# Magnetic flux quantum

CODATA values | Units | |
---|---|---|

Φ_{0} |
2.067833848...×10^{−15}^{[1]} |
Wb |

K_{J} |
483597.8484...×10^{9}^{[2]} |
Hz/V |

K_{J-90} |
483597.9×10^{9}^{[3]} |
Hz/V |

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 and so is **Φ**. However, if one deals with the superconducting loop or a hole in a bulk superconductor, it turns out that the magnetic flux threading such a hole/loop is quantized.
The (superconducting) **magnetic flux quantum** Φ_{0} = *h*/(2*e*) ≈ 2.067833848...×10^{−15} Wb^{[1]} 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.
The phenomenon of flux quantization was discovered experimentally by B. S. Deaver and W. M. Fairbank^{[4]} and, independently, by R. Doll and M. Näbauer,^{[5]} 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, 1/Φ_{0}, is called the **Josephson constant**, and is denoted *K*_{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 (since 1990) have been related to a fixed, conventional value of the Josephson constant, denoted *K*_{J-90}. With the 2019 redefinition of SI base units, the Josephson constant had an exact value of *K*_{J} = 483597.84841698... GHz⋅V^{−1},^{[6]} which replaced the conventional value *K*_{J-90}.

## Contents

## IntroductionEdit

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 any complex function Ψ can be written as Ψ = Ψ_{0}*e*^{iθ}, where Ψ_{0} is the amplitude and *θ* is the phase. Changing the phase *θ* by 2π*n* 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 *θ*_{0} to the value *θ*_{0} + 2π*n* 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.

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

It is important to note that the magnetic flux frozen in a loop/hole (plus its *λ*_{L}-layer) will always be quantized. However, the value of the flux quantum is equal to Φ_{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 (≤ *λ*_{L}) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from Φ_{0}.

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 **H**_{c1} and the second critical field **H**_{c2}, 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 *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 *λ*_{L}-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 Φ_{0}. Although theoretically, it is possible to have more than one flux quantum per hole, the Abrikosov vortices with *n* > 1 are unstable^{[note 1]} and split into several vortices with *n* = 1.^{[7]} In a real hole the states with *n* > 1 are stable as the real hole cannot split itself into several smaller holes.

## Measuring the magnetic fluxEdit

The magnetic flux quantum may be measured with great precision by exploiting the Josephson effect. When coupled with the measurement of the von Klitzing constant *R*_{K} = *h*/*e*^{2}, this provides the most precise values of Planck's constant *h* obtained to date. This may be counterintuitive, since *h* is generally associated with the behavior 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.

## See alsoEdit

## NotesEdit

**^**In mesoscopic superconducting samples with sizes ≃ ξ one can observe giant vortices with*n*> 1^{[citation needed]}

## ReferencesEdit

- ^
^{a}^{b}"2018 CODATA Value: magnetic flux quantum".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. Retrieved 2019-05-20. **^**"2018 CODATA Value: Josephson constant".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. Retrieved 2019-05-20.**^**"2018 CODATA Value: conventional value of Josephson constant".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. 20 May 2019. Retrieved 2019-05-20.**^**Deaver, Bascom; Fairbank, William (July 1961). "Experimental Evidence for Quantized Flux in Superconducting Cylinders".*Physical Review Letters*.**7**(2): 43–46. Bibcode:1961PhRvL...7...43D. doi:10.1103/PhysRevLett.7.43.**^**Doll, R.; Näbauer, M. (July 1961). "Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring".*Physical Review Letters*.**7**(2): 51–52. Bibcode:1961PhRvL...7...51D. doi:10.1103/PhysRevLett.7.51.**^**"*Mise en pratique*for the definition of the ampere and other electric units in the SI" (PDF). BIPM.**^**Volovik, G. E. (2000-03-14). "Monopoles and fractional vortices in chiral superconductors".*Proceedings of the National Academy of Sciences of the United States of America*.**97**(6): 2431–2436. arXiv:cond-mat/9911486. Bibcode:2000PNAS...97.2431V. doi:10.1073/pnas.97.6.2431. ISSN 0027-8424. PMC 15946. PMID 10716980.