Dynamics of a driven single flux line in superconductors.

We study the low temperature dynamics of a single flux line in a bulk type-II superconductor, driven by a surface current, both near and above the onset of an instability which sets in at a critical driving. We found that above the critical driving, the velocity profile of the flux line develops a discontinuity. 74.60.Ge, 02.60.Lj, 02.30.Mv Typeset using REVTEX 1 Dynamics of a driven elastic string have attracted much recent attention [1,2]. While most of the work has been focused on the interesting physics of pinning-depinning transitions in the case of bulk driving, the paper by Tang, Feng, and Golubovic [2] studied the case of a surface-current-driven flux line in a bulk type-II superconductor. They found a novel instability of the flux line motion at large driving currents. The instability sets in at a critical driving, where the line loses its steady state motion and (presumably) will be stretched longer and longer. Their finding depends crucially on the boundary condition they use. Physically, the surface driving current is within a boundary layer of thickness λ, where λ is the penetration depth. The boundary condition used in Ref. [2] is somewhat equivalent to taking the limit λ → 0 in a plausible but uncontrolled way. Since the instability sets in at or near the boundaries, it is necessary to examine the situation carefully using a more physical boundary layer. Also, it is important to see what happens when the driving current is larger than the critical driving – a question which can not be addressed by using the boundary condition in Ref. [2]. In this paper we analyze the flux motion with the more physical boundary layer Lorentz driving force. We first use the method of matching asymptotic expansions to study the steady state solutions. The lowest order matching condition justifies the form of the boundary conditions used in [2] and gives the relation of the driving force to the current. We then study, both numerically and analytically, the complete equation below and above the onset of instability. Let us first derive the equation for the flux line motion which involves the Lorentz force as a term in the equation, as opposed to just a boundary condition. As we will be mostly interested in fairly large driving forces, we neglect pinning effects. The Lorentz force on a flux line is just F = 1 c ∫ j × hdsdA where s is the arclength along the flux line and dA a section of infinitesimal area transverse to the flux line. If the applied current, j, is slowly varying in the direction transverse to the line, then the integration in these coordinates may be carried out to give