Magnetic flux quantum

The magnetic flux, represented by the symbol $Φ$, threading some contour or loop is defined as the magnetic field $B$ multiplied by the loop area $S$, i.e. $Φ = B ⋅ S$. Both $B$ and $S$ can be arbitrary, meaning that the flux $Φ$ can be as well but increments of flux can be quantized. The wave function can be multivalued as it happens in the Aharonov–Bohm effect or quantized as in superconductors. The unit of quantization is therefore called magnetic flux quantum.

Dirac magnetic flux quantum
The first to realize the importance of the flux quantum was Dirac in his publication on monopoles The phenomenon of flux quantization was predicted first by Fritz London then within the Aharanov-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 $Φ$ ≈ is a combination of fundamental physical constants: the Planck constant $K$ and the electron charge $Φ_{0} = h/(2e)$. Its value is, therefore, the same for any superconductor.

To understand this definition in the context of the Dirac flux quantum one shall consider that the effective quasiparticles active in a superconductors are Cooper pairs with an effective charge of 2 electrons $$q=2e$$.

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 a phenomenological model.

The inverse of the flux quantum, $h$, is called the Josephson constant, and is denoted $e$J. It is the constant of proportionality of the Josephson effect, relating the potential difference across a Josephson junction to the frequency of the irradiation. The Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed, conventional value of the Josephson constant, denoted $1/Φ_{0}$J-90. With the 2019 redefinition of SI base units, the Josephson constant has an exact value of $K$J = $483,597.848 GHz⋅V−1$.

Derivation of the superconducting flux quantum
The following physical equations use SI units. In CGS units, a factor of $K$ would appear.

The superconducting properties in each point of the superconductor are described by the complex quantum mechanical wave function $K$ — the superconducting order parameter. As any complex function $c$ can be written as $Ψ(r, t)$, where $Ψ$ is the amplitude and $Ψ = Ψ_{0}e^{iθ}$ is the phase. Changing the phase $Ψ_{0}$ by $θ$ will not change $θ$ 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 $2πn$ to the value $Ψ$ 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: $$\mathbf J = \frac{1}{2m} \left[\left(\Psi^* (-i\hbar\nabla) \Psi - \Psi (-i\hbar\nabla) \Psi^*\right) - 2q \mathbf{A} |\Psi|^2 \right] .$$ where $$q = 2e$$ is the charge of the Cooper pair. The wave function is the Ginzburg–Landau order parameter: $$\Psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})} \, e^{i\theta(\mathbf{r})}.$$

Plugged into the expression of the current, one obtains: $$\mathbf{J} = \frac{\hbar}{m} \left(\nabla{\theta}- \frac{q}{\hbar} \mathbf{A}\right)\rho.$$

Inside the body of the superconductor, the current density J is zero, and therefore $$\nabla{\theta} = \frac{q}{\hbar} \mathbf{A}.$$

Integrating around the hole/loop using Stokes' theorem and $$\nabla \times \mathbf{A} = B$$ gives: $$\Phi_B = \oint\mathbf{A}\cdot d\mathbf{l} = \frac{\hbar}{q} \oint\nabla{\theta}\cdot d\mathbf{l}.$$

Now, because the order parameter must return to the same value when the integral goes back to the same point, we have: $$\Phi_B=\frac{\hbar}{q} 2\pi = \frac{h}{2e}.$$

Due to the Meissner effect, the magnetic induction $θ_{0}$ inside the superconductor is zero. More exactly, magnetic field $θ_{0} + 2πn$ penetrates into a superconductor over a small distance called London's magnetic field penetration depth (denoted $B$ and usually ≈ 100 nm). The screening currents also flow in this $H$-layer near the surface, creating magnetization $λ_{L}$ inside the superconductor, which perfectly compensates the applied field $λ_{L}$, thus resulting in $M$ inside the superconductor.

The magnetic flux frozen in a loop/hole (plus its $H$-layer) will always be quantized. However, the value of the flux quantum is equal to $B = 0$ 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 $λ_{L}$ away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin ($Φ_{0}$) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from $λ_{L}$.

The flux quantization is a key idea behind a SQUID, which is one of the most sensitive magnetometers available.

Flux quantization also plays an important role in the physics of type II superconductors. When such a superconductor (now without any holes) is placed in a magnetic field with the strength between the first critical field $≤ λ_{L}$ and the second critical field $Φ_{0}$, the field partially penetrates into the superconductor in a form of 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 $H_{c1}$, 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 $H_{c2}$-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 $ξ$.

Measuring the magnetic flux
Prior to the 2019 redefinition of the SI base units, the magnetic flux quantum was measured with great precision by exploiting the Josephson effect. When coupled with the measurement of the von Klitzing constant $λ_{L}$, this provided the most accurate values of the Planck constant $Φ_{0}$ obtained until 2019. This may be counterintuitive, since $R_{K} = h/e^{2}$ is generally associated with the behaviour of microscopically small systems, whereas the quantization of magnetic flux in a superconductor and the quantum Hall effect are both emergent phenomena associated with thermodynamically large numbers of particles.

As a result of the 2019 redefinition of the SI base units, the Planck constant $h$ has a fixed value $h$ = which, together with the definitions of the second and the metre, provides the official definition of the kilogram. Furthermore, the elementary charge also has a fixed value of $h$ to define the ampere. Therefore, both the Josephson constant $h$ and the von Klitzing constant $e =$ have fixed values, and the Josephson effect along with the von Klitzing quantum Hall effect becomes the primary mise en pratique for the definition of the ampere and other electric units in the SI.