Magnetic Flux Quantum

The magnetic flux quantum, conventionally denoted $\Phi_0$, is a physical constant that sets the scale of magnetic flux quantization in superconductors. It equals $h/(2e)$, where $h$ is the Planck constant and $e$ is the positive elementary charge. Its value is approximately $2.067833848\times10^{-15}$ webers. More precisely, superconducting quantization applies to a quantity called the fluxoid, which combines magnetic flux with a contribution from circulating supercurrent; magnetic flux itself is quantized when that current contribution is negligible. (physics.nist.gov)

Definition and value

In the International System of Units (SI),

The weber is the unit of magnetic flux, with

Because $h$ and $e$ have exact defining values in the SI, $\Phi_0$ is also exact; the ellipsis indicates an uncompleted decimal expansion, not measurement uncertainty. (physics.nist.gov)

Magnetic flux through an oriented surface $S$ is

where $\mathbf B$ is the magnetic field. Flux quantization concerns this integrated quantity, not a universal minimum value of the field at each point. The flux quantum therefore does not mean that every magnetic field, or the flux through every arbitrarily chosen surface, must occur in discrete steps. (wmi.badw.de)

Quantum-mechanical origin

The factor of two in $h/(2e)$ reflects the charge magnitude of a Cooper pair, formed from two electrons. In conventional superconductors, these pairs establish a coherent collective state, described macroscopically by a complex order parameter

Its phase $\theta$ need not be spatially constant. However, when a closed path remains entirely within a region where $\Psi\neq0$, the order parameter must return to the same value after one circuit. Consequently,

The integer $n$ is the phase-winding number. This condition connects quantum mechanics with macroscopic quantum coherence: a restriction on the collective phase produces observable effects in an entire superconducting ring. (feynmanlectures.caltech.edu)

Combining phase winding with the electromagnetic relation between phase, vector potential, and supercurrent gives a quantization unit $h/|q|$ for a condensate of charge $q$. For Cooper pairs, $|q|=2e$, giving $\Phi_0=h/(2e)$. A charge-$e$ phase would instead give the scale $h/e=2\Phi_0$; distinguishing these charge scales was central to interpreting the first experiments. (feynmanlectures.caltech.edu)

Fluxoid versus magnetic flux

For a homogeneous, isotropic superconductor in the local London description, fluxoid quantization can be written

where $\mathbf j_s$ is the superconducting current density, $\lambda_L$ is the London penetration depth, and $\mu_0$ is the vacuum permeability. The contour $C$ lies within the superconducting material and bounds the surface used to calculate $\Phi$. (wmi.badw.de)

In a thick superconducting ring, the Meissner effect confines screening fields and currents near the surfaces. A contour sufficiently far inside the material can have negligible current along it, reducing the condition to

In a thin ring, the current contribution may remain appreciable, so the enclosed magnetic flux need not itself equal an integer multiple of $\Phi_0$. The distinction prevents the common but overly broad statement that all flux through a superconducting loop is necessarily quantized. (feynmanlectures.caltech.edu)

Experimental establishment

Fritz London predicted flux quantization before the microscopic theory of superconductivity was established, but his original charge assignment gave a flux unit twice the experimentally observed value. In 1961, Bascom Deaver and William Fairbank at Stanford, and independently Robert Doll and Martin Näbauer in Germany, observed quantized trapped flux in hollow superconducting cylinders. The observed unit corresponded to $h/(2e)$, supporting the paired-electron description. (feynmanlectures.caltech.edu)

Related experiments by William Little and Richard Parks found periodic changes in the superconducting transition temperature of thin-walled cylinders as applied flux varied. These Little–Parks oscillations demonstrated a consequence of fluxoid quantization even when the magnetic flux itself was not restricted to discrete values. (doi.org)

Applications

The magnetic flux quantum sets the scale for superconducting interference and electronic devices.

A superconducting quantum interference device (SQUID) combines a superconducting loop with one or more Josephson junctions. Its response is periodic in applied flux, with $\Phi_0$ providing the characteristic period. This enables magnetic measurements based on changes much smaller than one flux quantum; the quantum is not a lower bound on measurement resolution. (nobelprize.org)

The same constant appears in the Josephson effect. A junction at voltage $V$ has an oscillation frequency

When synchronized to an applied frequency $f$, its quantized voltage steps satisfy

This relation underlies Josephson voltage standards used in metrology. (nist.gov)

In single-flux-quantum electronics, a $2\pi$ change in junction phase produces a voltage pulse whose time integral is

Such pulses provide reproducible units for superconducting signal processing and waveform synthesis. Here, the flux quantum appears as a voltage–time area rather than simply as trapped magnetic flux. (nist.gov)

References

1. CODATA Value: magnetic flux quantumphysics.nist.gov
2. 2022 CODATA Recommended Values of the Fundamental Physical Constantsphysics.nist.gov
3. Applied Superconductivity: Chapter 1wmi.badw.de
4. Lecture 10: Supercurrent Equationweb.mit.edu
5. Fluxoid Quantization in a Multiply-Connected Superconductordoi.org
6. From SLUGs to macroscopic quantum phenomenanobelprize.org
7. Quantum Locking Rangesnist.gov
8. Quantum Voltage Projectnist.gov
9. US 11,557,708 B2nist.gov
10. Calibrating Next-Gen Telecom at 5G and Beyondnist.gov