Nonequilibrium dynamic phases and plastic flow of driven vortex lattices in superconductors with periodic arrays of pinning sites

C. Reichhardt, C. J. Olson, and Franco Nori
Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
(Received 29 December 1997)

We present results from an extensive series of simulations and analytical work on driven vortex lattices interacting with periodic arrays of pinning sites. An extremely rich variety of novel dynamical plastic flow phases, very distinct from those observed in random arrays, are found as a function of an applied driving force. Signatures of the transitions between these different dynamical phases appear as very pronounced jumps and dips in striking voltage-current V(I) curves that exhibit hysteresis, reentrant behavior and negative differential conductivity. By monitoring the moving vortex lattice, we show that these features coincide with pronounced changes in the microscopic structure and transport behavior of the driven lattice. For the case when the number of vortices is greater than the number of pinning sites, the plastic flow regimes include a one-dimensional (1D) interstitial flow of vortices between the rows of pinned vortices, a disordered flow regime where 2D pin-to-pin and winding interstitial motion of vortices occurs, and a 1D incommensurate flow regime where vortex motion is confined along the pinning rows. In the last case, flux-line channels with an incommensurate number of vortices contain mobile flux-discommensurations or "flux solitons," and commensurate channels remain pinned. At high driving forces, the 1D incommensurate paths of moving vortices persist with the entire vortex lattice flowing. In this regime, the incommensurate channels move at a higher velocity than the commensurate ones, causing incommensurate and commensurate rows of moving vortices to slide past one another. Thus there is no recrystallization at large driving forces. Moreover these phases cannot be described by elastic theories. Different system parameters produce other phases, including an ordered channel flow regime, where a small number of vortices are pinned and the rest of the lattice flows through the interstitial regions, and a vacancy flow regime, which occurs when the number of vortices is less than the number of pinning sites. We also find a striking reentrant disordered-motion regime in which the vortex lattice undergoes a series of order-disorder transitions that display unusual hysteresis properties. By varying a wide range of values for the microscopic pinning parameters, including pinning strength, radius, density, and the degree of ordering, as well as varying the commensurability of the vortex lattice with its pinning substrate, we obtain a series of interesting and novel dynamic phase diagrams which outline the onset of the different dynamical phases. We show that many of these phases and the phase boundaries can be well understood in terms of analytical arguments.
[S0163-1829(98)07934-X]

I. INTRODUCTION

Many condensed matter systems can be understood in terms of a periodic elastic lattice driven over a disordered rigid substrate. These systems include flux line lattices in superconductors, ^1-35 Josephson junctions,³⁶ charge-density waves,^37,38 magnetic bubble arrays,³⁹ Wigner crystals,⁴⁰ and models of friction. ⁴¹ Recently, intense interest has focused on the onset of motion, dynamic phases, and topology of lattices as a driving force is increased. For superconducting systems, experimental work in neutron scattering,¹⁷, voltage-current V(I) measurements,^18-23 and decoration experiments,^24,25 as well as theoretical work,^5,6,26-32,37 have suggested that, as an external driving force on the lattice is increased, three distinct dynamical phases appear: a pinned or immobile phase, an amorphous plastic flow phase, and at high drives an ordered uniform flow phase. The plastic flow phase begins at the onset of depinning when portions of the lattice break away and become mobile while other portions remain pinned. The flow-paths at this stage are characterized by winding channels or percolationlike paths. ^4,6,10,14,33 Due to the breaking and tearing during this phase, the vortex lattice can become very disordered. As the external driving force is further increased, larger portions of the lattice become mobile until, for high driving, the entire lattice enters a uniform flow stage in which the vortex lattice is much more ordered than in the plastic flow or pinned regimes.

These different flow phases have permitted the construction of a dynamic phase diagram as a function of pinning parameters, temperature, and driving force.^{6,13,27,31,32} Similar results have been obtained for charge-density-wave systems. ^37,38 It has been suggested that the onset of ordering at the uniform flow phase is sudden and represents a true phase transition to a moving solid or crystal with long-range order.⁶ Recently, other studies have suggested that this is not the case and that the driven lattice is instead a highly ordered moving glass.^{27,28,31,32,37} Transitions in the dynamical behavior of the vortex flow are believed to be directly related to the peak effect^18-23 where a transition to plastic flow is marked by a sudden enhancement of the critical current just below H_c2(T), near the vortex melting point.

B. Lattice Driven over a Periodic Substrate

Although dynamical transitions have been examined in superconducting systems where the pinning is random, they have not yet been carefully studied in systems with periodic pinning where very different dynamics are expected due to the symmetry of the underlying pinning and the tunability of the disorder. Periodic pinning occurs in numerous superconducting systems such as wire networks,^42,43 Josephson junctions^44,45 and layered superconductors.^46,47 High-T_c samples contain other forms of periodic pinning such as that produced by the periodic copper-oxide planes.⁴⁸

Recently, increasing attention has focused on samples with lithographically created well-defined periodic pinning structures such as a lattice of microholes^34,49-56 or magnetic dots.⁵⁷ In such samples the microscopic parameters of the pinning, such as the size, depth, periodicity, and density, can be carefully controlled. Using Lorentz microscopy techniques,^34,56 it has been possible to directly image a vortex lattice interacting with periodic arrays of pinning sites, revealing very interesting commensurability effects in the vortex configurations and dynamics.

Extensive work has been done on the interaction of vortices with a periodic "egg-carton" potential as in wire networks and Josephson-junction arrays. However, the dynamics of vortices in such systems differs significantly from the case of vortices in periodic pinning arrays we consider in this work. In our system the radius of the pinning sites can be made much smaller than the period of the pinning lattice, permitting interstitial vortices to appear. The distinctive interstitial vortex motion is not found in superconducting wire networks and Josephson-junction arrays. Periodic pinning arrays may also be of technological importance since the arrays can produce higher critical currents than an equal number of randomly placed pins.^10,11 This enhancement of critical currents using periodic arrays has recently been demonstrated for high-T_c systems.⁵⁵

C. Tunable commensurability

A particularly interesting aspect of periodic pinning arrays where the pin radius is small compared to the lattice spacing and where there is only one vortex per pinning site is that disorder can be fine tuned by changing the commensurability. At B/=1, where is the field at which the number of vortices equals the number of pinning sites, the vortex lattice is expected to be locked into the periodic pinning array. For B/ > 1, the vortex lattice can be thought of as containing two species of vortices: the pinned vortices that are commensurate with the array of pins, and the generally more weakly pinned interstitial vortices that are caged by vortices at the pinning sites. The very different dynamical behaviors of these two species has been observed both when the vortices are current-driven⁵³ and when they are field-gradient-driven.^10,11,51,52 These different dynamics have also been directly observed with Lorentz microscopy techniques.⁵⁶ The effect of interstitial vortices at B/ 1, as well as the commensurate effects at B/ = 1, are also of major importance in systems with randomly placed columnar pins, where theoretical^10,58 and experimental^59,60 magnetization studies have shown a substantial drop in the critical current associated with the appearance of interstitial vortices.

It is also possible to have a well-defined number of vacancies in the vortex lattice when B/ < 1. Here, the vortex lattice can be thought of as containing two species of vortices, where the second species, the vacancies, may have different dynamical properties than the first species, the commensurate vortices. This situation and terminology is reminiscent of the case of electrons and "holes" in semiconductors.

D. Nonequilibrium dynamic phase diagram

In order to observe the dynamics of current-driven lattices near commensurability interacting with a periodic array of pinning sites, we have performed a large number of current-driven molecular dynamics (MD) simulations in which we examine experimentally realizable parameters. Our work is distinct from previous current-driven simulations in that we specifically examine the effects of periodic pinning arrays rather than random arrays. Further, our work covers a much larger range of the microscopic pinning and system parameters than used in previous MD simulations, allowing us to construct a variety of detailed dynamical phase diagrams. A much shorter and less detailed account of some the results presented here can be found in Ref. 13.

We find that a vortex lattice driven on a periodic array of pinning sites exhibits far richer and more complex dynamical phase diagrams than the diagrams produced by systems with random pinning arrays. We observe numerous novel plastic flow states and vortex lattice structures distinct from those observed in random pinning arrays. For a certain range of parameters and when the driving force is increased, we find that there is a counterintuitive drop in the number of mobile vortices as the system undergoes a transition from one plastic flow phase to another. This drop in the number of mobile vortices also implies a negative-differential conductivity, a property that is found in several technologically important semiconductor devices.

We find that the vortex dynamics is highly dependent on the commensurability effects. When B/ > 1, the interactions between commensurate vortices and interstitial vortices give rise to a number of interesting phases, while when B/ = 1, a Mott insulator-like phase arises.⁶² By systematically disordering the positions of the pinning sites we recover, in the limit of strong disorder, the dynamical phases found in recent theoretical studies of random pinning arrays. ^{4-6,10,14,16,26,31-33} For B/ > 1, and at the onset of depinning in samples with large disorder, we find a distinct "channel phase" where long-lived stable channels, composed predominantly of interstitial vortices, flow around pinned vortices. Such intermittent channel phases where some channels form, freeze and reform have previously been noted in flux-gradient-driven simulations.^10,14,33 A phase with a network of flowing channels is also observed in current-driven samples. ^{5,6,16,26,31,32} Here, we examine in detail the evolution of the network of vortex channels. In particular, we show that, although some channels are short lived, others are robust under increases of the driving force.

E. Overview

This work is organized as follows. In Sec. II, we describe the samples used, our numerical algorithm, and the parameters varied. In Sec. III, we show in detail the results for a sample with a square array of pinning sites at a field slightly above commensurability. Current-voltage V(I) curves, individual vortex velocity signals, and images of the vortex motion are presented and correlated with each other, explicitly characterizing the different dynamical phases present.

In Sec. IV we obtain a dynamical phase diagram for the case B > and explain many of its features by using force balance arguments which take into account the interactions between interstitial vortices and vortices in the pinning sites. In Sec. V we show that significant hysteresis is found for certain phases but not for others, and that the hysteresis is not affected by the sample size. Sharp discontinuities in V(I), along with the hysteresis, suggest that some of the dynamical phase transitions are of first order.

In Sec. VI, we demonstrate the existence of a transverse threshold force, as well as the effect of the onset of motion in the transverse direction on the vortex lattice structure. We also show how the transverse force varies with driving force, and discuss the relevance of these results to the recent work by Giamarchi and Le Doussal.²⁸

In Sec. VII, the V(I) curves and the phase diagram for varying vortex-pin commensurability are presented. We find that the various dynamic phases strongly depend on the commensurability ratio B/. For < B< 2, we observe up to six distinct dynamic phases that arise from the interactions of two species of vortices. For B > 2 we observe several additional phases. At B/=1, only two phases are observed, and the critical depinning force reaches its highest level, reminiscent of the so-called "Mott-insulator" phase. For B/ < 1, we demonstrate that the initial plastic flow occurs by the motion of vacancies.

In Sec. VIII, we derive the dynamic phase diagram produced when varying the pinning radius. For pins of small radius, we observe a remarkable reentrant disordered dynamic phase. In between these disordered-motion phases there is a channel phase in which the majority of vortices flow in narrow well-defined interstitial channels. This motion is characterized both by a linear Ohmic response in the V(I) curve and by a reduction in voltage fluctuations. We also find that near the boundaries of this channel phase the system exhibits some interesting hysteresis effects. We show the effects of this reentrant behavior on the percentage of sixfold coordinated vortices in the sample. For certain pinning radii, some transitions among the various plastic flow regimes are marked by a large drop in the number of mobile vortices. We argue that this drop is a force driven dynamical analog of the thermally driven peak effect observed close to T_c. ^19-21,23

In Sec. IX, we show the effects on the dynamic phase diagrams of gradually increasing the spatial disorder of the pin array. In the case of large spatial disorder we recover results consistent with studies on totally random pinning arrays. We also present evidence for the existence of distinct regimes of predominantly interstitial channel flow near the initial depinning transition. The characteristics of the channels are also discussed.

In Sec. X we plot the phase diagram for increasing the vortex-vortex interaction strength by means of increasing the vortex density. In Sec. XI we present results for a system with a triangular pinning array. These qualitatively resemble most of the results from the earlier sections, although the flow of vortices in some of the dynamic phases differs from the flow found for the square pinning arrays. In Sec. XII, we summarize our results and discuss their relevance to experimentally realizable systems.

II. SIMULATION AND PARAMETERS

We model a transverse two-dimensional (2D) slice (in the x-y plane) of an infinitely long (in the z direction) parallelepiped with periodic boundary conditions in which stiff vortex lines are parallel to the sample edges (i.e, H=H). Inside the sample, the interacting vortices are driven by a Lorentz force over a quenched pinning background. We numerically integrate the overdamped equations of motion:
f_i=f_i^vv+f_i^vp+f_d=v_i. (1)
Here, f_i is the total force acting on vortex i, f_i^vv is the force on the ith vortex due to interactions with other vortices, f_i^vp is the vortex-pin interaction force, and f_d is the driving force; v_i is the net velocity of vortex i and is the viscosity, which is set equal to unity in this work. The vortex-vortex interaction between two vortices located at r_i and r_j is correctly modeled by a modified Bessel function. Thus the force acting on a vortex i due to other vortices is
(2)
Here

=hc/2e is the elementary flux quantum, is the penetration depth, N_v is the number of vortices, and =(r_i-r_j)/|r_i-r_j|. The force between two vortices decreases exponentially at distances larger than , and we cut off the then negligible force for distances greater than 6. We also place a cutoff on the logarithmic divergence of the forces for distances less than 0.1. These cutoffs were found to produce negligible effects on the dynamics for the range of parameters investigated. Throughout this work forces are measured in units of f₀, lengths in units of , and fields in units of /.

The pinning is modeled as N_p short-range parabolic wells located at positions r^(p)_k. The total force on a vortex from other vortices and pinning is
(3)
Here, is the Heaviside step function, r_p is the range of the pinning potential, f_p is the maximum pinning force of each well, measured in units of f₀, and =(r_i-r_k^(p))/|r_i-r_k^(p)|. The pinning sites are placed in periodic arrays with a lattice constant a. The driving force f_d is modeled as a constant force which is slowly increased or decreased linearly with time.

In implementing the parallelization of the code we take advantage of the cut-off in the vortex interaction range. Using a one-dimensional domain decomposition, we divide the simulation sample into strips that are multiples of the interaction range in width. Each strip is placed on a separate node, and message-passing techniques⁶¹ are used at the processor boundaries. Since the vortex-vortex interaction has a finite range, any one strip only needs to communicate with its two neighboring strips. Due to periodic boundary conditions the code is run on an even number of nodes. Load balancing is simplified by the repulsive nature of the vortex interaction which tends to spread the vortices evenly among the processors. With flexible domain decomposition the number of processors can be varied without affecting the results. The code is run on a IBM SP parallel computer.

B. Parameters

We typically increase the driving force by 0.0005f₀ every 400 MD steps for a range of f_d ranging from 0 to 0.85f₀; using slower rates produces negligible differences. For this work we consider the driving force to be in the x-direction along a symmetry axis of the periodic pinning array. We examine the average of the velocities in the x-direction
(4)
as a function of driving, writing out V_x every 100 to 400 MD steps. This quantity is related to a macroscopic measured voltage-current V(I) curve. Here, N_v is the total number of flux lines in the system.

Since MD simulations are computationally intensive, and we wish to vary many parameters in order to investigate several phase diagrams and generate a complete picture of the dynamic phases, considerable effort has been put into optimizing our algorithm. We use a cell-index method as well as force tables for the vortex-vortex and vortex-pin interactions so that excessive function calls are avoided during program execution. More importantly, the simulation also uses a high performance parallel processing technique.

We focus on a system of size 36 x 36; however, to study finite size effects, we have also computed several hysteresis runs with system sizes of up to 72 x 72. The parameters we vary include the vortex density n_v, the pinning site density n_p, radius r_p, strength f_p, and spatial distribution. We will consider only the case where the vortex lattice is driven along a symmetry axis of the periodic pinning array. Results for general driving driving angles will be presented elsewhere.¹⁵

We use pinning parameters that are experimentally accessible and are close to those used in recent experiments.^51,52 For the density of pinning sites we have investigated a range from n_p=0.1/ to 0.7/, and the corresponding number of pinning sites varies from N_p= 130 to N_p= 907, respectively. Experiments use samples with a fairly low density of pinning sites, so we focus on a square array with a pinning density of n_p= 0.25/, as found in the experiments described in Refs. 51 and 52. We focus on the vortex motion near the first matching field B/=1, and systematically examine the change in dynamics as B/ is varied.

In order to separate the different effects that each of the pinning and system parameters (i.e, H, f_p, ) have on the dynamics of the vortices, we fix all the parameters and vary only one in each of the sections of this paper. Moreover, in order to distinguish the effects of disorder caused by the pinning from that caused by thermal effects, we set T=0. Since we are dealing with square arrays of pinning sites near , the ground state vortex configuration is a square lattice commensurate with the pinning sites.

In Figs. 1(a) and 1(b), the positions of the vortices and pinning sites are shown for B/< 1, (a), and B/ > 1, (b). Vacancies appear in Fig. 1(a) and interstitial vortices are present in Fig. 1(b). The initial ground states are obtained in one of two ways. The first is simulated annealing, in which we start from a high-temperature vortex lattice and cool it down. In the second approach, we put down a commensurate vortex configuration and randomly add vortices to the interstitial regions for B/ > 1 or randomly remove vortices to create vacancies for B/ < 1. For B near , the resulting dynamics are identical using either method.

III. VOLTAGE-CURRENT CHARACTERISTICS AND PLASTIC FLOW REGIMES

In Fig. 2(b) for a sample with weaker pinning (f_p=0.37f₀) phase II is lost and the vortex flow jumps directly to phase III. There is also no sharp discontinuity in the transition from phase II to phase III; however, the transition from region III to region IV still shows a discontinuity. The onset of phase V now occurs at f_d=0.37f₀=f_p. In Fig. 2(c) for a sample with even weaker pinning, (f_p=0.187f₀), both regions II and III are lost. The vortex flow goes directly from phase I to IV and jumps to phase V at f_d=0.187f₀. Region V is the homogeneous flow phase where the entire vortex lattice is flowing but doing so plastically.

To show explicitly that the features in the V(I) curves correspond to different nonequilibrium dynamical flow phases, in Figs. 3 and 4 we show a series of snapshots of the vortex positions (left panels) as well as trajectory lines (right panels) indicating where the vortices have flowed during a short period of time for regions II through V of the voltage-current curve in Fig. 2(a). The vortex lattice structure for region I, where the applied driving force is too weak to depin any vortices, is shown in Fig. 1(b). In Figs. 5(a)-5(d) plots of the velocity of an individual vortex versus time for regions II-IV also show the very different types of vortex dynamics present in each phase. It is important to point out that the vortices in interstitial locations are pinned by the magnetic-repulsion caging effect of the vortices trapped at pinning sites.

A. Interstitial 1D vortex channels

Figure 3(a) presents a "snapshot" of the vortex locations at a single instant in region II of the voltage-current curve. This figure shows that the mobile interstitial vortices are located between the rows of pins. The vortex lattice structure is essentially the same for regions I and II. To approximate the percentage of flux lines which are mobile at a specific driving force, we define the following measure based on the net velocity in the x-direction:
(5)
For region I, has a value of =0.06, indicating that only the interstitial vortices are mobile since the percentage of vortices above is also (B-)/=0.06. Figure 3(b) shows the flow in region II of the V(I) curve with outlines of the paths the vortices have followed. This confirms that only the interstitial vortices flow in region II and that the motion is constrained to move in 1D channels between the rows of pinning sites due to the square symmetry imposed by the pinned vortices. Figure 5(a) shows that the velocity of the mobile vortices is never zero and that it has an oscillatory component⁶⁵ since the energy landscape imposed by the vortices pinned at the pinning sites has the same square periodicity as the pinning array.

B. Phase boundary I-II For the case of strong pinning when region II exists, as in Fig. 2(a), we can make a simple estimate of the threshold driving force needed to depin an interstitial vortex by considering the balance of forces when there are two species of vortices, interstitial and pinned. An interstitial or incommensurate vortex will feel a force from commensurate pinned vortices f_inc-c=f_incomm-comm, forces from other interstitial vortices f_inc-inc, and the driving force f_d. An interstitial vortex will thus remain immobile as long as
f_inc-c+f_inc-inc+f_d=0.
For the case when there are few interstitial vortices, as in Fig. 2(a), we consider only the interactions with the commensurate pinned vortices and the driving force and neglect the interactions between pairs of incommensurate vortices. When there is no driving force the interstitial vortices sit in the center of the interstitial locations, and each has four nearest-neighbor commensurate pinned vortices. As the driving force is increased the interstitial vortex will start to shift in the interstitial location and experience a restoring force from the commensurate pinned vortices. Since the vortex interactions decay exponentially with distance, the dominant restoring force on an interstitial vortex comes from the four nearest commensurate pinned neighbors. The maximum restoring force occurs when the interstitial vortex has shifted halfway between its zero driving position and the pinning site. An estimate for the threshold force is
(6)
Using this equation with the parameters from Fig. 2(a), we find that the onset of region II should occur at a driving force of f_d=0.14f₀, which is in very good agreement with the value of f_d=0.146f₀ obtained from the simulations.

C. Disordered 2D vortex motion

In Fig. 2(a), at the onset of region III we find a significant jump in V_x indicating a sudden increase in the number of mobile flux lines. The approximate percent of mobile vortices is now =0.44, which indicates that vortices are now being depinned from the pinning sites since the percentage of vortices above remains =0.06. Significant fluctuations for phase III in the V(I) curves can be seen, indicating that the number of mobile flux lines is rapidly fluctuating. Figure 3 presents the vortex positions (c) and trajectories (d) for region III of Fig. 2(a). We see that the vortex lattice structure and flow pattern is of a remarkably different nature than that observed in the interstitial flow phase II. The vortex lattice has now become disordered, and the vortex trajectories are no longer 1D but move in both the x and y directions. It can be seen that some pin-to-pin vortex motion now occurs. Unlike the motion in phase II, where only the interstitial vortices move and vortices at the pinning sites remain pinned, all the vortices in region III take part in the motion with any one vortex moving for a time and then being temporarily trapped. Figure 5(b) shows the velocity of a single vortex in region III. The velocity is not periodic as in phase II, Fig. 5(a), but shows irregularities, including sharp bursts followed by periods of no motion.

D. Phase boundary II-III: interstitial and disordered flow phases

To understand the crossover from the 1D interstitial flow phase II to the disordered 2D flow region III, we consider the balance of forces in a sample with two kinds of vortices, interstitial and pinned, just as in the crossover from region I to region II. Interstitial vortices exert a force f_inc-c=f_incomm-comm on a commensurate vortex (denoted by the sub-index c) while commensurate vortices exert a force f_c-c on this vortex. In the simple case when there are no interstitial vortices, commensurate vortices stay pinned as long as f_d< f_p. When interstitial vortices are present, they move between the rows of the lattice of pins and exert an additional force on the pinned vortices, resulting in an earlier depinning transition. A commensurate vortex at a pinning site remains pinned by the force from the pinning site as long as the following force balance inequality holds:
|f_p| > |f_d + f_inc-c + f_c-c| . (7)
Due to the symmetry of the underlying pinning lattice, it is clear that the net force from the other commensurate vortices will be f_c-c=0 when every pin is occupied by a vortex (or for other symmetrical configurations). Since f_inc-c is the sum of all the interactions from mobile interstitial vortices, f_inc-c is time-dependent and has a complicated form.

To estimate f_inc-c, we consider the case where B is only slightly higher than . Here, the interstitial vortices are on average sufficiently far apart that we can estimate f_inc-c from a single interstitial vortex interacting with commensurate vortices as it moves between the pinning rows. The force exerted by this interstitial vortex i on a commensurate vortex j has a maximum value when the interstitial vortex reaches the point of its closest approach to the pinned vortex, which for a square lattice occurs when the interstitial vortex is aligned with a column of pinning sites (see Fig. 6). The distance between vortices i and j at closest approach is r_ij=a/2, or half the pinning lattice spacing a. We also take into account the fact that the pinned vortex will be shifted slightly from the center of the well both because of the applied driving force and because of the repulsion from the interstitial vortex. This adds a small distance, which we approximate as r_p/2, to the distance between the interstitial vortex and the pinned vortex, giving a total "distance of closest approach" of approximately r_ija/2 + r_p/2. In Fig. 6 a schematic diagram of this is presented. In terms of magnitudes, a commensurate vortex remains pinned as long as the following inequality holds:
(8)
(9)
where f_ij=f₀K₁(r_ij/) is the force between the interstitial vortex i and the commensurate vortex j. Driving forces which satisfy this inequality result in phase-II-type interstitial flow. As the driving force f_d is increased, the inequality eventually ceases to hold, and an interstitially moving vortex can cause vortices in the pinning sites to depin. These depinned vortices move away with some component of their velocity in the transverse y direction. As these vortices move, they depin additional vortices in an irregular fashion, thus producing the transition to the disordered 2D-flow phase III. As long as f_p < f_d, the vortices in the sample can be temporarily trapped; therefore, not all of the vortices move simultaneously.

We predict that the transition from 1D interstitial to disordered 2D flow [i.e., from phase II to phase III in Fig. 2(a)] should occur when the inequality in (9) no longer holds. To verify this prediction we consider a sample with parameters equal to the ones presented in Fig. 2(a), namely, a=2.0, r_p=0.3, and f_inc-c=f₀K₁(1.15)=0.4695f₀. The upward jump in V_x, signaling the transition between phases II and III, occurs at f_d=0.406f₀. Using inequality (9), we find that the jump will occur at this value of f_d if f_p=0.621f₀. This is in very good agreement with the input value of f_p=0.625f₀ used for the simulations shown in Fig. 2(a). The fact that the calculated value is somewhat less than the actual value is probably due to the interactions of other interstitial vortices which were not taken into account in inequality (9).

E. Flux soliton motion in incommensurate 1D vortex channels

Upon further increasing the driving force f_d in Fig. 2(a), we observe a new nonequilibrium dynamic phase, which we label region IV. At the onset of this phase, a surprising sharp drop in V_x occurs, indicating a decrease in the number of mobile vortices. In particular, we find that a fraction 0.24 of the vortices are mobile, significantly lower than the fraction, 0.44, that were mobile in phase III.

In Figs. 4(a) and 4(b), we see that the vortex motion possesses a very different structure and flow pattern from that of the 1D interstitial flow of phase II and the random flow of region III. The flow in phase IV is entirely 1D with the mobile vortices flowing along the pinning rows, which is distinct from the 1D interstitial flow in phase II where mobile vortices flowed between the rows. We observe that the presence of additional vortices in certain rows creates incommensurate 1D structures in the form of discommensurations or flux solitons along the pinning row, and that it is in these rows where motion occurs⁶⁵ in the direction of the driving force f_d. Rows that are commensurate are immobile. Due to the 1D nature of the incommensurate flow in phase IV, the voltage fluctuations V_x for phase IV are considerably smaller than the V_x's found during the random flow of phase III.

It is interesting to point out that the drop in the fraction of moving vortices at the III-IV transition implies a negative differential conductivity
dV/dI < 0.
This behavior is often observed in semiconductors, and can be useful for certain devices.⁶³

The appearance of 1D motion exactly along the rows of pins might seem counterintuitive. This is so because for B/ > 1 and f_d=0, when the vortices are not moving, the only positions for extra vortices that are stable against perturbations in the transverse y-direction caused by other interstitial vortices are the interstitial positions between the rows of pinning, as in Fig. 1(b). For moving vortices, the situation is quite different because the extra vortices in incommensurate rows spend part of their time in the pinning sites. The pinning sites create a stabilizing force against perturbations in the transverse y direction, so that motion becomes confined to the pinning rows along the longitudinal direction. The vortex motion occurs only where one or more incommensurate vortices are located along the row. As the disturbance or flux soliton moves, it causes the vortex in front of it to depin and then this one moves to a point where it will depin the next vortex. From direct observations of the flux motion, as in Fig. 4(b), we find that every discommensuration is composed of four mobile vortices. The periodic-pulse nature of the discommensuration flow is clearly seen in Fig. 5(c) where the individual velocity of a single vortex in region IV from Fig. 2(a) is plotted. An individual vortex moves only for a short time as the discommensuration moves through and then is pinned again. Using the parameters presented in Fig. 2(a), where a fraction (B-)/=0.06 of the vortices are incommensurate, we find that the total fraction of mobile vortices is about four times higher, or =0.24.

F. Phase boundary III-IV

In order to understand when incommensurate vortex motion along the pinning rows, as seen in phases IV and V, is stable, we construct a simple force-balance argument. We consider a force f_y in the transverse y direction that deflects a vortex away from its longitudinal motion along the pinning row in the x direction. Suppose that under the influence of this force f_y, the vortex moves a distance d_y in the transverse y direction equal to r_p, the pinning radius, in time t. Thus,
d_y=f_yt=r_p. (10)
For a particular driving force f_d in the longitudinal x direction, if a vortex can move a distance d_x in the x direction equal to the pinning lattice constant, a, within the same time t=d_y/f_y, the vortex will feel a pinning force in a direction opposite to f_y that pulls the vortex back into the row a distance:
d_x=f_dt=a. (11)
We note that in actuality the vortex only spends a finite amount of time in the pinning site so it is not pulled all the way to the center of the pinning site; however, for convenience we assume here that the vortex is pulled all the way to the center. From Eqs. (10) and (11) it can be seen that motion along the row is stable if
(12)
For example, for the parameters in Fig. 2(a), with r_p=0.3 and a=2.0, if the vortex lattice is being driven with a force of f_d=0.5f₀ in phase IV with incommensurate 1D channels on pin rows, the 1D motion is stable against perturbations of size f_y < 0.075f₀. Note that inequality (12) indicates that as the size of the pinning sites is increased (i.e., increasing r_p) or the pinning density is increased (i.e., decreasing a) motion along the rows will be more stable. Inequality (12) also erroneously predicts that f_y increases linearly with f_d; however, the maximum possible value for f_y for stable vortex motion is f_p.

To derive an expression for the increasing driving force at which region IV appears, we take into account the fact that there is an average energy barrier for interstitial vortices to enter the pinning rows since vortices at pinning sites tend to keep interstitial vortices in between the rows. Once the vortices are in the random-flow phase III, many of the vortices are moving in interstitial regions and (see Sec. III) these vortices depin vortices at nearby pinning sites. As f_d increases, pinned vortices become easier to depin and there is an increase in the distance r_min from which an interstitial vortex can depin a pinned vortex, so that r_min > a/2 + r_p/2. When f_d is large enough, r_min reaches a limiting value such that a vortex located approximately halfway between two columns of pinning sites can depin one of the vortices in the row ahead of it. A schematic diagram of this situation is presented in Fig. 7. To verify this, we note that a vortex placed at the center of an interstitial site exerts a force approximately equal to f_ij(a/+r_p)cos(45^o) in the x-direction on a vortex in a pinning site. As a function of the driving force f_d, the pinning force f_p needed to keep a vortex pinned at a pinning site when an interstitial vortex is located at the center of an interstitial site is
, (13)
. (14)

After an interstitial vortex depins a commensurate vortex, it experiences a net force toward the vacated pinning site from the other pinned commensurate vortices. It must travel an approximate distance a/2 in the x direction before it is adjacent to the just vacated pinning site. During this time, however, the depinned vortex also moves a distance a/2 in the x direction as well as a/2 in the y direction and in principle should itself be able to depin another commensurate pinned vortex. The original interstitial vortex is just far enough from the first depinned vortex that it is able to be trapped by the vacant pinning site.

Thus, when f_d is large enough, vortices moving in the random 2D flow regime (phase III) will start to become confined along the horizontal pinning rows and, provided that inequality (12) is met, this motion along the pinning rows should be stable. Substituting in inequality (14) the parameters used in Fig. 2(a) and the value of f_d=0.406f₀ observed at the onset of the 1D incommensurate motion, we find a calculated value of f_p=0.624f₀, which is in very good agreement with the actual value f_p=0.625f₀ used in the simulations.

G. Large driving plastic flow phase: IV-V phase boundary

As f_d is increased to almost f_p, the entire vortex lattice starts to flow in a phase which we label region V. The vortex flow pattern for phase V for the system in Fig. 2(a) is presented in Figs. 4(c) and 4(d). The type of motion and vortex lattice structure in region V resembles that of phase IV with vortices moving in 1D channels along the pinning rows; however, in region V all the vortices are mobile. The velocity of an individual vortex is periodic [as shown in Fig. 5(d)] due to the underlying periodic pinning lattice. Individual vortices can slow down considerably but are never completely repinned, as indicated in Fig. 5(d) by the fact the the velocity is always greater than zero.

The transition from region IV-V in Fig. 2(a) is not as sharp as that of II-III and III-IV but instead occurs over the range of 0.595f₀f_d0.613f₀. This can be understood by considering that in region IV certain rows contain incommensurate mobile vortices. As these vortices move past pinned vortices in adjacent commensurate rows they will exert a small force in addition to the driving force that causes the adjacent commensurate rows to depin at a driving force lower than f_d=0.625f₀.

We also observe that in region V rows with incommensurate numbers of vortices move faster than commensurate rows so that rows with different numbers of vortices slide past one another. Some portions of the vortex lattice can be seen to have triangular order; however, there are portions where this does not hold. A Voronoi analysis gives a probability of sixfold coordinated vortices of P₆=0.83. This is less than P₆=1 because of the presence of the incommensurate rows. As f_d is increased further, the density of defects and structure of the vortex lattice do not change. This result is different from simulations^5,6 with random pinning arrays which show that for sufficiently high driving rates, the defects in the vortex lattice heal out and P₆=0.95.

IV. DYNAMIC PHASE DIAGRAM FOR VARYING PINNING FORCE

A. Onset of phase II

Figure 8 shows that the interstitial motion of phase II occurs only when f_p0.37f₀. This can be understood by considering that when f_p0.37f₀, inequality (9) cannot be satisfied for any driving force, so that phase II cannot occur, and thus the boundary between phases I and II stops.

For a pinning force of f_p=0.35f₀, an interstitial vortex at closest approach exerts a force f₀K₁(1.3)=0.3725f₀ on a vortex at a pinning site. This is greater than the pinning force, so as soon as the interstitial vortices move, they cause vortices at the pinning sites to depin, producing phase III motion. In this case, it is not possible for interstitial vortices to move between the pinning rows without depinning vortices trapped at the pinning sites, so the interstitial 1D motion of region II does not occur for these weak pins. Region II occurs in the phase diagram only for values of f_p and f_d that satisfy both inequality (9) and inequality (14).

B. Onset of phase III

As f_p is increased further, a stronger driving force is required for the interstitial vortices moving in phase II to depin commensurate vortices so that a transition to region III can occur. The phase boundary line separating phases II and III is expected to follow the equation
(15)
which for large drives, f_d >> f_ij(a/2+r_p/2), Eq. (15) reduces to the linear relation
f_pf_d, (16)
in good agreement with the phase diagram in Fig. 8. Phase III occurs for f_p and f_d values that satisfy inequality (14) but violate inequality (9). Under these conditions, commensurate vortices can be depinned, but phase IV motion is not yet possible.

C. Onset of phase IV

It is clear from the dynamic phase diagram in Fig. 8 that when f_p < 0.25f₀, motion begins immediately in phase IV, without passing through phases II and III. For values of f_p less than f_p=0.25f₀, inequality (14) remains invalid for all driving forces so that the initial motion appears as region IV. For example, for f_p=0.25f₀, motion starts at f_d=0.09f₀, the right side of inequality (14) is 0.277f₀, and the inequality is invalid.

The boundary between phases III and IV satisfies the relation (14). At high drives, inequality (14) behaves as f_pf_d, in good agreement with the III-IV boundary in the phase diagram in Fig. 8. Region IV flow occurs in the portion of the phase diagram where both inequalities (9) and (14) no longer hold but f_d < f_p.

Finally, the phase boundary between regions IV and V follows the linear relation of f_pf_d. When f_d > f_p, the flow is 2D and not uniform because rows slide past each other.

We point out that in Eq. (1) we have assumed f_d to be uniform on all the vortices, which is the case for thin-film superconductors where can be comparable or larger than the system size. Vortices in thin film superconductors will interact via a Pearl potential f_vv~1/r rather than the modified Bessel function which we use. The modified Bessel function is appropriate for bulk superconductors. By using these different vortex interactions, the phase boundaries we observe may be shifted slightly but the qualitative features should be the same. For bulk superconductors an applied current will form a gradient in the vortex density. If the pinning is weak, the gradient should be small and the dynamic phases should be the same throughout the sample. For very large flux gradients, different dynamic phases may occur in different regions of the sample. In Sec. VII we show how the dynamic phases are modified when considering samples with different vortex densities.

V. HYSTERETIC VOLTAGE-CURRENT RESPONSE

The fact that phase III appears at all for such a low value of f_d may appear to contradict the predictions of inequality (9), which indicates that for f_p=0.625f₀, region III can only arise when f_d > 0.41f₀. Inequality (9) is, however, only valid when there are interstitial vortices moving between the rows of pinning. This occurs for the ramp-up part of the hysteresis curve, when there is a transition from phase II to III. In contrast, when a transition from phase IV to III occurs, the vortices are moving along the pinning rows, so there are no interstitial vortices between the rows, and thus inequality (9) does not apply. It is possible for phase III to appear on the reverse path because the extra incommensurate vortices approach pinned vortices closely since they are moving along the pinning rows rather than between the rows.

Region IV persists for so long on the reverse leg because once the 1D incommensurate channels are formed, there is a barrier to perturbations in the transverse-direction, as noted in Sec. III and by inequality (12). This barrier is examined further in the next section. After region III appears on the reverse path, the vortex motion returns to the interstitial flow type seen in phase II.

The hysteresis and the sharp discontinuities in V_x suggest that the transitions from II-III and III-IV are first order. The lack of hysteresis and the absence of sharp discontinuous jumps in V_x suggest that the transitions I-II and III-IV are of second order.

VI. EFFECTS OF APPLYING A TRANSVERSE FORCE

Recently, studies^31,32 with random pinning arrays have shown that for large enough driving the moving vortex lattice is anisotropic with long-range order in the transverse direction, but not in the longitudinal direction, and have also provided evidence for a small transverse barrier. In our case, the strongly driven vortices in phase V travel in 1D channels that do not change with time in the transverse direction, as can be seen in Fig. 4(d). Because commensurate and incommensurate rows move at different speeds, the vortex lattice is not moving elastically but plastically. Furthermore, it can be seen that the vortex lattice in region V (strongly driven homogeneous flow) is anisotropic. Vortices are evenly spaced along the transverse direction a distance a, the pinning lattice constant, apart; but due to the presence of incommensurate vortices, the vortex lattice is not evenly spaced along the longitudinal direction.

The vortex motion of phase V is somewhat analogous to a series of weakly coupled 1D Frenkel-Kontorova chains. Note that this kind of transport with fast (incommensurate) and slow (commensurate) moving vortex lanes is significantly different from the elastic transport of the "moving crystal" predicted for systems with random arrays of pins.⁶

To test for the presence of a finite transverse barrier, a force f_y in the transverse direction was applied to a moving lattice for samples with the same parameters as in Fig. 2(a). The lattice was driven at f_d=0.8f₀, which puts the sample in phase V. In Fig. 10(a) we show the V(I) curve in which V_y,
,
the sum of the net velocities in the y direction at one instant, is plotted versus f_y. As f_y is increased, the lattice remains locked in the 1D paths and moves only along the longitudinal x direction until f_y0.075f₀, at which point the vortex lattice starts to move in the y-direction as well. This result clearly shows the existence of a finite transverse critical force for the strongly driven region V.

To further characterize the onset of motion in the y direction, in Fig. 10(b) we plot the fraction of sixfold coordinated vortices, P₆, as f_y is increased. For values of f_y that are too small to cause the vortices to move in the y direction, we find P₆0.82, with the defects occurring due to the incommensurate vortices. At the onset of motion in the transverse direction, the vortex lattice shows a sharp decrease in P₆ to a value of P₆0.7, which indicates a change in the vortex lattice structure to a more disordered state. This implies that, even though the vortex lattice is moving, the depinning in the transverse direction is similar to the depinning of a pinned static vortex lattice where the vortex lattice becomes more disordered at the initial depinning transition.^{6,13,17,26,32} Once the vortex lattice starts moving in the transverse direction it is no longer being driven along a symmetry axis of the pinning lattice, so the pinning lattice appears more disordered. As f_y is increased, the vortex lattice regains its order up to P₆0.96. This higher degree of order indicates that the defects caused by the incommensurate vortices that were locked into the vortex lattice in region V heal out once the symmetry of the pinning lattice is broken. Simulations^5,6,26 where the vortex lattice is driven over a random distribution of pinning sites find that at high drives P₆0.95 or higher. These results suggest that one way to experimentally observe the effects of a finite transverse barrier on a moving vortex lattice is by looking for a change in the vortex lattice structure at the onset of motion in the transverse direction.

We examine the dependence of the transverse force on the driving force in order to determine how the transverse force varies in each phase and to test the predictions of inequality (12). In Fig. 11, we plot the critical transverse force f_y^c versus driving force f_d for a sample with the same parameters as in Fig. 2(a). For driving forces that satisfy 0.2f₀ < f_d < 0.4f₀ the system is in phase II. Here the transverse critical force f_y^c linearly decreases as f_d approaches the II-III transition at f_d0.4f₀. This agrees well with inequality (9) that indicates that as f_d is increased in phase II, the force needed for an interstitial vortex to depin a commensurate vortex decreases. The transverse critical force reaches its lowest value of f_y^c0.01f₀ in the disordered-flow phase III, 0.406f₀ < f_d < 0.46f₀, and then very sharply increases as the system enters phase IV. The transverse critical force reaches a maximum of f_y^c0.09f₀ just before the onset of region V at f_d=0.61f₀. The response falls to f_y^c0.075f₀ for phase V. A more detailed study of how the barrier depends on the microscopic pinning parameters as well as how the vortex motion and structure changes with an applied transverse force will be presented elsewhere.¹⁵

VII. DYNAMIC PHASE DIAGRAM AS A FUNCTION OF COMMENSURABILITY

In Fig. 12 we present the phase diagram in which f_p=0.625f₀, r_p=0.3, and =0.25/ are kept fixed, while B/ is varied from 0.75 to 2.296, so that we can observe the vortex dynamical phases for B both above and below and 2. The dynamical phases labeled I through V correspond to the same phases discussed in section III. Figure 12 was obtained from a systematic study involving more than 25 separate simulations, each one spanning a wide range of driving forces.

A. Above the first matching field: < B < 2

First, let us consider the cases for B/ just above one. Here, Fig. 12 shows the five phases present from section III [see, e.g., Fig. 2(a)]. Phase III, the random 2D motion regime, starts to grow for increasing B/ while the more ordered-motion regions II and IV shrink. This is expected since an increase in B/ effectively introduces disorder via the addition of more interstitial vortices. As the density of interstitial vortices increases, second or higher-neighbor interstitial vortex interactions become important and cause the vortices at the pinning sites to depin at lower driving rates, leading to the shrinking of phase II. Similarly, the width of region IV will be reduced since for larger B/ there will be a larger V_y produced by the additional interstitial vortices. This inhibits confinement of the moving vortices along the pinning rows until higher drives are applied. For B/ > 1.3, phase V disappears and the vortices no longer flow strictly along the rows of pinning pinning sites. Instead, a considerable number of vortices flow in 1D channels through the interstitial areas, as shown in the snapshot in Fig. 13. We label this type of flow phase VI. The vortex flow in region VI is generally plastic and exhibits some properties similar to those of phase III; however, unlike region III, vortices are never pinned and always have a non-zero velocity. The transition from phase III to phase VI is not abrupt but occurs continuously as indicated by the dashed line in Fig. 12. No hysteresis in V(I) is observed at the III-VI phase boundary, further indicating the continuous nature of the transition. We also observe that in phase VI for drives f_d > f_p, the vortex lattice usually displays a higher degree of ordering. This ordering is highly dependent on B/; for example, when B/ = 1.7 the vortex lattice is completely ordered at high drives and flows elastically. The higher amount of ordering in region VI does not noticeably affect the V(I) curves.

Near B/1.7 and f_d0.25f₀ we find evidence for a new phase that we label VII. The onset of this phase appears in Fig. 14(a) as a clear jump in the V(I) curve near f_d0.21f₀. This jump is followed by a linear or Ohmic region which ends when another jump into phase III occurs at f_d0.3f₀. We verify the existence of a new dynamic phase by observing the vortex flow patterns plotted in Fig. 15. In phase VII whole rows of vortices that were commensurate with the pinning sites are depinned and begin to flow in the interstitial regions. These vortices follow stationary well-defined winding paths unlike the transient time dependent paths seen in phase III. Since the number of moving vortices remains constant as the driving force is increased throughout phase VII, the velocity of these vortices increases linearly and an Ohmic response is observed. The initial jump in the V(I) curve at f_d0.21f₀ occurs when a portion of the commensurate vortices are suddenly depinned. Further evidence that region VII is a distinct dynamical phase is presented in Fig. 14(b). Here, some hysteresis occurs in the phase transition from VII to II with region VII persisting to lower forces on the ramp down.

B. Second matching field: B=2

We observe only three dynamic phases at the second matching field B/=2, where the vortex lattice is commensurate with the pinning lattice and forms a square lattice at 45^o with respect to the pinning lattice. As seen in Fig. 12, the phase transition from I to II occurs at a driving force of f_d0.15f₀, approximately the value needed to depin a single interstitial vortex. For fields 1 < B/ < 2 the transition occurs at lower drives due to the presence of defects in the interstitial lattice. The interstitial flow for B/ = 2 in phase II differs slightly from that found for B/ > 2 and B/ < 2 since at B/ = 2 the interstitial vortices form a defect free lattice that flows elastically with respect to other interstitial vortices. At higher or lower fields, defects cause some rows of the interstitial lattice to flow at different speeds from other rows in a manner similar to phase V. Phases VII and III, which are observed for B/ < 2, are absent when B/2 and the transition to phase VI, which is continuous for fields B/ < 2, is marked by a sharp jump for B/ 2.

C. Above the second matching field: B > 2

Above the second matching field, the pinned phase I shrinks considerably as seen in the V(I) curves of Fig. 16(a). The shrinking occurs when extra interstitial vortices appear in the commensurate interstitial lattice present at B/=2 and begin to flow before the commensurate interstitial vortex lattice flows. This shrinking of phase I for B/ > 2 resembles the shrinking of phase I just above B/ = 1, when interstitial vortices appear in the commensurate pinned vortex lattice. An extra interstitial vortex in the B/ = 2 interstitial lattice exerts an additional force on the vortex just ahead of it, reducing the required depinning force. The flow of the interstitial vortices just above the depinning current is labeled phase VIII. Snapshots of region VIII in Fig. 17(a) show that only rows containing extra interstitial vortices move. The vortices do not move continuously as in phase II; instead, the vortices move in small localized pulses similar to those observed in phase IV.

The phase diagram in Fig. 12 shows that as B is increased further above 2 the transition from region I to region VIII occurs for lower driving forces f_d0.02f₀. At f_d0.146f₀ all the interstitial vortices start to flow and the system enters phase II, indicated by a jump in the V(I) curve in Fig. 16(a). At the onset of phase II (f_d0.146f₀), the interstitial vortices first start to flow in a disorganized way with rows moving at different speeds. Near f_d0.23f₀ the interstitial vortices begin flowing in a coherent manner, with all rows that contain an equal number of vortices flowing at the same speed. Rows containing extra interstitial vortices move at different speeds. In Fig. 17(b) the flow pattern for the coherently flowing vortices in phase II is illustrated. The onset of the more coherent flow appears in the V(I) curve as an increase in the amplitude of V_x.

Regions VII and III do not appear for B/ > 2. Instead the transition from phase II to phase VI is marked by a very pronounced jump in V(I) seen in Fig. 16(a). The strong hysteresis associated with the VI to II transition appears in Fig. 16(b). On the ramp down of f_d, region VI persists down to f_d0.25f₀, a driving force slightly greater than the force at which the coherent phase II motion appears on the ramp up. As the driving force is further decreased a sharp drop in the voltage occurs and the system enters a flow phase that resembles region VII, where whole rows of pinning sites are unoccupied as interstitial vortices flow continuously around a smaller portion of pinned vortices. Phase VII does not appear when increasing the driving force. As f_d is reduced further, another drop in V(I) occurs as vortices become pinned in the empty rows of pins. The vortex flow then returns to phase VIII.

D. At commensurability: B=

At B/ = 1, the commensurate case, we find only two phases: a pinned regime for f_d/f₀0.6, and a flowing regime for f_d/f₀0.6, with the depinning transition occurring approximately at f_df_p=0.625f₀. This is a direct consequence of the existence of only one species of mobile vortices rather than the two species required to generate the other phases. All critical driving force values reach their highest levels for this case. The enhancement of the critical force and reduction of the vortex mobility for this commensurate case is consistent with magnetization experiments,^{51,52,54,59,60} direct imaging experiments⁵⁶ and simulations¹⁰ as well as with resistance measurements in Josephson junction arrays.⁴⁴. This enhancement of the critical current can also be considered as a realization of the Mott-insulator phase.⁶²

The vortex motion in phase V is different at commensurability and away from commensurability. For B > and B< , the presence of incommensurate vortices causes certain rows of moving vortices to slide past each other. For B=, however, the vortex lattice flows elastically since all the rows move at the same speed. Thus for this commensurate case a "moving crystal" can be realized.

E. Below commensurability: B<

For B/< 1, the critical force for vortex motion drops well below the value at B=, and three dynamical phases appear plus the pinned phase. The pinned phase I extends to higher driving forces than for the B/ > 1 case since there are no weakly-pinned interstitial vortices present. The moving phases consist of an initial randomlike flow regime that resembles phase III and a vacancy-flow regime, which we label phase IV_vac. In the latter regime, the vortex motion consists of jumps from pinning site to pinning site along the pinning rows, which also corresponds to a vacancy or a hole moving in the opposite direction.

Figure 18 explicitly shows the vacancy motion of region IV_vac. This 1D motion is very similar to that seen in phase IV, where motion occurs for incommensurate rows and commensurate rows remain pinned. In this case the discommensurations are due to vacancies rather than interstitials. A vortex i to the west of a vacancy feels an extra force of f_ij(a), where f_ij is the vortex-vortex interaction, in the direction of the eastbound driving force, so these vortices become mobile at a lower f_d than vortices not located near vacancies. The vacancy flow regime should occur when the following inequalities hold:
f_d < f_p < f_d + f_ij(a), (17)
thus, when
f_p-f_d < f_ij(a).
For our parameters [with a=2, f_p=0.624f₀, and f_ij(a)=0.15f₀], we find from inequality (17) that the vacancy motion should first occur at a driving force of f_d0.47f₀, which is in good agreement with the value obtained from the simulation. The force needed to move a vacancy is much larger then the force needed to move an interstitial vortex since in order for a vacancy to move, a vortex must be depinned from a pinning site. This result has also been observed in recent imaging experiments.⁵⁶

At f_d=0.6f₀f_p, the entire vortex lattice becomes mobile. This phase, which we label V_vac, is very similar to the flow found in region V, where incommensurate rows move faster than commensurate rows. In this case, the incommensurate rows have fewer vortices than pinning sites. No hysteresis is found in the V(I) curves for B < .

VIII. DYNAMIC PHASE DIAGRAM AS A FUNCTION OF PINNING RADIUS

As r_p is increased, the ordered flow phases II and IV grow in size while the disordered flow region III shrinks. The increase in phase II is explained by the fact that in larger pinning sites the pinned vortices can move further away from the interstitial vortices, increasing the quantity f_min(a/2 + r_p/2) in inequality (8) that marks the II-III phase boundary. Thus, a higher driving force is now needed for the inequality to remain valid and for phase III to appear.

The growth of phase IV results from the fact that larger pinning sites better stabilize vortices moving along the pinning row since the vortices move a smaller distance a-2r_p between pinning sites. Region IV should also be more stable for larger pins since a higher transverse force is predicted by inequality (12).

A. Dynamic "peak" effect

In order to emphasize the very different behaviors that occur when decreasing r_p, in Fig. 20 we present the voltage-current curves for the same parameters used in Fig. 2(a) but with a smaller and larger pinning radius: r_p=0.2 (curve with larger arch around f_d/f₀0.3 - 0.5) and r_p=0.35 (curve with the much smaller phase III). It is interesting to note that the same number of vortices are mobile in both systems for region IV around f_d/f₀0.58, since the net velocities V_x are approximately equal. In Fig. 20 we can see the striking discontinuous large drop in the number of mobile vortices at the phase transition from phase III to IV around f_d/f₀0.56 for the r_p=0.2 system, indicated by the sudden decrease in V_x. For the r_p=0.35 case, this jump occurs much earlier around f_d/f₀0.44 and is significantly smaller.

The decrease in the number of mobile vortices at the transition from phase III (disordered flow) to phase IV (1D incommensurate channels) is reminiscent of the peak effect where an increase in J_c occurs as the temperature or field is raised. The dynamic "peak" effect we observe here occurs due to an increase in the driving force rather than an increase in the field or temperature. The usual peak effect is believed to be associated with a dynamical transition from elastic flow to plastic motion of vortices.^18-20 In Fig. 20 the phase transition III-IV is between two different plastic flow regimes: from random flow (III) to 1D incommensurate channels (IV). It is also possible to observe the transition from region III to IV in our system by maintaining a constant driving force and decreasing the field, as can be seen from the phase diagram in Fig. 12; by varying the pin radius, as indicated by the phase diagram in Fig. 19; or by changing the pinning force, as shown in Fig. 8.

B. Dynamic "winding interstitial" phase

At a pinning radius r_p=0.175, a new phase emerges that was not seen in our other phase diagrams. Figure 21 shows the voltage-current curve for this case. We increase the driving force and observe the dynamic phases which appear beyond phase II ( 1D interstitial flow). In Fig. 21 we can first see region I, the pinned phase, and phase II, the 1D interstitial flow phase. These are then followed by the onset of a disordered flow region III, which we also confirm by directly viewing the moving lattice. Above phase III, the fluctuations V_x in the voltage-current plot are clearly reduced and the V(I) curve becomes linear, suggesting the presence of different dynamics from the random flow regime. We label this new region IX. Beyond phase IX the motion once again becomes disordered and region III reappears. This is followed by a narrow phase IV, where the flow is 1D incommensurate along the horizontal pin rows, and then the whole lattice starts to flow in phase V.

To further characterize the motion in phase IX, we compute the evolution of the Voronoi construction and show in Fig. 21(b) the fraction of six-sided polygons P₆ as a function of the driving force f_d. Here we can see that the initial vortex lattice configuration has a very low fraction P₆0.38 of six-sided polygons, which is due to the fact that the underlying pinning lattice is square. Those six-sided polygons which appear in the pinned region are a result of both the small number of incommensurate vortices and the Voronoi algorithm used, as square polygons are very sensitive to small distortions or displacements and are likely to appear as polygons with more than four sides. As f_d is increased, a slight increase in P₆ occurs at the onset of phase II when the interstitial vortices depin. At the beginning of region III another increase in P₆ appears since the square symmetry is lost in the random-flow motion. At the onset of region IX there is a sharp increase in the number of six-sided polygons corresponding to the ordering of the lattice at this stage. In this region P₆ reaches a local maximum value of P₆0.8. At the transition to this reentrant disordered motion phase III, there is another drop in P₆ to an average value of P₆0.58. As the lattice finally moves out of region IV and into region V, the number of six-sided polygons once again rises to P₆0.9. It does not reach P₆=1.0 because of the 1D incommensurate structure of phase V.

To show conclusively that region IX in Fig. 21 represents a new dynamical phase, in Fig. 22 we plot a snapshot of the vortex lattice (a) and the vortex trajectories (b) in phase IX. Here we can see a remarkably different behavior from that observed earlier. The vortices travel in well defined flow channels with most of the vortices moving between the horizontal pinning rows and a small number of vortices pinned at the pinning sites. The motion is distinct from the interstitial flow of phase II where only a small number of interstitial vortices were mobile and all the pinning sites were occupied. The motion is not in straight 1D paths along the longitudinal-direction, as in phases I, IV and V, but wanders in the transverse direction. As can be seen in Fig. 22(b), the wandering is caused when the flowing interstitial vortices are deflected by a small number of pinned vortices. The trajectory lines show that the vortices travel in paths which are stationary in time. There is also no depinning of vortices from the pinning sites. This behavior is very different from that seen in phase III, where the channels are changing rapidly as a function of time, and vortices continuously depin and become pinned. Interestingly, the number of vortices pinned at the pinning sites is exactly the number of extra vortices above . After region IX, as the driving force is increased, the flow reenters phase III when the combination of the driving force and perturbations from interstitially moving vortices begin to depin the vortices at the pinning sites.

C. Hysteresis for winding interstitial phase IX

The phase IX displays some remarkable hysteresis properties. In Fig. 23 we examine the hysteresis of a system with the same pinning parameters as in Fig. 19 with r_p=0.125. In Fig. 23(a) the driving force is brought up to f_d=0.75f₀, driving the system into phase V, and then f_d is brought back to zero. The reverse curve follows the forward curve down to f_d0.625f₀ or f_df_p, and then drops down as the vortex flow enters phase IV. Note that region IV is not seen on the initial ramp up, as expected from Fig. 19. The appearance of phase IV on the ramp down occurs since the flow in region V is very similar to the flow in phase IV: the vortices move along the pinning rows and an effective transverse barrier forms (see Sec. IV) that keeps the vortices in the channels. It is interesting to note that the transition from V to IV is quite broad and some small steps appear in this transition region. We believe that the steps occur as individual commensurate rows become pinned. As each row is pinned V_x drops suddenly and then levels off until another row is pinned. The 1D incommensurate-flow phase IV persists down to about f_d0.22f_d and then jumps to phase III. The flow remains in region III almost down to the depinning force observed in the initial ramp up curve. At this point, it jumps briefly to phase II and finally reaches the region I pinned phase. The width of the hysteresis curve is quite large, e.g., at f_d=0.5f₀ the fraction of mobile vortices is =0.27 on the ramp down and =0.94 on the ramp up. Notice that region IX does not occur on the ramp down portion. If we ramp up the force again the system will follow the same curve as the initial ramp up curve.

In Fig. 23(b) we show a hysteresis plot of the same system shown in Fig. 23(a) except that the maximum driving force f_d=0.59f₀ places the system in region III during the initial part of the ramp down. The reverse curve follows the forward curve down to about f_d=0.4f₀ where it enters phase IX at the same point that region IX ends on the ramp up. The ramp-down curve in phase IX has a slightly higher number of mobile vortices, causing this curve to lie above the ramp-up curve. The reverse curve is smoother and lacks the small jumps seen during the ramp up phase. During the ramp down, region IX persists below the driving force at which phases III and II appear on the ramp up, and then abruptly ends when the vortices repin at approximately the driving force needed for initial motion on the ramp up (i.e; the I-II ramp-up phase boundary phase). This strong hysteresis suggests that phase IX is a first order transition.

To understand the appearance of a distinct phase IX for small r_p we must consider that, prior to region IV, the sample is in the random flow phase III. In this state, the vortices are being pinned and depinned at random. A large portion of configuration phase space can be explored and many different configurations are possible. One possible configuration has most of the vortices flowing in between the vortex rows while a small number of vortices remain pinned at the pinning sites. The vortices at the pinning sites cause a deflection of the vortices that are moving in between the pinning rows. It can be seen that if the pinning radii are too large, the deflected vortices will be trapped at the pinning sites. If the pinning sites are small, vortices can pass the pinned vortices without being trapped by a pin, and continue to flow between the pinning rows. We write a minimum criteria for this continuous flow by assuming that a pinned vortex deflects a flowing vortex by a distance of 0.5=a/4 [which is close to the deflection distance observed in Fig. 22(b)]. Since the center of each pinning site is located a distance a/2 from the interstitially moving vortices at their point of close proximity, it is expected that continuous winding interstitial motion (phase IX) will occur for pinning sites of radius r_p < a/4. This can be seen in Fig. 22(b) where some winding paths almost collide with the pins. If the latter were slightly larger, the winding path would be blocked. Contrary to this, the onset of continuous flow did not occur until the pinning radius was reduced to r_p0.175. To understand this, we note that in Fig. 22 there are several places (third and eighth rows) where two vortices are pinned in a row, causing additional deflection in the y-direction and preventing continuous flow until smaller pins are used.

IX. DYNAMIC PHASE DIAGRAM FOR GRADUALLY VARYING THE DEGREE OF DISORDER

In Fig. 24 we present the phase diagram for varying disorder, where maximum displacements up to r=a/2= are used, and in Fig. 25 we present the V(I) curves for the case with B/=1.062, f_p=0.625f₀, =0.25/, a=2, and r_p=0.3. As the amount of disorder is gradually increased from zero, the disordered flow region III grows while the ordered flow phases II and IV shrink correspondingly. Region II shrinks since the displacement of pinning sites into the interstitial areas allows interstitial vortices to approach pinned vortices more closely, resulting in a lower depinning transition.

The V(I) curves in Fig. 25 also emphasize the change in the flow behavior. For disorder greater than r/2a > 0.05 the phase II to III transition is no longer discontinuous. The size of the jump down in the voltage at the phase boundary from III to IV transition also decreases. For large enough disorder, as in Fig. 25(c), only a small dip can be seen marking the III-V phase transition. For disorder greater than r/2a=0.125, as seen in Fig2. 25(a) and 25(b), phase IV disappears and there is no longer a sharp boundary between regions III and V, as indicated by the dotted line in Fig. 24. For r/2a > 0.125 an increase in P₆ near f_d=f_p indicates that a transition from a less ordered to a more ordered vortex lattice still occurs even in the presence of disordered pinning. This has also been observed in other simulations with random pinning. ^5,6,26,27 We label this flow region V_r to distinguish it from the 1D incommensurate channels seen in phase V.

From Fig. 24 is can be seen that at r0.08a, phase IV disappears. This can be understood when we consider that in region IV, vortices flow in straight 1D paths along the pinning rows. As the pinning sites are disordered, it becomes increasingly difficult to create 1D straight paths that follow the pinning sites. For the system in Fig. 24, when r=0.08a and r_p=0.3, the maximum transverse displacement of two pinning sites in a row is 0.16a if one pin is displaced a distance y=+0.08a and the other a distance y=-0.08a. When this occurs, we have
2yr_p
[because here: 2y=0.16ar_p=0.3=0.3(a/2)=0.15a]. In this case, a vortex cannot travel in a straight path along the longitudinal direction and still intersect two consecutive pinning sites, so the 1D incommensurate motion of phase IV is lost. From Fig. 24 we find that the disappearance of region IV occurs near a maximum transverse displacement of y=0.08a=0.16, giving 2y0.32 > r_p=0.3, which is in agreement with the inequality 2yr_p that signals the disappearance of phase IV.

A. Formation of vortex channels in samples with a random distribution of pins

For r > 0.15a there are only three dynamical phases: regions I, III, and V, with region II only occupying a negligible part of the dynamic phase diagram. These three regions are consistent with results obtained theoretically and experimentally with random pinning arrays. As the disorder increases, the nature of the flow in disordered flow phase III changes due to the presence of randomness in the location of the pins. In particular, the dynamics of the vortices at the onset of region III is distinct from that of the higher drives. We have found that in the regime of higher disorder there are two different plastic flow phases: a single-channel flow phase, and an intermittent channel flow phase where some channels form, freeze, and flow again.^10,14,33

We observe that, for large values of the disorder, the initial depinning of vortices occurs by the formation of specific channels that are stationary in time. In Fig. 26 we show a small section of the voltage-current curve for the initial depinning of vortices for the case of r=1.0 on the dynamic phase diagram. A running average over a period of time was performed to remove small time scale fluctuations from the data. Here it can be seen that in the initial part of the V(I) curve there is a jump in V_x at f_d0.2f₀ and a plateau for 0.19f₀f_d0.206f₀. The value of V_x at f_d0.207f₀ rises again to a second plateau at which point V_x again becomes roughly constant. These two jumps in V_x correspond to the opening of single 1D channels of mobile vortices. In Fig. 27(a) we show the vortex trajectories for the first plateau in the V(I) curve, labeled (a) in Fig. 26. Here we can see that a single channel has appeared. This channel consists of sixteen interstitial vortices flowing around the vortices pinned at the pinning sites. For Fig. 27(a), the driving force in the x direction is f_d=0.206f₀, and thus we estimate that if the vortices were moving freely we would have
V_x/N_v=16f_d/N_v=16(.206)f₀/344=0.0096f₀.
The value from the V(I) curve is V_x=0.0092f₀. This actual value is lower as a result of the slower vortex motion in the channels caused by the existence of points along the channels where vortices are temporarily immobile or are slowed down. The features of the channel motion shown in Fig. 27(a) do not change with time. To test whether this motion is truly c