Phase-locking of driven vortex lattices with transverse ac force and periodic pinning

Charles Reichhardt,¹ Alejandro B. Kolton,² Daniel Dominguez,² and Niels Gronbech-Jensen^3,4
1Applied Physics Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
²Centro Atomico Bariloche, 8400 S. C. de Bariloche, Rio Negro, Argentina
³Department of Applied Science, University of California, Davis, California 95616
⁴NERSC, Lawrence Berkeley National Laboratory, Berkeley, California 94720
(Received 23 May 2001; published 5 September 2001)

For a vortex lattice moving in a periodic array we show analytically and numerically that a new type of phase locking occurs in the presence of a longitudinal dc driving force and a transverse ac driving force. This phase locking is distinct from the Shapiro step phase locking found with longitudinal ac drives. We show that an increase in critical current and a fundamental phase-locked step width scale with the square of the driving ac amplitude. Our results should carry over to other systems such as vortex motion in Josephson-junction arrays.

DOI: 10.1103/PhysRevB.64.134508 PACS number(s): 74.60.Ge, 74.60.Jg

I. INTRODUCTION

An important difference of vortex arrays from charge-density-wave systems and single-degree-of-freedom systems (like a small Josephson junction) is that the displacement field acting on vortices is a two-dimensional (2D) vector. This means that displacements can be induced in two different directions, and therefore a new kind of phase locking is possible when an ac force is applied transverse to the direction of the dc force.

In this work we show analytically and numerically how phase-locking phenomena may occur when vortices are moving in a periodic potential and the ac force is applied transverse to the direction of the longitudinally applied dc force. The type of phase locking observed in this case is qualitatively different from the Shapiro-type phase locking observed when dc and ac forces are in parallel. We first show analytically the possibility of the phase locked states for certain commensurate vortex configurations and predict that the critical current increases quadratically with the ac amplitude and that the width of some of the phase-locked steps scale as the square of the ac amplitude. We also find scaling of the critical current as well as the first phase-locked region as a function of the ac frequency and the pinning geometry. The predictions of the perturbation analysis are confirmed with numerical simulations. We analyze the validity of the perturbation approach and demonstrate how deviations occur. Our results suggest that more pronounced transverse phase locking may be observed at lower commensurate magnetic fields, such as B=1.5 or B=1.25. The general model and perturbation results are easily generalizable to other systems which exhibit Shapiro-step-like phase locking such as Josephson-junction arrays and Frenkel-Kontorova-type models.

Our studies are directly relevant for several contemporary efforts since vortex matter interacting with nanostructured pinning arrays of holes^10-12 and dots^13,14 has been attracting increasing attention due to the easily tunable pinning properties. Pronounced commensuration effects are observed in these systems when the density of vortices matches to integer or fractional multiples of the density of pinning sites. In addition to square pinning arrays, recent experiments have been conducted on rectangular pinning arrays.¹⁴ Simulations have shown several interesting dynamical phases of dc-driven vortices in periodic pinning systems.^15,16 Imaging¹² and transport experiments^4,11 along with simulations¹⁵ have found that the vortex motion above the first matching field can occur by the flow of interstitial vortices between the pinning sites. These vortices still experience a periodic potential created by the vortices located at the pinning sites. Recently, phase locking was observed for dc-and ac-driven vortices interacting with periodic pinning at B=2, where is the field at which the number of vortices equals the number of pinning sites,⁴ where the dc and ac drives were in the same direction. The values of the voltage response in Ref. 4 strongly suggest that is is the interstitial vortices that are mobile while the vortices on pinning sites remain immobile.

II. MODEL

We are performing simulations with periodic boundary conditions (periodicity L_x, L_y), and with a logarithmic interaction potential [u_vv(r_ij)=-ln r_ij] valid when is of the order of the sample size.

Combining the logarithmic interactions with the periodic boundary conditions results in¹⁷
(4)
where (x_ij,y_ij) are the Cartesian coordinates, r²_ij=x²_ij+y²_ij, and C(.) is a function determined by the energy normalization of the long-range interactions.

We have previously demonstrated that phase locking of the vortex motion can exist if a certain periodicity of the global pinning potential is present. This was shown for driving forces of the form, f_d=(+sint), where is a unit vector in the x direction. Analogous to the well-known Shapiro steps¹ in the current-voltage characteristics of an ac- and dc-driven Josephson junction, it was demonstrated that nonzero intervals of will result in phase locking of vortex motion with vortex velocities resonating with the external ac field. We will in this paper demonstrate that one can also expect phase locking in the transversely ac-driven case, i.e., for
(5)
where and are the magnitudes of dc and ac forces and is the frequency of the ac drive. The relevant equations of motion are, therefore,
, (6)
. (7)

III. ANALYTICAL RESULTS

A. Critical dc force without ac drive

The first result to evaluate for the purely dc-driven case is the "critical dc current", , below which the interstitial vortex is trapped by the pinned vortex lattice. As is increased from =0, the interstitial vortex will find an equilibrium point for which the pinned vortex lattice provides a cancelling force to the dc bias. However, for a given (critical) value of , namely, , the pinned vortex lattice can no longer provide enough force to resist the motion of the interstitial, which therefore begins to propagate in the x direction. Assuming that the pinned vortices do not move and that an interstitial vortex only interacts with pinned vortices, this critical force can be found as the maximum gradient value of Eq. (4) for y_ij=ml_y. The corresponding equation of motion is given by Eq. (6):
(8)
This equation can produce a critical current by requiring =0 and optimizing the left hand side through varying the position x_i. In general, this must be done numerically. The result is shown in Fig. 1, where the critical dc force (thick solid) is shown together with the optimized value of the equilibrium value of x_i=x_*. In the limit of sech(l_y/l_x)1, we can derive the approximate expression of the critical current by noticing that only two terms (n=-1,0) are required to describe the critical current, which is then always given by x_i=x_*=(3/4)l_x:
(9)
(10)

The approximate expression, Eq. (9), is shown in Fig. 1 as a thin solid curve, validating that the interaction potential between interstitial and pinned vortices can be simplified to the terms n=-1,0 for l_x1.5l_y. One can also see that in the extreme opposite limit l_xl_y, the critical dc force can be approximated by /l_y. This is shown as a dashed line in Fig. 1.

It is here important to remember that the analysis, which is based on a magnetic field of "matching field plus one flux quantum," is actually valid for magnetic fields < B 2, for which the vortex configuration results in forces between interstitial vortices cancelling due to symmetry; such magnetic fields can be, e.g., B=2, (3/2), (5/4), ...

B. Transverse ac drive included

Let us assume that sech(l_yl_x)1, so that we can write the approximate equations of motion for the interstitial vortices as
+[1+(1/2)k²y_i²] sin(x_ik)= (11)
(12)
where we have retained terms in y_i only to lowest order. We have required |y_i|l_x in order to obtain Eqs. (11) and (12).

The solution to the second of those equations is easily found,
y_i = sin(t + ), (13)
with
, (14)
. (15)

Inserting Eq. (13) into Eq. (11) yields
+ [ 1 + (1/4)k²] sin k x_i-(1/4)k² cos 2(t + ) sin k x_i = . (16)
This equation is the effective equation of motion for the vortex behavior in the longitudinal (x) direction given an ac force in the transverse (y) direction, derived within the approximations listed above. The equation is equivalent to that of an overdamped pendulum ( k x_i being the pendulum phase) with dc torque () and a pivot vertically oscillating with frequency 2 and amplitude proportional to - i.e., an overdamped equivalent of the classic Kapitza problem¹⁸ for underdamped and parametrically driven pendulums.

We will now consider a few separate cases of vortex responses to the transverse ac force.

(i) < > 0 x_i(t)=x₀. We will omit the last term on the left-hand side of Eq. (16). The remaining (dc) terms are
sin k x₀ = (17)
|| =[1+(1/4) k²]=+
, (18)
, (19)
where is the critical dc force. It is here worthwhile to notice that the critical current increases ( positive) quadratically with the transverse ac amplitude . This is in direct contrast to the case when the ac drive is longitudinal, in which case the critical dc force decreases quadratically.² We remember that validity of the expressions requires l_x.

(ii) < > > 0 || > . Without the parametric term in Eq. (16), the solution, for , is given by²
(20)
(21)
where is the dc force when the parametric term in Eq. (16) is neglected. Let us assume the following simple ansatz for the vortex motion:
x_i(t) = v₀ t + x₀, (22)
where x₀ is a constant. Inserting this ansatz into the effective equation of motion, Eq. (16), while only maintaining dc terms, yields
v₀- t + ) sin k x_i(t) > = = + . (23)
With the ansatz of Eq. (22) we can therefore expect the term <.> to contribute to if kv₀=2 (see Fig. 2), and the resulting relationship between the internal phase k x₀ - 2 and the dc current + is
2/k- (1/2) sin(k x₀-2) = +. (24)
As the phase k x₀-2 can be adjusted to balance this equation for different choices of , we can argue for a nonzero range in dc force for which the average speed of the interstitial vortices is unchanged. This phase-locking range has the magnitude
= . (25)
Thus, we can predict phase locking in the transversely ac-driven vortex system and the predicted total range in phase locking is equal to the increase in the critical dc force due to the ac drive. This prediction is correct up to and including terms .

This range in bias current for which the average speed (voltage) of the interstitial vortex is constant will manifest itself as a step in the dc current-voltage characteristics of the system. It is important to emphasize that this step is very different in origin from the Shapiro steps⁵ recently demonstrated for longitudinal ac drive. The essential difference lies in the fact that the Shapiro steps arise from a "direct" driving term in an effective pendulum equation, whereas the present phase locking for the transverse ac drive arises from an effective parametric ac-driving term in the longitudinal equation of motion.

If one chooses a better (and more complicated) ansatz for the vortex motion [e.g., Eq. (21) instead of Eq. (22)], it becomes evident that an phase-locked step also exists for kv₀= and that higher-order phase-locked steps may exist at any subharmonic and superharmonic of the driving frequency. However, we will not go into detail with other phase-locked modes in this presentation.

IV. NUMERICAL SIMULATIONS

In order to validate the analysis of Sec. III predicting some basic effects of a transverse ac force on vortices, we have conducted numerical simulations of driven interstitial vortices in a rectangular lattice of pinned vortices.

Figure 3 shows a series of simulated (normalized) current (dc force) voltage (average vortex velocity v₀) for different values of the transverse ac drive. The system is a square array of 4 x 4 pinned vortices with periodicity (l_x,l_y)=(2,2) in a computational simulation box of size (L_x,L_y)=4l_x,4l_y). The simulated magnetic field corresponds to B/=1.5 such that each pinning center is occupied by a vortex and every other plaquette has an interstitial vortex. The applied normalized frequency is =4. The four curves (shifted vertically for clarity) in Fig. 3 are for (from top) =0,1,2,3. According to the analysis above, the critical dc force resulting in the onset of voltage (vortex transport) should increase quadratically for increasing . This is clearly visible from Fig. 3. A phase-locked step in dc bias current should develop for increasing around k v₀=2. This is also clearly visible for 2.5. As the ac amplitude is increased we observe more steps in the IV characteristics. However, while the simplest of these, k v₀=, can be analyzed in some detail, we will be concerned with only the steps at kv₀=0 and kv₀=2 here.

Two characteristically different vortex evolutions are shown as insets to Fig. 3. At bottom right we show a snapshot () of a vortex configuration within the parameter range of the phase-locked step for =2 together with a long time trace (thin line) of the vortex trajectories. The vortex paths are obviously periodic and the interstitial vortices seem to move in a geometrically simple configuration where all interactions between the interstitials cancel due to symmetry. The top left inset shows the traces outside of the phase-locked region. Here we observe the pinned vortices as static dots, while the interstitials move in a chaotic or quasiperiodic band between the pinning sites.

A detailed investigation of the validity of the above perturbation results was performed for several different sets of system parameters. For a system of a single interstitial vortex, we show in Fig. 4 the detailed comparisons between the analytical predictions (lines), Eq. (19), and numerical simulations (markers) of the increase in critical dc force as a function of ac amplitude. Figure 4(a) shows data for square pinning arrays of different plaquette areas (l_xl_y=4,16) and with different driving frequencies (=4,8), while Fig. 4(b) shows comparisons for rectangular systems (l_x/l_y=1/2,2) of one plaquette area (l_xl_y=4) for different frequencies (=4,8). The details of the parameter sets are given in the caption. The agreement between simulation and analysis is obviously good overall for these different parameters. Certainly the quadratic relationship between the increase of the critical current and the ac amplitude is confirmed. The quantitative agreement is also rather good. The simulation results seem underestimated by the predicted values. However, this is to be expected since the analysis only includes the transverse vortex motion as moving in a harmonic potential. The vortex model will provide a stronger nonlinearity for larger ac amplitudes, resulting in a larger than predicted increase of the critical current.

For the same parameter sets as shown in Fig. 4, we have performed simulations of the phase-locked step at kv₀=2 and compared the results to the predicted range given in Eq. (25). The results, shown in Fig. 5, are also here in good agreement with the predictions. Thus, we are confident that the above analysis of the single interstitial behavior can be a useful tool for understanding phase locking in this system.

Figure 6 shows, again for the same parameter sets as in Fig. 4, the comparison between the magnitude of the phase-locked step, , and the predicted value, but in this case for B=1.5, i.e., for one interstitial vortex per two plaquettes of pinned vortices. Once again we find rather good agreement between simulations and the simple prediction. However, we notice that rectangular systems with l_x=2l_y [top curves in Fig. 6(b)] cannot produce a measurable phase-locked step for very small ac amplitudes.

If one simulates the system for the same parameter values, but for B=2 (not shown), the agreement will get rather poor for all rectangular cases where l_x=2l_y and even for some of the square array cases as well. The reason for these discrepancies is of course found in the assumptions behind the perturbation analysis. Particularly, the validity of the assumption of noninteracting interstitial vortices will break down for densely populated (B=2) systems since the vortices are mutually repulsive. The result is sketched in Fig. 7, where the vortex repulsion is deforming the geometrically simple lattice into a configuration where the effective interstitial vortex-vortex interaction is nonzero. The lattice deformation is characterized by the two parameters, and . Figure 8 shows simulation results for L_x=4l_x=L_y=4l_y=8, =4, and B=2. The circular markers () show the simulated range in phase locking and the solid straight line represents the prediction of Eq. (25). It is apparent that the agreement between simulations and prediction is only fair () for relatively large values of . Measuring the average (in time and space) "phase difference" < || > and < || >, we can correlate the magnitude of lattice deformation with the comparison between simulation and analysis of phase locking. The measures of lattice deformation are defined as
= (x_i mod l_x), (26)
= (y_i mod l_y), (27)
, (28)
, (29)
< || > = < || >_t,i,j (30)
< || > = < || >_t,i,j, (31)
, (32)
. (33)

Finally, these averages are averaged over the phase-locked step to provide a single measure for a given ( < || > and are typically smallest near the center of the locking range). Thus, 0 < || > l_x,y/2 is a measure of the average spatial lattice deformation and is a measure of the average temporal fluctuations in the lattice deformation, ||.

The minimum averaged lattice deformation parameters for a given phase-locked step are shown in Fig. 8 as solid markers (, < || >; , < || >). We observe large lattice deformations at small ac amplitudes, where the agreement between the observed range of phase locking is poor, while the deformations "collapse" for larger values of , thereby giving rise to better agreement between simulations and analysis. We have also shown the average of the temporal fluctuations and in order to illustrate that the lattice deformations depend on time when < || > and < || > are nonzero. Thus, it indicates that internal modes of the lattice deformation are excited and may contribute significantly to the phase-locking range in the low ac amplitude cases. For large ac amplitudes we observe a saturation of the range in phase locking accompanied by the reappearance of the deformation lattice. This is consistent with chaotic behavior which are expected for strong ac-driven nonlinear systems.

V. CONCLUSION

We have developed a simple analysis of the behavior of interstitial vortices in systems with periodic pinning and transverse ac drive. The result indicates that the dc-driven longitudinal motion can phase lock to the transverse ac signal and that the range of phase locking (in dc current) is quadratic in the ac amplitude and we have developed a quantitative expression providing detailed dependencies of also other relevant system parameters, such as pinning geometry and driving frequency. We have further demonstrated that the critical current increases quadratically with the ac amplitude. The perturbation results have been validated by numerical simulations which show good agreement with the analytical predictions in the expected range of system parameters. The mechanism of phase locking discussed in this paper is distinct from phase locking to a longitudinally applied ac signal. The latter case was studied in Ref. 5 and exhibits qualitatively different responses to ac perturbations, such as decreasing critical dc current with increasing ac amplitude and phase-locked steps that grow linearly with ac amplitude. The results shown in this paper are for commensurate fields. Incommensurate fields, where no simple geometrical relationship can exist between the pinned and interstitial lattices, are characterized by noncancelling interactions between interstitial vortices. Thus, phase-locking ranges for the noncommensurate fields usually have magnitudes less than predicted by the single interstitial analysis.

Our results and analyses indicate that the most important phase locking appears at relatively small magnetic fields (with respect to the first matching field) where the inter-vortex repulsion is not deforming the interstitial vortex lattice.

The predictions in this paper should be directly applicable for experimental verification in superconductors (or Josephson-junction arrays) where dc and ac fields are orthogonal.

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