Flux Quantization and Generalized Pairing in Superconducting Cylinders

Abstract

The observation that flux is quantized in units of $\frac{\mathrm{ch}}{2e}$ in multiply connected superconductors can be understood if the paired electrons in the ground-state wave function have identical angular momentum. It is possible, however, to generalize the pairing condition and obtain a superconductor in which the ground-state paired electrons have different momentum. Under these circumstances, the relation between superconductivity and flux quantization is obscured and in this paper we reinvestigate the question. The Gibbs free-energy per unit volume of the superconductor can be separated into three parts: A magnetic term which is a minimum when the external and internal fields are equal; a kinetic term, which after suitable averaging is found to be independent of the enclosed flux; and the superconducting correlation energy, which is periodic in the enclosed flux, with minima at integral multiples of $\frac{\mathrm{ch}}{2e}$. We obtain the result that flux quantization in units of $\frac{\mathrm{ch}}{2e}$ occurs when the minima of the correlation energy at integral flux values determine the behavior of the free energy; and this is the usual case. By choosing an appropriate sample (one which is not rotationally invariant and in which the inner hole is very small), the magnetic energy might be made to dominate in the Gibbs function. Then one would expect a breakdown of quantization, with the flux approaching that due to the external field. Nonintegral flux values would correspond to states of mixed pairing.