# Non-equilibrium Dynamic Phase Diagram for Vortex Lattices

## C.J. Olson, C. Reichhardt, and Franco Nori, Physical Review Letters81, 3757 (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. Fig. 1: transport [ V(I) and dV/dI ], voltage noise, velocity distributions, and defect density (i.e., topological order) versus driving.

(a): dV/dI curves for vortex-vortex interactions (right to left) A_v = 0.01, 0.1, 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 5, 6. The peak in dV/dI increases in magnitude with decreasing A_v. Inset to (a): Voltage-current V(I) curves for the A_v values listed above. The depinning transition shifts to higher driving currents f_d and becomes more abrupt as A_v decreases. (b): Voltage noise power S_0 versus f_d for the A_v values listed above. In each case the noise power peaks during the plastic flow regime. Inset to (b): Maximum noise power S_{max} as a function of A_v. The peak value of S_{max} corresponds to A_v \approx 0.75, the same A_v at which the system crosses between smectic and coupled channel behavior at high drives. (c): Fraction of six-fold coordinated vortices P_6 as a function of f_d for values of A_v listed above. At zero drive and strong VL coupling, P_6 \sim 1 since the VL is field-cooled. The lowest value of P_6 at each A_v corresponds exactly to the peak in dV/dI. The VL eventually reorders to P_6 \sim 1 only when A_v \geq 1. For A_v \leq 0.75, P_6 saturates at a value below 1. Inset to (c): Velocity distribution functions D(v_x) for (1) plastic, (2) smectic, and (3) coupled channel phases.  Fig. 2: Structure Factor and Snapshots versus Driving Force

Structure factor S(q) (a,c,e) and Delaunay triangulations (b,d,f) of the VL for: (a,b) A_v = 1.5, f_d = 1.12 f_0, in the plastic flow regime. S(q) is liquid-like, and the VL is filled with defects (marked with circles). (c,d) A_v = 0.5, f_d = 3.04 f_0, in the uncoupled channel regime. S(q) has smectic peaks, and the defects in the VL are oriented transverse to the x-direction driving force. (e,f) A_v = 4, f_d = 3.04 f_0, in the coupled channel regime. S(q) has slightly anisotropic crystalline-like peaks, and the VL contains almost no defects. Fig. 3 Dynamic Phase Diagram for Vortex Lattices

Dynamic phase diagram for different driving currents f_d and vortex-vortex interaction strengths A_v. At very low drives, the VL is pinned. In the plastic flow phase, we observe peaks in dV/dI (crosses) as well as peaks in the voltage noise power S_0 (circles). When A_v \leq 0.75, the VL flows in uncoupled channels for all high driving currents. For larger A_v, the VL passes through a transition region (UC) in which the channels progressively couple, until reaching a reordered state at high drives. A: Typical voltage noise spectrum S(\omega) \sim 1 / \omega observed near S_{max} for A_v = 0.1. B: Washboard frequency observed in the coupled channel regime for A_v = 2. The magnitude of the narrow band signal decreases in samples larger than the 36 \lambda \times 36 \lambda sample shown here. The letters A,B refer to inset plots shown here; a,c refer to the plots shown in Fig. 2(a) and 2(b).  Dynamic Phase Diagrams for strong pinning (left) and weak/medium pinning (right). Increasing the temperature would also turn a strong-pinning-landscape into one with medium or weak pinning (right panel).

Postcript Figures are available here (some in color, and some unpublished yet):

Manuscript is available here:  