Superconductivity of networks. A percolation approach to the effects of disorder

Solutions of the linearized Landau-Ginzburg equations on networks of thin wires are studied. We derive linear-difference equations for the value of the order parameter at the junctions of the net with the use of the explicit form of the solutions on the wires. The technique is shown to be applicable to the diffusion equation, to harmonic lattice vibrations, and to the Schr\"odinger equation and results in equations similar to tight-binding equations. The equations are solved and the upper critical field is determined for some simple finite nets, for the infinite square net, and for the triangular Sierpinski gasket. Dead-end side branches are shown to lead to a mass renormalization. On the square net the equations map on the Azbel-Hofstadter-Aubry model. When the coherence length is small, vortex cores can be accommodated in the holes of the net and there is no upper critical field. The equations on the Sierpinski gasket are solved by an iterative decimation process. The process determines a new length scale proportional to a power of the bare coherence length. The upper critical field is studied for a finite gasket and for a lattice of gaskets. With the use of scaling arguments the results are applied to percolation clusters. Far from the percolation threshold the results are described by a renormalized correlation length of standard form. When this length becomes shorter than the correlation length for the percolation problem the critical field is shown to be constant or decreasing as the threshold is approached. Existing experiments are discussed and the importance of high-field-susceptibility measurements is emphasized.

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