Flux quantization in type-I superconductors ↩

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created: 2022-01-10 05:32:41
modified: 2022-01-10 06:05:29

Due to the Meissner-Ochsenfeld effect inside a type-I superconductor, the magnetic field inside the material is zero. Imagine a superconducting ring where inside the ring there is a perpendicular magnetic field to the plane of the ring. Experiments show that the closed flux is quantized.

In order to understand the phenomenon, we have to use the specific properties of type-I superconductors. Namely, the superconductor is an ideal diamagnet, i.e. inside the superconductor, the magnetic field is zero (Meissner effect). In the stationary case, the Maxwell equation:

Where is the magnetic field, is the electric field, is the vacuum permittivity and is the current density. This equation implies that inside the superconductor, the current density is also zero. The current flows only at the surface of the superconductor within the London penetration depth. One can show that in the superconducting phase the charge density is homogeneous, so the wave function in the superconducting state can be written as:

Where is the density of superconducting particles (pretty much charge density) and is the phase of the superconducting state.

The current density can then be expressed as:

Using the kinetic momentum operator, where is the magnetic vector potential. Since , it follows:

Now we are going to use the single valuedness of the wave function, meaning that traveling along a closed loop in the superconducting ring, the wavefunction can just pick up a phase of , where :

Where is the magnetic flux, which became quantized:

Where is the Planck constant.

Since the superconducting media is a condensate of Cooper pairs with charge , the magnetic flux closed by the type-I superconductor is a multiple of .