The following physical equations use SI units. In CGS units, a factor of would appear. The superconducting properties in each point of the
superconductor are described by the
complex quantum mechanical wave function – the superconducting order parameter. As with any complex function, can be written as , where is the amplitude and is the phase. Changing the phase by will not change 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 may continuously change from some value to the value as one goes around the hole/loop and comes to the same starting point. If this is so, then one has 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 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 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 inside the superconductor is zero. More exactly, magnetic field penetrates into a superconductor over a small distance called the
London penetration depth (denoted and usually ≈ 100 nm). The screening currents also flow in this -layer near the surface, creating magnetization inside the superconductor, which perfectly compensates the applied field , thus resulting in inside the superconductor. The magnetic flux frozen in a loop/hole (plus its -layer) will always be quantized. However, the value of the flux quantum is equal to 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 away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin () superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from . 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 and the second critical field , 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 , 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 -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 . A recent research work highlights that the concept of photon emerges directly from the Faraday law of induction of classical electromagnetism while assuming magnetic flux quantization. == Measuring the magnetic flux ==