Physica C 432, 125 (2005)

Rectification and Flux Reversals for Vortices Interacting with Triangular Traps

C.J. Olson Reichhardt and C. Reichhardt

Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Received 11 May 2005; received in revised form 25 July 2005; accepted 27 July 2005
Available online 22 September 2005

Abstract
We simulate vortices in superconductors interacting with two-dimensional arrays of triangular traps. We find that, upon application of an ac drive, a net dc flow can occur which shows current reversals with increasing ac drive amplitude for certain vortex densities, in agreement with recent experiments and theoretical predictions. We identify the vortex dynamics responsible for the different rectification regimes. We also predict the occurrence of a novel transverse rectification effect in which a dc flow appears that is transverse to the direction of the applied ac drive.
Vortices interacting with periodic pinning structures have been attracting considerable attention since they are an ideal system in which to study the static and dynamical behaviors of collectively interacting particles coupled to periodic substrates. In these systems, a variety of commensurability effects and novel vortex crystals can occur as a function of the ratio of the number of vortices to the number of pinning sites [1,2,3,4,5,6,7,8,9,10,11,12]. The vortex crystals are composed of one or more vortices trapped at each pin with any remaining vortices located in the interstitial regions between the pinning sites. A variety of dynamical flow phases occur when a drive is applied, including the flow of interstitial vortices between the pinning sites [13,14,15,16,17] as well as channeling along rows of pinning [13,14,15]. Much of the physics of vortices interacting with periodic pinning is also observed for repulsively interacting colloids in periodic optical trap arrays [18,19,20,21,22,23,24]. In addition to the basic science issues, vortices interacting with periodic pinning arrays are relevant to possible technological applications of superconductors, such as critical current enhancement or controlled motion of flux for new types of devices.
Several methods have been proposed for using periodic pinning or controlled disorder in superconductors to create vortex ratchets, rectifiers, and logic devices [25,26,27,28,29]. As in general ratchet systems, which can be deterministic or stochastic in nature [30], a vortex ratchet transforms an ac input into a dc response. Vortex ratchets can be created via asymmetric periodic pinning lines [25,26], asymmetric channels, or the asymmetry introduced by multiple ac drives in a system with a symmetric substrate [27,28]. Ratchets constructed of periodic two-dimensional (2D) arrays of asymmetric pinning sites have also been proposed [29]. Recently, both positive and negative vortex rectification have been experimentally realized in periodic arrays of triangular pinning sites [31]. In Ref. [31], the triangular pins are arranged in a square lattice with the tips of the triangles oriented in the +y direction. When an ac force is applied in the y direction at low matching fields, rectification of the vortex motion in the +y direction occurs over a range of ac amplitudes, with a peak rectification at a particular amplitude. For higher matching fields, there is a −y rectification at lower ac drives, followed by a +y rectification at higher ac drives. This change in the rectification direction is explained in Ref. [31] as originating from the separate motion of interstitial vortices at low drives, giving −y rectification, and the motion of the pinned vortices at higher drives, producing +y rectification. It is not clear how the motions of these two vortex species can be fully separated, especially in the presence of thermal fluctuations, and thus a clearer picture of the vortex dynamics producing the rectification is needed.
In this work we present simulations of vortices interacting with 2D square arrays of triangular pinning sites for parameters corresponding to the recent experiments of Ref. [31]. We find a +y rectification at low ratios n of the number of vortices to the number of pinning sites, with a maximum +y dc flux as a function of ac amplitude. When interstitial vortices are present, the initial rectification is in the −y direction, and is followed by a crossover to +y rectification at higher ac amplitude, as seen in experiments. We find that for large ac amplitudes, when the motion of the vortex lattice becomes elastic, the rectification switches to the −y direction, which is explained in terms of asymmetric drag. We also predict a novel transverse ratchet effect where a dc motion is generated in the direction transverse to the applied ac drive. All of the effects we describe here should also occur for repulsive colloidal particles interacting with 2D triangular traps.
Fig1.png
Figure 1: Pinning sites (triangles) and vortex positions (dark circles) at the matching fields n= (a) 1, (b) 2, (c) 3, and (d) 4.
We consider a thin film superconductor containing a square array of Np=64 triangular pinning sites with periodic boundary conditions in the x and y-directions. The sample contains Nv=nNp vortices (where n is an integer), each of which obeys the overdamped equation of motion
η d ri
dt
 = fivv +fp + fAC + fT  .
(1)
Here the vortex-vortex interaction force, appropriate for a thin film superconductor, is fivv = ∑j ≠ iNv(Φ02/μ0πΛ)∧r/rij, where Φ0 is the elementary flux quantum and Λ is the thin film screening length [32]. We use a fast summation method to evaluate this long-range interaction [33]. The pinning force fp comes from a square array of equilateral triangles with one vertex pointing in the +y direction, as shown in Fig. 1. Every triangular pin is modeled as three half-parabolas of equal strength, each of which attracts the vortex to a line passing through the center of the triangle and parallel to one of the sides. The pinning force is cut off at the edge of the triangle; however, note that interstitial vortices still experience an effective pinning potential due to the repulsion from the surrounding pinned vortices. The applied ac force fAC = Asin(ωt)∧r acts in the direction perpendicular to an applied ac current. Here we consider either fACy=Asin(ωt)∧y or fACx=Asin(ωt)∧x, with no mixtures of ac drives. Temperature is modeled as random thermal kicks with the property < fT(t) > = 0 and < fT(t)fT(t′) > = 2ηkBTδ(t − t′).
We match our parameters to those of the experiment in Ref. [31], performed in Nb films with η = 1.4×10−12 Ns/m. We take T/Tc=0.98, giving fT=0.46f0, where f0=ϕ02/2πμ0λ3=1.09×10−5 N/m. At this temperature, the London penetration depth λ = λ(0)/(1−(T/Tc)2)1/2=368 nm, so our pin spacing is 2.09λ in the x direction and 2.03λ in the y direction, and the pins are 1.68λ on a side. In the experiment, each pin held a maximum of three vortices, so to match this we set the pinning strength to fp=1.05f0. We fix ω = 78 kHz, higher than experiment due to the limitation of simulation time; however, since the behavior of the system is controlled by the ratio A/ω, we compensate by taking A larger than in experiment. We note that we find the same generic behaviors for other parameters, including a pinning array that is triangular rather than square.
In Fig. 1 we show the vortex positions and pin locations at A=0 for a system with a square array of triangular pins at fillings n=1, 2, 3, and 4. A global symmetry breaking occurs at n=2 between the two possible arrangements of the vortices allowed by the square pinning lattice. All of the vortices sit in the pins for n ≤ 3, but for n ≥ 4 the vortex-vortex interaction is strong enough that some vortices move to the interstitial regions.
Fig2.png
Figure 2: Net dc velocity < Vy > in units of m/s as a function of applied ac current A in units of f0 for n= (a) 1, (b) 2, (c) 3, (d) 4, (e) 5, and (f) 6. Inset to (a): dc depinning force |fdp| for +y (circles) and −y (x's) directions.
We first consider the dc depinning force fdp in the positive and negative y directions by applying a dc driving force of increasing magnitude and measuring the average vortex velocity < Vy > . In the inset to Fig. 2(a) we plot |fdp| as a function of n. At n=1 the depinning is symmetric in the ±y directions, as expected given our model for the pinning. For n=2, |fdp−y| is twice as large as |fdp+y|, due to the fact that the vortices align vertically for +y driving, and horizontally for −y driving. For n=3 and 4, |fdp+y| is 25% larger than |fdp−y|. Here, the vortex at the top of each pin acts to assist in the depinning of the vortices along the bottom of the pin for −y driving. For n > 4, the depinning is dominated by the interstitial vortices and the asymmetry in the depinning is lost. The depinning asymmetry at n=2 to 4 should be large enough to be observable experimentally. Depinning in the ±x directions is symmetric for all n. Asymmetric depinning at noninteger n was observed in Ref. [34].
Fig3.png
Figure 3: Pins (triangles), vortices (dots), and vortex trajectories (lines for all vortices in the right half of the panels, and for two selected vortices in the left half of the panels) during 1/10 of the driving period for: (a) N=2, A=1.0, +y portion of drive cycle, (b) −y portion of same cycle; (c) N=4, A=0.35, +y portion of drive cycle, (d) −y portion of same cycle; (e) N=4, A=0.75, +y portion of drive cycle, (f) −y portion of same cycle.
We next consider the effect of an applied ac drive fACy. We monitor the net dc velocity < Vy > at fixed ω and sweep the ac amplitude A, averaging < Vy > for 20 periods at each increment of A. In Fig. 2(a) we plot < Vy > vs A for n = 1. For A < 0.9, there is no net flow in x or y, indicating that the vortices are still pinned. For A = 0.9 a finite +y dc velocity appears with no net x dc velocity, corresponding to a +y rectification. As A increases further, < Vy > increases, reaching a maximum at A = 0.99 and then dropping back to zero at A=1.05. This behavior matches that seen in the experiments of Ref. [31]. We find no steps in the +y rectification regime, in agreement with the experiments, but in disagreement with recent T=0 simulations of a very small system for asymmetric pinning sites that predicted steps in the velocity vs fAC curve [35]. The transport does not occur in an organized step like flow, but is instead a stochastic process, as we will show in more detail in Fig. 3. In Fig. 2(b) and Fig. 2(c) we plot < Vy > for n = 2 and 3, respectively. These curves are similar to the n = 1 case, with an initial pinned phase followed by +y rectification that goes through a maximum with increasing A.
In Fig. 2(d) we show < Vy > for the case of n = 4 when interstitial vortices are present, as seen in Fig. 1(d). Here there is an initial negative rectification regime in the −y direction, followed by a crossover to a +y rectification as seen in experiments [31]. In Fig. 2(e,f) we plot the cases for n = 5 and n = 6, respectively, which show a similar rectification to that in Fig. 2(d). In all of these cases, a combination of interstitial and pinned vortices are present. As n is increased further, the ratcheting effects are gradually reduced and become indistinguishable from thermal noise.
Fig4.png
Figure 4: Schematic illustrating pins (triangles), vortices (dots), and vortex trajectories (dotted lines) for different cycles of the rectifying motion. (a) +y portion of the driving period for the case when net +y rectification occurs. The vortex trajectories are deflected by the pin and focused toward the tip of the triangle; as a result of the focusing of neighboring vortex trajectories, both vortices jump out of the pin due to their mutual repulsion before reaching its tip, and experience less than the maximum possible pinning force in the y direction. (b) −y portion of the drive for the same +y rectification case. Vortex trajectories are not strongly focused and each vortex feels the full pinning force from the bottom section of the triangle. (c) −y portion of the drive for the case when net −y rectification occurs. A vortex passing close to the triangle tip can depin and replace the vortex pinned at the triangle tip. Vortices pinned at the tips of neighboring triangles are too far away to prevent depinning. (d) +y portion of the drive for the same −y rectification case. A vortex arriving at the base of a triangle is unable to displace the pinned vortex due to the fact that other vortices pinned at the triangle bases are nearby and can prevent depinning.
To understand the role of interstitial and pinned vortices in the rectification process, we examine the vortex dynamics. 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 trajectories during 1/4 of the drive cycle when the drive is in the +y direction and Fig. 3(b) shows the −y portion of the cycle. The existence of similar motion for n=4 is illustrated in Fig. 3(e,f). At these temperatures, none of the vortices remain pinned. The triangle shape focuses the vortex trajectories during the +y motion and causes the vortices to move into the tip of the triangle; for n > 1, at most one vortex per plaquette can avoid the triangle tip altogether by passing through the interstitial region, as shown in the left half of Fig. 3(a). The focused vortices deviate out the sides of the triangle near its tip due to the mutual repulsion that occurs when two vortices are simultaneously focused, as schematically illustrated in Fig. 4(a), and experience a maximum pinning force of  ∼ fd. During the −y motion, shown schematically in Fig. 4(b), there is no focusing of the vortex trajectories and all vortices pass through the triangle bases where they experience a pinning force of fp. The net drag on the vortices during the −y cycle is n fp, while during the +y cycle it is (n−1)fd. The greater net drag during −y driving produces the +y rectification, which ends when
fd= n
n−1
 fp .
Taking an effective pinning force of fp=0.8 (due to thermal fluctuations) gives good agreement with the simulation results for the end of the +y rectification.
At n=4 and above, when interstitial vortices are present, −y rectification occurs at low drives, as shown in Fig. 2(d-f). In Fig. 3(c-d) we illustrate an example of vortex motion in the −y rectification regime for n=4 and A=0.35. It is clear that none of the vortices remain pinned; instead, all of the vortices are participating in the motion, with pinned and interstitial vortices switching places frequently. This can be understood by referring to Fig. 1(a) and noting that the interstitial vortices are offset in the +y direction with respect to the vortices in the tips of the triangles at zero drive. When we apply a −y drive, the interstitial vortices pass between the vortices at the triangle tips. For the vortex density and pin strength considered here, a single interstitial vortex can exert enough force on the vortex at the triangle tip while passing to depin it in the x direction, as schematically shown in Fig. 4(c). This depinning is prevented due to the fact that a second interstitial vortex is simultaneously passing on the opposite side of the pinned vortex, and counterbalances the force. Thermal fluctuations of the interstitial vortices are large enough to destroy this counterbalance, and as a result the vortices at the triangle tips are frequently depinned and replaced with interstitial vortices during the −y drive. In contrast, during the +y drive, the interstitial vortex moves past a vortex pinned at the base of a triangle, illustrated schematically in Fig. 4(d). This pinned vortex is stabilized by the presence of a second pinned vortex. Since thermal fluctuations of the pinned vortices are reduced relative to those of the free interstitial vortices, the probability that the vortex at the base of the triangle will be depinned and replaced by the interstitial during the +y drive is much lower than the probability for depinning of a triangle tip vortex during the −y drive. As a result, there is a larger amount of vortex motion in the −y portion of the drive cycle, producing a net −y rectification.
For n=1 in Fig. 2(a), at higher A > 1.05, < Vy > becomes negative, indicating −y rectification at high drives. This reversal occurs for all fillings but is most pronounced for n=1 and n=4, when the onset of the reversal falls at lower values of A. At higher drives, the focusing effects of the pinning are lost, and the vortices experience an averaged pinning force determined by the area of pinning over which they pass. During the +y portion of the drive, the entire area of each triangle exerts a drag on the vortices, whereas during the −y half of the ac cycle, only the lower 3/4 of each triangle exerts a drag due to the nature of the pinning. As a result of the greater effective pinned area for the +y drive, the net drag is higher, and a net rectification in −y results. As fd is further increased, the vortices sample a larger number of triangles during each drive cycle, and the difference in pinned area between +y and −y driving is enhanced, leading to a larger −y rectification.
Fig5.png
Figure 5: Transverse rectification for fACx. (a) < Vx > for n=2; (b) corresponding < Vy > with −y rectification. (c) < Vx > for n=4; (b) corresponding < Vy > with +y rectification.
In Fig. 5 we illustrate < Vx > and < Vy > when an ac drive fACx is applied in the x direction for n = 2 and 4. The net x velocity is zero but we find a transverse rectification in the y direction: −y for n=2 and +y for n=4. The magnitude of the rectification is comparable to that seen for fACy, and it should thus be experimentally observable. The transverse ratchet effect is produced by the interaction of the vortices with the pinning sites. For n=2, the vortices are deflected downward as they channel along the rows of pinning sites, producing −y rectification. Similar motion occurs for n=3. For n=4 to 6, the interstitial vortices pass through channels along the tips of the triangles and are deflected upward by the pinned vortices, producing +y rectification. This motion dominates even though all vortices are being pinned and depinned. Since the transverse motion does not require breaking of the x direction symmetry of the system, it can also be observed for a dc x direction drive.
In conclusion, we have conducted simulations of vortices in thin film superconductors interacting with triangular pinning sites for parameters relevant to recent experiments. We show that both a +y and −y rectification can occur at depinning depending on whether interstitial vortices are present, in agreement with experiment. We find dc depinning anisotropy for n=2 to 4, but observe that even when the dc depinning is not anisotropic, rectification can still occur due to the fact that the vortex dynamics differs under ac and dc drives. Rectification in the +y direction occurs when the ac drive overcomes the pinning strength and vortex channels form that flow in the +y direction. When interstitial vortices are present at zero ac drive, an initial −y rectification occurs at lower ac drives due to the easier exchange of pinned and interstitial vortices during −y motion. This occurs for n = 4 and above, in agreement with experiments. We also predict that a novel transverse ratchet effect with rectification in the y direction should occur when the ac drive is applied in the x direction. Our results should be generic to any type of repulsively interacting particles moving through triangular traps and may have potential applications for transport in colloidal systems.
This work was supported by the U.S. DoE under Contract No. W-7405-ENG-36.

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