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 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 Φ₀ = h/(2e) ≈ 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[6], but was predicted earlier by Fritz London in 1948 using a phenomenological model.[7][8]

CODATA values		Units
Φ0	2.067833848...×10^−15[1]	Wb
KJ	483597.8484...×10^9[2]	Hz/V
KJ-90	483597.9×10^9[3]	Hz/V

The inverse of the flux quantum, 1/Φ₀, 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,[9] which replaced the conventional value K_J-90.

Introduction

The following uses 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 any complex function Ψ can be written as Ψ = Ψ₀e^iθ, 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[8], as shown below:

Per minimal coupling, the probability current of cooper pairs in the superconductor is:

${\displaystyle \mathbf {J} ={\frac {1}{2m}}\left[\left(\Psi ^{*}(-i\hbar \nabla )\Psi -\Psi (-i\hbar \nabla )\Psi ^{*}\right)-2q\mathbf {A} |\Psi |^{2}\right]\,\!.}$

Here, the wave function is the Ginzburg–Landau order parameter:

${\displaystyle \Psi (\mathbf {r} )={\sqrt {\rho (\mathbf {r} )}}\,e^{i\theta (\mathbf {r} )}.}$

Plugging into the expression of probability current, one obtains:

${\displaystyle \mathbf {J} ={\frac {\hbar }{m}}(\nabla {\theta }-{\frac {q}{\hbar }}\mathbf {A} )\rho .}$

While inside the body of the superconductor, the current density J is zero; Therefore:

${\displaystyle \nabla {\theta }={\frac {q}{\hbar }}\mathbf {A} .}$

Integrating around the hole/loop using Stokes' theorem and ${\displaystyle \nabla \times \mathbf {A} =B}$ gives:

${\displaystyle \Phi _{B}=\oint \mathbf {A} \cdot d\mathbf {l} ={\frac {\hbar }{q}}\oint \nabla {\theta }\cdot d\mathbf {l} .}$

Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:

${\displaystyle \Phi _{B}={\frac {\hbar }{q}}2n\pi =n{\frac {h}{2e}}.}$

Due to the 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.

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 Φ₀ 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 Φ₀.

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 Φ₀. 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.[10] 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 flux

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/e², this provides the most precise values of Planck's constant h obtained until 2019. 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.

After the 2019 redefinition of the SI base units, Planck's constant h has a fixed value ${\displaystyle h=}$ 6.62607015×10⁻³⁴ J⋅s,[11] which, together with definition of second and metre, provides the official definition of kilogram. Furthermore, elementary charge also takes a fixed value of e = 1.602176634×10⁻¹⁹ C[12] to define Ampere. Therefore, both Josephson constant K_J=(2e)/h and von Klitzing constant R_K = h/e² have fixed values, and Josephson effect along with von Klitzing quantum Hall effect becomes the primary mise en pratique[13] for the definition of the ampere and other electric units in the SI.

See also

• Brian Josephson
• Committee on Data for Science and Technology
• Domain wall (magnetism)
• Flux pinning
• Ginzburg–Landau theory
• Husimi Q representation
• Macroscopic quantum phenomena
• Magnetic domain
• Magnetic monopole
• Quantum vortex
• Topological defect
• von Klitzing constant

Notes

1. In mesoscopic superconducting samples with sizes ≃ ξ one can observe giant vortices with n > 1

References

1. "2018 CODATA Value: magnetic flux quantum". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
2. "2018 CODATA Value: Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
3. "2018 CODATA Value: conventional value of Josephson constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
4. 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.
5. 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.
6. Parks, R. D. (1964-12-11). "Quantized Magnetic Flux in Superconductors: Experiments confirm Fritz London's early concept that superconductivity is a macroscopic quantum phenomenon". Science. 146 (3650): 1429–1435. doi:10.1126/science.146.3650.1429. ISSN 0036-8075. PMID 17753357.
7. London, Fritz (1950). Superfluids: Macroscopic theory of superconductivity. John Wiley & Sons. pp. 152 (footnote).
8. "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". www.feynmanlectures.caltech.edu. Retrieved 2020-01-21.
9. "Mise en pratique for the definition of the ampere and other electric units in the SI" (PDF). BIPM.
10. 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.
11. "2018 CODATA Value: Planck constant". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
12. "2018 CODATA Value: elementary charge". The NIST Reference on Constants, Units, and Uncertainty. NIST. 20 May 2019. Retrieved 2019-05-20.
13. "BIPM - mises en pratique". www.bipm.org. Retrieved 2020-01-21.