Discrete Models of Flux-Lines in Type II Superconductors

A general question points at what are the mechanisms that induce such slope dependence in the depinning threshold characteristic to the KPZ universality class with quenched disorder. In this regard, Tang et al. argue that an anisotropic random force, characteristic for an anisotropic disordered medium, is a possible mechanism that generates such slope dependence. They propose the flux line in superconductors as a typical example where the proposed mechanism is present. The motion of the flux line is the result of three different forces: (i) an smoothing line tension term, (ii) a Lorentz force perpendicular to the line, and (iii) the anisotropic random force due to the disorder in the medium.

To study the effects of the anisotropy of the medium, we introduce a model for the anisotropic flux line that incorporates the above considerations. [H. A. Makse, Singularities and Avalanches in Interface Growth with Quenched Disorder, Phys. Rev. E 52, 4080-4086 (1995); H. A. Makse, A.-L. Barabasi, and H. E. Stanley, Elastic String in a Random Medium, Phys. Rev. E 53, 6573-6576 (1996)].

We consider the line to be described by an internal coordinate s that parametrizes the position of the flux line r(s) confined in a two-dimensional plane.

The three terms contributing to the local velocity of the flux line are:

(i) A line tension that minimizes the line energy.

(ii) A Lorentz force

(iii) An anisotropic random force that describes the effects of the impurities.

We propose an equation of motion for the anisotropic flux line and also a cellular automaton model, which we investigate numerically. The proposed flux line model can be thought as a set of L beads.

We identify the internal coordinate s with the label k of each bead. We consider only integer values of x_k and y_k, and we imposed periodic boundary conditions. For each bead we calculate a vector force according to the discretization of the discretization of the line energy of the flux line. The strength of the external force is kept constant, and its direction is determined by the local unit normal at each position. A random force is defined for every point in the lattice. We start with a flat configuration at time t=0, and for a given time t the position of the k-th bead is updated when the total force overcomes the pinning effect of the impurities.

We have simulated this anisotropic flux line model for two different sets of the strengths of the anisotropic random forces The figure below shows a typical ``interface'' at the depinning transition obtained with the proposed flux line model (see also here) Notice the similarity between the QEW interface and the flux line model: both models develop large slopes at the depinning transition. This similitude is due to the fact that both models are free of constraints in the local slopes. A rather different interface is obtained when constraints to the growth of the local slopes are imposed, as can be observed for the DPD interface.

Our simulations sindicate that the anisotropic flux line model considered in this study does not belong to the DPD universality class. However, we notice that the value of the roughness exponent is smaller than the expected value for the QEW universality, for which \alpha ~ 1 is found. Therefore, we cannot conclude that the flux line model belongs to this universality class, either.

We wish to point out that the fact that we did not find a DPD behavior in the proposed anisotropic flux line model, does not rule out the possibility that other forms of anisotropy might generate a diverging lambda term at the depinning transition, or a slope dependence in the threshold force as argued in Tang. A potentially important difference between the discrete flux line model studied here and the continuum model considered by Tang et al. is that in their paper only the normal motion to the interface plays a role, while the discrete flux line has an internal structure, and we have explicitly included tangential motions. A more general study is needed, and we propose that other models of interface growth in disordered media which might be suitable to include anisotropic effects, such as the random field or random bond Ising model, or the fluid invasion model, should be considered as well.