Φ₀ = 2.068 * 10⁻¹⁵ T m²
= h / (2 e),
or its multiple to pass through the loop.
How can we explain this phenomenon?
A model where the charge carrier has the same charge e as the electron
Let us first assume that the charge carrier has the charge e and an arbitrary mass m. Initially the magnetic field is zero. We have a superconducting loop whose radius is r.
____
/ \ superconducting
\_____/ loop, radius r
e -->
charge carrier
does two loops
We require that the wave function of the carrier returns to its original value after two rounds around the loop.
The angular momentum L of the carrier is a multiple of
ħ / 2.
The momentum
p = L / r
and the velocity
v = L / (r m).
The cycle time is
t = 2 π r / v
= 2 π r² m / L.
The current running in the circle is
I = e / t
= e L / (2 π r² m).
The magnetic moment of the current loop is
μ = I A
= I * π r²
= e L / (2 m).
Let us compare the energy difference of two successive states 0 and 1 of the charge carrier, where
L₀ = N ħ / 2,
L₁ = (N + 1) ħ / 2.
We assume that N is a large positive integer. The kinetic energy
E = p² / (2 m)
= L² / (2 m r²).
If we increase the value of L by ħ / 2, the square of L grows approximately by L ħ. The energy difference between successive states is approximately
ΔE = L ħ / (2 m r²).
We want to add or reduce the energy of the carrier by one such step. We add a magnetic field B which is normal to the loop. The magnetic moment then contributes to the energy of the carrier, or reduces it.
ΔE = μ B
<=>
B = ΔE / μ
= L ħ / (2 m r²) * (2 m) / (L e)
= ħ / (e r²).
The magnetic flux through the loop is then
B π r² = ħ π / e
= h / (2 e)
= Φ₀.
Note that we did not need to use Berry's phase, the magnetic vector potential, or the minimal coupling in the calculation. It is enough to treat μ B as a potential under which the charge carrier moves.
The model allows the mass m of the charge carrier to be arbitrary. It only set the charge strictly to e.
How to explain magnetic flux quantization?
Let us start from a zero magnetic field. Charge carriers in the loop might be in states L = N ħ / 2, where N is an integer. This would be analogous to the Bohr atomic model, but we allow also a half integer orbital angular momentum.
Does the superconductor prefer to order the charge carriers in the way in which they would be if B = 0? That might be true. If B ≠ 0, then symmetric orbits to the opposite directions, angular momenta L and -L, along the loop will have different energies, because their magnetic moments are opposite. It might be that the superconductor prefers symmetry.
What if we add a magnetic field
B = n Φ₀ / A,
where n is a "smallish" integer? Then the orbits L and -L no longer have the same energy. Their energy difference is
2 n ΔE,
where ΔE is the energy difference of successive states close to L or -L. It may be that the superconductor tolerates this asymmetry. That is, the asymmetric state has essentially the same energy as the original state.
What if the magnetic field B is not an integer multiple of Φ₀ / A?
Then the energy levels of symmetric orbits will have an energy difference which may be an arbitrary fraction of the energy difference ΔE between successive states. It may be that this asymmetry is too ugly for the superconductor. The state would have a much higher energy, and the superconductor rather generates a current which sets the magnetic flux to an integer times Φ₀.
Conclusions
Flux quantization offers us a clue about how large is the electric charge of the carrier in superconductivity.
We introduced a model where the carrier has the charge e, and its wave function must return to the original value after two loops. Usually, people assume a model where the carrier has the charge 2 e, and its wave function returns after one loop.
We explained magnetic flux quantization from a hypothesis that the superconductor prefers a symmetry in the energy levels of symmetric orbits L and -L.
Flux quantization may help us to design a time crystal model for superconductivity.
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