Magnetic Flux Quantization

Magnetic flux quantization is a phenomenon of superconductivity in which the magnetic flux trapped by a sufficiently thick superconducting ring occurs in integer multiples of a universal unit, the magnetic flux quantum, $\Phi_0=h/(2e)$ . More generally, the quantized quantity is the fluxoid, which combines magnetic flux with a contribution from the circulating supercurrent. The phenomenon connects quantum mechanics with macroscopic electromagnetic behavior and provides evidence that the superconducting condensate carries charge in units of $2e$ . (feynmanlectures.caltech.edu)

Magnetic flux and its quantum

Magnetic flux measures the magnetic field passing through an oriented surface $S$ :

\Phi=\int_S\mathbf B\cdot d\mathbf S.

For a superconducting ring in the limit where the current contribution to the fluxoid is negligible, the allowed flux values satisfy

\Phi=n\Phi_0,\qquad n\in\mathbb Z.

The integer can be positive, negative, or zero, according to the orientation and amount of trapped flux. This is a constraint on integrated flux, not a requirement that the local magnetic field have discrete values everywhere. (feynmanlectures.caltech.edu)

In the International System of Units,

\Phi_0=\frac{h}{2e} =2.067833848\ldots\times10^{-15}\ \mathrm{Wb},

where $h$ is the Planck constant, $e$ is the positive elementary charge, and Wb denotes the weber. The defining expression is exact because $h$ and $e$ have exact SI values; the displayed decimal is truncated. (physics.nist.gov)

Origin in superconducting phase coherence

A superconducting condensate is described macroscopically by a complex order parameter,

\Psi(\mathbf r)=|\Psi(\mathbf r)|e^{i\theta(\mathbf r)}.

Its coherent phase extends around the ring. Because $\Psi$ must return to the same value after traversing a closed contour $C$ , its phase can change only by an integer multiple of $2\pi$ :

\oint_C\nabla\theta\cdot d\mathbf l=2\pi n.

The integer $n$ is the phase winding number. This single-valuedness condition is the quantum-mechanical origin of fluxoid quantization. (wmi.badw.de)

For condensate carriers of signed charge $q$ and effective mass $m^\ast$ , the superfluid velocity obeys

m^\ast\mathbf v_s=\hbar\nabla\theta-q\mathbf A,

where $\mathbf A$ is the magnetic vector potential. Integrating around the contour and applying Stokes’s theorem gives

m^\ast\oint_C\mathbf v_s\cdot d\mathbf l+q\Phi=nh.

Thus the quantum depends on the magnitude of the condensate charge. For Cooper pairs, $|q|=2e$ , yielding $h/(2e)$ ; the sign can be absorbed into the definition of the winding number. (feynmanlectures.caltech.edu)

Fluxoid versus magnetic flux

In the London description, for a homogeneous superconductor, the quantization condition can be written

\Phi_{\mathrm f} =\Phi+\mu_0\lambda_L^2 \oint_C\mathbf j_s\cdot d\mathbf l =n\Phi_0,

where $\mathbf j_s$ is the superconducting current density, $\lambda_L$ is the London penetration depth, and $\mu_0$ is the vacuum permeability. The expression on the left is the fluxoid. (wmi.badw.de)

If the ring wall is much thicker than $\lambda_L$ , a contour can be chosen deep inside the material where the supercurrent is negligible. The fluxoid then reduces approximately to magnetic flux. In thin wires or films, the current contribution may remain substantial, so magnetic flux itself need not equal $n\Phi_0$ . The flux in this condition is the total flux, including both the externally applied field and the field generated by circulating currents. (wmi.badw.de)

This distinction also separates superconductivity from perfect classical conductivity. A classical perfectly conducting ring could preserve an arbitrary initial flux. Superconducting quantization additionally requires the single-valued coherent condensate phase. (feynmanlectures.caltech.edu)

Rings, vortices, and changes of winding number

The geometry matters through topology. A contour surrounding a ring’s hole cannot be continuously contracted to a point within the superconducting material. It can therefore support nonzero phase winding. A contractible contour in a region with a nonvanishing, nonsingular order parameter has zero winding. (wmi.badw.de)

In type-II superconductors, magnetic flux can penetrate through Abrikosov vortices. Each elementary vortex has a core where superconductivity is suppressed, surrounded by circulating supercurrent. The phase winds by $2\pi$ around the core, and an isolated elementary vortex carries one flux quantum when its full field distribution is included. Quantization does not mean that its magnetic field is spatially uniform or confined entirely to the core. (webapps.ps.uci.edu)

A ring can remain in a metastable winding state as the applied field changes. Transition to another winding state occurs through a phase slip, during which the order parameter is locally suppressed and its phase winding changes. Such transitions can result from thermal activation, quantum tunneling, or disappearance of the barrier separating metastable states. Their rates depend on temperature, geometry, and the system’s energy landscape; quantization alone does not determine when a transition occurs. (nature.com)

Experimental discovery

Direct evidence was reported independently by Bascom Deaver and William Fairbank, and by Robert Doll and Martin Näbauer, in papers published on July 15, 1961. Both experiments used small hollow superconducting cylinders and measured trapped magnetic flux. The observed spacing corresponded to $h/(2e)$ , rather than the $h/e$ scale associated with single-electron charge. (physics.aps.org)

Fritz London had previously predicted flux quantization from the requirement of a single-valued superconducting wave function. The measured factor of two supported electron pairing, a central feature of BCS theory. Analyses by Nina Byers and C. N. Yang, and by Lars Onsager, clarified the relation between quantized flux and paired charge carriers. (physics.umd.edu)

Applications and modified quantization conditions

A superconducting quantum interference device (SQUID) combines a superconducting loop with one or more Josephson junctions. Fluxoid quantization links the junction phase differences to the enclosed flux. For an ideal symmetric two-junction SQUID with negligible loop inductance,

I_{c,\mathrm{SQUID}}(\Phi) =2I_{c,\mathrm j} \left|\cos\left(\frac{\pi\Phi}{\Phi_0}\right)\right|,

where $I_{c,\mathrm j}$ is the critical current of either junction. This periodic response enables sensitive flux measurements; it does not require the externally applied flux itself to take discrete values. (wmi.badw.de)

The familiar integer rule assumes no additional intrinsic phase shift around the loop. A ring containing an odd number of junctions with an intrinsic $\pi$ phase shift can instead support half-integer flux states, approximately

\Phi=\left(n+\frac12\right)\Phi_0

when current corrections are negligible. Measurements of spontaneous half-flux states in appropriately designed cuprate rings have been used to investigate the sign-changing symmetry of the superconducting order parameter. These states modify the loop’s phase condition rather than changing the value of $\Phi_0$ . (wmi.badw.de)

References

Applied Superconductivity, Chapter 1wmi.badw.de
CODATA Value: magnetic flux quantumphysics.nist.gov
2022 CODATA Recommended Values of the Fundamental Physical Constantsphysics.nist.gov
Introduction to Superconductivitywebapps.ps.uci.edu
Deterministic phase slips in mesoscopic superconducting ringsnature.com
Landmarks—Superconductor Quantizes Magnetic Fieldphysics.aps.org
Experimental Evidence for Quantized Flux in Superconducting Cylinders; Theoretical Considerations Concerning Quantized Magnetic Flux in Superconducting Cylindersphysics.umd.edu
Christoph Scheuer, Master’s Thesiswmi.badw.de
Superconductivity and Low Temperature Physics I, Chapter 7wmi.badw.de