Journal of Physics: Condensed Matter 24, 225702 (2012)

Structural Transitions and Dynamical Regimes for Directional Locking of Vortices and Colloids Driven over Periodic Substrates

C. Reichhardt and C.J. Olson Reichhardt

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 USA

E-mail: reichhardt@lanl.gov
Received 13 October 2011, in final form 27 March 2012
Published 4 May 2012
Online at stacks.iop.org/JPhysCM/24/225702
Abstract
We examine collective dynamical locking effects for superconducting vortices and colloids interacting with square and triangular substrate arrays under a slowly rotated dc drive. A rich variety of lattice configurations associated with a series of steps in the velocity-force curves occur during the locking transitions. These include triangular, square, smectic, and disordered particle arrangements that can be identified using the structure factor. We show that the step widths vary with the ratio of the number of particles to the number of pinning sites. Unlike a static system, where matching effects occur at simple integer commensuration ratios, we find dynamical commensuration effects that arise when an integer number of particle chains flow between rows of pins. We identify two distinct types of locking as a function of substrate strength distinguished by whether the particles flow along or between the pinning rows.
1. Introduction
2. Simulation
3. Structural Transitions and Directional Locking
4. Density Dependence and Dynamic Commensuration Effects
5. Substrate Strength and Different Locking Regimes
6. Colloidal Particles Moving Over Triangular and Square Substrates
7. Discussion
8. Summary
References

1.  Introduction

Directional or kinetic locking of driven particles interacting with periodic substrates has been studied for superconducting vortices as well as colloidal particles [1,2,3,4,5]. Here, some or all of the particles move along a symmetry direction of the substrate instead of in the direction of the applied drive. None of the numerous works on this subject have analyzed in detail the rich variety of structural transitions between different moving lattice symmetries produced by collective effects, the associated transport signature changes, or the effect of substrate strength on collective flow. In [1], vortices were driven over a square pinning array under a fixed dc drive FDx while a perpendicular drive FDy was gradually increased to rotate the driving direction. As the motion locked to different symmetry directions, the y-component of the response formed a devil's staircase with steps at FyD/FxD=n/m with integer n, m. Similar locking was later predicted for classical charged particles moving over a two-dimensional (2D) periodic potential [2], while the locking of colloids driven over an optical substrate was proposed as a novel fractionation method for separating colloidal species in [3,4]. Symmetry locking for particles or nanoparticles moving on periodic substrates has since been observed in numerous experiments and simulations [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], but most of these studies involved noninteracting particles and did not consider collective effects. In very recent work [21], structural transitions of colloids driven over patterned substrates has been reported.
Symmetry locking of vortices has recently been observed both in numerical simulations [22,23] and experiments [22,24,25,26], and occurs even on a random substrate provided that the vortex-vortex interactions are strong enough to overcome the substrate and produce a triangular vortex lattice [27,28]. On the locking steps, the transverse depinning threshold for motion perpendicular to the lattice symmetry directions is enhanced. Vortices and colloids driven over quasicrystalline arrays show directional locking effects even though the substrate has no long-range translational order [29].
In this work we consider collective effects in symmetry locking as the substrate strength and filling fraction is varied. Ref. [1] considered only fillings near B/Bϕ=1.0, where Bϕ is the field at which the number of vortices equals the number of pinning sites. We show that as the filling changes, dynamical commensurability effects produce oscillations in the strength of the locking effects. Ref. [1] also considered only relatively weak pinning, but here we show that there is a distinct strong pinning regime that arises when some vortices become permanently pinned. The remaining interstitially confined vortices depin at a low threshold and can exhibit dynamical locking, but for some driving directions they cause a portion of the vortices at the pinning sites to depin, altering or destroying the locking steps.
An understanding of collective effects is important for applications of directional locking such as sorting of different colloid species. If the collective interactions cause the particles to form a crystalline state, this could destroy the separation effect. Additionally, we show that collective effects produce several regimes of chaotic or disordered flow, where separation would become difficult or impossible. We find that the regions of maximum random flow appear near transitions between two ordered flow regimes. Superconducting vortices fall naturally into the strongly interacting regime so it is important to understand features of collective locking as well as to identify which magnetic fields would maximize the locking effects.

2.  Simulation

We perform molecular dynamics simulations of a 2D system with periodic boundary conditions. We consider a sample of size Lx ×Ly with Lx=24λ, where length is measured in units of the London penetration depth λ. For square pinning arrays, Ly=Lx, and for triangular arrays, Ly=1.097 Lx. The motion of an individual vortex evolves according to the overdamped equation
η dRi
dt
 = Fivv + Fpi + Fexti .
(1)
Here Ri is the location of vortex i, η = ϕ02d/2πξ2ρN, d is the sample thickness, ξ is the superconducting coherence length, ρN is the normal state resistivity, and ϕ0=h/2e is the flux quantum. The vortex-vortex interaction force is given by Fivv = ∑Nvj ≠ if0K1(Rij/λ)∧Rij where K1 is the modified Bessel function, f0=ϕ02/(2πμ0λ3), Rij=|Ri−Rj|, and ∧Rij=(Ri−Rj)/Rij. The periodic substrate is modelled as Np pinning sites of radius Rp and maximum strength Fp with Fpi = ∑k=1Np(FpRik(p)/Rp)Θ((Rp − Rik(p))/λ)∧R(p)ik. Here Θ is the Heaviside step function, Rk(p) is the location of pinning site k, R(p)ik = |Ri −R(p)k|, and ∧R(p)ik = (Ri − R(p)k)/R(p)ik. Unless otherwise noted, Rp = 0.3λ and the pinning density is np=0.279/λ2. As an example of physical units, for the Al films considered in Ref. [44] with Tc=1.313K, ρN=1.7 ×10−8Ωm, d=50 nm, ξ(0)=145 nm, and λ(0)=120 nm, at T=1.278K we have λ = 735 nm, η = 2.54×10−18 Ns/m, and f0=1.37 ×10−6N/m.
The initial vortex configurations are obtained by simulated annealing. We then apply an external force representing the Lorentz force from a current at an angle that changes very slowly, Fext = Asin(θ(t))∧x + Acos(θ(t))∧y . Here A=2.0, θ(t)=δθ⎣t/δt ⎦, δθ = 0.0228°, and δt=10000 simulation time steps. We measure the velocity response 〈Vx〉 = Nv−1∑i=1Nvv·∧x and 〈Vy〉 = Nv−1∑i=1Nvv·∧y, averaged over only the final 4000 time steps for each value of θ in order to avoid any transient effects. We also measure the structure factor for the vortex positions, S(k) = (1/L2)∑i,jexp(ik ·[Ri(t) − Rj(t)]), that could be compared with scattering data.
For the colloidal simulations we use particle-particle interactions of the Yukawa form Fivv=−∑j ≠ iNv∇V(Rij)∧Rij with V(Rij) = q2(E0/Rij)exp(−κRij), where E0 = Z*2/4πϵϵ0, ϵ is the solvent dielectric constant, Z* is the unit effective charge of each colloid, q is the dimensionless colloid charge, and 1/κ = a0/4 is the screening length. The strength of the colloid-colloid interaction force can be altered by varying q2.

3.  Structural Transitions and Directional Locking

Fig1.png
Figure 1: (a) V vs θ for vortices moving on a triangular pinning array with Fp=1.85 at B/Bϕ=1.5. Dark line: Vx; light line: Vy. Locking steps occur at θ = tan−1(√3m/(2n +1)), where m,n are integers. The locking steps (m,n) at (0,0), (1,2), (1,1), (2,1), (1,0), and (3,0) are highlighted. There is also a locking step at θ = 90°. (b) The corresponding fraction of sixfold coordinated particles P6 vs θ. Changes in P6 are correlated with the velocity locking steps in (a). (c,e,g,i,k,m) The vortex positions and (d,f,h,j,l,n) corresponding S(k). (c,d) The smectic structure along the (0,0) step. (e,f) Square lattice formation on the (1,2) step. (g,h) Partial triangular ordering in a non-step region at θ = 36°. (i,j) The smectic structure on the (1,0) step. (k,l) An anisotropic liquid at the transition out of the (1,0) step. (m,n) A distorted square lattice at the locking regime for θ = 90°
On a triangular pinning substrate, locking occurs at driving angles θ = tan−1(√3m/(2n +1)) when the distance a vortex travels between consecutive pinning sites is minimized. The dominant locking angles fall at θ = 0° for m=0, n=0; θ = 60° for m = 1, n = 0; and θ = 30° for m = 1, n = 1. We use the notation (m,n) to denote the locking regions. In figure 1(a) we plot Vy and Vx versus θ for a sample with Fp = 1.85 at B/Bϕ = 1.5. In the absence of pinning, the velocities would follow a smooth sinusoidal curve. Figure 1(a) shows that instead, both Vy and Vx pass through a series of pronounced steps and jumps. For θ < 10°, the motion is locked in the x-direction. The value of the y component of Fext at the end of the (0,0) step is the critical transverse depinning force FTrc. A large (1,1) locking step is centered at θ = 30°, and a second strong step in both Vx and Vy occurs at θ = 60° at the (1,0) locking. In figure 1(b) we plot the fraction of sixfold coordinated particles P6 as a function of θ. Here P6=Nv−1∑i=1Nvδ(zi −6), where the coordination number zi of each particle is obtained from a Voronoi construction. On the (0,0) step, P6 = 0.8, indicating that some dislocations are present. Along the (1,1) step, P6=0.9, but P6 dips at the start and the end of the step. In general, the vortex lattice shows abrupt changes in ordering at either end of a step.
To better characterize the different vortex structures in the locking regimes, in figure 1(c) we plot the vortex positions on the (0,0) step and in figure 1(d) we show the corresponding S(k). Here the particles are moving in one-dimensional (1D) channels along the pinning rows in the x-direction. S(k) has smectic features with well spaced peaks along the ky-axis indicating that periodic spacing of the particles along y is induced by the pinning, while particles in adjacent rows slip past each other. Similar smectic ordering was observed for vortices moving over random pinning arrays [30,31,32,33] or vortices moving at finite temperature over periodic pinning arrays [34,35], and is associated with a transverse depinning barrier [30,31,36]. In figure 1(c), the vortices are confined to move only along the pinning rows. Since B/Bϕ = 1.5, this compresses the vortex lattice spacing along x, producing an anisotropic vortex-vortex interaction. Some moving channels contain more vortices than others, creating dislocations in the vortex lattice. Along the (1,1) step, the system has a smectic ordering similar to that shown in figure 1(c,d) but tilted by 30° with respect to the x axis. On the (1,2) locking step, figure 1(e,f) shows that the vortex lattice has square symmetry with disordered regions that produce some smearing in S(k). The P6 measurement cannot detect square ordering, so in figure 1(b) P6 shows large fluctuations on the (1,2) step even though the S(k) measurement indicates a consistent square ordering throughout the step. In figure 1(g,h) we show the vortex positions and S(k) for θ = 36°, a non-step region with P6 = 0.95 where the vortex channeling effect is lost and where for weak pinning the vortices revert to a mildly disordered triangular lattice. In figure 1(i,j) at θ = 60° on the (1,0) step, the particles form 1D channels and move along the pinning rows in a rotated version of the ordering found on the (0,0) step. Some dislocations in the vortex lattice produce smectic ordering. At the dip in P6 at the end of the (1,0) step, figure 1(k,l) shows that the vortices form an anisotropic liquidlike structure. Another locking region centered at θ = 90° has Vx = 0.0 and a cusp feature in Vy. Along this step P6 = 0.78, and in figure 1(m,n) a distorted square lattice appears. As θ increases above θ = 60°, the system cycles back through the same structural transitions and steps.

4.  Density Dependence and Dynamic Commensuration Effects

Fig2.png
Figure 2: (a) Vortex velocities Vy vs θ for a triangular pinning array with Fp = 1.85 at B/Bϕ = 0.852 (black), 1.26 (red), 1.5 (light green), 2.0 (blue), 2.78 (purple), and 4.07 (dark green), from bottom right to top right. The velocities are normalized by Bϕ. (b) A blowup of panel (a) in the region near the (0,0) step showing that the width of the (0,0) step decreases to zero at B/Bϕ = 2.0 and increases again for B/Bϕ = 4.07. (c) The width FcTr=Acos(θ) of the (0,0) step vs B/Bϕ for the same system. A series of minima and maxima appear that do not coincide with the commensuration effects expected at integer multiples of B/Bϕ. (d,e,f,g) The positions of the pinning sites (large open circles) and vortices (small filled circles) in a portion of the sample along the (0,0) step. (d) B/Bϕ = 1.5. The dashed lines indicate the motion of the vortices in 1D channels aligned with the pinning rows. (e) B/Bϕ = 2.037, a filling at which FcTr=0 in panel (c). Here the 1D channel structure shown in panel (d) is lost. (f) B/Bϕ = 4.07, where there are 1D channels of motion both along and between the pinning sites, as indicated by the dashed lines. (g) B/Bϕ = 5.93, where FcTr is small. The 1D channel structures are lost.
In figure 2(a) we plot Vy for different vortex densities B/Bϕ. The number of resolvable steps is high for low B/Bϕ and low for high B/Bϕ. Different steps respond differently to changes in θ. For example, for B/Bϕ < 2.0 in figure 2(a), the (0,0) and (1,0) steps are present, but at B/Bϕ = 2.0, these steps are lost. In contrast, the step at (1,1) is present at all the values of B/Bϕ. The (0,0) and (1,0) steps reappear for B/Bϕ ≥ 2.78 and grow in width with increasing vortex density up to B/Bϕ = 4.07. Figure 2(b) shows a blowup of the region around the (0,0) step indicating that at B/Bϕ = 2.0 the width of the (0,0) step drops to zero but that at B/Bϕ = 4.07 the step reappears, producing a crossing in the Vy versus θ curves. In figure 2(c) we plot the width FcTr=Acos(θ) of the (0,0) step versus B/Bϕ. For low B/Bϕ, the vortex-vortex interactions are weak so the system is in the single particle limit and FcTr is large. As B/Bϕ increases, FcTr decreases to a local minimum at B/Bϕ=0.75 before increasing and decreasing once again. This nonmonotonic behavior of FcTr contrasts with the critical depinning force observed in a system with random pinning, which monotonically decreases to a saturation level with increasing vortex density. The maxima and minima in FcTr generally do not fall at integer multiples of B/Bϕ, where static matching effects are expected to occur [37,38,39,40]. This is because the dynamical ordering that occurs just below a transverse depinning transition differs from the static structure that forms just below the longitudinal depinning transition and has a distinct field dependence. Smectic ordering appears close to the peak in FcTr at B/Bϕ = 1.5, as shown in figure 1(c,d), as well as within the broad peak in FcTr centered around B/Bϕ=4. In the regions where FcTr drops to zero, near B/Bϕ=2.0 and above B/Bϕ=5.9, the lattice is disordered and S(k) has an anisotropic ringlike structure.
In studies of vortex matter confined to narrow channels, the critical depinning force oscillates as a function of vortex density depending on how many rows of vortices can fit inside the channel [41,42]. More recent studies of 2D periodic pinning arrays show a dynamical commensuration effect that occurs in the limit where the vortices at the pinning sites remain pinned but the number of interstitial vortices increases with B/Bϕ [43]. The transverse depinning force FTrc passes through local maxima as a function of B/Bϕ whenever an integer number n of vortex rows fit between the pinned vortices and the vortex trajectories are highly ordered. Local minima of FTrc appear when a buckling instability of the rows occurs and produces a disordered moving vortex structure. The oscillations in FcTr shown in figure 2(c) are very similar in nature to this effect; however, a key difference is that there are no pinned vortices in the rotating drive system. Along the (0,0) step, some of the vortices slide over rows of pinning sites and the rest slide through the interstitial regions. When an integer number of interstitial sliding rows fits between adjacent rows of pinning sites, the vortex lattice has smectic structure and there is a local maximum in FcTr. At other values of B/Bϕ where an integer number of interstitial sliding rows is unable to form, the vortex lattice is more disordered and FcTr is low or zero.
We illustrate the effect of commensurate or incommensurate numbers of rows of flowing vortices in figure 2 (d-g) at different values of B/Bϕ on the (0,0) step. Near the local maximum in FcTr at B/Bϕ=1.5, figure 2(d) indicates that the vortices flow only in 1D channels along the pinning rows. At B/Bϕ = 2.037 in figure 2(e), the vortices are no longer aligned with the pinning rows and some vortices flow through the interstitial regions. Near another local maximum in FcTr at B/Bϕ = 4.07, figure 2(f) shows that all of the vortices are flowing in 1D rows with half of the rows passing through pinning sites and the other half passing through interstitial regions. At B/Bϕ = 5.93, near a local minimum in FcTr, figure 2(g) illustrates that the 1D channel structure is lost. We expect that for even higher values of B/Bϕ, additional local maxima in FcTr will occur for fields at which two, three, or higher integer numbers of rows of vortices can be accommodated in the interstitial regions.
Fig3.png
Figure 3: Vy vs θ for a square pinning array with Fp = 1.85 at B/Bϕ=1.0. (a) A sample with Rp=0.35λ for pinning lattice constants ap/λ = 1.414, 1.6, 2.0, 2.4, 2.67, 3.0, 3.428, 4.0, and 4.8, from top to bottom. For increasing ap the locking effects are reduced. (b) A sample with ap/λ = 3.428 and varied Rp/λ = 1.9, 1.6, 1.3, 1.0, 0.7, and 0.35, from top to bottom. (c) A sample with ap/λ = 4.8 and varied Rp/λ with the same values as in panel (b). In (b) and (c), for larger ap the locking effects are significantly enhanced at larger Rp.
The same general behavior found for the triangular pinning array also occurs for vortices moving over square pinning arrays. Locking steps appear when θ = tan−1(m/n) with m and n integers. The most prominent steps fall at (0,0) for θ = 0° and (1,1) for θ = 45°. One distinction is that for square pinning, there is a much larger number of locking steps where the vortices form a square moving lattice structure. For superconducting systems, the pinning lattice constant ap and pin radius Rp are important parameters since smaller structures are more difficult to fabricate. To investigate the robustness of the dynamical locking effects, we consider a system with a square pinning array with Fp=1.85 and B/Bϕ=1.0 at fixed Rp=0.35λ and varied ap. In figure 3(a), the directional locking effects in Vy vs θ are much stronger for smaller ap, as indicated by the highlighted (0,1), (1,1), and (1,0) steps. Strong directional locking effects can still be achieved even for larger ap provided that Rp is sufficiently large, as illustrated in figure 3(b) for ap/λ = 3.428 and in figure 3(c) for ap/λ = 4.8 where the locking steps become more pronounced as Rp increases. These results indicate that directional locking is strongest for large, dense pinning sites, but that in arrays with larger lattice spacing, locking effects can still be observed at large pinning radii.

5.  Substrate Strength and Different Locking Regimes

Fig4.png
Figure 4: A sample with a triangular pinning array at B/Bϕ=1.11. (a,b,c,d) Vy vs θ. (a) Fp = 2.35. There are only two locking regions separated by strongly fluctuating regions where some guided motion of the particles occurs. (b) Fp = 2.5. The locking regions are almost completely absent. (c) Fp = 2.75. Several locking regions reappear. (d) Fp = 2.85. There are a larger number of locking steps intermixed with randomly fluctuating regions. (e,f,g) Pinning site (large open circles) and vortex (small filled circles) positions as well as vortex trajectories (lines) over a fixed time interval in a portion of a sample on the (1,1) locking step at θ = 30°. (e) In the weak pinning regime at Fp = 1.75, an ordered flow occurs with all the vortices moving along the θ = 30° direction. (f) In the strong pinning regime at Fp = 2.75, a portion of the vortices are pinned and the flow occurs by a pulse motion or flowing kink. (g) At Fp = 2.75 for θ = 15°, a non-locking fluctuating flow phase occurs. (h) The dynamic phase diagram of θ vs Fp. The regions in which the (0,0), (1,1), (1,0), and 90° locking steps appear are marked. The left side of the diagram at lower Fp is the weak pinning regime where structural transitions of the vortex lattice can occur. Near Fp = 2.5 a large portion of the vortex dynamics falls in the random fluctuating phase rather than on locking steps. In the strong pinning regime the second type of locking steps, associated with solitonlike pulses of moving vortices, appear. For Fp ≥ 3.9 all the vortices are pinned for all θ.
We next consider the effect of substrate strength on the locking regimes for a system with a triangular pinning lattice at fixed B/Bϕ=1.11. We find that the step widths initially grow with increasing Fp, but for large Fp the step widths diminish again and the nature of the ordered flow on the steps changes. In figure 4(a) we plot Vy vs θ for Fp=2.35, where only the (0,0) and (1,0) steps are present and the remaining steps have been replaced by a strongly fluctuating regime that still exhibits directional guidance effects in the disordered vortex flow. The trajectories of the particles do not form channels but mix strongly, and vortices pin and depin at random. On the locking steps motion occurs in the form of a pulse of depinned vortices that passes through a background of pinned vortices. A similar soliton or incommensurate flow of vortices along pinning rows was observed previously for vortices driven along a principle axis of periodic pinning arrays [38]. A transition from a higher velocity random or turbulent vortex flow to a lower velocity flow state with smaller fluctuations was reported in Ref. [38], a feature that resembles the transition in and out of the (1,0) step in figure 4(a). Figure 4(b) shows that for Fp = 2.5, all the steps are replaced by local minima in Vy and the flow is always in the strongly fluctuating regime. At Fp = 2.75 in figure 4(c), the (0,0), (1,1), and θ = 90° locking steps reappear, while for Fp = 2.75 in figure 4(d), the (0,0), (1,2), (1,1), (2,1), and θ = 90° steps are restored and between the steps the flow rapidly fluctuates while Vy is significantly enhanced. For Fp > 2.9, the vortices transition directly from step to step and there are no longer regions of random flow between the steps, while for Fp ≥ 3.9 the vortices become pinned for all θ.
To demonstrate the differences in the flow for samples with strong and weak pinning, in figure 4(e,f) we plot the vortex trajectories on the (1,1) locking step at θ = 30° over a fixed time interval. For Fp = 1.75 in figure 4(e), all the vortices move along 1D paths oriented along the θ = 30° driving direction. For Fp = 2.75 in figure 4(f), only a portion of the vortices move in 1D paths along the pinning rows while the remaining vortices are pinned. In figure 4(g) for the Fp = 2.75 sample at θ = 15°, a fluctuating flow phase occurs and there is no longer any 1D channeling of the vortex motion. In the strong pinning regime, along the locking steps the vortex lattice is largely triangular since most vortices are pinned, and the large scale structural transitions are lost. By conducting a series of simulations for varied Fp we map the transition between the strong and weak pinning regimes, as shown in figure 4(h) where we highlight the widths of the (0,0), (1,1), (1,0), and θ = 90° locking steps. The (0,0) step increases in width with increasing Fp up to Fp = 2.45 and then vanishes at the transition from the weak pinning locking regime to the random fluctuating phase. For Fp > 2.55 the (0,0) step reappears in the strong pinning regime and its width saturates for Fp > 3.0. The higher order steps show similar features.
Fig5.png
Figure 5: (a) Vy vs Fp for a sample with a triangular pinning array at B/Bϕ=1.11 at θ = 90° (upper curve) and θ = 30° (lower curve) showing that in the strong pinning regime the particle velocity saturates. (b) Vy vs Fp for the same system at θ = 60° on the (1,0) step (upper curve) and at θ = 17° (lower curve). A local maximum in Vy occurs in the random fluctuating regime that separates the weak and strong pinning regimes. (c) A scaling collapse of the curves in (a) near the crossover at Fcp from the weak pinning to the strong pinning regime. The line is a fit to Vy − Vcy ∝ (Fp − Fcp)α with α = 1/3, where Vcy is the value of the Vy in the strong pinning regime. (d) Vy vs θ for colloidal particles on a square pinning array with Fp = 1.85 for q2=0.001 and varied Nc/Np=0.432, 0.778, 1.125, 1.82, and 2.85, from bottom to top. (e) FcTr from the square pinning array sample in panel (d) at Nc/Np=0.432 vs q2.
The plots of Vy versus Fp at fixed θ shown in Figure 5(a) for θ = 90° and θ = 30° illustrate the crossover between the two pinning regimes. As Fp increases, Vy monotonically decreases until at Fp=Fcp=2.5 the system enters the strong pinning regime in which Vy remains constant for increasing Fp. In the weak pinning regime Vy can be fit to the functional form Vy − Vcy ∝ (Fp − Fpc)α, where Vcy is the saturation value of the velocity in the strong pinning regime. In figure 5(c) we show this scaling with Vy normalized by sin(θ). Here the θ = 30° and θ = 90° curves collapse on each other and the solid line indicates a fit with α = 1/3. The behavior of the Vy curves varies on different sets of locking steps, so a straightforward scaling such as that shown in figure 5(c) is not always possible. For example, in figure 5(b) we plot Vy versus Fp on the (1,0) step as well as for a driving angle of θ = 17°. In both cases Vy decreases with increasing Fp for Fp < 2.5. Near Fp = 2.5, Vy for both driving angles passes through a local maximum when the system enters the random fluctuating flow regime. We have also studied the behavior of FcTr in the strong pinning regime as a function of B/Bϕ and find features similar to those observed for the weak pinning regime. We find the same transition from a weak to a strong pinning regime for square pinning arrays.
An ideal system for exploring directional locking effects would exhibit a significant flux flow regime and no multi-quanta vortex states, such as the recently studied thin-film Al samples with periodic pinning [44]. We expect that in this system, the weak pinning regime where all the vortices are flowing could be explored at T = 1.278K with a rotating dc drive of amplitude J = 10kA/cm2 just above B/Bϕ, where steps of width 2−3kA/cm2 should occur. The strong pinning regime where only incommensurate flow occurs should appear near J = 5kA/cm2.

6.  Colloidal Particles Moving Over Triangular and Square Substrates

Another question is how general our results are for other types of particle interactions, particularly colloidal particles where there is currently considerable interest in kinetic locking effects. We have tested all of our major predictions for colloidal particles interacting with a screened Yukawa potential. In figure 5(d) we plot Vy vs θ for colloidal particles on a square pinning array of strength Fp=1.85 for varied colloid densities Nc/Np at q2 = 0.001. Here we find the same locking step features observed for the vortex system, with the same oscillations in the width of the first and higher order steps. At Nc/Np = 1.82 FcTr drops to zero, but the first step reappears at Nc/Np = 2.85, similar to what was observed for the vortex system. We also find the same behaviors for colloids on triangular pinning arrays.
The colloid-colloid interaction strength can be altered by changing the effective colloid charge q. Figure 5(e) shows FcTr as a function of q2 for colloids on a square pinning lattice from the system in figure 5(d). As q2 increases, FcTr gradually decreases to zero. There is no clear transition between weak and strong pinning regimes when q2 is varied, unlike the transition found in the vortex system for varied Fp. This is because we are in the regime Fp < A, so all the colloids are moving even when the particle-particle interaction strength is negligible. In order to observe the strong pinning regime discussed earlier, it is necessary for a portion of the particles to be pinned and for the motion to occur in the form of depinned incommensurations.
Recent experiments on optical light arrays trapping charged polystyrene spheres which interact via a Yukawa potential [21] provide an ideal system for testing the locking effects. Near the first commensurate filling, the locking effects could be studied in the weak pinning regime, where all the particles flowed under a drive of 80fN in [21]. At incommensurate fillings the strong pinning regime would correspond to 10−25 fN drives where only incommensurate particles flow and the remaining particles are pinned.
Region Particle Flow Structure Transport
Weak pinning All particles flowing Dynamic lattices Steps
Crossover Fraction of particles flowing Disordered Guidance effects
Strong pinning Only incommensurate particles flowing Lattice with symmetry of substrate Steps, dips
Table 1: Summary of strong and weak pinning locking regimes and the behaviors associated with them.

7.  Discussion

In Table 1 we summarize our results and list the basic features of the weak, strong, and transitional pinning regimes. For vortices, the best candidate system to explore the directional locking effects would have a large flux-flow regime. Recent experiments on Al thin films with periodic pinning arrays showed a large flux flow regime, and the features in the transport curves are in excellent agreement with those predicted in simulations of vortices driven in one direction over periodic pinning arrays [44]. The different pinning regimes could be probed by varying the applied field and applied drive. Direct vortex imaging [37] would provide information on the individual vortex level for driving vortices along different angles with respect to the pinning axis. In the colloidal system, the most straightforward experimental realization would be strongly interacting charged colloids driven over a periodic substrate. Recent experiments at commensurate and incommensurate fillings for collectively interacting colloids obtained velocity-force curves and images of colloid motion for colloids driven in only one direction [21]. It should be straightforward to drive the system at different angles to explore the collective direction locking effects, while the strong and weak pinning regimes can by accessed by varying the laser power to change the substrate strength. An experimental realization of directional locking for colloids driven over quasicrystalline arrays [45] has already shown excellent agreement with predictions from simulations [29].

8.  Summary

In summary, we have shown that a rich variety of dynamical locking effects associated with structural transitions can occur for collectively interacting particles such as vortices and colloids moving over triangular and square substrate arrays. We identify two different locking regimes as a function of the substrate strength and of the ratio of particle density to substrate minima density that have not been previously studied. For weak substrates, at certain driving angles all the particles flow along 1D channels through the pinning sites, generating a series of constant velocity steps on which the motion remains locked to a certain direction over a range of driving angles. As the pinning strength decreases, the width of the locking steps decreases and there are larger nonlocking regions in which the particle-particle interactions dominate and a triangular particle lattice forms.
In the weak substrate regime, the widths of the steps including the initial transverse depinning barrier pass through local minima and maxima as a function of the ratio of particle density to substrate minima density. In contrast to the longitudinal depinning threshold for periodic substrates, which shows peaks at commensurate fields, the local maxima for the width of the first locking step are not correlated with the 2D periodicity of the pinning array but are instead related to a dynamical commensuration effect caused by the formation of 1D channels of moving particles. A local maximum in the width of the first locking step occurs when integer numbers of rows of moving vortices can fit in the interstitial areas between the pinning sites. For fillings at which the moving rows are unable to fit without buckling, the particle structure becomes disordered and the width of the first locking step is small or zero.
As a function of substrate strength for fixed particle density, we identify two distinct locking regimes: a weak pinning regime where all the particles flow along the pinning sites, and a strong pinning regime where the flow occurs by means of an incommensuration or a pulse passing through a background of pinned particles. Between these two regimes the locking steps are lost and are replaced by a strongly fluctuating regime where the particle motion does not lock to a particular direction. For a fixed driving angle, the average particle velocity drops sharply at the crossover between these two regimes, and the velocity saturates to a plateau value in the strong substrate limit. In the strong substrate regime the width of the first step displays commensurate peaks when the number of particles is a fractional matching ratio of the number of substrate minima, in addition to showing peaks at the incommensurate fields as in the weak pinning regime. We expect these effects to be relevant to a wide class of collectively interacting particles moving over periodic substrates.

9.  Acknowledgments

This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396.

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