Magnetic flux quantum
2010 CODATA values | Units | |
---|---|---|
Φ_{0} | 833758(46)×10^{−15} 2.067 | Wb |
K_{J} | 597.870(11)×10^{9}^{[1]} 483 | Hz/V |
K_{J–90} | 597.9×10^{9} 483 | Hz/V |
The magnetic flux, represented by the symbol Φ, threading some contour or loop is defined as the magnetic inductance B multiplied by the loop area S, i.e. Φ = B · S. Obviously, 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/(2e) ≈ 833758(46)×10^{−15} Wb 2.067^{[2]} 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^{[3]} and, independently, by R. Doll and M. Näbauer,^{[4]} 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}.
Contents
- Introduction 1
- Measuring the magnetic flux 2
- See also 3
- References 4
Introduction
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. It is obvious that 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 θ_{0} to the value θ_{0} + 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.
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 play in important role in the physics of a 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^{[5]} and split into several vortices with n = 1. 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/e^{2}, this provides the most precise values of Planck's constant h obtained to date. This is remarkable 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 collective phenomena associated with thermodynamically large numbers of particles.
See also
- Domain wall (magnetism)
- Flux pinning
- Ginzburg–Landau theory
- Husimi Q representation
- Magnetic domain
- Quantum vortex
- Topological defect
- Macroscopic quantum phenomena
- Committee on Data for Science and Technology
- Brian Josephson
- Dirac flux quantum
- von Klitzing constant
References
- ^ "_{J}K"Josephson constant . 2010 CODATA recommended values. Retrieved 10 January 2012.
- ^ "_{0}Φ"magnetic flux quantum . 2010 CODATA recommended values. Retrieved 10 January 2012.
- ^ Deaver, Bascom; Fairbank, William (July 1961). "Experimental Evidence for Quantized Flux in Superconducting Cylinders". Physical Review Letters 7 (2): 43–46.
- ^ Doll, R.; Näbauer, M. (July 1961). "Experimental Proof of Magnetic Flux Quantization in a Superconducting Ring". Physical Review Letters 7 (2): 51–52.
- ^ In mesoscopic superconducting samples with sizes ≃ ξ one can observe giant vortices with n > 1