{{safesubst:#invoke:Unsubst||date=__DATE__|$B= Template:Ambox }}Template:Short description
The magnetic flux, represented by the symbol Template:Math, threading some contour or loop is defined as the magnetic field Template:Math multiplied by the loop area Template:Math, i.e. Template:Math. Both Template:Math and Template:Math can be arbitrary, meaning that the flux Template:Math can be as well but increments of flux can be quantized. The wave function can be multivalued as it happens in the Aharonov–Bohm effect or quantized as in superconductors. The unit of quantization is therefore called magnetic flux quantum.
Dirac magnetic flux quantumEdit
The first to realize the importance of the flux quantum was Dirac in his publication on monopoles<ref>Template:Cite journal</ref>
ISQ | CGS units |
---|---|
<math>\Phi_0 = \frac{2\pi \hbar }{q}= \frac{h }{q}</math> | <math>\Phi_0 = \frac{2\pi \hbar c}{q}= \frac{h c}{q}</math> |
The phenomenon of flux quantization was predicted first by Fritz London then within the Aharonov–Bohm effect and later discovered experimentally in superconductors (see Template:Slink below).
Superconducting magnetic flux quantumEdit
CODATA values | Units | |
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Template:Math0 | Template:Physconst | Wb |
Template:MathJ | Template:Physconst | Hz/V |
If one deals with a superconducting ring<ref>Template:Cite journal</ref> (i.e. a closed loop path in a superconductor) or a hole in a bulk superconductor, the magnetic flux threading such a hole/loop is quantized.
The (superconducting) magnetic flux quantum Template:Nowrap is a combination of fundamental physical constants: the Planck constant Template:Math and the electron charge Template:Math. Its value is, therefore, the same for any superconductor.
To understand this definition in the context of the Dirac flux quantum one shall consider that the effective quasiparticles active in a superconductors are Cooper pairs with an effective charge of 2 electrons Template:Math.
The phenomenon of flux quantization was first discovered in superconductors experimentally by B. S. Deaver and W. M. Fairbank<ref name=Deaver:1961:FluxQuantum /> and, independently, by R. Doll and M. Näbauer,<ref name=Doll:1961:FluxQuantum /> in 1961. The quantization of magnetic flux is closely related to the Little–Parks effect,<ref>Template:Cite journal</ref> but was predicted earlier by Fritz London in 1948 using a phenomenological model.<ref>Template:Cite book</ref><ref name=":0" />
The inverse of the flux quantum, Template:Math, is called the Josephson constant, and is denoted Template:MathJ. It is the constant of proportionality of the Josephson effect, relating the potential difference across a Josephson junction to the frequency of the irradiation. Template:AnchorThe Josephson effect is very widely used to provide a standard for high-precision measurements of potential difference, which (from 1990 to 2019) were related to a fixed, conventional value of the Josephson constant, denoted Template:MathJ-90. With the 2019 revision of the SI, the Josephson constant has an exact value of Template:MathJ = Template:Val.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Derivation of the superconducting flux quantumEdit
The following physical equations use SI units. In CGS units, a factor of Template:Math would appear.
The superconducting properties in each point of the superconductor are described by the complex quantum mechanical wave function Template:Math – the superconducting order parameter. As with any complex function, Template:Math can be written as Template:Math, where Template:Math is the amplitude and Template:Math is the phase. Changing the phase Template:Math by Template:Math will not change Template:Math 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 Template:Mvar may continuously change from some value Template:Math to the value Template:Math as one goes around the hole/loop and comes to the same starting point. If this is so, then one has Template:Mvar magnetic flux quanta trapped in the hole/loop,<ref name=":0">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> as shown below:
Per minimal coupling, the current density of Cooper pairs in the superconductor is: <math display="block">\mathbf J = \frac{1}{2m} \left[\left(\Psi^* (-i\hbar\nabla) \Psi - \Psi (-i\hbar\nabla) \Psi^*\right) - 2q \mathbf{A} |\Psi|^2 \right] .</math> where Template:Math is the charge of the Cooper pair. The wave function is the Ginzburg–Landau order parameter: <math display="block">\Psi(\mathbf{r})=\sqrt{\rho(\mathbf{r})} \, e^{i\theta(\mathbf{r})}.</math>
Plugged into the expression of the current, one obtains: <math display="block">\mathbf{J} = \frac{\hbar}{m} \left(\nabla{\theta}- \frac{q}{\hbar} \mathbf{A}\right)\rho.</math>
Inside the body of the superconductor, the current density J is zero, and therefore <math display="block">\nabla{\theta} = \frac{q}{\hbar} \mathbf{A}.</math>
Integrating around the hole/loop using Stokes' theorem and Template:Math gives: <math display="block">\Phi_B = \oint\mathbf{A}\cdot d\mathbf{l} = \frac{\hbar}{q} \oint\nabla{\theta}\cdot d\mathbf{l}.</math>
Now, because the order parameter must return to the same value when the integral goes back to the same point, we have:<ref> R. Shankar, "Principles of Quantum Mechanics", eq. 21.1.44</ref> <math display="block">\Phi_B=\frac{\hbar}{q} 2\pi = \frac{h}{2e}.</math>
Due to the Meissner effect, the magnetic induction Template:Math inside the superconductor is zero. More exactly, magnetic field Template:Math penetrates into a superconductor over a small distance called London's magnetic field penetration depth (denoted Template:Math and usually ≈ 100 nm). The screening currents also flow in this Template:Math-layer near the surface, creating magnetization Template:Math inside the superconductor, which perfectly compensates the applied field Template:Math, thus resulting in Template:Math inside the superconductor.
The magnetic flux frozen in a loop/hole (plus its Template:Math-layer) will always be quantized. However, the value of the flux quantum is equal to Template:Math 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 Template:Math away from the surface. There are geometries where this condition cannot be satisfied, e.g. a loop made of very thin (Template:Math) superconducting wire or the cylinder with the similar wall thickness. In the latter case, the flux has a quantum different from Template:Math.
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 Template:Math and the second critical field Template:Math, 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 Template:Math, 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 Template:Math-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 Template:Math.
Measuring the magnetic fluxEdit
Prior to the 2019 revision of the SI, the magnetic flux quantum was measured with great precision by exploiting the Josephson effect. When coupled with the measurement of the von Klitzing constant Template:Math, this provided the most accurate values of the Planck constant Template:Math obtained until 2019. This may be counterintuitive, since Template:Math is generally associated with the behaviour 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.
As a result of the 2019 revision of the SI, the Planck constant Template:Math has a fixed value Template:Nowrap which, together with the definitions of the second and the metre, provides the official definition of the kilogram. Furthermore, the elementary charge also has a fixed value of Template:Nowrap to define the ampere. Therefore, both the Josephson constant Template:Math and the von Klitzing constant Template:Math have fixed values, and the Josephson effect along with the von Klitzing quantum Hall effect becomes the primary mise en pratique<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> for the definition of the ampere and other electric units in the SI.
See alsoEdit
- Aharonov–Bohm effect
- 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
ReferencesEdit
Further readingEdit
- Aharonov–Bohm effect and flux quantization in superconductors Template:Citation
- David tong lectures: Template:Citation