Europhysics Letters 61, 221 (2003)

Depinning and Dynamics of Systems with Competing Interactions in Quenched Disorder

C. Reichhardt, C.J. Olson, I. Martin, and A.R. Bishop

Center for Nonlinear Studies, Theoretical Division and Applied Physics Division
Los Alamos National Laboratory - Los Alamos, NM 87545, USA

(received 2 July 2002; accepted in final form 11 November 2002)

PACS. 64.60.Cn - Order-disorder transformations; statistical mechanics of model systems.
PACS 73.20.Qt - Electron solids.
PACS 71.45.Lr - Charge-density-wave systems.

Abstract. - We examine the depinning and driven dynamics of a system in which there is a competition between long range Coulomb repulsive and short range attractive interactions. In the absence of disorder the system forms Wigner crystal, stripe and clump phases as the attractive interaction is increased. With quenched disorder, these phases are fragmented and there is a finite depinning threshold. The stripe phase is the most strongly pinned and shows hysteretic transport properties. At higher drives beyond depinning, a dynamical reordering transition occurs in all the phases, which is associated with a characteristic transport signature.

Two-dimensional (2D) systems with a competition between long-range repulsion and short range attraction exhibit a remarkable variety of patterns such as stripes, bubbles, and labyrinths [1]. Such systems include magnetic films [2], Langmuir monolayers, polymers, gels, and water-oil mixtures [3]. It has been proposed that similar competing interactions can arise in 2D electron systems leading to stripes, clumps [5,6] and liquid crystalline electron states [7]. Stripe and other charge-ordered phases in metal oxides are sometimes modeled as systems with competing long range repulsion and short range attraction [8,9]. In a doped itinerant antiferromagnet (AF), there is a short-range attraction between mobile carriers since fewer AF bonds are broken when the carriers spatially separate from the insulator, while the quantum kinetic energy creates a non-local repulsion. Many of these systems may contain quenched disorder from the underlying substrate; however, it is not known how this disorder would affect the structure and dynamics of these systems. Quenched disorder can strongly alter the transport properties, producing a pinning effect in which a finite driving force must be applied before net motion occurs.

The effects of quenched disorder on driven elastic media have been studied extensively in the context of vortices in superconductors [10,11,12,13,14,15,16,17,18,19,20], Wigner crystals [21] and charge density waves (CDW's) [22]. A remarkable wealth of nonequilibrium behaviors arise, one of the most intriguing of which is dynamic reordering. At low drives above depinning, the elastic system can break up and flow plastically. For increasing drive, the effects of the quenched disorder are partially reduced, allowing the elastic interactions to dominate. The system then orders or partially orders to a moving crystal or smectic. Dynamical reordering has been studied theoretically [10,11,12,13], experimentally [19,20] and in simulations [14,15,16] in vortex matter, CDW systems [22] and driven Wigner crystals [21]. An open question is whether dynamic reordering is universal to other types of systems with quenched disorder, particularly those with competing interactions.

The study of transport in systems with competing interactions and quenched disorder is of considerable value since in many physical systems it is not possible to probe the microscopic behavior directly. Instead transport signatures such as I−V characteristics and conduction noise are measured. Recently, highly nonlinear and hysteretic transport was observed for 2D electrons in the reentrant integer quantum Hall state, and it was suggested that this is a signature of the depinning of some form of charge ordered state [23]. We note that simulations for driven vortex matter [14,15,16] and Wigner crystal states [21] in 2D have produced non hysteretic transport curves; however, it is not known if the depinning of other ordered states such as stripes or clumps is hysteretic.

In order to examine the transport characteristics as well as to determine whether dynamic reordering occurs in this class of system, we have conducted numerical simulations of systems of particles that have a competing long-range Coulomb repulsion and a short range attraction. Our model is similar to one introduced by Stojkovic et al. [8,9] to examine charge ordering in metal oxides. In our model, as a function of increasing attractive interaction, we find three generic phases in the absence of quenched disorder similar to those observed in [8,9]: Wigner crystal, stripe, and clump. The addition of disorder affects the stripe phase most strongly, fragmenting the stripes and producing a large depinning threshold and nonlinear I−V characteristics. The I−V curves are hysteretic in the stripe region and in a portion of the clump region where the disorder fragments the clumps. The initial depinning of the fragmented stripe and clump phase is plastic, but at higher drives there is a dynamical reordering, where ordered stripes that are aligned with the drive form and the clumps reform. Characteristic signatures of the reordering appear as features in the I−V curves as well as the conduction noise, which shows a 1/f² characteristic in the moving fragmented stripe and clump phase, and a washboard periodic signal in the moving reordered states.

We model a 2D system of overdamped interacting particles that have a long-range Coulomb repulsion and a short range exponential attraction. The equation of motion for a particle i is f_i = f_ii + f_p + F_d = ηv_i, where the damping term η = 1. The force from the other particles is f_ii = −∇U(r), where

U(r) = 1/r − B exp(−κr).

Here κ = 0.25 is the inverse range, and the parameter B is used to vary the relative strength of the attractive interaction. The repulsive Coulomb term, treated with a summation method [24], dominates at small and large r. The particle density n=0.64κ² and spacing a₀=1.25κ⁻¹, and the system size ranges from 9a₀ to 25a₀ on a side. The quenched disorder is modeled as N_p randomly placed attractive parabolic pins of radius r_p=0.125κ⁻¹, density n_p=1.92κ², and strength f_p=0.2, giving f_p=(f_p/r_p)(|r_i−r_k^(p)|) Θ(r_p−|r_i−r_k^(p)|)∧r_ik^(p) where r_k^(p) is the location of pin k, Θ is the Heaviside step function, and ∧r_ik^(p)=(r_i−r_k^(p))/|r_i−r_k^(p)|. The driving term F_d is increased from 0 to 0.08 in increments of 0.0002, and the sample is held at each drive increment for 8×10⁴ time steps to ensure a steady state. We measure the time-averaged particle velocity < V_x > and its derivative dV/dF, and average over 5 disorder realizations. The system is initially prepared in a high temperature molten state and annealed to T = 0. After annealing the driving force is applied. For the work presented here we fix κ and n, and vary B. Changing n and κ do not affect the qualitative results. Similarly, using a smoother disorder potential, such as that present in a quantum Hall system, does not change the qualitative results but merely shifts the depinning thresholds and phase boundaries.

Figure 1: Individual snapshots of the system without disorder for: (a) Wigner crystal, (b) stripe, (c) clump phases.

Figure 2: (a) Velocity V vs applied drive F_d in a sample with f_p=0.2 for B = 0.29 (solid line) stripe phase, B=0.0 (doted line) Wigner crystal, B=0.35 (dashed) clump phase, and B=0.4 (long-dashed) clump phase. (b) The corresponding dV/dF curves. (c) Depinning force vs B. Labels indicate where the Wigner crystal, stripes, and clumps form in the absence of quenched disorder.

We first investigate the transport properties for different phases as a function of B. In the absence of disorder we find a Wigner crystal for B = 0 [Fig. 1(a)]. As B is increased the lattice becomes increasingly distorted until for B = 0.25 to B < 0.325 stripes form [Fig. 1(b)], and for B ≥ 0.325 clumps form [Fig. 1(c)]. In Fig. 2(a) we show representative velocity vs driving force curves for the system with quenched disorder for the three phases. The curves for the Wigner and stripe phases are highly nonlinear and S-shaped. In addition the pinning is much more effective for the stripe phase than the Wigner crystal as indicated by the larger depinning threshold. In Fig. 2(b) we present the corresponding dV/dF curves. The initial depinning for B < 0.35 is plastic which coincides with the onset of strong peaks in the dV/dF curves. Plastic depinning is distinct from elastic depinning: in elastic depinning, each particle maintains the same nearest neighbors, whereas in plastic depinning, particles may move arbitrarily far away from their initial nearest neighbors and the lattice tears. The stripe phase shows a second small peak in the dV/dF curve, in addition to the large plastic peak.

In Fig. 2(c) we show the depinning threshold vs B, obtained from the velocity curves where we take depinning to occur at a velocity threshold of 0.002. Using a different threshold value does not significantly change the results. The figure also indicates where the Wigner, stripe, and clump phases occur as a function of B without disorder. The susceptibility of the stripe state to the quenched disorder and the corresponding increase in the depinning threshold can be viewed as a consequence of the softening of the system due to self-induced disorder. Recently Schmailian and Wolynes [4] proposed that for systems with competing interactions without quenched disorder, in particular stripe generating systems, self-generated glass or disordering transitions occur due to frustration [4]. A disordered system is soft, allowing individual particles to be displaced without costing elastic energy, so that particles can find optimal pinning sites. Conversely a system with strong elastic interactions such as a crystal is very stiff, preventing particles from finding optimal pinning sites. In our system, for low B the Coulomb interaction dominates and a distorted Wigner crystal state appears. As B increases the repulsive and attractive interactions compete more strongly. In the stripe phase the elasticity is lowest, and the system can most easily adjust to the pinning. This implies that well defined stripe structures will be easily destroyed in systems where quenched disorder is present. For increasing B the short-range attraction becomes dominant and clumps form. The pinning is strongly reduced in the clump region due to the increase in elasticity from three effects: once the particles are in the clump phase, individual particles cannot leave the clump to take advantage of the pinning; within the clumps the particles form crystal structures which create an intraclump elasticity; additionally, since Coulomb interactions dominate at long range, the clumps form a partially ordered Wigner crystal with an additional elasticity. An increase in the pinning due to the softening of an elastic media has also been evoked to explain the peak effect phenomena observed in type II superconductors where an increase of the vortex pinning occurs as a function of field or temperature [19]. Simulations for vortex matter where the lattice is softened have shown that this softening will lead to a smooth increase in the depinning threshold as well as an onset and enhancement of the plastic flow region [17,18].

Figure 3: Individual snapshots for increasing applied drive for B = 0.29. (a) F_d = 0.0, (b) F_d = 0.021, (c) F_d = 0.029, (d) F_d = 0.04, (e) F_d = 0.045 and (f) F_d = 0.06.

In Fig. 3 we show snapshots for different values of F_d for the system at B=0.29 in the stripe phase to indicate the correlation between the velocity force curve features and the microstructure of the moving system. Fig. 3(a) shows the F_d = 0 disordered stripe phase. Here only stripe fragments can be seen. Just above depinning, at F_d = 0.021 in Fig. 3(b), the system is in plastic flow and the stripe structures are almost completely destroyed. For drives up to the first peak in the dV/dF curve in Fig. 2(b), the system is strongly disordered, while above the peak the system begins to reorder, as seen in Fig. 3(c,d) for F_d = 0.029 and 0.04 where stripe segments reform. For F_d = 0.045 [Fig. 3(e)] an aligned stripe phase forms, and becomes very well ordered for F_d=0.06, beyond the second peak in dV/dF, as shown in Fig. 3(f). The second peak in dV/dF appears because the reordering transition changes the effective pinning, from stronger pinning (and slower-moving particles) in the disordered state to weaker pinning (and faster-moving particles at the same drive) in the ordered stripe state.

Figure 4: Individual snapshots for increasing applied drive for B = 0.325 for (a) F_d = 0.0, (b) F_d = 0.02, and (c) F_d = 0.06.

In Fig. 4(a-c) we show the reordering dynamics for B = 0.325, where in the absence of quenched disorder the systems forms the clump state. For this value of disorder the F_d = 0.0 clump state is fragmented as seen in Fig. 4(a). Above depinning, as in Fig. 4(b) (F_d = 0.02), the clumps are further fragmented. For F_d = 0.03 and 0.035 the clumps began to reorder. In Fig. 4(c) for F_d = 0.06 the clumps are completely reordered.

For B > 0.35, the clumps are no longer fragmented by the quenched disorder at F_d=0, and the depinning transition is elastic, with the entire clump structure depinning as a unit, and with greatly diminished peaks in the dV/df curves. For B < 0.25 the system forms a disordered Wigner crystal which depins plastically and forms a moving smectic, as previously studied in Ref [21].

Figure 5: (a) The hysteretic response in velocity force curves for the stripe phase B = 0.29. (b) Noise curves for driven stripe phase B = 0.29 (thick line) in the plastic flow phase F_d = 0.021 showing a broad band signature. The dashed line is the 1/f² curve. Thin line is the noise curve for the reordered state F_d = 0.06.

In Fig. 5(a) we show that only certain phases have hysteretic properties. The velocity-force curve for the stripe phase shows hysteresis which corresponds to the ordered stripe structure, as seen in Fig. 3(e), "supercooling" on the downward drive sweep. We do not observe any hysteresis for the elastically depinning clump phases at B > 0.35, since the moving state and the pinned state are structurally the same. Additionally we find no hysteresis for the Wigner crystal regions B < 0.25, which is in agreement with previous work on driven Wigner crystals as well as previous work for vortex matter in 2D.

We have measured the power spectra S(f) of the velocity fluctuations at fixed drive values, and find two general forms. In the moving disordered or plastic flow phases, we observe a broad band signal with a 1/f^α characteristic. In the reordered moving phases we observe narrow band peaks. In Fig. 5(b) we show S(f) for the stripe phase in the plastic flow region and the reordered region. For the plastic flow regime a 1/f² signal can be seen, while for the reordered phase the noise power is considerably reduced and a series of narrow band peaks are observable. For the moving Wigner crystal we observe only a single narrow band peak, produced by the washboard signal of the moving lattice of frequency ω = v/a, where v is the velocity of the lattice and a is the lattice constant. This has also been seen for vortices moving in an ordered phase [14,15,16]. For the reordered stripe and clump phase, we observe a series of peaks, as shown. The series of peaks rather than one washboard signal arise due to the combination of the macro-structural ordering of the moving stripe and clump phases, as well as the internal ordering of the particles within the stripes and clumps. These additional internal frequencies are at much higher frequencies than the washboard signal since the internal structure lattice constant is much smaller than the macrostructure lattice constant.

We briefly comment that we have also performed simulations for different values of disorder. We find that the peak in the pinning force always corresponds to the stripe phase. For larger disorder the peak broadens gradually. For different values of κ or for different density of particles we observe that the onset of the different phases occurs for different values of B than presented here; however, the qualitative behaviors are unchanged. An example of one possible experimental system in which to observe the dynamical behaviors described here is superconducting materials such as perovskites in the non-superconducting state, where a uniform driving force can be applied to the sample. Similar effects should also be observable in a fractional quantum Hall system.

In conclusion we have used numerical simulations to examine the depinning and dynamics of a system with competing long-range repulsion and short range attraction where in the absence of quenched disorder we find a Wigner crystal, stripe and clump phase. With quenched disorder these phases become partially disordered. The stripe phase is most strongly affected by the quenched disorder and has the highest depinning threshold due to the self-induced frustration. It may be difficult to observe well defined stripe structures in systems with quenched disorder. For increasing drives we find that these phases can exhibit a dynamical reordering transition. For the fragmented stripe phase the transition is hysteretic with the stripes becoming highly ordered and aligned with the drive. The enhanced pinning and hysteresis we observe resemble those found recently in experiments on 2D electron systems [23]. We also demonstrate that the reordering transitions can be inferred from characteristics in the I−V curves and noise spectra. The dynamics and the mesoscopic patterns we observe should be generic to the general class of systems with competing long-range and short-range interactions in quenched disorder.

We thank A. Castro-Neto for useful discussions. This work was supported by the US Department of Energy under Contract No. W-7405-ENG-36.

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