Flux quantization

As flux quantization is referred to the effect that the magnetic flux through a ring of superconducting material only integer multiples of the flux quantum can be. The river quantization is a consequence of the Meißner-Ochsenfeld effect . The terms fluxon and fluxoid are also used instead of flux quantum .

The term fluxon is also used in the discretization of magnetohydrodynamics using the finite element method .

Discovery and Evidence

Experimental set-up by Doll and Näbauer: A hollow cylinder made of lead (B) vapor-deposited on a quartz rod (A) hangs on a quartz thread (D) and is excited to torsional vibrations by an external measuring field . The resonance of the rod as a function of the frozen field is measured using a mirror (C) with a light pointer. ${\displaystyle H_{x}}$ ${\displaystyle H_{y}}$

As early as 1957, the existence of Cooper pairs was predicted in the BCS theory : however, the experimental proof of flux quantization was not provided until 1961. At the commission for low temperature research of the Bavarian Academy of Sciences , Robert Doll and Martin Näbauer dealt with river quantization; At Stanford University Bascom Deaver and William Fairbank Sr. Both groups cooled hollow cylinders made of lead and tin with diameters in the range of below the critical temperature. Doll and Näbauer's group used a resonance method to measure the torque of the hollow cylinder attached to a quartz thread, which is proportional to the magnetic flux and external field. Their structure can be seen in the figure on the right. ${\displaystyle 10\,\mathrm {\mu m} }$

The group at Stanford University vibrated the cylinder and measured the field with pickup coils. The results of both groups showed discrete values ​​for the captured flow.

Flux quantum in the superconductor

The quantization of the magnetic flux can be determined by the quantum mechanical consideration of the current flow distributed in the superconductor :

${\displaystyle \Phi _{n}=n\,{\frac {h}{2\,e}}\;}$ With ${\displaystyle n\in \mathbb {N} }$

The magnetic flux is always an integral multiple of the flux quantum

${\displaystyle \Phi _{0}={\frac {h}{2\,e}}=2{,}067\,833\,848\dotso \,\cdot 10^{-15}\,\mathrm {Wb} \,,}$

where Wb stands for the unit Weber . Since the constants h ( Planck's quantum of action ) and e ( elementary charge ) were precisely defined in the definition of the International System of Measurement Units (SI) , Φ _{0 also has} an exact value.

The factor in the denominator of the formula denotes a double electron charge. The BCS model, which sees the so-called Cooper pairs as the cause of superconductivity , is based on this double electron charge. ${\displaystyle 2\,e}$

The distribution of the magnitude of the magnetic field of a single flux tube in space is given by the equation ${\displaystyle {\vec {B}}}$

${\displaystyle B(r)={\frac {\Phi _{0}}{2\,\pi \,\lambda ^{2}}}\,K_{0}\left({\frac {r}{\lambda }}\right)\approx {\sqrt {\frac {\lambda }{r}}}\,e^{-{\frac {r}{\lambda }}}}$

with the field pointing in the direction of the axis of the flow tube and being the modified Bessel function . ${\displaystyle K_{0}(z)}$

Abrikosov turbulence

A flux quantum in the sense of the Abrikossow turbulence is a needle-shaped single crystal (core) in a superconductor of the 2nd type , which is surrounded by supercurrents.

The magnetic field through such a single crystal and its neighborhood has an order of magnitude of approximately and is quantized by the phase properties of the magnetic vector potential in quantum electrodynamics . ${\displaystyle \lambda _{L}\approx 100\,\mathrm {nm} }$

Josephson Turbulence

The Josephson turbulence is the counterpart to the Abrikossow turbulence in circling supercurrents without a physical core in a superconductor of the 2nd type. The core in this case is the mathematical center of the circle.

The inverse of the flux quantum is the Josephson constant :

${\displaystyle K_{J}={\frac {1}{\Phi _{0}}}={\frac {2\,e}{h}}=4{,}835\;978\;484\;4...\cdot 10^{14}\,{\frac {\mathrm {Hz} }{\mathrm {V} }}}$.

Their value is also exact.

Derivation of the flux quantization

The superconducting state is a quantum mechanical state that extends over macroscopic length scales. It can therefore be described by a macroscopic wave function :

${\displaystyle \psi ({\vec {r}})=\psi _{0}\,\exp(i\,S({\vec {r}}))}$

It is assumed (in a quasi-classical, i.e. macroscopic approximation) that it has a constant amplitude and that only the phase S is location-dependent. The London equation applies to this wave function ${\displaystyle \psi }$ ${\displaystyle \psi _{0}}$

${\displaystyle {\vec {j}}={\frac {n\,q\,\hbar }{m}}\operatorname {grad} S-{\frac {n\,q^{2}}{m}}{\vec {A}}.}$

As a result of the Meißner-Ochsenfeld effect , the magnetic induction inside a superconductor disappears . For the static case (one of Maxwell's equations ) applies, which also applies to the interior of the superconductor. It therefore applies ${\displaystyle {\vec {B}}}$ ${\displaystyle \operatorname {rot} {\vec {B}}=\mu {\vec {j}}}$ ${\displaystyle {\vec {j}}=0}$

${\displaystyle {\frac {n\,q\,\hbar }{m}}\operatorname {grad} S={\frac {n\,q^{2}}{m}}{\vec {A}}.}$

If one summarizes the constants and integrates both sides along a closed path C through the interior of the superconductor, one obtains

${\displaystyle \oint _{C}\operatorname {grad} S\cdot {\text{d}}{\vec {l}}={\frac {q}{\hbar }}\oint _{C}{\vec {A}}\cdot {\text{d}}{\vec {l}}.}$

The left side describes the change in phase when walking through the closed path . Since the wave function is unique, the phase change can only be an integer multiple of 2  . So it applies ${\displaystyle S}$ ${\displaystyle C}$ ${\displaystyle \pi }$

${\displaystyle \oint _{C}\operatorname {grad} S\cdot {\text{d}}{\vec {l}}=2\pi s\quad ,s\in \mathbb {Z} .}$

According to Stokes' theorem,

${\displaystyle \oint _{C}{\vec {A}}\cdot {\text{d}}{\vec {l}}=\iint _{F}{\text{rot}}\,{\vec {A}}\cdot {\text{d}}{\vec {F}}=\iint _{F}{\vec {B}}\cdot {\text{d}}{\vec {F}}=\Phi ,}$

where is a bounded area and the magnetic flux through this area. is the vector with the magnitude and direction of the outer normal on the surface element under consideration. It turns out overall ${\displaystyle F}$ ${\displaystyle C}$ ${\displaystyle \Phi }$ ${\displaystyle {\rm {d{\vec {F}}}}}$ ${\displaystyle \mathrm {d} F}$ ${\displaystyle {\vec {n}}}$

${\displaystyle \Phi ={\frac {h}{q}}s.}$

The flow through a superconducting ring is thus quantized. Experiments show what suggests that the electrons form pairs, the so-called Cooper pairs . ${\displaystyle q=-2e}$

Fluxon in magnetohydrodynamics

In magnetohydrodynamics (MHD), fluxon is used to describe a discretized magnetic field line of finite magnitude in a finite element model . In doing so, the attempt is made to preserve the topology of the examined facts, taking into account limited computing capacities.

See also

• Fluxtronics
• Physical constants
• Josephson Effect
• Flow hose

Individual evidence

1. ↑ Ch. Kittel: Introduction to Solid State Physics . Oldenbourg, ISBN 978-3-486-57723-5 , p. 306. Quotation: “We now show that the entire magnetic flux through a superconducting ring can only assume quantized values, and only integer multiples of the flux quantum”.
2. ↑ biogram
3. ↑ Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring . In: Physical Review Letters . August. doi : 10.1103 / PhysRevLett.7.51 . , Phys. Rev. Lett., Volume 7, 1961, p. 51
4. ↑ Experimental Evidence for Quantized Flux in Superconducting Cylinders . In: Physical Review Letters . August. doi : 10.1103 / PhysRevLett.7.43 . , Volume 7, 1961, p. 43
5. ^ Rudolf Gross, Achim Marx: Solid State Physics . 2nd Edition. De Gruyter, Berlin / Boston 2014, ISBN 978-3-11-035869-8 , pp. 785 ff .
6. ↑ Fundamental Physical Values. National Institute of Standards and Technology, accessed July 8, 2019 .  Value for the quantum of the magnetic flux.
7. ↑ In connection with the quantum Hall effect , a similarly formed quantity Φ _J  : = h / e (= 2Φ ₀ ) occurs as an elementary flow , which is formed directly with the elementary  charge e of the electron.
8. ↑ CGPM 2018, video of the open session, 0:24:00. Accessed December 30, 2018
9. ↑ Fundamental Physical Values. National Institute of Standards and Technology, accessed July 8, 2019 .  Value for the quantum of the Josephson constant.
10. ^ Calculation based on Ch. Kittel: Introduction to Solid State Physics . Oldenbourg, ISBN 978-3-486-57723-5 , pp. 299-300, 306-308.