Propagating fingerlike patterns in type-II superconductors are studied through a boundary layer model that takes into account the coupling with the temperature of the sample. By formulating an approach based on an interfacial description for a domain of vortices, we determine the shape-preserving fronts and study the properties and scale of the patterns, such as the fingers' shape and width. We show that the formation and the characteristics of these instabilities are strictly related to the local overheating of the material and depend on the substrate temperature, in agreement with the experiments and suggestions from linear stability calculations. The dynamics of vortices in type-II superconductors exhibits a wide variety of instabilities of thermomagnetic origin.1 Beyond phenomena such as avalanches and flux jumps, recent experiments have revealed interesting out-ofequilibrium patterns involving the formation of ramified dendritic or finger-shaped domains of vortices in Nb and MgB 2 thin films and dropletlike patterns in NbSe 2 single crystals. [2][3][4][5][6] It is generally accepted that the nonuniform penetration of the magnetic flux is a thermomagnetic effect due to the local overheating produced by the dissipative motion of vortices. As a consequence of the increased local temperature, the pinning barrier is lowered, leading to a large-scale flux invasion and to a final nonuniform magnetic flux distribution. 7The thermomagnetic nature of the instability underlying the evolution of a flat front between the vortex and the superconducting states into narrow fingers and dendrites has been proposed in some recent theoretical models, [8][9][10] augmented by numerical simulations and linear stability analysis. However, the shapes of the fingers, their remarkably well-defined widths between 20 and 50 m, and their dependence on the substrate temperature were not obtained explicitly in this earlier work. In this paper we concentrate particularly on these finger-type growth forms and propose that they are self-organized propagating shapes with a relatively high temperature and mobility at the tip and a low temperature and mobility on the sides.A detailed analysis for the shape of the fingers requires a more tractable mathematical model than the ones proposed previously. In particular, the formulation of an interfacial description for the vortex front is an effective and simple method to study the problem in its essential features. Local growth models have proven to be a useful tool to analyze front propagation in other physical systems, such as dendrites in crystal growth, and also magnetic flux penetration in type-I superconductors.11-14 The sharp interface limit is appropriate when the vortex density and temperature change rapidly in a layer whose thickness is thin in comparison to the radius of the curvature of the front.In the case of a type-II superconductor, the coarse-grained density of vortices can be represented by a continuous field that decays near the interface with the vortex-free superconducting state...

The dynamics of vortices in type-II superconductors exhibit a variety of patterns whose origin is poorly understood. This is partly due to the nonlinearity of the vortex mobility, which gives rise to singular behavior in the vortex densities. Such singular behavior complicates the application of standard linear stability analysis. In this paper, as a first step towards dealing with these dynamical phenomena, we analyze the dynamical stability of a front between vortices and antivortices. In particular, we focus on the question of whether an instability of the vortex front can occur in the absence of a coupling to the temperature. Borrowing ideas developed for singular bacterial growth fronts, we perform an explicit linear stability analysis which shows that, for sufficiently large front velocities and in the absence of coupling to the temperature, such vortex fronts are stable even in the presence of in-plane anisotropy. This result differs from previous conclusions drawn on the basis of approximate calculations for stationary fronts. As our method extends to more complicated models, which could include coupling to the temperature or to other fields, it provides the basis for a more systematic stability analysis of nonlinear vortex front dynamics.