# Mode-locking in ac-driven vortex lattices with random pinning

###### Abstract

We find mode-locking steps in simulated current-voltage characteristics of ac-driven vortex lattices with random pinning. For low frequencies there is mode-locking above a finite ac force amplitude, while for large frequencies there is mode-locking for any small ac force. This is correlated with the nature of temporal order in the different regimes in the absence of ac drive. The mode-locked state is a frozen solid pinned in the moving reference of frame, and the depinning from the step shows plastic flow and hysteresis.

###### pacs:

PACS numbers: 74.60.Ge, 74.40.+k, 05.70.Fh[

]

In 1971 A. T. Fiory [1] observed steps in the current-voltage (IV) characteristics of ac-driven superconducting thin films, analogous to the Shapiro steps found in Josephson junctions [2]. The steps were observed for voltages such that , with the average vortex velocity, the triangular vortex lattice period, the frequency of the external ac-drive, and integers. This is a particular case of mode-locking, where an internal frequency of the system (in this case ) locks to a rational multiple of the external frequency. Several other systems with many degrees of freedom, such as charge density waves [3, 4], spin density waves [5], Josephson junction arrays[6] and superconductors with periodic pinning arrays [7, 8], also exhibit mode-locking behavior experimentally. Recently, Harris et al. [11] observed the Fiory steps in YBCO and found that they vanish when the vortex melting line is crossed. Historically, the experiment of Fiory motivated the landmark works of Schmid and Hauger [9] and Larkin and Ovchinikov [10] on the dynamics of moving vortex lattices. In [9, 10] it was assumed that at large velocities vortices form a perfect triangular lattice in which the effect of random pinning could be treated perturbatively. However, it is now clear that, instead of a perfect triangular lattice, there are several other dynamical phases of driven vortices: plastic flow [12], moving smectic, transverse moving glass and moving Bragg glass [13, 14, 15]. These regimes have been observed experimentally [16] and in numerical studies [17, 18]. Thus, it may be of interest to study how the Fiory steps can arise in different moving vortex phases. Moreover, a peak in the voltage noise power spectrum at the “washboard” frequency has recently been observed experimentally in driven superconductors [19, 20]. This is a signature of temporal order in the high-current steady states of moving vortices [15, 21]. We here present a numerical study of mode-locking for the different regimes of vortex velocities and its relationship with temporal order.

The equation of motion of a vortex in position is:

(1) |

where is the distance between vortices , is the distance between the vortex and a pinning site at , is the Bardeen-Stephen friction and is the driving force due to an alternating current superimposed to a constant current . A 2D thin film superconductor of thickness with , has an effective penetration depth . Since is of the order of the sample size, the vortex-vortex interaction is considered logarithmic: , with [18]. The vortices interact with a random distribution of attractive pinning centers with , being the coherence length. Length is normalized by , energy by , and time by . We consider vortices and pinning centers in a rectangular box of size , and the normalized vortex density is . Moving vortices induce a total electric field , with .

We study the response of the vortex lattice to an external ac+dc force of the form [22] at , solving Eq. (1) for different values of and . The simulations are for constant vortex density in a box with , and (we show results for ), and we consider weak pinning strength of with a density of pinning centers being . Periodic boundary conditions are imposed on the simulation box and the resulting long-range interaction is determined by Ref. [23]. The equations are integrated using a time step of , averages are evaluated during steps after steps for equilibration (when the total energy reaches a stationary mean value).

Let us first review the behavior for . There are three dynamical regimes when increasing above the critical depinning force, , in the case : plastic flow for , smectic flow for

, and a transverse solid for (see [18]). The characteristic forces in our case are , and . Since the nature of mode-locking is related to the existence of temporal order at the washboard frequency we analyze the voltage noise in each dynamical regime. In the insets of Fig. 1(a-c) we show the spectral densities corresponding to each regime and we indicate the corresponding . In the inset of Fig. 1(a) we see that there is no temporal order at the washboard frequency, and only the typical broad band noise of plastic flow is observed[12, 20]. For the smectic flow regime, we see in the inset of Fig. 1(b) that there is a small and broad peak at . Only for large forces, , in the transverse solid regime well developed peaks appear at the washboard frequency [20] and harmonics (see the inset of Fig. 1(c)).

We now study the response of each of these dynamical regimes with velocity to the superimposed ac-force , for varying values of for a given . We expect the main interference step () to occur when (i.e., ) if there is mode-locking. We therefore choose the values of such that the expected step, , would correspond to velocities belonging to a given dynamical regime

of the limit . Each simulation is started at with an ordered triangular lattice up to values such that by slowly increasing the dc force with . For low , for which we have plastic flow when , we find that there are no interference steps in a wide range of (shown in Fig. 1(a) for (left curve) and (right curve)). This is consistent with the observed lack of temporal order in the inset of Fig. 1(a) and indicates that very large are possibly needed in order to induce mode-locking. For intermediate , for which we have smectic flow when , we find that there are no steps for small amplitudes, , while there are steps for , as shown in Fig. 1(b) in the left and right curves, respectively. This means that the small washboard peak observed in the inset of Fig. 1(b) is not large enough to induce mode-locking steps for small . However, this short-range temporal order can be amplified for intermediate values of giving place to steps in this case. For high , corresponding to a transverse solid regime when we find that there are steps both for small and large values of the ac amplitude, as we can observe in Fig. 1(c). This is in agreement with the spectral response

observed in inset of Fig. 1(c). In this case, the temporal order is robust enough for mode-locking to be produced by small values of .

Let us now examine in detail the dynamics and the structural order within and in the vicinity of a mode-locked step, in the transverse solid case. In Fig. 2(a) we show a typical curve around the step. In the inset of Fig. 2(a) we show a finite size analysis of the step width for , where the error bars are due to the observed dependence of the width in three different realizations of disorder. We observe that for the step width tends to saturate in a size-independent value. To analyze the dynamical behavior we define the quadratic mean displacements of vortices in directions parallel and perpendicular to the external force, calculated from the center of mass position as: and , where y . In Fig. 2(b) we show the time average of these quantities, and , as a function of . Outside the step we observe that the transverse mean displacement (TMD) is limited , while the longitudinal mean displacement (LMD) is unbounded . This corresponds to a state with only longitudinal diffusion (i.e., a transverse solid [18]). Noticeably, in the transition to the synchronization, the LMD freezes in a value, , while the TMD remains practically constant. This mode-locking longitudinal freezing can also be observed as a dramatic decrease of the low frequency voltage noise in both directions. The mode-locked state is therefore a frozen solid. To study the translational order we calculate the structure factor as . In this regime there are smectic order peaks of magnitude with

and longitudinal peaks of magnitude with and . In Fig. 2(c) we plot and . We do not observe any important change in the transition to the mode-locked state. Inside the steps we observe jumps in indicating that there are different metastable mode-locked structures. In Fig. 3 we analyze in detail the process of depinning from the step. We show a detailed view of the curve in the transition from the synchronized regime. Varying back and forth, we observe a clear hysteresis cycle. In the lower right inset of Fig. 3 we show the evolution of with time inside the step and outside the step. While for all inside the step, outside there is ballistic diffusion . To visualize the spatial structure in the transition we define a coarse-grained vortex density seen from a system of reference moving with velocity as follows: , where . We take a coarse-graining scale . In the upper left insets of Fig. 3 we show the temporal average of the density, inside (mode-locked) and outside (mode-unlocked) the step. This quantity shows the stationary trajectories of vortices seen from a moving frame of reference with velocity . Inside the step we see that the vortices are localized, oscillating around their equilibrium positions in a moving lattice with velocity . This is in agreement with the mode-locked frozen solid inferred from the result shown in Fig. 2(b), which is “pinned” in the moving reference frame with velocity . Just above the depinning from the step we see that some vortices delocalize following straight trajectories parallel to the force around “pinned” vortices, producing coexistence of mode-locked and mode-unlocked channels of flow. This could be interpreted as a one-dimensional “plastic” depinning from the step.

Mode-locking of the steps in the curve can be characterized qualitatively by how the dc current range in mode-locking depends on and , (as it was done for example in [6, 7, 8]). In Fig. 4(a-b) we show the range (width) for the case and respectively. The error bars and the mean values were estimated by repeating the simulation for three different disorder realizations. In Fig. 4(a) we show for vs , which corresponds to the smectic flow regime for . We see that there is mode-locking above a finite critical value (see Fig. 1(b)). For larger amplitudes we were not able to obtain a systematic dependence on amplitude because the step widths depend strongly on the disorder realization in this case. In Fig. 4(b) we show for two frequencies vs , which correspond to the transverse solid in the limit. We can collapse (approximately) both curves into a single curve if we plot vs . A dependence, , was found by Schmid and Hauger [9] (and also found in other elastic models) where and is the shear modulus. This result was obtained using a perturbative approach in an elastic model for the vortex lattice. Strikingly, our results seem to follow more closely a dependence of the form with being a constant. This dependence is the same found for the one-dimensional problem of an overdamped single Josephson junction or a particle moving in a periodic potential. In our case, a linear dependence of the mode-locking intensity with would be a consequence of the existence of temporal order in the limit. This was not taken into account in the perturbative calculation of Schmid and Hauger where mode-locking arises as a second order effect.

In conclusion, it is possible to have mode-locking in driven vortices with random pinning for high enough frequencies, in agreement with the experiments of Fiory [1] and Harris et al [11]. The mode-locked state can be viewed as a frozen solid pinned in the moving frame of reference, and the depinning from mode-locking is plastic and hysteretic. Also, the response to an ac drive for different frequencies can be an interesting experimental probe of the dynamical regimes of driven vortices.

We acknowledge discussions with L. Balents, P. S. Cornaglia, M. F. Laguna, V. I. Marconi. This work has been supported by CONICET, Fundación Antorchas and ANPCYT (Argentina) and by the Director, Office of Advanced Scientific Computing Research, Division of Mathematical, Information and Computational Sciences of the U.S. Department of Energy under contract number DE-AC03-76SF00098.

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