Spatiotemporal dynamics and plastic flow of vortices in superconductors with periodic arrays of pinning sites

C. Reichhardt, J. Groth, C. J. Olson, Stuart B. Field,^* and Franco Nori
Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
(Received 30 May 1996)

We present simulations of flux-gradient-driven superconducting rigid vortices interacting with square and triangular arrays of columnar pinning sites in an increasing external magnetic field. These simulations allow us to quantitatively relate spatiotemporal microscopic information of the vortex lattice with typically measured macroscopic quantities, such as the magnetization M(H). The flux lattice does not become completely commensurate with the pinning sites throughout the sample at the magnetization matching peaks, but forms a commensurate lattice in a region close to the edge of the sample. Matching fields related to unstable vortex configurations do not produce peaks in M(H). We observe a variety of evolving complex flux profiles, including flat terraces or plateaus separated by winding current-carrying strings and, near the peaks in M(H), plateaus only in certain regions, which move through the sample as the field increases. Several short videos, illustrating several particular cases of the type of dynamics described here, are available at http://www-personal.engin.umich.edu/~nori. [S0163-1829(96)03142-6]

I. INTRODUCTION

Commensurability effects are also important in superconducting systems with random arrays of columnar pinning sites, where a Mott insulator phase is predicted¹³ at the matching field, where the number of vortices equals the number of pinning sites. In this phase the vortex lattice locks into the pinning array preventing further vortices from entering the sample and creating a Meissner-like effect.

Nonsuperconducting systems also exhibit magnetic-field-tuned matching effects, notably in relation to electron motion in periodic structures where unusual behaviors arise due to the incommensurability of the magnetic length with the lattice spacing. ¹⁴ A recent example of such a system is provided by the anomalous Hall plateaus of "electron pinball"¹⁵ orbits scattering from a regular array of antidots. Commensurate effects also play central roles in many other areas of physics, including plasmas, nonlinear dynamics, ¹⁶ the growth of crystal surfaces, domain walls in incommensurate solids, quasicrystals, and Wigner crystals, as well as spin and charge density waves.¹⁷

To investigate the microscopic vortex dynamics in systems with a PAPS we have performed extensive molecular dynamics simulations using a wide variety of relevant parameters which are difficult to continuously tune experimentally, such as disorder in the pinning lattice geometry, sample size, vortex density n_v, as well as pinning density n_p, radius , and maximum pinning force (or strength) f_p. We find a rich variety of behavior in which all these parameters play an important role.

Recent experiments and theories (see, e.g., Refs. 1-8) involving periodic and commensurate pinning effects in superconductors have raised questions that can be systematically explored by our computer simulations. One such question addressed in this work involves the topological ordering and dynamics of the vortices at commensurate and incommensurate fields and how this microscopic order relates to bulk macroscopic experimentally measurable quantities, such as the magnetization M(H). At the matching fields, the vortex lattice is expected to be topologically ordered; however, most experiments have not been able to directly image the vortex lattice to determine the precise vortex arrangements for varying external magnetic fields. In particular, at the matching field most experiments cannot directly discern whether several vortices sit at each pinning site or the vortices sit at the interstitials (regions between pinning sites). Interstitially pinned vortices are those which are trapped by other vortices, and not by defects. Interstitial pinning refers to "magnetic caging", in contrast to the standard, and stronger, "core pinning" by defects. Our direct knowledge of the vortex positions allows us to determine the degree of topological disorder in the vortex lattice at the incommensurate fields and compare it with the ordered phases at the matching fields.

Another important question that can be directly studied using computer simulations is the vortex lattice dynamics near and above the first matching field. Recent theories ^18,19 and experiments ^20-23 suggest that beyond the first matching field, weakly pinned interstitial vortices should be more mobile than vortices at the defect sites; thus, quantities such as critical currents and magnetization should drop considerably. Since these interstitial vortices should be flowing around the strongly pinned immobile vortices, plastic flow should occur in these systems. Plastic flow of vortices in superconductors has recently attracted considerable attention^24-26 and superconductors with a PAPS are an ideal system in which to study it. Our simulations allow us to explicitly show, in a quantitative manner, how dominant interstitial vortex motion is beyond the matching . Direct imaging of the vortex lattice can also determine exactly how different periodicities in the underlying pinning geometry affect the ordering of the vortex lattice.

It is important to emphasize that our simulations take into account the critical state^27,28 in which a gradient in the flux profile arises as flux penetrates a superconductor. Recently, Cooley and Grishin¹² presented a 1D static model of the Bean critical state with a PAPS, in which a terraced flux profile arises. In this work we will focus on a 2D dynamical model of flux-gradient-driven vortices; this 2D dynamical descripton produces a far more complex and richer behavior than that predicted by 1D static models.¹²

Section II describes the numerical algorithm used here and compares our system to other superconducting structures with periodic pinning. In Sec. III we present calculated values for the magnetization M(H(t)) for superconducting samples with either a square or a triangular array of pinning sites. This computed macroscopic average, M(H), is then related to microscopic quantities which describe the evolving topological order of the vortex lattice [e.g., the density of n-coordinated defects P_n(H) in the flux lattice]. Section IV shows specific vortex configurations and describes their dynamics close to the first commensurate peak in M(H). Section V presents vortex lattices at high matching field values and compares them with the low matching field configurations. Section VI shows the effects on the magnetization of varying the pinning strength, density, and disorder in the pinning lattice. Section VII contains a summary of our results and a discussion of how they compare to recent theories and experiments.

II. SIMULATION

We measure all forces in units of f₀=/8, fields in units of /, and lengths in units of . Our simulation correctly models the driving force as a result of local interactions; no artificial "uniform" external force is applied to the flux lines. As the vortices enter the sample (i.e., the pinned region) we can compute experimentally measurable quantities, including the density of flux lines B(x,y,H(t)), and the local J_c(x,y,H(t)) and global currents J_c(H(t)). Here, we will mainly focus on the most commonly measured macroscopic quantity, the global magnetization

and relate it to the spatiotemporal dynamics of the vortices.

Our simulation procedure and the samples used here have several differences that distinguish them from recent experiments. First, our samples have a slab geometry, unlike the films used in Refs. 3-5, where demagnetization effects are important. Second, the size and strength of our pinning sites are much smaller than those considered in Refs. 3-6. Thus, at higher matching fields, interstitial pinning will be important rather than multiple vortices per pinning site, as in the experiments.^3-6 Moreover, the relatively small radius of our pinning sites, among other reasons, makes our system very different from wire networks, where interstitial vortices cannot exist. Finally, and most importantly, our simulations are dynamical. The vortices are driven by the flux gradient as the external magnetic field is increased, and we can observe how the vortices are moving at the various external field values. This is considerably different from the static experimental results where the vortex motion has not been imaged.

III. RELATION BETWEEN MACROSCOPIC AVERAGES AND THE MICROSCOPIC TOPOLOGICAL ORDER OF INDIVIDUAL FLUX LINES

In order to correlate the behavior of this macroscopic quantity M(H) with the actual microscopic dynamics of individual vortices, it is useful to monitor the field-dependent fraction (or probability) P_n(H) of vortices with coordination number n. This quantity is obtained from the Voronoi (or generalized Wigner-Seitz) construction for the individual vortices and is shown in Figs. 1(b)-1(e) and 1(g)-1(j). The lengths of each side of a given Voronoi cell are compared, and if any side is less than one-tenth of the lengths of the other sides, it is ignored. This permits us to examine square lattices, which can be difficult to deal with using standard Voronoi algorithms. In the absence of any pinning, the flux lines form a perfectly hexagonal lattice [P₆(H)=1]; the presence of pinning sites introduces disorder into the flux lattice. The fraction of fourfold-coordinated vortices P₄(H), in Fig. 1(b), quantifies the degree of matching of the vortex lattice with the square pinning array at the commensurate fields n_v/n_p1/2, 1, and 2. At these fields where P₄ peaks, all the other probabilities show a clear dip. Notice also the increase in P₄(H) near the 1/4 and 1/8 rational matching fields; these are not obviously reflected in M(H), due to the usual large peak at HH^*. Moreover, rational matching peaks higher than one (e.g., 3/2, 5/2) are not observed, due to the presence of a flux gradient. P₆(H) greatly increases between the MF's, since there the vortex-vortex forces dominate over the vortex-pinning interactions. For very weak fields, the low numbers of vortices makes the Voronoi construction ill-defined.

Figure 1(f) shows M(H) for a triangular PAPS. Here the MF's at n_v/n_p1 and 3 produce peaks in M(H). The n_v/n_p2 peak is missing for this weak pinning case (although we observe a small peak for strong pinning f_p=2.0f₀) because this vortex configuration is less stable than the triangular VL at the same n_v. Figures 1(g)-1(j) show the corresponding fraction of n-fold coordinated vortices. Here P₆(H) increases at the MF's 1 and 3, and also has a smaller increase near 1/2. Notice that P₄(H), P₅(H), and P₇(H) decrease at the MF's 1 and 3. P₆(H) tends to increase with field for both square and triangular PAPS. Since here the five to seven disclinations are paired, P₅(H) and P₇(H) follow each other closely.

Figures 2(a) and 2(b) [2(c) and 2(d)] presents the actual positions of the vortices and the square [triangular] PAPS for a small part of the sample used in Fig. 1(a) [1(f)] at the n_v/n_p 1, 2 [1, 3] MF's. Due to the field gradient noticeable here, the VL is only commensurate over this small range. In Fig. 2(a) all the vortices are pinned at the defect sites; these vortices are said to experience "core pinning", instead of "magnetic pinning". The additional vortices that enter the sample when the external field is increased will not be trapped by the already-occupied pinning sites, but will still feel a magnetic "caging" effect due to the repulsion produced by other vortices. These are known as interstitially-pinned, or "magnetically-pinned", vortices. Here, we do not consider "magnetic pinning" due to magnetic impurities in the sample (e.g., iron dots); vortices are pinned either by defects or by other pinned vortices. In Fig. 2(b), half the vortices are interstitially pinned between the vortices located at the pinning sites, forming a square vortex lattice. In Fig. 2(c) and 2(d) the same cycle is observed only now there are two interstitially pinned vortices for every vortex trapped at a defect site.

A recent experiment⁵ finds that the particular type of pinning geometry [i.e., square, triangular] produces little difference in the matching fields observed (i.e., peaks in M(H) appear at all integer matching fields). This is in strong contrast to our results where the particular kind of pinning geometry clearly affects where matching peaks occur. Our results can be understood in terms of interstitial vortices being constrained to form a lattice with the same symmetry as the underlying pinning lattice. For a triangular array of weak pinning sites, a triangular vortex lattice with two vortices per pin, in which one vortex is in a pinning site and the other is interstitially confined, is unstable as can be shown by a simple linear stability analysis. Thus at the second matching fields, we do not observe a peak in either the sixfold coordination number or the magnetization. Similarly, at the third matching field of a square array of pinning sites, a square vortex lattice is unstable and a peak in the both the fourfold coordination number and the magnetization is not observed there. For the large and strong pinning sites used in certain experiments, e.g., Ref. 5 however, the onset of multiple vortices per pinning site leads to reduced sensitivity to the underlying geometry of the pinning lattice.

IV. VORTEX DYNAMICS AND PLASTIC FLOW CLOSE TO A COMMENSURATE FIELD

Since the commensurate portions of the vortex lattice are uniform in density, a terrace in the flux gradient arises with large currents at its boundary, where the flux density abruptly changes. In larger systems, we observe a flux profile with several separate commensurate regions (plateaus or terraces) inside the sample. For increasing H, the commensurate portions or terraces gradually move towards the center of the sample, reaching the center, where they eventually disappear. We point out that we do not assume a priori any particular form of the profile; we compute it dynamically by ramping up the external field H from zero. This dynamically-generated 2D terraced profile is somewhat similar to the static 1D one obtained in Ref. 12. However, while all integer MF's produced peaks in that 1D model, in our 2D model, for a given pinning geometry, only certain MFs actually produce peaks in M(H). We find, for instance, that a Kagome and honeycomb PAPS do not have a peak at n_v/n_p1 for weak pinning (f_p0.1f₀); a 1D analysis cannot distinguish these important geometrical effects. We have also found that the terraced ordered regions of the VL [corresponding to consecutive matching fields that produce commensurability peaks in M(H)] are separated by distinct domains where the flux lattice is disordered.

V. COMPLEX PATTERNS OF MAGNETIC FLUX: ISLANDS AND STRIPED DOMAINS

VI. EFFECTS OF THE MICROSCOPIC PINNING PARAMETERS AND CONTROLLED DISORDER ON THE MAGNETIZATION

Figure 5(b) shows M(H) for maximum pinning forces between f_p=0.1f₀ and f_p=0.4f₀, with all other parameters fixed and n_p=0.86/). The overall magnetization width increases and the peaks at n_v/n_p1 and 2 [indicated by the arrows in Figs. 5(b) and 5(c)] have sharper drop offs right after the MF's. The sharp dropoff after the first matching peak can be understood as a crossover in the pinning mechanism from vortices strongly pinned at defect sites to much weaker interstitial pinning. Such a crossover has been considered theoretically^18,19,31 and observed experimentally^20-23 for random pinning arrays. Since J_c is directly related to the width of M, these results indicate that enhancements of J_c may be restricted to fields less than the first matching field in the case when only one vortex can be trapped by each pinning site.

To examine the effects of disorder we displaced the pins from their ordered positions in random directions up to a maximum distance r. Figure 5(c) shows the effects on the first two peaks of M of gradually increasing the disorder in a square PAPS. As the disorder is increased from r=0 to r=/8 (bottom two curves), the second peak at n_v/n_p2 is suppressed, while the first peak remains. This occurs because the second peak is caused by weaker interstitial pinning - thus more susceptible to disorder. As the disorder further increases to r=/4 (top curve), the first peak disappears as well. Wire networks⁹ in which disorder is gradually introduced show similar effects since, in both cases, commensurate peaks decrease with increasing disorder.

VII. SUMMARY AND DISCUSSION

Our simulations explicitly show that at certain matching fields a subset of the vortex lattice becomes commensurate with the underlying array of pinning sites. Because of the gradient in the flux density due to the critical state, the commensurate portions of the vortex lattice first start at the edge of the sample and then gradually move to the center. These configurations persist for a finite range of increasing external field, creating a peak in M(H). By monitoring the vortex coordination number as a function of external field, P_n(H), we have shown that the vortex lattice goes through a series of ordered-disordered states with the higher matching (n_v > n_p) vortex configurations having interstitially pinned vortices rather than multiple vortices per pinning site. For high matching fields, domains of different orientational order appear - including coexisting stripes and islands of approximately constant field (see Fig. 4). These novel striped and island-type domains are separated by current-carrying winding loops and paths made of fivefold to sevenfold defect pairs. These unexpected and complex phases could be imaged by using, for instance, scanning Hall probes and Lorentz microscopy.

We have also shown in a quantitative manner that beyond the first matching field interstitially pinned vortices are much more mobile, leading to plastic flow of these weakly pinned vortices around more strongly pinned ones trapped at defect sites. This result is consistent with recent experimental work with randomly placed columnar pinning,^21,22 in which a sharp drop off in M(H) was observed; this drop was interpreted as a consequence of the weak pinning felt by interstitial vortices.

It is of interest to compare our results for commensurability at the first matching field to those predicted by the Bose glass theory.¹³ Here, a Mott-insulator-like state occurs in which the vortex lattice is locked into the pinning lattice for a certain field range, preventing further vortices from entering and creating a Meissner-like rise in the magnetization. With a PAPS, we observe such a Mott-insulator-like peak in the magnetization near the first matching field; however, this peak is gradually lost as the pinning array is distorted [see Fig. 5(c)]. Experiments with random columnar pinning also do not observe a peak in the magnetization at the first matching field.^21,22 This suggests that because of the symmetry of the periodic pinning array all the pinning sites are equally accessible so that the vortex lattice can easily lock into the pinning sites. Conversely, for a random distribution of defects, fluctuations in the locations of pinning sites can make some sites inaccessible due to screening of pins by repulsion from nearby occupied pinning sites.³¹

For samples with a random distribution of pinning sites, fluctuations in the distribution can create regions of low pinning density which can give rise to easy-flow percolating paths for the vortices ³¹ and thus reduce J_c. With a PAPS such fluctuations in pin density are minimized, thus maximizing J_c. Since J_c is related to M(H), our results quantitatively show that the enhancement of the critical current for certain field regions is greater for periodic arrays of pinning sites than for random arrays.

ACKNOWLEDGMENTS

^*Present address: Department of Physics, Colorado State University, Fort Collins, CO 80523.
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