Abstract
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
I. INTRODUCTION
Flux pinning in type-II superconductors is of both technological and scientific interest. While most experiments focus on the effects of random pinning distributions, some investigations have been carried out on periodic arrays of pinning sites ͑PAPS͒. 1 These find striking peaks in the magnetization [2] [3] [4] [5] and critical current J c . These peaks are believed to arise from the greatly enhanced pinning that occurs when parts of the vortex lattice ͑VL͒ become commensurate with ͑i.e., match͒ the underlying PAPS. Under such conditions, high-stability vortex configurations are produced which persist under an increasing current or external field. Other important vortex-matching effects have also recently been observed in a variety of different superconducting systems, [6] [7] [8] [9] [10] [11] [12] including Josephson junctions, 8 superconducting networks, 9, 10 and the remarkable matching of the VL to the crystal structure of YBa 2 Cu 3 O 7 due to intrinsic pinning. 11 Commensurability effects are also important in superconducting systems with random arrays of columnar pinning sites, where a Mott insulator phase is predicted 13 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-fieldtuned 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, and vortex density n v , as well as pinning density n p , radius p , and maximum pinning force ͑or strength͒ 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. 1-8͒ 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 sites͒. 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] [21] [22] [23] suggest that beyond the first matching field, weakly pinned interstitial vortices should be more mo-bile 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] [25] [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 B . 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 Grishin 12 presented a one-dimensional ͑1D͒ 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 lattice͔. 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 consider a 2D transverse slice ͑in the x-y plane͒ of an infinite 3D slab at zero temperature containing rigid vortices and columnar defects, all parallel to both the sample edge and the applied field HϭHẑ. Pinning is placed in the central 75% of the system ͑which is our actual sample͒ with the outer 25% free of pinning sites. We simulate the ramping of an external field as described in Refs. 29 and 30 by the slow addition of vortices to the unpinned region. These vortices attain a uniform density n, allowing us to define the external field Hϭn⌽ 0 , where ⌽ 0 ϭhc/2e. As the external density of the flux lines builds, they will, through their own repulsion, force their way into the sample where they interact with the lattice of columnar defects. We correctly model the vortexvortex force by a modified Bessel function K 1 (r/), where is the penetration depth. For the vortex-pin interactions, we assume the pinning potential well to be parabolic. For the simulations presented here, the pinning range ͑i.e., the radius of the parabolic well͒ is p ϭ0.25, far less than our vortex- 
Here ⌰ is the Heaviside step function, r i (v i ) is the location ͑velocity͒ of the ith vortex, r k p is the location of the kth pinning site, r i j ϭ(r i Ϫr j )/͉r i Ϫr j ͉, r ik ϭ(r i Ϫr k p )/͉r i Ϫr k p ͉, f p is the maximum pinning force ͑or strength͒, and we take ϭ1. Unlike previous simulations which have investigated systems with random pinning arrays, [29] [30] [31] [32] here we place the pinning sites in square as well as triangular arrays.
We measure all forces in units of f 0 ϭ⌽ 0 2 /8 2 3 , fields in units of ⌽ 0 / 2 , 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 region͒ 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] [4] [5] [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
Previous simulations of flux-gradient-driven vortices conducted in 1D ͑Ref. 29͒ and 2D ͑Ref. 30͒ with dense randomly placed pins produced a modified Bean-Kim model in which the flux profile had no terraces and the magnetization curves were very similar to those observed by standard experiments. However, with a PAPS, a very different behavior arises. Figure 1͑a͒ presents the first quarter of the initial magnetization curve for a square PAPS. Although our simulations produce complete hysteresis loops, 30 ,31 the ramp-up phase contains the relevant information. As the external field H is increased from zero, the flux fronts move towards the center of the sample and M (H) rises; when the two flux fronts meet at the center of the sample, M (H) peaks. This initial peak is thus unrelated to the PAPS, and is seen even in random arrays. 30 However, as the field is increased further, we find three cusps near the matching fields ͑MF's͒ given by n v /n p Ϸ1/2, 1, and 2, with the peak at n v /n p Ϸ3 missing since the VL cannot form a stable configuration commensurate with the underlying square geometry for n v /n p Ϸ3. The VL can form a stable configuration commensurate with the underlying square geometry for n v /n p Ϸ2. The peaks occur slightly after the exact matching due to the flux gradient in the sample. For the range of fields studied here (n v /n p Շ4), we observe that the radius of the pinning wells p is too small to permit multiple vortices per well; thus, the higher MF's form lattices with vortices pinned at interstitial sites. In flux creep experiments with a square PAPS, 3 it was suggested that multiple vortices were trapped in a single large well at the n v /n p Ϸ3 MF. Indeed, our results indicate that for a square PAPS any experimentally observed 3 MF's near n v /n p Ϸ3 must be due to multiple vortices per well.
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 probability͒ P n (H) of vortices with coordination number n. This quantity is obtained from the Voronoi ͑or generalized Wigner-Seitz͒ construction for the individual vortices and is shown in Figs. 1͑b͒-1͑e͒ and 1͑g͒-1͑j͒. 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 ͓ P 6 (H)ϭ1͔; the presence of pinning sites introduces disorder into the flux lattice. The fraction of fourfold-coordinated vortices P 4 (H), in Fig. 1͑b͒ , quantifies the degree of matching of the vortex lattice with the square pinning array at the commensurate fields n v /n p Ϸ1/2, 1, and 2. At these fields where P 4 peaks, all the other probabilities show a clear dip. Notice also the increase in P 4 (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 HϷH*. Moreover, rational matching peaks higher than one ͑e.g., 3/2, 5/2) are not ob-FIG. 1. Initial magnetization M "H(t)… for two samples with a pinned region of size 27ϫ36 for a square ͑a͒ and triangular ͑f͒ periodic array of pinnings sites. These are embedded in a 36ϫ36 system in which the external field is brought from zero to a final value of Hϭ2.6⌽ 0 / 2 , corresponding to a total of N v ϭ3400 flux lines ͑2550 vortices in the pinned region͒ and n v Ϸ2.6/ 2 . All panels on the left ͑right͒ side refer to the square ͑triangular͒ periodic array of pinning sites. The pin densities are n p ϭ0.86/ 2 (N p ϭ834) for the square PAPS and n p ϭ0.81/ 2 (N p ϭ784) for the triangular PAPS. N p is the total number of pinning sites. For both of these samples f p ϭ0.3f 0 , p ϭ0.25. For the square array, peaks in M (H) near the 1/2, 1, and 2 MF's can be observed in ͑a͒. For the triangular array, peaks in M (H) near the 1 and 3 MF's can be observed in ͑f͒. The fraction of n-fold coordinated vortices P n (H) exhibit clear peaks in P 4 , ͑b͒, and dips in the other P n (H)'s, near the MF's for the square case. The fraction of n-fold coordinated vortices P n (H) exhibit clear peaks in P 6 , ͑i͒, and dips in the other P n (H)'s, near the MF's for the triangular case. For increasing field in both PAPS, P 6 becomes close to one when HϾ2.5⌽ 0 / 2 ; meanwhile all the other P n 's decrease.
served, due to the presence of a flux gradient. P 6 (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 1͑f͒ shows M (H) for a triangular PAPS. Here the MF's at n v /n p Ϸ1 and 3 produce peaks in M (H). The n v /n p Ϸ2 peak is missing for this weak pinning case ͑al-though we observe a small peak for strong pinning f p ϭ2.0f 0 ) because this vortex configuration is less stable than the triangular VL at the same n v . Figures 1͑g͒-1͑j͒ show the corresponding fraction of n-fold coordinated vortices. Here P 6 (H) increases at the MF's 1 and 3, and also has a smaller increase near 1/2. Notice that P 4 (H), P 5 (H), and P 7 (H) decrease at the MF's 1 and 3. P 6 (H) tends to increase with field for both square and triangular PAPS. Since here the five to seven disclinations are paired, P 5 (H) and P 7 (H) follow each other closely.
Figures 2͑a͒ and 2͑b͒ ͓2͑c͒ and 2͑d͔͒ present the actual positions of the vortices and the square ͓triangular͔ PAPS for a small part of the sample used in Fig. 1͑a͒ ͓1͑f͔͒ at the n v /n p Ϸ1,2 ͓1,3͔ MF's. Due to the field gradient noticeable here, the VL is only commensurate over this small range. In Fig. 2͑a͒ 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 dots͒; vortices are pinned either by defects or by other pinned vortices. In Fig. 2͑b͒ , half the vortices are interstitially pinned between the vortices located at the pinning sites, forming a square vortex lattice. In Fig.  2͑c͒ and 2͑d͒ the same cycle is observed only now there are two interstitially pinned vortices for every vortex trapped at a defect site.
A recent experiment 5 finds that the particular type of pinning geometry ͓i.e., square, triangular͔ produces little difference in the matching fields observed ͑i.e., peaks in M (H) appear at all integer matching fields͒. 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 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
To investigate further the VL spatiotemporal dynamics as M (H) crosses the first commensurate (n v /n p Ϸ1) peak for the system in Fig. 1͑a͒ , we present snapshots in Figs. 3͑a͒-3͑d͒ of the real-space time evolution of the VL Voronoi cells. The locations of these snapshots on the magnetization curve are indicated in Fig. 1͑a͒ by the letters a -d . Figure  3͑a͒ shows the flux distribution just before the n v /n p Ϸ1 peak in M (H). Commensurability, as indicated by the presence of mostly square Voronoi cells, starts at the edges of the sample as the magnetization begins to rise. Figure 3͑b͒ shows the flux distribution precisely at the magnetization peak. The VL is strongly commensurate with the PAPS, but only in regions close to the sample edges. Beyond the peak, Fig. 3͑c͒ , the commensurability between the VL and the PAPS moves from the edges of the sample to the center, and the edges start to become disordered as vortices move ͑by flowing between the vortices trapped at the pins͒ into the interstitial sites, indicated by the smaller ͑lighter͒ polygons oriented 45°from the sample edge. This clearly shows that plastic flow of interstitial vortices dominates vortex transport when n v Ͼn p . Finally, Fig. 3͑d͒ shows the vortex configuration near the minimum between the two commensurate peaks, where the flux distribution is disordered throughout the sample. As the external field is further increased, the vortex lattice remains disordered until the second MF, where   FIG. 2 . Snapshots of a small part of the sample showing the actual location of vortices ͑solid dots͒ and pinning sites ͑open circles͒ at the M (H) peaks n v /n p Ϸ1,2, in ͑a͒, ͑b͒ ͓n v /n p Ϸ1,3, in ͑c͒, ͑d͔͒ for the square ͓triangular͔ PAPS indicated in Fig. 1͑a͒ ͓1͑f͔͒. Note the interstitially pinned vortices in ͑b͒ and ͑d͒.
the cycle is repeated; however, this time the VL also includes interstitially pinned vortices.
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 terraces͒ 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 MF's actually produce peaks in M (H). We find, for instance, that a Kagomé and honeycomb PAPS do not have a peak at n v /n p Ϸ1 for weak pinning ( f p Ϸ0.1f 0 ); 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
Figures 4͑a͒ and 4͑b͒ present the Voronoi cells for a system with a lower pin density n p ϭ0.49/ 2 , but about 4 times
FIG. 3. ͑a͒-͑d͒
Voronoi ͑Wigner-Seitz͒ cell construction indicating the location of the vortices as they move through the first matching peak n v /n p Ϸ1, for the square PAPS system described in Fig. 1͑a͒ . Larger ͑smaller͒ cells correspond to low ͑high͒ densities of flux lines and are indicated in dark ͑light͒ shading. The sample has periodic boundary conditions on the left and right sides, and the vortices penetrate at the top and bottom edges of the sample. ͑a͒-͑d͒ correspond to locations on M (H) indicated in Fig. 1͑a͒ by the letters a -d . Before the matching peak n v /n p Ϸ1 ͓frame ͑a͔͒ and right at the matching peak ͓frame ͑b͔͒, the PAPS and the VL are commensurate only near the edges of the sample, as indicated by the squares. ͑b͒ At the peak in M (H), 1D stringlike defects which are perpendicular to the edges ͑or along 60°a ngles for the triangular case͒ originate from the boundaries. These gradually destroy the commensurability between the PAPS and the VL at the sample edges, shifting it towards the center of the sample ͓in ͑c͔͒ when H is increased. In ͑c͒, just past the peak in M (H), the smallest ͑white͒ diamond-shaped cells ͑which look like squares rotated 45°from the horizontal͒ are mobile interstitial vortices whose plastic flow destroys the commensurability. These are also clearly present in ͑d͒, further down in M (H), where commensurability effects disappear. This loss of commensurability can be quantified by observing in Fig. 1͑c͒ the drastic drop in P 4 right after the commensurate peak.
higher pin strength f p ϭ1.4f 0 , than the n p and f p used for Fig. 1͑a͒ . The magnetization ͑not shown͒ is initially very high when n v Ͻn p , because the strong pinning forces create a very high J c ; however, after the first MF ͑i.e., when n v Ͼn p ), there is a very rapid drop off in M (H) and J c (H). This rapid drop occurs because the characteristic pinning mechanism changes from individual columnar defect pinning to much weaker interstitial pinning. This result, obtained via a dynamical simulation, is consistent with previous work using a time-independent formalism. 18, 19 At high fields intricate vortex patterns appear with islands and stripes even though no noticeable change in the magnetization can be seen. The extremely weak interstitial pinning found at these high fields means that ordered regions of vortices with different orientations are close enough energetically that different local configurations can coexist in the sample. Notice that, in general, the field distribution is not 1D terraced as in Ref. 12 . For instance, contours of constant B indicate that the ''islands'' and ''stripes'' in Figs. 4͑a͒ and 4͑b͒ have nearzero current (ٌϫBϷ0) and are surrounded by currentcarrying winding strings composed mostly of small current loops.
VI. EFFECTS OF THE MICROSCOPIC PINNING PARAMETERS AND CONTROLLED DISORDER ON THE MAGNETIZATION
Submatching peaks ͓indicated by arrows at n v /n p Ϸ1/4 and 1/2 in Fig. 5͑a͔͒ in M (H) become evident for samples with a large density of weak pinning sites. At the half matching field a portion of the vortices in the sample are arranged in a checkerboard pattern similar to that seen in wire networks; 10 however, submatching configurations at n v /n p ϭ1/3, 2/5, 2/3, and 3/4, which are found for wire networks, are not observed here. Our result for the submatching configurations are in agreement with the experiments in Ref. 3 , which use a system resembling ours. As the pinning force is increased the width of M (H) increases, making it difficult to identify the submatching fields below 1/2. As the flux gradient increases, the submatching vortex configurations can only form on smaller regions of the sample. It is thus easier to observe submatching field configurations in a sample with a higher density (n p ϭ1.83/ 2 ) of weak pinning sites ͓e.g., Fig. 5͑a͔͒ than in a sample with a lower density (n p ϭ0.86/ 2 ) of pins ͓see Fig. 5͑b͔͒ . This is the reason why experiments ͑e.g., Ref. 3͒ can distinguish these peaks in M (H) more clearly when the temperature is near T c , where pinning is weak and submatching vortex configurations can occupy a larger portion of the sample. Figure 5͑b͒ shows M (H) for maximum pinning forces between f p ϭ0.1f 0 and f p ϭ0.4f 0 , with all other parameters fixed and n p ϭ0.86/ 2 . The overall magnetization width increases and the peaks at n v /n p Ϸ1 and 2 ͓indicated by the arrows in Figs. 5͑b͒ and 5͑c͔͒ 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] [21] [22] [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. Vortex arrangements of different orientations form complex domains of various shapes including ͑a͒ islands and ͑b͒ stripes, both surrounded by current-carrying winding strings of five to seven disclination pairs. These unexpected and striking flux domain patterns continuously evolve as a function of the applied field. These results show that, in general, the current distribution is not terraced. This is in contrast to predictions by static 1D models. We also note that the islands and stripes of ͑approximately͒ constant B, surrounded by curved domain boundaries carrying large currents, contain weak currents because in them ٌϫBϷ0.
To examine the effects of disorder we displaced the pins from their ordered positions in random directions up to a maximum distance ␦r. Figure 5͑c͒ 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 ␦rϭ0 to ␦rϭ/8 ͑bottom two curves͒, the second peak at n v /n p Ϸ2 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 ␦rϭ/4 ͑top curve͒, the first peak disappears as well. Wire networks 9 in which disorder is gradually introduced show similar effects since, in both cases, commensurate peaks decrease with increasing disorder.
VII. SUMMARY AND DISCUSSION
In summary, we have presented simulations of fluxgradient-driven superconducting vortices interacting withvarious PAPS, on systems of varying pinning strength, density, geometry, and disorder. The comparison between M (H) and the P n (H)s, shown in Fig. 1 , quantitatively correlates how the geometry of the pinning lattice affects the underlying microscopic dynamics of individual vortices and its relation to macroscopic measurable quantities like M (H) and J c .
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 Ͼn 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. 4͒ . These striped and islandtype 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-insulatorlike peak in the magnetization near the first matching field; however, this peak is gradually lost as the pinning array is distorted ͓see Fig. 5͑c͔͒ . 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.

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