Editing Magnetic flux quantum (section)

== Derivation of the superconducting flux quantum ==
The following physical equations use SI units. In CGS units, a factor of {{math|''c''}} would appear.

The superconducting properties in each point of the [[superconductor]] are described by the ''complex'' quantum mechanical wave function {{math|Ψ('''r''', ''t'')}} – the superconducting order parameter. As with any complex function, {{math|Ψ}} can be written as {{math|1=Ψ = Ψ<sub>0</sub>''e''<sup>''iθ''</sup>}}, where {{math|Ψ<sub>0</sub>}} is the amplitude and {{math|''θ''}} is the phase. Changing the phase {{math|''θ''}} by {{math|2''πn''}} will not change {{math|Ψ}} and, correspondingly, will not change any physical properties. However, in the superconductor of non-trivial topology, e.g. superconductor with the hole or superconducting loop/cylinder, the phase {{mvar|θ}} may continuously change from some value {{math|''θ''<sub>0</sub>}} to the value {{math|''θ''<sub>0</sub> + 2''πn''}} as one goes around the hole/loop and comes to the same starting point. If this is so, then one has {{mvar|n}} magnetic flux quanta trapped in the hole/loop,<ref name=":0">{{Cite web| url=https://feynmanlectures.caltech.edu/III_21.html |title=The Feynman Lectures on Physics Vol. III Ch. 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity, Section 21-7: Flux quantization | website=feynmanlectures.caltech.edu| access-date=2020-01-21}}</ref> as shown below:

Per [[minimal coupling]], the [[current density]] of [[Cooper pair|Cooper pairs]] in the superconductor is:
<math display="block">\mathbf J = \frac{1}{2m} \left[\left(\Psi^* (-i\hbar\nabla) \Psi - \Psi (-i\hbar\nabla) \Psi^*\right) - 2q \mathbf{A} |\Psi|^2 \right] .</math>
where {{math|1=''q'' = 2''e''}} is the charge of the Cooper pair.
The wave function is the [[Ginzburg–Landau theory|Ginzburg–Landau order parameter]]:
<math display="block">\Psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})} \, e^{i\theta(\mathbf{r})}.</math>

Plugged into the expression of the current, one obtains:
<math display="block">\mathbf{J} = \frac{\hbar}{m} \left(\nabla{\theta}- \frac{q}{\hbar} \mathbf{A}\right)\rho.</math>

Inside the body of the superconductor, the current density '''J''' is zero, and therefore
<math display="block">\nabla{\theta} = \frac{q}{\hbar} \mathbf{A}.</math>

Integrating around the hole/loop using [[Stokes' theorem]] and {{math|1=∇ × '''A''' = '''B'''}} gives:
<math display="block">\Phi_B = \oint\mathbf{A}\cdot d\mathbf{l} = \frac{\hbar}{q} \oint\nabla{\theta}\cdot d\mathbf{l}.</math>

Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:<ref> R. Shankar, "Principles of Quantum Mechanics", eq. 21.1.44</ref>
<math display="block">\Phi_B=\frac{\hbar}{q} 2\pi = \frac{h}{2e}.</math>

Due to the [[Meissner effect]], the magnetic induction {{math|'''B'''}} inside the superconductor is zero. More exactly, magnetic field {{math|'''H'''}} penetrates into a superconductor over a small distance called [[London penetration depth|London's magnetic field penetration depth]] (denoted {{math|''λ''<sub>L</sub>}} and usually ≈&nbsp;100&nbsp;nm). The screening currents also flow in this {{math|''λ''<sub>L</sub>}}-layer near the surface, creating magnetization {{math|'''M'''}} inside the superconductor, which perfectly compensates the applied field {{math|'''H'''}}, thus resulting in {{math|1='''B''' = 0}} inside the superconductor.

The magnetic flux frozen in a loop/hole (plus its {{math|''λ''<sub>L</sub>}}-layer) will always be quantized. However, the value of the flux quantum is equal to {{math|Φ<sub>0</sub>}} only when the path/trajectory around the hole described above can be chosen so that it lays in the superconducting region without screening currents, i.e. several {{math|''λ''<sub>L</sub>}} away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin ({{math|≤ ''λ''<sub>L</sub>}}) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from {{math|Φ<sub>0</sub>}}.

The flux quantization is a key idea behind a [[SQUID]], which is one of the most sensitive [[magnetometer]]s available.

Flux quantization also plays an important role in the physics of [[type II superconductor]]s. When such a superconductor (now without any holes) is placed in a magnetic field with the strength between the first critical field {{math|'''H'''<sub>c1</sub>}} and the second critical field {{math|'''H'''<sub>c2</sub>}}, the field partially penetrates into the superconductor in a form of [[Abrikosov vortex|Abrikosov vortices]]. The [[Abrikosov vortex]] consists of a normal core – a cylinder of the normal (non-superconducting) phase with a diameter on the order of the {{math|''ξ''}}, the [[superconducting coherence length]]. The normal core plays a role of a hole in the superconducting phase. The magnetic field lines pass along this normal core through the whole sample. The screening currents circulate in the  {{math|''λ''<sub>L</sub>}}-vicinity of the core and screen the rest of the superconductor from the magnetic field in the core. In total, each such [[Abrikosov vortex]] carries one quantum of magnetic flux {{math|Φ<sub>0</sub>}}.