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The stability of the normal state of superconductors in the presence of electric currents. (English) Zbl 1165.82029

The paper analyzes the stability of the normal state of superconductors in the condition of an experiment when a diminishing of the current below a critical one leads to that the sample would abruptly become purely superconducting. The developed model in the paper is constructed on the base of the time-dependent Ginzburg-Landau model. There are considered cylindrical-like domains with insulating side boundaries into a typical 2-D sample where the current flows into the sample from one part of its interface and exists from another part, disconnected from the first one (the domain to being proper for most wires). In order to simplify the original problem the author treats a large domain limit, i.e., the case when the domain size has to be much larger than the coherence length but also much smaller than the penetration depth (the last is the length-scale characterizing variations in magnetic field). After linearizing the considered initial-boundary problem near the normal state, the author considers the reduced problem in 3-D settings. By this all obtained results are equally valid for 2-D objects. The formulated main Theorem states that the normal state remains stable in the large domain limit as long as the current on the boundary, at points where it is perpendicular to it, is greater than the critical current in the 1-D case. If the current is nowhere perpendicular to the boundary, then as long as it doesn’t vanish there, the normal state must be stable. In order to prove the theorem, first there are considered two different 1-D settings on R and on R+. In R+, the first critical current for which a steady state solution exists is found. There are discussed the cases of infinite and semi-infinite 1-D domain. Then, some of these results are extended to unbounded 3-D domains. For this, it is analyzed the existence of eigenvalues with non-positive real part of the elliptic operator in the right-hand side of the reduced problem. The eigenfunctions are considered for case of the perpendicular current, and the steady solutions are found for non-perpendicular current. In the final stage of proof of the main theorem, first it is shown that any eigenfunctions of the elliptic operator in the reduced problem must decay exponentially fast away from the boundary in the large domain limit. Then, the author proves that any eigenfunction corresponding to a non-positive eigenvalue must decay exponentially fast away from the boundary. These results allow ones to prove the main theorem in total. Finally, it is demonstrated short-time instability when the current is both perpendicular to the boundary and smaller than the 1-D critical current.

MSC:

82D55 Statistical mechanics of superconductors
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
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