscillations in thedepinningforce also occur.
−1A xˆ if 0≤x moda<l+
Fs = 2 p i
i (cid:26)Apxˆ if l+ ≤xi moda<a.
Here, Ap = 1.0f0 and xi =ri·xˆ. The width of the long versus Fdc for Feixt = xˆFdc for samples with Nv rang-
or “easy” side of the pinning potential is l+ = (2/3)a ing from nv = 1.04/λ2 to 3.82/λ2. The bottom set of
and the width of the short or “hard” side is l− =(1/3)a curves show hVxiNv versus Fdc for the same samples
such that l+ +l− = a. The term Feixt in Eq. 1 is the with Feixt = −xˆFdc. A clear diode effect occurs for
force from a dc or ac external driving current. For the nv <2.08/λ2whenthenegative(hard)criticaldepinning
case of an applied dc drive, Feixt = Fdcxˆ, where Fdc is force fc− is larger than the positive (easy) critical depin-
increased from zero in increments of δFdc = 0.01 every ning forcefc+. Here,fc−/f0 .1,closeto the value ofthe
104 simulation time steps. For the case of an applied ac pinning force in the hard direction, while fc+/f0 ≈ 0.5,
drive,Fext =±F xˆ,wherethepositivesignisuseddur- nearlythesameastheeasydirectionpinningforce. Thus,
i ac
ingthefirsthalfofeachperiodτ andthe negativesignis for nv < 2.08/λ2, the depinning force in each direction
used during the second half period, resulting in a square isprimarilydeterminedbythepinningsubstrateandthe
wave centered about zero. The initial vortex configura- vortex-vortex interactions are only weakly relevant. A
tions are obtained through simulated annealing, where pronounced drop in fc− from fc−/f0 ≈ 1 to fc−/f0 ≈ 0.4
wesetFeixt =0andapplyathermalforcewiththeprop- occurs above nv =2.08/λ2. For nv >2.08/λ2, the diode
erties hFTi i = 0 and hFTi (t)FTj(t′)i = 2ηkBTδijδ(t−t′). effect is reversed since now fc− <fc+.
The simulated annealing begins at a high temperature When n = 2.08/λ2, the negative depinning curve in
v
FT = 5.0 which is well above the melting tempera- Fig. 2 shows a clear two step depinning process with an
ture of the vortex lattice, and then FT is slowly re- initial depinning near F =0.45f followed by a second
dc 0
duced to zero. After annealing, we set FT = 0, ap- jumpinhV inearF =0.8f . Atthelowfirstdepinning
i x dc 0
ply a dc or ac drive, and measure the average velocity threshold, only a portion of the vortices depin, while at
hV i = h(N−1) Nv v ·xˆi. For dc driving, we identify the second threshold near F = 0.8f , the entire lattice
x v i=0 i dc 0
the critical currPents in the positive and negative driving depins.
directions, fc+ and fc−, which we define as the drive at The values of fc+ and fc− undergo changes as a func-
which |hVxi|=0.1f0. tionofnv thatare difficult to discerninFig.2. InFig. 3
we plot both f+ and f− versus n for systems with up
c c v
to 7000 vortices and n = 12.15/λ2. There is a posi-
v
III. DC DEPINNING AND VORTEX LATTICE tive diode effect for n < 1.91/λ2 when f− > f+. The
v c c
STRUCTURES diode effect reverses for 1.91/λ2 ≤ n ≤ 9.90/λ2, where
v
f− < f+. A second reversal occurs for n > 9.90/λ2,
c c v
We first compare the dc response of the system for when f+ drops back below f−. The high n rever-
c c v
driving in the positive (easy) and negative (hard) x di- sal is shown in more detail in Fig. 4, where we plot
rection. The top set of curves in Fig. 2 show hV iN hV i versus F for two different samples. In Fig. 4(a),
x v x dc
4
numberofrowsineachtroughbutinsteadcausesadisor-
dering of the rows relative to one another, resulting in a
smectic structure. This occurs because the shear modu-
lus is smaller thanthe compressionmodulus ofthe rows.
0.4
nv=8.68 Aslargernumbersofvorticesareadded,thecompression
of each row grows until the rows begin to buckle. The
0.2 buckling transition destroys the alignment perpendicu-
>
x lar to the vortex rows, causing the vortex configuration
v
< 0.0 to become more isotropically disordered. This is similar
to the commensuration effects found for vortex lattices
in multilayer systems [21]. In this case, the field is ap-
-0.2
pliedalongthedirectionofthelayerssothatthevortices
Positive driving force effectively interact with a potential that is periodically
-0.4 Negative driving force modulated in one dimension. These experiments found
oscillationsinthemagnetization,whichisproportionalto
0.1 0.2 0.3 0.4 0.5
thedepinningforce. Theoscillationswerealsoattributed
totransitionsinthevortexlatticestructurefromnchains
0.4
n =10.42 per layer to n+1 chains per layer [21].
v
The vortex configuration for n < 1.04/λ2 has only
v
0.2 one column of vortices per pinning trough modulation,
>
as illustrated in Fig. 5(a). The presence of triangular
x
v
< 0.0 ordering in the vortex lattice is clearly indicated by the
structure factor S(k),
-0.2 2
Nv
Positive driving force S(k)=Nv−1(cid:12) exp(ik·ri)(cid:12) , (3)
-0.4 Negative driving force (cid:12)(cid:12)Xi=1 (cid:12)(cid:12)
(cid:12) (cid:12)
0.1 0.2 0.3 0.4 0.5 plotted in Fig. 5(b). On(cid:12)average, the vo(cid:12)rtex-vortex in-
teractions cancel along the x direction since the vortices
F
dc are equally spaced a distance a apart in this direction.
Near n = 2.0/λ2, a transition in the vortex configura-
v
FIG. 4: hVxi versus Fdc curves for positive driving, Feixt = tionoccurs inwhichthe vortexcolumns beginto buckle,
xˆFdc(upperdashedline),andnegativedriving,Feixt =−xˆFdc as shown in Fig. 5(c). Here, the vortex-vortex interac-
(lower solid line). (a) A sample with nv = 8.68/λ2 showing tionsalongthexdirectionnolongercancel,andatpoints
a negative diode effect with fc− < fc+. (b) A sample with inthevortexcolumnswhereabuckledvortexisadjacent
nv =10.42/λ2 showing a positivediode effect with fc− >fc+. to the column, a vortex within the column experiences
both the force from the pinning substrate, which has
a maximum value of A , as well as an additional force
p
at n = 8.68/λ2, f− < f+ and we find a negative from the buckled vortex, which has a maximum value of
v c c
diode effect. The situation is reversed in Fig. 4(b) at K (a)f . A simple estimate of the depinning force in a
1 0
n = 10.42/λ2, where f− > f+ and a positive diode buckledregion,f− givesf− =A −K (a)f =0.395f ,
v c c c,b c,b p 1 0 0
effect occurs. which is close to the value of f− shown in Fig. 2 for
c
Fig. 3 indicates that f+ and f− both undergo oscil- n > 2.08/λ2. For values of n just above the buck-
c c v v
lations. A simple 1D picture for the vortex structure ling onset, buckling occurs in only a small number of
would predict maxima to appear in the depinning force placesthatarewidelyseparated. Vorticesnearthebuck-
at the commensurate fields at which an integer number led locations depin at f−, while vortices outside of the
c,b
of vortex rows fit inside each pinning trough, given by buckled locations do not depin until F is close to A .
dc p
n = n2/λ2, where n is an integer. We do observe a The result is the two stage depinning process shown in
v
maximum in both f+ and f− near n = 4.0/λ2, corre- Fig. 2 for n = 2.08/λ2. As the vortex density is fur-
c c v v
sponding to the presence of two ordered commensurate ther increasedaboven =2.08/λ2, the distance between
v
vortex rows per channel. Although f− has a very broad buckled regionsdecreases,smearing out the two step de-
c
plateau centered around n =9.0/λ2 which could be in- pinning process.
v
terpreted as the n=3 matching field, the inadequacy of For 1.91/λ2 < n < 5.90/λ2, y direction spatial cor-
v
a simple 1D picture is indicated by the fact that for f+, relations in the vortex configurations are lost due to the
c
thecorrespondingmaximumfallsatthemuchlowerfield bucklingofthevortexcolumns,butthepinningsubstrate
n =7.81/λ2. For vortex densities just above and below maintains the spatial correlations in the x direction, re-
v
eachofthecommensuratefields,theadditionorsubtrac- sulting in smecticlike vortex structures. This is illus-
tion of a small number of vortices does not change the trated in Fig. 5(c,d) for n = 2.08/λ2 and in Fig. 5(e,f)
v
5
FIG.5: (a)Vortexlocations (blackdots)and edgesofthepinningtroughs(blacklines) in a12λ×12λ subsection ofasystem
with nv = 1.04/λ2 and no applied drive. A single column of vortices is confined within each trough. (b) Corresponding
structure factor S(k) for nv = 1.04/λ2. Here and in other plots of S(k), we show a greyscale heightmap of S(kx,ky) with
white values higher than dark values, and with the origin of coordinates at the center of the panel. kx is along the vertical
axis, and ky is along the horizontal axis. (c) Vortex and pinning trough locations for a system with nv = 2.08/λ2 and no
applied drive. In some regions, the columns of vortices have buckled and a smectic type of ordering begins to appear. (d)
S(k) for nv = 2.08/λ2. (e) Vortex and pinning trough locations for a system with nv = 3.47/λ2 and no applied drive. Each
trough contains two columns of vortices. (f) S(k) for nv = 3.47/λ2. A double smectic ordering appears, as indicated by the
two prominent lines in S(k).
for n = 3.47/λ2. At n = 2.08/λ2 in Fig. 5(c), a fit across the pinning trough at a 45◦ angle so that the
v v
second column of vortices begins to form in each pin- overall vortex lattice symmetry is square. This is sup-
ning trough by means of a buckling transition, as pre- ported by the four prominent peaks that appear in S(k)
viously discussed. The corresponding structure factor in Fig. 6(f). There is a competition between minimiz-
in Fig. 5(d) shows some smectic type smearing, indicat- ing the vortex-vortex interaction energy, which favors a
ingthatthevorticesarelesscorrelatedinthe y-direction triangular lattice, and maximizing the vortex-substrate
thaninthex-direction. AsN increases,thecorrelations interaction, which favors commensurate vortex configu-
v
in the y direction are further decreased, as seen in S(k) rations. Since the difference in energy between a trian-
for n =3.47/λ2 in Fig. 5(f). At this density, the vortex gular and square vortex lattice is quite small, the com-
v
configuration shown in Fig. 5(e) indicates that there are mensuration effect is favored at n = 10.42/λ2, and a
v
twocolumnsofvorticespertrough,whichisalsoreflected squarestructureappears. Sincethevortexlatticeissym-
by the double band feature in S(k). At n = 5.21/λ2, metrical, the vortex-vortex interactions are strongly re-
v
shownin Fig. 6(a), the vortex configurationstill has two duced and the depinning transition is similar to the case
columns of vortices in each pinning trough, and the two of n < 1.91/λ2 where f− > f+ and the vortices depin
v c c
band feature in S(k) is more prominent, as illustrated more readily in the easy direction. In this case, the de-
in Fig. 6(b). As N increases further, the two column pinning response is elastic, but since there are multiple
v
structure is destroyed and the vortex configuration be- vortexcolumnsineachtroughthereisapronounceddis-
comes highly disordered in both the x and y directions, tortion in the lattice when a dc force is applied, causing
as seen at n = 6.94/λ2 in Fig. 6(c). Here S(k) has a f− to be significantly lower than A .
v c p
ring structure, as shown in Fig. 6(d). The effect of the
one-dimensionalsubstratecanstillbeseenintheformof Asafunctionofvortexdensity,thesystemthuspasses
small modulations along k in Fig. 6(d). through a sequence of triangular, smectic, disordered,
x
and square vortex arrangements. These results show
Atn =10.42/λ2,plottedinFig.6(e),anewcommen- similarities to studies performed on magnetic particles
v
surate crystalline structure appears where four vortices interacting with simple symmetric periodic modulated
6
FIG. 6: (a) Vortex locations (black dots) and edges of the pinning troughs (black lines) in a 12λ ×12λ subsection of a
system with nv = 5.21/λ2 and no applied drive. Each trough captures two columns of vortices. (b) Corresponding S(k) for
nv = 5.21/λ2 showing evidence of smectic ordering. (c) Vortex and pinning trough locations for nv = 6.94/λ2. The vortex
configurations are disordered. (d) S(k) for nv = 6.94/λ2 has a clear ring structure. (e) Vortex and pinning trough locations
for nv =10.42/λ2. The vortices form a tilted square lattice. (f) S(k) for nv =10.42/λ2 shows four-fold peaks.
latticethatwefinddidnotappearinthemagneticparti-
cleexperimentsincetherelativeparticledensitiesconsid-
eredinRef.[19]weremuchlowerthantherelativevortex
350 densitiesweusehere. Additionally,theasymmetryinour
<vx>Nv potential substrate may help to stabilize certain phases
300 that are not seen with simple symmetric potentials.
0
0.343
250
0.776
1.209
200 IV. RATCHET AND REVERSE RATCHET
1.642
EFFECT
2.075
150 2.508
2.941 We next examine the response when an external ac
100 3.374
drive is applied. When the vortex density is low, the
3.807
vortex-vortex interactions are negligible and the system
50 4.240
responds in the single particle limit. In Fig. 7 we show a
0.5 1.0 1.5 2.0 2.5 3.0 contour plot of the average velocity hVxi obtained from
simulationswithn =0.087/λ2forvariedacperiodτ and
F v
ac
ac amplitude F . A series of tongues where a positive
ac
ratcheteffectoccursareclearlyvisible. Atsmallτ,there
FIG. 7: (Color online) Contour plot of hVxi as a function is not enough time for the vortices to respond to the ac
of ac drive period τ (in units of simulation time steps) and drive so there is no ratchet effect. Similarly, at low F ,
ac
ac drive amplitude Fac (in units of f0) for a system with the vortices are unable to cross the potential barrier of
nv =0.087/λ2. the trough, and no rectification occurs. At intermediate
values of τ and F , a ratchet effect can occur. In the
ac
first tongue, which falls at the lower left edge of Fig. 7,
substrates, where crystalline, smectic, and partially dis- the vortices move by +a in the x direction during the
orderedstructureswereobservedasafunctionofparticle positivehalfofthe drivingperiod, butdo notmoveback
density [19]. The double smectic phases and the square during the negative half of the period, resulting in a net
7
0.15
0.3
nv=1.39
0.10 Fac=0.6 nv=1.74
Fac=0.4 0.2 nv=2.43
> 0.05 > nv=3.13
<vx 0.00 <vx 0.1
0 5 10
-0.05 nv 0.0
0 1
F
-0.10 ac
-0.1
-0.15
FIG. 9: hVxi vs Fac for the same system as in Fig. 8 for
FIG. 8: hVxi vs nv for the system in Fig. 7 at fixed τ =104 varied nv. Filled squares: nv = 1.39/λ2; open squares:
simulation steps for (black circles) Fac = 0.6f0 and (open nv = 1.74/λ2; filled circles: nv = 2.43/λ2; open circles:
circles) Fac =0.4f0. Here both positive and negative ratchet nv = 3.13/λ2. The positive ratchet effect is preceded by a
effects occur as a function of vortex density. negative ratchet effect for nv >1.74/λ2.
rectification. On the second tongue, vortices translate
the system passes through a maximum negative value of
by +2a during the first half of the period and by −a
hV i just below n = 5.21/λ2. Above n = 5.21/λ2,
during the second half of the period, again giving a net x v v
hV i gradually increases back toward zero before cross-
x
rectification. As τ and F increase, the ratchet effect
ac ing zero and reversing to a small positive ratchet effect
on the nth tongue is characterized by motion of +na
near n =10.42/λ2. The features in these curvesappear
v
during the positive half period of the drive and motion
to be correlated with the vortex lattice structures pre-
of−(n−1)aduringthe negativehalfperiodofthe drive.
sented in Figs 5 and 6. At low n , the vortices form a
v
The tongue structure emerges since for some values of τ
triangularorsmecticstructurewithonlyasinglecolumn
andF ,vorticesundergoequaltranslationsduringboth
ac of vortices in each pinning trough, as in Fig. 5(a,b). In
halves of the driving period, leading to no net motion.
this case,since the symmetryofthe vortexconfiguration
The boundary of the nth tongue where ratcheting can
cancels or strongly reduces the vortex-vortexinteraction
occur is given by
force, a positive ratchet effect occurs that is determined
nl+ (n−1)l− only by the pinning forces of the substrate. The onset
τn =ηF −F+ +ηF +F− (4) of the negative ratchet effect corresponds to the appear-
ac ac anceofbucklingofthecolumnsofvorticesinthepinning
The weak pinning force F+ associated with the trough troughs, as seen in Fig. 5(c,d). In the double smectic
side of length l+ is F+ = −A /2 in our case, while the phase illustrated in Fig. 5(e,f) and Fig. 6(a,b), the neg-
p
strong pinning force F− associated with the trough side ative ratchet effect persists. The gradual disappearance
of length l− is F− = Ap. For the first tongue in Fig. 7, of the negative ratchet effect for nv > 5.21/λ2 in Fig. 8
Eq. 4 predicts τ ∝ (F +A /2)−1, which matches the is associated with the crossoverfrom the double smectic
1 ac p
simulation results. The results in Fig. 7 and Eq. 4 agree state to the disordered state shown in Fig. 6(c,d). The
well with the tonguelike features found in Ref. [5] as a small positive ratchet effect near nv =10.42/λ2 is corre-
function of τ and F . This confirms that at low vor- lated with the formation of the ordered square lattice as
ac
tex densities,anordinaryratcheteffectoccurswherethe seen in Fig. 6(e,f).
vortices move in the easy direction. In Fig. 9 we plot hVxi versus Fac for a series of sim-
We next consider the effect of increasing nv. Since ulations performed at varied nv. For low nv ≤ 1.74/λ2,
the ratchet effect in Fig. 7 is enhanced for larger peri- there is no ratcheteffect until a thresholdvalue of Fac is
ods τ, we fix the period at τ = 104 simulation steps. reached. hVxi then increases to a peak value and gradu-
In Fig. 8 we plot hVxi versus nv for Fac = 0.6f0 and allydecreaseswithincreasingFac. Fornv >1.74/λ2,the
0.4f0. For nv < 2.60/λ2, there is a positive ratchet thresholdvalueofFac abovewhicharatcheteffectoccurs
effect for Fac = 0.6f0 and no ratchet effect for Fac = is lowerthanfor nv ≤1.74/λ2,andthe initialratchetef-
0.4f . For n > 2.60/λ2 a negative ratchet effect oc- fect is in the negative direction rather than the positive
0 v
curs for Fac =0.4f0, while the positive ratchet effect for direction. As Fac is increased,the negativeratcheteffect
F = 0.6f decreases in size and reverses to a negative reverses to a positive ratchet effect.
ac 0
ratcheteffect near n =3.65/λ2. For both values of F , Figure 10 illustrates hV i in a contour plot as a func-
v ac x
8
<v >
x
1.5
-0.126
-0.012
1.0
c
a
F 0.102
0.5 0.216
0.330
0.0
2 4 6 8 10 12
n
v
FIG.10: (Coloronline)AcolorscalemapofhVxiasafunction
ofFacandnv. ForlowFacthereisnoratcheteffect. Athigher
Fac there is a positive ratchet effect for low nv, a reversal
to a negative ratchet effect for intermediate nv, and another
reversal at high nv to a positive ratchet effect.
FIG. 11: (a) Vortex locations (black dots) and edges of the
pinning troughs (black lines) in a 12λ×12λ subsection of a
tion of F and n . The positive and negative ratchet system with nv = 2.08/λ2 at Fac = 0.5f0 when the nega-
ac v
tiveratcheteffect occurs. Thepositiveportion ofthedriving
effect regimes are highlighted. For low F there is no
ratchet effect, while for high F /f > 1a.c0 all ratchet period, with Feixt = xˆFac, is shown. The vortices form two
ac 0 columnsineverytrough. (b)Thecorrespondingstructurefac-
effects are gradually reduced. The maximum positive
torS(k)forpositivedrive. Theoveralldisorderinthevortex
ratchet effect occurs for Fac/f0 close to 1.0, the value configuration is indicated by the ring structure. (c) Vortex
of the pinning force in the strong direction. The posi- and pinningtrough locations for thesame system duringthe
tive ratchet effect for nv < 2.60/λ2 does not occur un- negative portion of the driving period, with Feixt = −xˆFac.
less F /f > 0.5, which is the maximum pinning force The vortices are arranged in buckled columns. (d) The cor-
ac 0
in the weak direction. The negative ratchet effect ap- responding S(k) for negative drive shows a smectic type of
pears for n >2.08/λ2 and extends up to n =8.68/λ2, ordering.
v v
with the maximum negative rectification occurring for
3.47/λ2 < n < 5.21/λ2 at 0.45 < F /f < 0.5. For
v ac 0
n > 9.72/λ2, a reentrant positive ratchet effect ap- ture. Thefirstreversaloftheratcheteffectforincreasing
v
pearsthatismuchweakerinamplitudethanthepositive Nv is related to the vortex-vortex interactions. This is
ratchet effect region found for low n . In general, the more clearly seen from Fig. 11, where we plot the vor-
v
ratchet effect in either direction is reduced for increas- tex positions and corresponding structure factor during
ing nv since the effects of the substrate are gradually both halves of the driving cycle for nv = 2.08/λ2 and
washed out by the increasing vortex-vortex interaction Fac =0.5f0 where a negative ratchet effect occurs. Dur-
forces. This result shows that vortices interacting with ingthepositivedriveportionoftheaccycle,thevortices
simple asymmetric pinning potentials can exhibit multi- are roughly evenly spaced in a disordered arrangement,
ple ratchet reversals similar to those seen for the more as shown in Fig. 11(a,b). When the negative drive is
complex 2D pinning array geometries [17]. It is likely applied, the vortices form a clear smectic structure with
thatforhighern ,higherorderreversalswilloccurwhen roughly one vortex column per pinning trough, as illus-
v
thevorticesagainformasmecticphasewithfiveormore trated in Fig. 11(c,d). In this case, the vortex configura-
vortexchainsper pinning trough. We arenotable to ac- tion is asymmetric due to the appearance of buckling in
cessthehighvortexdensitieswherethesereversalswould the vortexcolumns. At a buckledlocation,anextra vor-
be expected to appear. The higher order ratchet effects tex appears on the positive x side of the vortex column
would likely continue to decrease in amplitude with in- (to the right of the column) in the same trough. These
creasing field. extravorticeseffectivelypushtheadjacentvorticesinthe
Thepositiveorregularratcheteffectthatoccurswhen columnoverthe potentialbarrierinthe hardornegative
thevorticesmoveintheeasydirectionisstraightforward x direction.
tounderstandatlowfieldsbasedonasingleparticlepic- If the extra vortex is located a distance a from the
9
column, then a portionof the vorticesinthe column can
escape over the potential barrier in the hard direction
when the following condition is met: 150 Fac=0.3
Fac=0.4
Fac+K1(a)f0 >Ap. (5) Fac=0.5
100 Fac=0.6
When A /f = 1.0 and K (a) = 0.6, a negative ratchet
effect isppre0dicted to occu1r when F /f > 0.4, which Nv Fac=0.7
ac 0 > Fac=0.8
is what is observed in Fig. 9. It could also be argued x50
v
that when the drive is in the positive direction, the ex- <
tra vortices should be pushed by the column of vortices
and should more easily move in the positive direction, 0
allowing the conditions of Eq. 5 to be met in the posi-
tive direction as well. In Fig. 11(a) we show that this
does not occur because the vortex arrangement is quite
-50
different during the positive drive portion of the cycle.
0.0 0.5 1.0 1.5 2.0
The vortex-vortex interaction term is not present since
A
the vortices do not form the smectic structure that is p
seen when the driving is in the negative direction. In-
stead,thevorticesformtwocolumnswithineachpinning
FIG. 12: hVxiNv vs Ap for the same system as in Fig. 10
trough,allowingthe individual vorticesto remainevenly with nv = 3.13/λ2. Squares: Fac = 0.3f0; circles: Fac =
separated in both the x and y directions, and prevent-
0.4f0;uptriangles: Fac =0.5f0;downtriangles: Fac =0.6f0;
inganindividualvortexfrompushinganothervortexout diamonds: Fac = 0.7f0; stars: Fac = 0.8f0. For large Ap
of the trough. The vortex structure for positive drive is only a positive vortex ratchet effect occurs as the vortices
more symmetrical, as seen in S(k) in Fig. 11(b) which form a single column in each pinningtrough. Asthepinning
has a smeared crystalline or disordered structure. This strength is reduced, buckling of the columns begins to occur
symmetry reduces the vortex-vortexinteraction forces. and a negative ratchet effect appears.
A ratchet effect in the positive direction occurs when
the following equation is satisfied:
The negative rectification is enhanced for both low A
p
Fac >Ap/2. (6) and low Fac. Changing the strength of Ap in a single
superconducting sample is difficult; however,ageometry
If F /f = 0.45 and A /f = 1.0, then the conditions similar to that shown in Fig. 1 could be created using
ac 0 p 0
of Eq. 5 are met but the conditions of Eq. 6 are not, so tunable optical traps for colloidal particles driven with
the negativeratcheteffectcanoccurbutnotthe positive an ac electric field.
ratchet effect. Once F /f > 0.5, it is also possible for
ac 0
the positive ratchet effect to occur and the two ratchet
mechanisms will compete, decreasing the magnitude of V. SUMMARY
the negative ratchet effect as seen in the phase diagram
of Fig. 10 for higher F . As n is increased, the smectic In conclusion, we have shown that the vortex ratchet
ac v
structure is graduallylostas shownin Fig.6(c,d), so the system proposed by Lee et al. in Ref. [5] for vortices in-
vortex-vortexinteractions required to produce the nega- teracting with simple periodic asymmetric pinning sub-
tive ratchet effect are reduced and the negative ratchet strates can exhibit a series of ratchetreversalssimilar to
effectisdiminished. Oncethevortexlatticeisagainsym- those observed for 2D asymmetric periodic pinning ar-
metrical, as in Fig. 6(e,f), the positive ratchet effect ap- rays. Unlike most vortex ratchet geometries considered
pears. This result suggests that in regions where a re- todate,theratcheteffectspresentedherecanonlybeex-
versedratcheteffectappears,thevortexlatticestructure plained by a 2D description of the vortex configuration,
must exhibit some asymmetry when driven in the hard and not by a simple 1D model. As a function of vortex
direction for the reverse ratchet effect to occur. density, a richvariety ofvortexphases canbe realizedin
We have also investigated the reversal of the ratchet thissystemincludingtriangular,smectic,disordered,and
effect as a function of the pinning strength A for fixed squarelattices. Thedccriticaldepinningforcesexhibita
p
n = 3.13/λ2 and τ = 104. In Fig. 12 we plot hV iN reversiblediode effect asa functionofvortexdensity. At
v x v
versus A for F ranging from 0.4f to 0.8f . For all low magnetic fields, the depinning force is higher in the
p ac 0 0
values of F , as A increases a positive ratchet effect harddirection,correspondingtothe regulardiode effect,
ac p
appears. This is because the stronger A forces the vor- while at higher magnetic fields, the depinning force is
p
tices to form completely 1D unbuckled columns inside lowerintheharddirection. Anotherreversalofthediode
the pinning troughs. For lower A , the vortex columns effect occurs at even higher fields. The reversed diode
p
canbuckle,providingtheunbalancedvortex-vortexinter- effect appears when the symmetry of the vortex config-
actions required to produce the negative ratchet effect. uration changes for different directions of dc drive, as in
10
the case of the smectic and disordered phases. Here, the ratchet effect on ac amplitude, vortex density, and pin-
importance of vortex-vortex interactions differs for the ning strength. This type of ratchet could be also real-
two drive directions. At densities where the vortex lat- izedforotherkindsofsystemscomposedofcollectionsof
ticeishighlyordered,aregulardiodeeffectoccurswhere repulsively interacting particles and simple asymmetric
the vortex depinning force is lower in the easy direction. substrates, such as colloid systems.
The vortex lattice structures are also directly correlated
with the sign of the ratchet effect that occurs under an
ac drive. The highly symmetric structuresratchetin the
VI. ACKNOWLEDGMENTS
easy direction since the vortex-vortex interactions effec-
tively cancel, causing the sign of the ratchet effect to be
determined only by the asymmetry of the pinning sub- This work was carried out under the auspices of the
strate. In the smectic states, the vortex-vortex interac- NationalNuclearSecurityAdministrationoftheU.S.De-
tionsbecomerelevantandcanproduceanegativeratchet partment of Energy at Los Alamos National Laboratory
effect, which can be predicted with force balance argu- underContractNo.DE-AC52-06NA25396. Q.L.wasalso
ments. We have shown the dependence of the reversible supported in part by NSF Grant No. DMR-0426488.
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