Rectification of Vortex Motion in a Circular Ratchet Channel

# Rectification of Vortex Motion in a Circular Ratchet Channel

## Abstract

We study the dynamics of vortices in an asymmetric (i.e., consisting of triangular cells) ring channel driven by an external current in a Corbino setup. The asymmetric potential can rectify the motion of vortices and cause a net flow without any unbiased external drive, which is called ratchet effect. With an applied ac current, the potential can rectify the motion of vortices in the channel and induce a dc net flow. We show that the net flow of vortices in such a system strongly depends on vortex density and frequency of the driving force/current. Depending on the density, we distinguish a “single-vortex” rectification regime (for low density, when each vortex is rectified individually) determined by the potential-energy landscape inside each cell of the channel (i.e., “hard” and “easy” directions of motion) and “multi-vortex”, or “collective”, rectification (high density case) when the interaction between vortices becomes important. The frequency of the driving ac current determines a possible distance that a vortex could move during one period. For high frequency current, vortices only oscillate in the triangular cell. For low frequency, the vortex angular velocity increases nearly linearly until the driving force reaches the maximum friction force in the hard direction. Furthermore, the commensurability between the number of vortices and the number of cells results in a stepwise curve, which means that the average angular velocity of the vortices is a discontinuous function of the applied current. Besides the “integer” steps, i.e., the large steps found in the single vortex case, we also found fractional steps corresponding to fractional ratio between the numbers of vortices and triangular cells. The principal and fractional frequencies for different currents are found, when the net flow of vortices reaches the maximum that is proportional to the frequency when the density of vortices is low. We have performed preliminary measurements on a device containing a single weak-pinning circular ratchet channel in a Corbino geometry and observed a substantial asymmetric vortex response.

###### pacs:
74.25.Uv, 74.78.Na, 05.70.Ln

## I Introduction

A net flow of particles under unbiased external fluctuations/drive due to an asymmetric potential, which is called ratchet effect, has received much attention during the last decades. The transport and dynamical properties of particles on asymmetric potential have been widely studied, e.g., in physics and biology (1); (2); (3); (5); (4); (6); (7); (8). Random motion of particles can be rectified in such an asymmetric system, which can be used for, e.g., controlling particles motion, separating different types of particles (i.e., molecular sieves), for both underdamped and overdamped particles (4) and for molecular motors (5). Vortices in a type II superconductor often (e.g., for magnetic field close to ) can be treated as classical overdamped “particles”. Most of the experiments on vortex motion rectification used arrays of asymmetric pinning sites (e.g., nanoengineered antidots or triangular magnetic dots/inpurties) to create an asymmetric potential, which rectifies the motion of vortices (8); (9); (10); (11); (12); (13); (27); (14). The rectified vortex motion was directly observed in experiments by imaging vortices via Lorentz microscopy (15). Periodic arrangement of point defects of a gradual density or periodic square array of ferromagnetic dots of decreasing size, i.e., varying the density of pinning sites or the size of pinning sites, were shown to result in a ratchet potential (16); (17). When vortices are trapped by pinning sites, the repulsive vortex-vortex interaction creates a higher energy barrier near the area with higher density of pinning sites. Therefore, an asymmetric potential can be created by the gradient of the density of pinning sites (16). Even without spatial asymmetry (i.e., without any asymmetric walls/boundaries or asymmetric pinning sites), the motion of vortices still can be controlled by time-asymmetric driving force (18); (19); (20). Due to the possibility of controlling their motion, the dynamical behavior of vortices in such systems has attracted considerable interest. A series of elastic and plastic vortex flow phases were found (21); (22); (23). Besides the liquid-like and solid lattice phase, vortex motion also revealed a jamming behavior (23); (24). When the density of vortices is changed, the vortex flowing direction can change to the opposite (9); (10); (17); (25), which means vortices can drift in either the “hard” direction or the “easy” direction of the ratchet, depending on the vortex density. By controlling the motion of vortices, it is possible to remove vortices or reduce the vortex density by using a combination of two opposite oriented ratchet arrays (26). The order of vortices and commensurability between vortices and cells also play an important role in vortex dynamics (17); (23); (25); (27). In two-dimensional (2D) ratchets, the dynamics in the transverse ratchet was also studied in theory (4) and in experiments (28); (29).

In the present paper, we study the dynamics of vortices in a circular channel formed by asymmetric triangular (funnel) cells (TCs) (see Fig. 2) [note that earlier this approach to form asymmetric channels in experiment (i.e., using weak-pinning channels) was employed in a stripe geometry (6)]. Due to the radially flowing current in a Corbino setup, the driving force is not uniform inside the cell which is different from linear ratchet channels. Such a geometry (i.e., asymmetric in the azimuthal direction and in the radial direction), as will be shown, leads to a specific dynamical behavior, for example, a vortex located near the inner corner of a TC (which is closer to the center), experiences a stronger driving force and moves to the next TC while a vortex located near the outer corner of TC does not move. The circular geometry of the ratchet channel is convenient for studying commensurability and step-motor (phase locking) behavior (13); (23). We analyze in detail low and high density regimes of rectification, i.e., “single-vortex” and “multi-vortex” regimes. We demonstrate that the mechanism of rectification are qualitatively different for these two cases. In addition, we have performed preliminary measurements using a single nanofabricated weak-pinning ratchet channel of a-NbGe with strong-pinning NbN channel edges in a Corbino set-up, and we observed a substantial asymmetric vortex response.

This paper is organized as follows. First in Sec. II, the model of our systems will be presented. Then in Sec. III, we will show how the density of vortices and the frequency of driving current influences the dynamical behavior of vortices. Commensurability effects of vortex density and of the frequency of current will be discussed in Sec. IV. We present the measurements of a vortex ratchet in a Corbino geometry in Sec. V. Finally, the conclusions will be given in Sec. VI.

## Ii Model and simulation

We consider a ring-like weak-pinning channel constructed of partially overlapping TCs as shown in Fig. 2. In our simulations, the radius of the ring is typically set as , where is the magnetic field penetration depth, the wider part of the channel (i.e., the base of TCs) , and the width of the narrow part (the neck) is typically 0.1 of the wider part (i.e., the ratio ). We also performed simulations in a channel with a wider neck part, e.g., , for comparison. The weak-pinning channel (where vortices can move freely) is surrounded by a strong-pinning superconducting material (30) which is modeled by a medium where vortices cannot move. An external current radially flows from the center of the disk to the edge, resulting in the density of current . Therefore, the closer a vortex to the center of the disk, the stronger the Lorentz force that acts on it. The driving force due to the radial current is (34); (32); (31); (33); (35); (36)

 fdi=Φ0I2πρid ^θ=f0I0ρi ^θ, (1)

where is the unit vector in the azimuthal direction in the disk plane, is the unit of force, is the dimensionless driving current, and is the flux quantum. We perform Langevin-type molecular dynamics (MD) simulations and numerically integrate the overdamped equations of motion (35); (36); (37); (38):

 ηdridt=fi (2)

with

 fi=∑jfvvij+fTi+fdi+fbi, (3)

where is the dimensionless viscosity coefficient which is set here to unity. We note that in practice the viscosity coefficient (39) varies between different superconductors. As an example, parameters for a-NbGe, a typical weak-pinning material, correspond to  Ns/m (40). Using this value of in our calculations results in typical maximum values of vortex linear velocity  m/s (for a 1m-thick film), which approaches, but does not exceed, the Larkin-Ovchinnikov critical velocity for this material (41); (42); (40).

The vortex interaction is described by a first order of the modified Bessel function of the second kind , and the thermal force obeys

 ⟨fTi(t)⟩=0, (4)

and

 ⟨fTi(t)fTj(t′)⟩=2ηkBTδijδ(t−t′). (5)

To model the vortex-boundary interaction, we assume an infinite potential wall at the boundary (i.e., vortices cannot leave the channel) which decays inside the triangular cell with the same dependence on position as the vortex-vortex interaction potential. The total interaction of a vortex with the channel boundaries is calculated by integrating the vortex-wall interaction potential over the geometrical boundary of the channel. The resulting potential due to the boundary is shown in Fig. 2, and the vortex-boundary force is directly calculated from the potential.

In our simulations, we first set , when no current is applied, and then gradually decrease temperature to let the system to relax to the ground state. Then we set and apply an external driving, i.e., an ac current resulting in oscillating Lorentz force with frequency and amplitude that acts on vortices, to study the dynamics of the system.

## Iii Rectification of vortex motion

### iii.1 Density of vortices

The vortex density, i.e., the number of vortices per TC, plays an important role in the dynamics of the system. For very low vortex density, the vortex-vortex interaction is small as compared to the interaction with the boundary, and therefore it can be neglected. With increasing the vortex density, the interaction between vortices becomes important. Therefore, we roughly distinguish two regimes, i.e., a single-vortex regime and a multi-vortex regime, respectively, corresponding to low and high vortex density.

In order to characterize the dynamical behavior of the system of vortices, we calculate the net angular velocity of each vortex and take average over all the vortices for, e.g., 100 ac periods. The resulting average angular velocity (called further “angular velocity”) is denoted as which is analyzed for different parameters of the system and drivings. We note that angular velocity can be related in a straightforward way to the flux-flow voltage (30); (31) that one would typically measure in an experiment.

One system contains vortices and triangles in a circular chain, e.g., in our simulations we take . If there is less than one vortex per cell, i.e., in our system (see Fig. 2), then the density of vortices is low enough and one can neglect the interaction between vortices. One can imagine that all the vortices are far away from each other and weakly interact with other vortices but they are strongly influenced by the ratchet potential induced by the boundary. Therefore, the dynamical behavior in the low density case is similar to that of the system with just one vortex [shown in Fig. 4 (a)], which is considered in the single-vortex regime. Thus, for , the maximum of the angular velocity is the same value [see Fig. 4 (a)]. When the density increases and therefore the interaction between the vortices becomes appreciable, the maximum of the angular velocity first decreases to 0.05 for [see Fig. 4 (b)] and then starts to increase [see Fig. 4 (c)] when the system turns to the regime of high vortex density.

To understand this non-monotonic behavior of the maximum of the angular velocity [shown in Fig. 4(d)], we studied the trajectories of moving vortices. When the system is in the single-vortex regime, where , all the vortices move along a nearly circular trajectory no matter whether the applied current is small or large, which can be considered as a one-dimensional (1D) motion (shown in Fig. 4). For low driving currents, each vortex oscillates near its initial position inside a TC [e.g., see Fig. 4 (a)], and when the driving force reaches some critical value, all the vortices move with a net angular velocity in the easy direction [e.g., see Fig. 4 (b)]. However, in the multi-vortex regime, i.e., the high density case (e.g., ), the motion of vortices is not 1D any more. The ac current drives vortices to pass through the narrow part (i.e., neck) of the channel in the easy direction and when current is alternated, some of vortices are forced to move into the corner by the others and “freeze” thus blocking the motion in the hard direction [shown in Fig. 5]. Due to the asymmetry in the radial direction and radially decreasing current density in the Corbino setup, a vortex near the inner corner, which is closer to the center of the disk, moves faster than the one near the outer corner. Therefore, at a specific value of the current [e.g., at , as shown in Fig. 5(a)], the vortex near the inner corner has a larger angular velocity. When the current increases, more and more vortices move along the circular trajectory and the distribution of vortices becomes strongly inhomogeneous [shown in Fig. 5(b)]. With further increasing current, this 2D trajectory becomes narrow and all the vortices move along the circle, i.e., the 2D motion becomes 1D [shown in Fig. 5(c)]. With increasing density, more and more vortices remain in the area near the corners of TCs and block other vortices moving in the hard direction. Therefore, the maximum of vortex angular velocity increases. Even in the descending-velocity region of the single-vortex regime [see e.g., when in Fig. 4(c)], the angular velocity is still large.

The maximum of the angular velocity for different vorticities is shown in Fig. 4(d). In the single-vortex regime, where there is less than one vortex per TC, remains almost the same value because of the weak interaction between vortices. When becomes larger, there are more than one vortex per TC and the repulsive vortex-vortex interaction in the same TC makes the motion in either easy or hard directions much easier. Therefore, first decreases when . However, for , the maximum starts to increase. The reason for this behavior is the transition from 1D motion to 2D motion. When vortices move along 2D trajectories (see Fig. 4), the repulsive interaction between vortices still “helps” a vortex to move to the next TC in the easy direction but blocks the vortex motion in the hard direction until applying a large enough driving current.

The radial asymmetry facilitates the motion of vortices from inner/outer corners (driven by stronger/weaker force) one by one. This is different from the case of a linear ratchet channel where two vortices in the corners are driven by the same force and, as a result, arrive simultaneously at the neck region leading to jamming. In an asymmetric channel, the symmetry is broken, and jamming occurs only for rather large driving force.

### iii.2 Frequency dependence

The dynamical behavior of vortices in the considered ratchet system is also strongly influenced by frequency of the ac current. In this subsection we study frequency dependence of the rectified vortex motion. For different frequencies , the curves are shown in Fig. 6.

When the frequency is low, the dynamical behavior is similar for different vortex densities. As the drive amplitude is increased, the angular velocity first increases, reaches a maximum, and then decreases to zero. The curves, especially for , in Fig. 6(a) are similar to the analytic result in a ratchet potential (e.g., see Fig. 2 in Ref. (26)). In the case of , the first critical value of current [shown in Fig. 6(a)] corresponds to the maximum friction in the easy direction and the second one corresponds to the case when the driving force reaches the maximum friction in the hard direction . When , the first critical value of current decreases to , which means vortices are easier to move in the easy direction. If the density increases further, decreases and increases, for [shown in Fig. 6(a)]. Therefore, the interaction between vortices allows vortices to move even easier in the easy direction and harder in the hard direction.

For an intermediate frequency [e.g., as shown in Fig. 6(c)], when the distance a vortex moves during one period is comparable to the size of a TC, the dynamical behavior of vortices will depend not only on the driving force but also on the vortex density. This is explained by the fact that the confinement force due to the boundary and the vortex-vortex interaction are also comparable. In this case we obtain different dynamical behavior in single-vortex and multi-vortex regimes which were discussed in the previous subsection. Due to the commensurability between the numbers of TCs and vortices, a step structure of the curve is revealed in the low density case (e.g., and ), which will be discussed in more detail in Sec. IV.

As one can predict, the angular velocity becomes zero [see Fig. 6(e)] when the frequency is high enough because a vortex oscillates near its initial position inside a cell and thus its motion cannot be rectified. Therefore, the variation of the angular velocity is much smaller than that in the low/intermediate frequency case. Each vortex is localized in a specific TC, i.e., the vortex is only influenced by a single potential well but not by the ratchet potential.

For comparison, we also calculate curves for the case when a TC has a wider connection part () [shown in Fig. 6(b), (d), (f)]. The function in general shows a similar behavior as for . However, for the intermediate frequency () of the applied current, we obtain more steps in curve in the case of [shown in Fig. 6(d)] than that for [shown in Fig. 6(c)]. This relates to the commensurability effect that will also be discussed in Sec. IV.

## Iv Commensurability effect

### iv.1 Commensurability of vortex density

As we defined above, the system contains vortices and triangles in a circular chain. If there is a common integer (except one) in terms of which two numbers and can both be measured, then they are commensurate. Otherwise, they are incommensurate. Fig. 7) shows the average angular velocity as a function of for different commensurate ratios. The ascending part of the curves (i.e., where versus increases) is stepwise. Besides the large steps of angular velocity in the curve for , which we refer to as “integer steps”, we also found smaller steps for some specific vorticities (shown in Fig. 7). If and , where and are incommensurate integers, the small steps can be found in the curves [in Fig. 7(b)-(d)]. When , we observe only integer steps of the angular velocity [in Fig. 7(a)]. The difference in the angular velocity between two adjacent steps is always . If , then we can find a fractional step whose magnitude is . For example, for [shown in Fig. 7(b)], the smallest step is in the system with 72 vortices (i.e., ), which is a half of that for (e.g., when ). In the case of , a unit cell contains only one vortex. [The “unit cell” (UC) is a minimum repeatable set of TC(s) containing an integer number of vortices. For example, for the UC is the entire channel with one vortex, and for the UC is one TC with one vortex.] If one vortex can overcome the potential barrier and move in the easy direction, all the vortices can do so at the same time, i.e., collectively. This results in integer steps of the average angular velocity in the curve. However, fractional steps appear when there are more than one vortex in each UC. For instance, a UC contains one TC with two vortices when , and if , the UC is constructed by three TCs with two vortices. In general case, the unit cell contains TCs with vortices inside. If , there are more than one vortex in the UC, and those vortices are not equivalent, i.e., they are not located at the equivalent position in the TC and/or they experience different interactions with the boundary. Therefore, they move with different angular velocities in each period. Let us take , for example (see Fig. 8). As shown in Fig. 8(a), the vortices located in a TC that is followed by an empty TC in the easy direction (type-A) can overcome the potential barriers in a period of alternating current and move to its neighbor TC, while the others (type-B) are still localized in its original TC. When [shown in Fig. 8(b)], type-A vortices move from C2 to C3 but type-B vortices only oscillate in C1. In the view of the whole circular channel, only a half of vortices (i.e., type-A vortices) move to another TC while the other half of vortices (i.e., type-B vortices) do not move. After that, the type-B vortices are located at similar position as type-A vortices in the previous cycle, i.e., the previous type-B vortices now become type-A vortices and they will move to next TC in the anticlockwise direction during the next period. Therefore, when we calculate the average angular velocity for all the vortices, the value is a half of the angular velocity of type-A vortices, i.e., . If the current increases, every vortex can overcome the barriers and move to the neighbor TC [e.g., as shown in Fig. 8, type-A vortices move from C2 to C3 and type-B vortices move from C1 to C2 when ]. Then average angular velocity becomes which is the same as that for when all vortices move to the neighbor TC.

### iv.2 Commensurability effect of frequency

Further we analyze the angular velocity evolution while varying the ac drive frequency (shown in Fig. 9). For varying frequency , the curves are characterized by peaks and/or oscillations. Let us clarify this behavior. As discussed in Sec. III, if , vortices can move in the easy direction and be frozen/blocked in the hard direction when the current alternates. Let us introduce a time scale, , to characterize the motion of a vortex over the entire TC. One TC occupies an angle . If we assume that a vortex moves over the entire TC to the equivalent position in the next TC, then should satisfy the following condition:

 ∫T00ωdt=∫T00[fd(I0)+fb+fvv]⋅^θdt/ηr=θ0, (6)

where is the unit vector in the azimuthal direction. In the single vortex case (), we set , and is a constant. Considering the circular trajectory of the moving vortex, the integral of is also a constant. Therefore, only depends on the current and we use the notation instead of in order to show the dependence on the driving current.

If and the ac period , then the rectified signal will be maximum since all the vortices coherently move over the entire TC in the easy direction during the first half period and do not move backwards in the next half period. Then we can obtain the principal period for each value of the current and roughly estimate forces and from the or curves [as shown in Fig. 9]. The maximum angular velocity for different current and vorticities is shown in the insets of Fig. 9. For example, in the case of [see Fig. 9(a)], the angular velocity is always zero when and becomes non-zero for , i.e., the vortex starts to move in the easy direction. Therefore, . However, for , the maximum angular velocity decreases as compared to the case for , which means . For when driving current increases, vortices move backwards (i.e., the motion in the hard direction) therefore resulting in the decreasing net angular velocity. For example, when [shown in Fig. 9(a)], the maximum becomes smaller and the jumps in the curves become smoother, which means the effect of boundary becomes weaker under a strong driving force. Therefore, the angular velocity should be zero when the driving force goes to infinity. It explains why the angular velocity decreases and goes to zero when the driving current increases above some critical value.

For the single-vortex case, the frequency dependence of the first local maximum in the curves versus the corresponding drive frequencies, , is plotted in Fig. 10 which shows a linear behavior. When the density is increased up to one vortex per cell (), the curve is similar to the one for for both incommensurate [e.g., see Fig. 9(b)] and commensurate cases [see Fig. 9(c)] because the interaction between vortices is weak. For a higher density, the (in-)commensurability effect becomes pronounced. The sharp jumps of the angular velocity are obtained only in the commensurate case [e.g., shown in Fig. 9(e)] but not in the incommensurate case [e.g., shown in Fig. 9(f)]. Comparing the high density cases with the low density cases, we can conclude that sharp jumps in curve, which are both found in single-vortex and multi-vortex regimes, are due to the effect of periodically repeated boundaries of the ratchet potential and the increasing local maximum of the angular velocity when increases in the multi-vortex regime. This is different from the single-vortex regime, because of the strong interaction between vortices induced by the high vortex density. If the principal period (i.e., the time period for the first jump) is denoted by , the others will be harmonics of , .

## V Experimental detection of vortex ratchet effect in a Corbino geometry

We have performed preliminary measurements on a device containing a single weak-pinning circular channel for guiding vortex motion in a Corbino geometry. This is based on a technique for using weak-pinning channels with tailored edges to produce asymmetric vortex confining potentials. Such an arrangement resulted in substantial asymmetric vortex response for linear channels on a strip geometry (6). The Corbino sample consists of a Si substrate with a 200-nm thick film of weak-pinning a-NbGe and a 50-nm thick film of strong-pinning NbN on top. The fabrication followed the scheme of previous weak-pinning channel devices (6); (30), with the 1.5-mm diameter Corbino disk pattern etched through the entire superconducting bilayer. The 500-m diameter circular chain of triangular cells was etched through the NbN layer (Fig. 11), thus defining the weak-pinning a-NbGe channel region for vortex flow.

Wirebonds were attached between the center and perimeter of the Corbino disk for injecting a bias current with a radial flow. Because of the rather small flux-flow voltages for vortex motion in a single channel, it was necessary to use a custom picovoltmeter based on a dc SQUID operated in a flux-locked loop (30). Measurements of the noise power at different temperatures were used to calibrate the value for the series resistance at the SQUID input, thus allowing for a measurement of the system gain, as described in Ref. (30). During the measurements, the sample and SQUID were immersed in a pumped liquid helium bath. Shielding of external magnetic fields was achieved with a -metal shield surrounding the dewar and a superconducting Pb shield around the sample and SQUID on the bottom of the cryogenic insert. Vortices were introduced into the channel by temporarily raising the sample above the helium bath, heating to 6 K – between = 2.88 K and 10 K – while applying a small magnetic field with a Helmholtz coil on the insert, then cooling back down below .

Upon reaching the desired measurement temperature, the bias current was varied incrementally and the flux-flow voltage sensed by the SQUID was recorded for each current value. Such current-voltage characteristics (IVC) exhibited substantial asymmetries between the critical current for vortices to begin to move through the channel for the two directions. This asymmetry persisted down to the lowest measurement temperature, 1.60 K, well below the onset of superconductivity in the channel at (Fig. 12). The larger critical current corresponded to the sense of vortex motion in the hard direction through the ratchet cells. Due to experimental wiring limitations on these preliminary measurements of a Corbino ratchet channel, it was not possible to sweep the bias current with an oscillatory drive to study the average flux-flow voltage over a cycle. Nonetheless, the large asymmetry between the two senses of critical current demonstrates the potential for a weak-pinning ratchet channel in a Corbino geometry to rectify vortex motion.

## Vi Conclusions

A vortex moving in an asymmetric circular channel in a Corbino setup experiences the confinement of the boundary, the repulsive interaction due to other vortices and the gradient (i.e., radially decreasing) driving force when an external current is applied. The combination of these factors determines the vortex motion. Different dynamical behavior is observed in low and high density systems, which are referred to as “single-vortex” and “multi-vortex” regimes, respectively. For low density, i.e., in the single-vortex regime, there is no more than one vortex per triangular cell so that the vortex-vortex interaction is negligible. Therefore, the ratchet potential due to the boundary dominates and all the vortices follow circular 1D trajectories. Since all the vortices still move in 1D when there is more than one vortex per cell (i.e., a higher density), a vortex can escape even easier either in the easy or in the hard direction due to the interaction between vortices in the same cell. This results in decrease of the rectified net current. However, when the number of vortices increases further (i.e., in the“multi-vortex” regime of rectification when the rectified net current increases with the vortex density), the trajectories of vortex motion for low driving currents are 2D while these trajectories squeeze and turn to 1D circle with increasing driving current. Because of the circular geometry of the channel, the density of vortices first becomes inhomogeneous [i.e, see Fig. 5(b)] during the transition from 2D motion to 1D motion in the multi-vortex regime and then becomes homogeneous again when all the vortices move in a circular trajectory.

Considering the asymmetry in the radial direction, vortices near the inner/outer corner of the triangular cells (TCs) (i.e., closer/further to the center of the disk) are driven by different Lorentz forces and for some specific value of driving current the vortex in the outer corner moves to the inner corner while the one in the inner corner moves to the next TC [e.g., see Fig. 5(a)]. This kind of motion prevents vortices from arriving simultaneously at the narrow part that would lead to jamming which occurs in case of a linear channel. When the density increases, the maximum net angular velocity remains the same in the single-vortex regime and then decreases until reaching the multi-vortex regime, and then increases.

The frequency of driving current also strongly influences the vortex dynamical behavior. The ac frequency determines the possible distance a vortex moves during an ac period. For high frequency, a nearly zero net angular velocity is obtained for different values of driving current . Each vortex is unable to overcome the energy barrier and is confined by a single potential well instead of the periodic ratchet potential. When the frequency is low, the ratchet effect is clearly observed in the curve but the commensurability effect is not present. For an intermediate frequency of driving current under which the distance of moving vortex in a period is comparable to the size of the TC, both the ratchet effect and the commensurability effect have been observed.

Besides the vortex density and the frequency of current, the commensurability between the numbers of vortices and TCs also plays an important role in the dynamical behavior of vortices, which leads to jumps in the angular velocity with increasing driving current (i.e., steps in the curve). Therefore, under some specific conditions, the average angular velocity of vortices is not a continuous function of the driving current. The commensurability also influences the minimum difference of the angular velocity for different steps in the curve, and results in integer steps (i.e., the large steps which are found, e.g., for vorticity ) and fractional steps (i.e., the smaller steps whose magnitude are fractions of the magnitude of the integer steps, e.g., for vorticity ) in a certain range of the current. We also obtained several peaks (sharp jumps) in the curve, which correspond to the principal period (during which the vortex can move over one TC in the easy direction), and harmonics periods (during which vortices move over TCs in the easy direction). The net flow of vortices is enhanced when the ac period is one of the harmonics periods, i.e., the average angular velocity reaches a local maximum.

## Acknowledgement

We thank Peter Kes and Marcel Hesselberth for providing the superconducting films from which the Corbino ratchet sample was fabricated. This work was supported by the “Odysseus” Program of the Flemish Government and the Flemish Science Foundation (FWO-Vl), the Interuniversity Attraction Poles (IAP) Programme — Belgian State — Belgian Science Policy, and the FWO-Vl (Belgium). T.W.H., K.Y., and B.L.T.P acknowledge support from the National Science Foundation under Grant DMR-0547147, as well as the use of the Cornell NanoScale Facility, a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation (Grant ECS-0335765).

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