The magnetic flux through some contour is defined as the magnetic field multiplied by the loop area. In some physical systems, such as in superconductors, the magnetic flux can only take certain fixed (or, quantized) values. The values allowed to be taken by the magnetic flux are integer multiples of the magnetic flux quantum. This restriction on the values of the magnetic flux is called magnetic flux quantization. The quantization of the magnetic flux originates from the requirement that the wave function describing a particle must be single-valued. Because a magnetic field induces a phase shift in the wave function when integrating on a closed contour, the wave function can only remain single-valued if the magnetic flux takes on a quantized value for which the phase shift is equal to 2 π {\displaystyle 2\pi } . The phenomenon of flux quantization was predicted first by Fritz London then within the Aharonov–Bohm effect and later discovered experimentally in superconductors.
Superconducting magnetic flux quantum
If one deals with a superconducting ring (i.e. a closed loop path in a superconductor) or a hole in a bulk superconductor, the magnetic flux threading such a hole/loop is quantized. The (superconducting) magnetic flux quantum Φ0 = h/(2e) ≈ 2.067833848...×10−15 Wb is a combination of fundamental physical constants: the Planck constant h and the electron charge e. Its value is, therefore, the same for any superconductor. To understand this definition in the context of the Dirac flux quantum, one considers that the supercurrent in a superconductor is carried by Cooper pairs which have twice the charge of an electron q = 2 e {\displaystyle q=2e} . The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank, and independently by R. Doll and M. Näbauer, in 1961. The quantization of magnetic flux is closely related to the Little–Parks effect, but was predicted earlier by Fritz London in 1948 using the London equations, a phenomenological model of superconductivity.
Derivation The following physical equations use SI units. In CGS units, a factor of c would appear.
The superconducting properties in each point of the superconductor are described by the complex quantum mechanical wave function Ψ(r, t) – the superconducting order parameter. As with any complex function, Ψ can be written as Ψ = Ψ0eiθ, where Ψ0 is the amplitude and θ is the phase. Changing the phase θ by 2πn will not change 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 θ may continuously change from some value θ0 to the value θ0 + 2πn as one goes around the hole/loop and comes to the same starting point. If this is so, then one has n magnetic flux quanta trapped in the hole/loop, as shown below: Per minimal coupling, the current density of Cooper pairs in the superconductor is:
J = 1 2 m [ ( Ψ ∗ ( − i ℏ ∇ ) Ψ − Ψ ( − i ℏ ∇ ) Ψ ∗ ) − 2 q A | Ψ | 2 ] . {\displaystyle \mathbf {J} ={\frac {1}{2m}}\left[\left(\Psi ^{*}(-i\hbar \nabla )\Psi -\Psi (-i\hbar \nabla )\Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right].}
where q = 2e is the charge of the Cooper pair. The wave function is the Ginzburg–Landau order parameter:
Ψ ( r ) = ρ ( r ) e i θ ( r ) . {\displaystyle \Psi (\mathbf {r} )={\sqrt {\rho (\mathbf {r} )}}\,e^{i\theta (\mathbf {r} )}.}
Plugged into the expression of the current, one obtains:
J = ℏ m ( ∇ θ − q ℏ A ) ρ . {\displaystyle \mathbf {J} ={\frac {\hbar }{m}}\left(\nabla {\theta }-{\frac {q}{\hbar }}\mathbf {A} \right)\rho .}
Inside the body of the superconductor, the current density J is zero, and therefore
∇ θ = q ℏ A . {\displaystyle \nabla {\theta }={\frac {q}{\hbar }}\mathbf {A} .}
Integrating around the hole/loop using Stokes' theorem and ∇ × A = B gives:
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