Physical Review Letters 89, 024101 (2002)

Rectification and Phase Locking For Particles on Symmetric Two Dimensional Periodic Substrates

C. Reichhardt, C.J. Olson, and M.B. Hastings

CNLS, Theoretical, and Applied Physics Divisions, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 16 October 2001; published 21 June 2002)

We demonstrate a rectification phenomenon for overdamped particles interacting with a 2D symmetric periodic substrate when driven with a dc and circular ac drive. As a function of longitudinal dc amplitude, the longitudinal velocity increases in a series of quantized steps distinct from Shapiro steps with transverse rectification occurring near these transitions. The rectification phenomenon is explained using symmetry arguments and a simple model.
There has been great interest in dissipative nonequilibrium systems capable of producing a ratchet effect. For a particle moving in an asymmetric potential, a net dc drift in one direction can occur in the absence of a dc drive when an applied ac drive is combined with ac flashing of the potential. This ratchet or rectification phenomenon has been studied extensively in the context of biological motors [1], particle segregation [1,2], atom transport in optical lattices [3], and fluxon motion in superconductors or SQUIDs [4,5]. In these effectively one dimensional (1D) systems the asymmetric potential is the key ingredient leading to rectification. In 2D systems, there are many new ways to introduce an asymmetry which are not accessible in 1D, such as the choice of a clockwise ac drive. There has been considerable work on separating different particle species through rectification in the flow of biomolecules driven with time-varying fields through 2D arrays of obstacles [7]. Particles transported through symmetrical periodic potentials under an ac drive exhibit a wide variety of nonlinear behavior, including phase-locking that occurs when the external ac frequency matches the frequency of motion over the periodic potential. This produces the well-known Shapiro steps observed in the V(I) curves of Josephson-junction (JJ) arrays [6]. In all these phase locking systems the ac and dc drives are in the same direction. Almost nothing is known about what type of phase locking can occur when the ac and dc drives are not in the same direction.
In this work we study rectification and phase locking for an overdamped particle moving in a 2D symmetrical periodic substrate under an applied longitudinal dc drive (fDC) and two ac drives: fACx in the longitudinal direction 90 degrees out of phase from fACy in the transverse direction. We find that the longitudinal velocity increases in a series of steps as fDC increases. Near the transition between two steps we find a rectification of the particle motion in the transverse direction. The steps correspond to drives at which the particle motion forms orbits commensurate with the substrate period. The rectification, which can be understood on symmetry grounds, is predominantly in one direction; however, we also observe a number of reversals of the rectification as well. Our results should be directly applicable to vortices in superconductors with periodic pinning, as well as colloids and biomolecules interacting with 2D arrays of obstacles.
As a model for vortices in superconductors or colloids in solution, we consider an overdamped particle moving in 2D interacting with a repulsive periodic substrate according to the equation of motion: fi = fs + fDC + fAC = ηvi, with η = 1. The force from the substrate, a square array of side a, is fs = −∇U(r); the form of U(r) is discussed below. We consider a system of size 8a ×8a. The dc drive fDC is applied along the symmetry axis of the pinning array, in the longitudinal (x) direction. The ac drive is fAC = Asin(ωAt)∧x + Bcos(ωBt)∧y. Note that there is no dc driving component in the y or transverse direction. We fix wA/wB = 1.0 and A = B, and examine both the longitudinal time averaged particle velocity < Vx > and the transverse velocity < Vy > . fDC is increased from 0 to 1.0 in increments of 0.0001, with 3 ×105 time steps spent at each drive to ensure a steady state.
To model specific systems, we consider periodic substrate potentials created by pinned particles, such as vortices in a periodic array of holes [8] or magnetic dots [9]. Once all the holes are filled with a vortex, any vortices in the interstitial regions between holes experience a smooth periodic substrate created by the interactions with the pinned vortices. For fDC=0, two crossed ac drives with small amplitudes A cause the vortex to move in a circle in the interstitial regions. At increasing A, there are stable vortex orbits which encircle one pinned vortex, then four, nine, and so on. We focus on A large enough to generate orbits encircling one or more pinned vortices. For the pinned particles, we use the potential for vortices in a thin film superconductor, U(r) = −ln(r), and employ a summation technique [10] for the long range interaction. We have also considered potentials for unscreened or screened charges with interaction of U(r) = 1/r and U(r)=e−κr/r, respectively.
Fig1.png
Figure 1: (a) Longitudinal velocity Vx/aω vs driving force fDC which is applied in the x direction. (b) Corresponding transverse velocity Vy/aω vs fDC. Inset to (a): Simulated Vy for the system shown in Fig. 1 at temperatures T=0.005 (top), T=0.01, T=0.03, and T=0.05 (bottom).
In Fig. 1(a) we show the longitudinal velocity Vx/(a ω) versus fDC for a particle with A = B = 0.36. For these parameters, at fDC=0 the particle encircles one pin for 0.29 < A < 0.4. As shown, Vx increases in quantized steps with step heights of ∆Vx = aω. We label a step n according to the value of Vx on the step, Vx=naω. The widths of the integer steps varies with A and ω in an oscillatory manner. There are also some fractional steps with heights ∆Vx = (p/q)aω where p and q are integers such that p/q < 1.0.
In Fig. 1(b) we show the transverse velocity Vy/(aω) versus fDC. Since there is no net dc force in the transverse direction, Vy = 0 for most values of fDC. Near the steps in Vx, however, Vy is nonzero, indicating that rectification is occurring. The first rectification, in the positive y direction, occurs at the n = 0 → 1 step, while at the n = 1 → 2 step the rectification is in the negative y direction. For the higher steps, positive rectification regions with increasing widths appear, while near the n = 7 → 8 step there are regions of negative rectification. Some of the rectification phases have very well defined heights of ∆Vy = ωa. These include the positive rectification phases at 3 → 4, 4 → 5, and 5 →6, as well as the negative rectification phases at 6 → 7 and 7 → 8. On these rectification plateaus, we find that the particle moves in only one type of orbit. The other rectification phases do not have a well defined height, including 0 → 1, 2 → 3, as well as some portions of the 6 → 7 and 7 → 8 regions. In these phases, for any fixed fDC the particle jumps intermittently between different rectifying orbits with transverse velocities (p/q)aω.
For all of the rectifying regions, if the polarity of the ac drive is reversed, Vx remains unchanged while Vy changes to −Vy. If the waiting time between dc drive increments is increased, the results are unchanged. The phases described here remain stable when the system is started from a fixed fDC value, and are not transient phenomena. As the system size varies, certain orbits become incommensurate with the system length and precess spatially; however, the velocity curves are not affected by the system size.
Fig2.png
Figure 2: Trajectories for fixed fDC for different non- rectifying regions seen in Fig. 1. The black dots denote the location of the fixed particles or the potential maxima of the periodic substrate. Shown are steps with: (a) n = 0, (b) n = 1, (c) n = 2, (d) n = 4, (e) n = 5, and (f) n = 8.
In Fig. 2 we illustrate representative non-rectifying particle orbits for Vx steps of n = 0, 1, 2, 4, 5, and 8. For n = 0 [Fig. 2(a)], Vx=0, and the particle moves in a confined square orbit which encircles one pin. For n = 1 [Fig. 2(b)] there is a net motion in Vx, and the particle circles around one pin before moving over to the next plaquette. For n = 2 [Fig. 2(c)] the orbit does not encircle a pin but forms a small loop which repeats every second plaquette. For n = 4 and n=5 [Fig. 2(d,e)], orbits similar to n = 2 occur, with the loop motion now repeating at every fourth or fifth plaquette, respectively. A similar process continues up through the n = 7 step. For the n = 8 step [Fig. 2(f)] and above, the particle is moving fast enough that the transverse width of the orbit is less than a, and no loops appear.
Fig3.png
Figure 3: Trajectories for fixed fDC for different rectifying regions seen in Fig. 1. The black dots denote the location of the fixed particles or potential maxima of the periodic substrate. (a) right before n=3 → 4 step, (b) right after n=3 → 4 step (c) right before n=4 → 5 step, (d) negative rectification phase below the n=6 → 7 step (fDC = 0.454), (e) negative rectification region on the n=7 step (fDC = 0.465), (f) negative rectification region on n = 8 step (fDC = 0.514).
Figure 3 shows representative rectifying orbits. The particle orbits differ below and above a given step, as illustrated in Fig. 3(a,b) for the n=3 → 4 step. Below the step [Fig. 3(a)], the particle moves 3a in the x direction and a in the positive y direction in a single period. A loop forms when the particle moves in the y direction. Above the step [Fig. 3(b)], the particle moves 4a in the x direction but still only a in the y direction in one period; therefore, Vy does not change at the Vx jump. In Fig. 3(c), below the n = 4 → 5 step, an orbit similar to that of Fig. 3(a) appears. The rectifying orbit above the 4 → 5 step resembles that of Fig. 3(b). In Fig. 3(d) (fDC = 0.454) we show the negative rectification phase below the n = 7 → 8 step. Here the particle jumps a in the negative y direction every seventh plaquette through a small kink. We do not observe loops in the trajectories for the negatively rectifying phases. Figure 3(e) (fDC = 0.465) illustrates trajectories for a particle in the negative rectification region close to the n = 7 → 8 step. The particle does not move in a specific orbit, but jumps intermittently over time between different orbits with Vy = (p/q)aω; however, Vx remains fixed. Similar intermittent trajectory patterns appear in the rectifying phases near n = 0 → 1, 1 → 2, and 2 → 3. Intermittent patterns occur only on the lower steps; above the n=8 step, only stable rectifying trajectories occur, as illustrated in Fig. 3(f) (fDC = 0.514).
Rectifying phases occur for any A large enough that the particle trajectory at fDC=0 encircles more than one pin. We have measured Vy/aω for the same system shown in Fig. 1, but with an orbit at A = 0.42 that encircles four pins. As in Fig. 1, steps in Vx and transverse rectification in Vy appear, but the step heights are now ∆Vx = 2aω. We find that as A is further increased, orbits that stably encircle p2 pins, with p integer, produce steps of height ∆Vx=paω. We obtain very similar results for substrates with a 1/r or e−κr/r interaction.
We next consider the effects of finite temperature. We add a noise term fT to the equation of motion with the property < fT(t) > = 0.0 and < fT(t)fT(t′) > = 2ηkB T δ(t−t′). In the inset of Fig. 1(a) we plot Vy for 0.18 < fDC < 0.22 at increasing T. For low T there are still regions where is Vy is near zero within our resolution. The value of Vy should not be zero but exponentially small for low temperatures. For higher T the maximum Vy decreases, the width of the Vy peaks are smeared, and the regions where Vy = 0 are lost. The transverse rectification still occurs well into the high T regime where the particle is diffusing about rapidly. We note that the T=0 approximation should still be a good approximation for vortex motion in thin film superconductors for T/Tc < 0.9. We have also found that the effects of temperature can be reduced for stronger interactions between the particle and the substrate, which are increased for smaller a.
We now consider a more formal argument employing nonlinear maps to understand the nature of the longitudinal phase locking and transverse rectification. Define a map (x,y)→ (x′+nx a,y′+ny a), from the position of the particle at the start of a period to that at the end, where we may restrict to 0 ≤ x,y,x′,y′ ≤ a, with nx,ny integer. If there is a stable fixed point, (x,y)=(x′,y′), then the particle translates by (nx a,ny a) in time ω−1 and so has average velocity Vx,Vy quantized in multiples of a ω, as found above. If the q−th power of the map has a stable fixed point, there are instead steps of fractional heights (p/q) a ω.
As fDC increases, the periodic orbit becomes unstable, and a different periodic orbit with larger Vx appears. This new orbit will be the next stable periodic orbit at higher drive. The transition to the new orbit can occur in one of three ways. (1) If both periodic orbits are stable simultaneously, the particle velocity will depend on the initial conditions in the transition regime. This was not observed. (2) The second periodic orbit could become stable at the same time that the first orbit becomes unstable. This behavior, which gives rise to infinitely sharp jumps in Vx, is not generic and hence not expected. (3) There can be a finite range of drive containing no stable periodic orbits. Over this range, the average velocity is not quantized. If, however, some orbits are close to stable, the particle will spend long times in these orbits, giving rise to intermittent behavior. This behavior is consistent with what we observe.
The rectification can be understood on symmetry grounds. The dc drive breaks Ry, the reflection symmetry across the y-axis, but preserves Rx, reflection across the x-axis. The ac drive breaks both Rx and Ry, but preserves the combined symmetry Rx Ry. The combined drives break all such symmetries, leading to rectification.
Fig4.png
Figure 4: (a) Time averaged longitudinal velocity Vx obtained from model. (b) Corresponding transverse velocity Vy.
We now turn to a specific toy model illustrating some of these ideas. Consider a particle in a lattice of repulsive sites with a=1, where the potential minima between repulsive sites are at integer x and y values. The y position of the particle is constrained to take only integer values, but the x position can be any real value. To model the translation of the particle through the lattice, the particle moves first (i.) right, then (ii.) down, then (iii.) left, then (iv.) up. (i.): We apply the rule x → x+vr. (ii.): If x is within 0.25 of an integer, x is set to that integer and y is decremented by one. (iii.): Apply x → x−vl. (iv.): As in (ii.) except y is incremented by one. Here vr and vl are the velocity of the particle in the rightward and leftward parts of the cycle, respectively. In steps (ii.) and (iv.), the particle will only move to a new y position if it reaches the minima between sites at the correct phase of the driving period, when transverse motion is possible. In this case, the particle slips into the next row and the x coordinate of the particle is set to midway between the pinning sites. In Fig. 4 we show the time-averaged velocities Vx and Vy obtained with this model for fixed vl=1.15 and increasing vr, representing increasing fDC. This simple model produces both plateaus and ratcheting behavior. Much of this behavior is specific to two or more dimensions. Consider a map x→ x′, subject to x+a → x′+a and dx′/dx ≥ 0, true for overdamped motion in one dimension. It can be shown that it is not possible to have stable periodic orbits with different values of the current: Shapiro steps cannot exhibit jumps.
The ratcheting behavior occurs near transitions in Vx when the number of pinning centers the particle passes in one period changes, making it possible for the particle to interact asymmetrically with the pinning sites. For a clockwise orbit, the particle moves rapidly on the upper portion of the orbit, and is likely to scatter off the pinning site below when the orbit does not quite match na. On the lower part of the orbit, however, the particle is moving more slowly, and is likely to slip between the pinning sites above in spite of a small mismatch, tending to ratchet in the positive y direction.
The phases we have described should be experimentally observable for vortex motion in superconductors with a dc and crossed ac applied currents with periodic pinning arrays where vortices are located in the interstitial regions, as well as fluxon motion in 2D JJ arrays at rational filling fractions where vortex-vortex interactions are reduced, and in biomolecules moving through 2D arrays of posts [7]. In these active ratchets the rectified velocities can be controlled by nωa, unlike thermal ratchets which rely on Brownian motion. The longitudinal velocity steps and transverse rectification may also be observed in electrons undergoing classical cyclotron orbits in anti-dot arrays [11] for orbits where electrons encircle at least one anti-dot with an additional external dc drive.
In conclusion, we find that a novel form of rectification can occur for an overdamped particle driven by a dc and circular ac drive in a system without an asymmetric potential. The longitudinal velocity increases in a series of steps of height nωa. Along these steps the particle moves in commensurate orbits. Near the steps, rectification in the transverse direction occurs. We have specifically demonstrated this model for vortices in superconductors with periodic pinning and overdamped charged particles such as colloids. Using symmetry arguments we explain the origin of the rectification phenomenon. In addition using nonlinear maps we show that the phase locking phenomenon is distinct from Shapiro steps found for 1D systems. With a simple toy model we have shown that the qualitative features of the phase locking and rectification can be captured indicating that the results are not specific to a particular system. Our results can be relevant to vortices in superconductors with periodic pinning arrays, colloids and biomolecules moving through arrays, and classical electron motion in anti-dot arrays.
Acknowledgments-We thank M. Chertkov and Z. Toroczkai for useful discussions. This work was supported by US DOE W-7405-ENG-36.

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