Flux quantization

Consider a superconducting ring that encloses a magnetic flux density B. The magnetic flux that is enclosed is l = JjB ds = JJVxA ds = /A -dL, where A is the vector potential. The supercurrent density from Equation 26.1 is J = (p). 

However, in the presence of a magnetic field, the momentum operator is p = —ihV — eA. Therefore, to get the momentum, we use 

We integrate inside of a superconducting ring containing magnetic flux 1 at a sufficient depth where / = 0, 

For the wavefunction iff to be single valued when integrated around a closed path, the phase change dL = AL = 2im where u is an integer. Since A dL = I , we see that the flux inside the ring is quantized in imits of 

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With e as a given elementary electric charge, there is also a condition on the quantization of magnetic flux. This could be reinterpreted as a subsidiary condition in an effort to quantize the electron charge and deduce its absolute value by means of the present theory [ 13,18,20], but the details of such an analysis are not yet available. Magnetic flux quantization is discussed in further detail in Appendix B. 

Using the spin condition (42) in combination with expressions (B.41), (B.44), and (B.54), the condition for magnetic flux quantization is reduced to... 

Superconducting Properties. By means of flux quantization and Josephson tunneling experiments, it has been firmly established that superconductivity in the cuprate corrqtounds is based on electron pairs similar to that of conventional Bardeen-Cooper-Schreiffer (BCS) superconductors. However, there are currently widely... 

Doll, R. and Nahbauer, M. (1961) Experimental proof of magnetic flux quantization in a superconducting ring. Phys. Rev. Lett., 7 (2), 51-52. 

Figure 10.7. Edge dislocations in a smectic liquid crystal. The value of the path integral of n - ds along a closed loop is always an integer which corresponds to the number of inserted layers. According to de Gennes (14), this is the liquid crystal analog of the magnetic flux quantization in superconductors.

Some years later a more thorough discussion of the motion of pairs of electrons in a metal was given by Cooper,7 as well as by Abrikosov8 and Gor kov,9 who emphasized that the effective charge in superconductivity is 2e, rather than e. The quantization of flux in units hc/2e in superconducting metals has been verified by direct experimental measurement of the magnetic moments induced in thin films.10 Cooper s discussion of the motion of electron pairs in interaction with phonons led to the development of the Bardeen-Cooper-Schrieffer (BCS) theory, which has introduced great clarification in the field of superconductivity.2... 

The present model of charged particle states is connected with two fundamental questions in EM held theory. The first of these concerns the electric charge quantization, partly through its relation with the magnetic flux in non-Abelian... 

X. Quantization of Magnetic Flux Enclosed by a Superconducting Path... 

X. QUANTIZATION OF MAGNETIC FLUX ENCLOSED BY A SUPERCONDUCTING PATH... 

According to Abrikosov (1957), magnetic flux penetrates type-II superconductors as a periodic arrangement of quantized magnetic flux lines or vortices, i.e. a vor-... 

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