Nonequilibrium Dynamic Phase Diagram for Vortex Lattices

C. J. Olson, C. Reichhardt, and Franco Nori
Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120
(Received 23 April 1998)

The new dynamic phase diagram for driven vortices with varying lattice softness we present here indicates that, at high driving currents, at least two distinct dynamic phases of flux flow appear depending on the vortex-vortex interaction strength. When the flux lattice is soft, the vortices flow in independently moving channels with smectic structure. For stiff flux lattices, adjacent channels become locked together, producing crystalline-like order in a coupled channel phase. At the crossover lattice softness between these phases, the system produces a maximum amount of voltage noise. Our results relate spatial order with transport and are in agreement with experiments. [S0031-9007(98)07292-5]

PACS numbers: 74.60.Ge

Nonequilibrium problems involving elastic lattices and disordered media, such as the nature of depinning transitions or the behavior of rapidly driven lattices, appear in a wide variety of systems including superconducting vortex lattices, charge-density waves, and solid-on-solid friction. Much recent interest has been devoted to the motion of a vortex lattice (VL) across a disordered substrate under applied currents both at and well beyond depinning. The reordering of a rapidly-driven VL is supported by simulations [1-5] as well as neutron scattering [6] and decoration experiment [7], but the nature of the order that appears remains a subject of debate. In addition, the relationship of the voltage noise observed near depinning to the microscopic vortex motion at higher currents has not been addressed.

Recent work on the behavior of a VL driven by a large current produced conflicting pictures of the VL order, ranging from a crystalline structure [2] to a moving smectic state [8]. In Ref. [9], the VL does not fully recrystallize, but instead enters a moving Bragg glass state with channels. Moreover, Ref. [8] describes the VL in terms of independently moving channels of vortices with overall smectic order. Other theories also focus on channels of vortices [10]. Smectic structure factors [S(q)] of the VL were observed in simulations of vortices moving over strong pinning [3,4], in agreement with Ref. [8].

Very recent decoration experiments [11] produced both crystallinelike and smectic order of the moving VL, depending on the magnitude of the applied magnetic field. Smectic order appears at low fields, when the vortices interact weakly, and crystallinelike peaks in S(q) appear at high fields, when the vortex interactions are stronger. This suggests that the softness of the VL is important in determining the vortex behavior at high driving currents.

In this paper, we propose a new dynamic phase diagram in which both smectic and crystallinelike order appear as the VL softness is varied. Using simulations of current-driven vortices, we clearly define regions of the phase diagram based on S(q), V(I) curves, voltage noise, velocity distributions, direct observation of the lattice, and defect density calculations. We compute experimentally relevant voltage noise spectra [12] at all currents from depinning to high drives, and find evidence for a washboard frequency in the stiff VL as well as broad-band noise in the plastic flow phase. As the VL softens, we observe a novel crossover to a regime in which smectic order is never destroyed even at high drives. At the crossover, the amount of voltage noise generated by the system during depinning is maximum, indicating that experimental voltage measurements can probe the VL order. Our results are in excellent agreement with experiments, and have implications for noise measurements in the peak effect regime [13].

We model a transverse two-dimensional slice (in the x-y plane) of a T=0 superconducting infinite slab containing rigid vortices that are parallel to the sample edge (H=H). The vortex-vortex repulsion is correctly represented by a modified Bessel function, K₁(r/), where is the penetration depth. The vortices are driven through a sample with periodic boundary conditions, filled with randomly placed columnar defects, by a uniform Lorentz force f_d, representing an applied current. The columnar pins are nonoverlapping, short-range parabolic wells of radius =0.30. The maximum pinning force f_p of wells has a Gaussian distribution with a mean value of f_p=1.5f₀ and a standard deviation of 0.1f₀, where f₀=/8. The pin density n_p=1.0/ is higher than the vortex density n_v=0.7/. We concentrate on samples 36 x 36 in size, containing 864 vortices and 1295 pins. We check for finite size effects using samples that range in size from 18 x 18 to 72 x 72 and contain up to 2484 vortices and 3600 pins.

The overdamped equation of vortex motion is f_i=f_i^vv+f_i^vp+f_d=v_i, where the total force f_i on vortex i (due to other vortices f_i^vv, pinning sites f_i^vp, and the driving current f_d) is given by
Here, is the Heaviside step function, r_i (v_i) is the location (velocity) of the ith vortex, r_k^(p) is the location of the kth pinning site, is the pinning site radius, N_p (N_v) is the number of pinning sites (vortices), =(r_i-r_j)/|r_i-r_j|, =(r_i-r_k^(p))/|r_i-r^(p)|.

Using the monotonic relationship between the shear modulus c₆₆ and the dimensionless prefactor A_v of the vortex-vortex interaction term, we model VLs with varying degrees of softness by changing A_v over 3.5 orders of magnitude, from A_v=0.001 to A_v=6.0. This is in contrast to previous simulations [3-5,14,15] that considered only an extremely soft VL. For each value of A_v, we simulate a voltage-current V(I) curve by initially placing the vortices in a perfect lattice, slowly increasing the driving force f_d from zero to 3.0f₀ and measuring the resulting voltage signal .

In order to identify the phase boundaries, each time the driving current f_d increases by 0.08f₀, we halt the increases in f_d, check that the voltage signal V_x is stationary over time, and then collect detailed velocity and position information for a long time interval, 2 x 10⁵ molecular dynamics (MD) steps, at a single current. For a 36 x 36 sample containing 864 vortices, each V(I) curve requires 8 x 10⁶ to 10⁷ MD steps to complete, corresponding to about ten days of CPU time on a SPARC Ultra. At each current, we are able to compute accurate voltage noise spectra S() with a frequency window extending into relatively low frequencies on the order of the vortex time of flight across the sample. We also determine the voltage noise power S₀ in one frequency octave, , where . The units of frequency are inverse MD time steps. Here, =0.027 and =0.054 were chosen to fall in the middle of our frequency window. The results are not affected if nearby values are used. To measure the order in the VL, we use the Delaunay triangulation of the instantaneous vortex positions to determine the fraction of six-fold coordinated vortices, P₆, and we compute the structure factor, , of the VL.

The depinning of the VL appears in our simulated V(I) curves, shown in the inset to Fig. 1(a). Since a soft VL is able to deform and take better advantage of the available pinning sites, as A_v decreases the depinning transition shifts to higher driving forces f_d, from f_d=0.48f₀ at A_v=6.0 to f_d=1.36f₀ at A_v=0.01. The depinning also becomes more abrupt for softer VLs, producing a peak in dV/dI that grows in magnitude with lower A_v, as seen in Fig. 1(a). The fact that a soft VL has a higher depinning current agrees well with experiments [12,13].

The VL also produces the greatest voltage noise at currents just above depinning, when the vortices are in a plastic flow regime. As seen in Fig. 1(b), for each value of A_v the noise power S₀ reaches its peak value S_max at a driving current f_d above the current at which the peak in dV/dI occurs. The current f_d at which the peak occurs is independent of the sample size. The spectra S() near S_max are of the form S()~1/, as illustrated in Fig. 3(A). As A_v decreases and the VL softens, we observe a peak in S_max at A_v~0.75, shown in the inset to Fig. 1(b). It is important to point out that this result agrees well with experiments conducted in the peak effect regime [13], in which an observed peak in voltage noise power with changing magnetic field is interpreted to occur due to the softening of the VL as the externally applied magnetic field increases.

We quantify changes in the VL structure by calculating the fraction of sixfold coordinated vortices P₆, shown in Fig. 1(c), and the structure factor S(q), shown in Fig. 2. The vortices are initially field-cooled, so P₆1 at f_d=0. At driving currents below the depinning transition, the VL relaxes into the pinning sites, causing a gradual decrease in P₆. The relaxation is most pronounced for soft VLs with low values of A_v. A plastic flow state appears just above depinning, producing a liquid-like structure factor S(q), shown in Fig. 2(a). The Delaunay triangulation in Fig. 2(b) reveals the large number of defects in the VL in this regime. As shown in Fig. 1, we find that the peak in dV/dI corresponds exactly to the driving force at which P₆ reaches its lowest value. Thus dV/dI can be used to probe the order in the VL.

As seen by the rise of P₆ in Fig. 1(c) for A_v0.1, the VL regains order as the driving force f_d is increased and the interactions with pinning sites become less important. In stiff VLs with large values of A_v, the defects quickly heal out and P₆ returns to a value near 1 for driving forces f_d not much larger than the pinning force f_p. As A_v is lowered and the VL softens, higher and higher drives f_d must be applied to bring P₆ back to 1. In Fig. 1(c), for A_v0.75, we see that P₆ saturates at a value less than 1, and does not increase further even after applying driving forces f_d considerably larger than those shown in the figure (f_d~6f₀). The permanent presence of a significant number of defects represents an important change caused by the VL softness in the behavior of the system at high driving currents.

Computations of the structure factor S(q), shown in Fig. 2, verify that two different types of VL order appear at high driving currents depending on the lattice softness. When A_v is low and the VL is correspondingly soft, the structure factor S(q) [Fig. 2(c)] has a smectic appearance, with well-defined peaks only along the q_x=0 axis. A Delaunay triangulation of the moving VL, shown in Fig. 2(d), reveals that the vortices are flowing in channels oriented in the x-direction (parallel to f_d) and approximately regularly spaced in the y-direction. (This is in contrast to the plastic flow regime, in which vortices that remain pinned act as barriers and result in vortex motion transverse to the direction of flow, preventing the formation of longitudinal, straight channels.) Vortices in adjacent channels interact so weakly that each channel is able to move independently of nearby channels. All of the defects remaining in the VL have their Burgers vectors aligned transverse to f_d, and each channel passes through at least one defect. Thus the channels can easily slip past each other as the defects glide, resulting in an uncoupled channel phase.

If A_v is large and the VL is correspondingly stiff, S(q) has a crystallinelike structure, shown in Fig. 2(e), at high driving currents. The peaks in S(q) are anisotropic, with slightly higher peaks at q_x=0 than at q_x0. For larger samples, the same behavior is observed in S(q), except the peaks are better resolved. The VL is nearly defect-free, as seen in the Delaunay triangulation of Fig. 2(f). The vortices still move in channels oriented in the direction of drive, but these channels are now locked together by the strong vortex-vortex interactions. Thus, the system is in a coupled channel dynamic state.

We summarize the transitions between different states of the moving VL when A_v and f_d are varied in the phase diagram shown in Fig. 3. The boundary between the pinned and plastic flow phases reflects the increase in depinning current as A_v decreases, noted earlier. The plastic flow regime, identified by its liquid-like S(q) and by its bimodal vortex velocity distribution [inset to Fig. 1(c)], disappears above f_d~f_p. For f_d > f_p, the softness of the VL determines the behavior of the system. When A_v0.75, the structure factor has a smectic appearance for all drives f_df_p, corresponding to an uncoupled channel (UC) flow regime. For 1A_v3, the system enters this same smectic UC state immediately after leaving the plastic flow state. At slightly higher driving currents, however, the defect-filled VL enters a transition regime marked by the appearance of weak peaks in S(q) at q_x0. These peaks are significantly smaller than the smectic peaks at q_x=0, but are at least twice as large as the background observed in the smectic phase. Throughout the UC transition regime, the channels gradually couple as the number of defects in the VL drops. The unimodal vortex velocity distribution becomes less broad, as in the inset to Fig. 1(c). When A_v4, the system goes directly into this transition state without ever passing through a truly smectic state. Eventually, for all VLs with A_v1.0, the system reaches a coupled channel flow phase for large enough values of f_d. The boundary of this phase is identified as the current at which P₆ reaches a value of 1. At the transition A_v~0.75 below which the coupled channel phase never appears, the greatest maximum noise power S_max is observed, indicating that the noise power could be used as an experimental probe of this transition. At A_v~0.75, vortex-vortex and vortex-pin interactions are nearly balanced, allowing the system to sample the largest number of metastable states. The boundaries shown on the phase diagram mark crossover points rather than sharp phase transitions.

In the coupled channel regime, we find a washboard frequency [16] in the voltage noise signal, shown in Fig. 3(B), corresponding to the time required for a vortex to move one lattice constant. The magnitude of this washboard signal decreases when the system size is increased, indicating that the signal appears only when the region of the VL sampled is locked into a single domain. Thus experimentally it would be necessary to probe the voltage over a very small area of a sample to observe a washboard frequency, such as with local Hall probes.

In conclusion, we have obtained a new vortex dynamic phase diagram as a function of lattice softness and driving current. At high driving currents two distinct phases of flux flow appear: in soft lattices, uncoupled channels with a smectic structure, and in stiff lattices, coupled channels with crystallinelike order. A signature of the crossover is observed in the voltage noise, which is largest at the transition between the two phases. In the coupled channel phase, a washboard frequency appears in the voltage noise spectrum for small sample sizes. Our results are in agreement with experiments.

We acknowledge helpful discussions with F. Pardo and S. Bhattacharya, and help from the UM CPC, funded by NSF Grant No. CDA-92-14296. C. O. was supported by NASA.

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