The bifurcation structure of a thin superconducting loop swith small variations in its thickness
Author:
G. Richardson
Journal:
Quart. Appl. Math. 58 (2000), 685-703
MSC:
Primary 82D55; Secondary 70K50
DOI:
https://doi.org/10.1090/qam/1788424
MathSciNet review:
MR1788424
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Abstract: We study bifurcations between the normal and superconducting states, and between superconducting states with different winding numbers, in a thin loop of superconducting wire, of uniform thickness, to which a magnetic field is applied. We then consider the response of a loop with small thickness variations. We find that close to the transition between normal and superconducting states lies a region where the leading-order problem has repeated eigenvalues. This leads to a rich structure of possible behaviours. A weakly nonlinear stability analysis is conducted to determine which of these behaviours occur in practice.
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X. Zhang and J. C. Price, Susceptibility of a mesoscopic superconducting ring, Phys. Rev. B 55, 3128 (1997)
J. Berger and J. Rubinstein, Bifurcation analysis for phase transitions in superconducting rings with nonuniform thickness, SIAM J. Appl. Math. 58, 103–121 (1998)
V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Soviet Physics JETP 20, 1064 (1950)
R. P. Groff and R. D. Parks, Fluxoid quantisation and field induced depairing in a hollow super-conducting microcylinder, Phys. Rev. 176, 568 (1968)
W. A. Little and R. D. Parks, Observation of quantum periodicity in the transition temperature of a superconducting cylinder, Phys. Rev. Lett. 9, 9 (1962)
V. M. Fomin, V. R. Misko, J. T. Devreese, and V. V. Moshchalkov, On the superconducting phase boundary for a mesoscopic square loop, Solid State Comm. 101, 303 (1997)
V. V. Moshchalkov, L. Gielen, C. Strunk, R. Jonckheere, X. Qiu, Van Haesendonck, and Y. Bruynseraede, Effect of sample topology on the critical fields of mesoscopic superconductors, Nature 373, 319 (1995)
G. Richardson and J. Rubinstein, A one-dimensional model of superconductivity in a thin wire of slowly varying cross-section, Proc. Roy. Soc. 455, 2549 (1999)
J. Rubinstein and M. Schatzman, Asymptotics for thin superconducting rings, J. Math. Pure Appl. 77, 801–820 (1998)
X. Zhang and J. C. Price, Susceptibility of a mesoscopic superconducting ring, Phys. Rev. B 55, 3128 (1997)
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© Copyright 2000
American Mathematical Society
