Ginzburg–Landau theory

The Ginzburg - Landau theory, also called Glag theory ( after the initials of the inventor Vitaly Lasarevich Ginsburg, who for 2003 was awarded the Nobel Prize in Physics, and Lev Davidovich Landau, Alexei Abrikossow, Lev Gorkov ), is a theory describing superconductivity.

In contrast to the BCS theory, which seeks a declaration at the microscopic level, it examines the macroscopic properties of superconductors using general thermodynamic arguments. It is a phenomenological theory, which was correct at the time of its formation in 1950, only that was originally chosen rather than the charge of the Cooper pairs of charge 2e of the general parameter q. 1959 was the Ginzburg - Landau theory are derived from the BCS theory, where one particular recognized the identification.

The Ginzburg - Landau theory is a gauge theory and arises as a scaling limit (s: scaling limit) of the XY model.

• 4.1 coherence length
• 4.2 depth
• 4.3 Ginzburg -Landau parameter
• 4.4 flux tubes

Mathematical formulation

Based on Landau's theory of second order phase transitions and Ginsburg Landau argued that the free energy F of a superconductor near the phase transition by a complex order parameter can be expressed as ψ. This describes the extent to which the system is in the superconducting state; ψ = 0 corresponds to the normal state with no superconductivity.

The free energy is then:

With

• Fn: the free energy in the normal state,
• α and β: phenomenological parameters,
• : The vector potential and
• The magnetic induction, which is related to the relationship.

The minimization of the free energy with respect to the fluctuations of the order parameter and the vector potential leads to the two Ginzburg -Landau equations:

While j denotes the electric current density and the real part Re.

In the mathematical treatment of the Ginzburg -Landau model Fabrice Bethuel, Frédéric Hélein, Haim Brezis and Sylvia Serfaty made ​​significant progress. They showed, among other things, that the vortex is defined for large values ​​of the order parameter by the values ​​of a renormalized energy.

Interpretation of a special case

Considering a homogeneous superconductors without external magnetic field, then the first Ginzburg -Landau equation simplifies to:

The trivial solution of this equation is the normal state of the metal ( non- superconducting state), present at temperatures above the transition temperature.

Below the critical temperature of a non-trivial solution is expected. Under this assumption, the above equation can be transformed into:

The magnitude of the complex number on the left side of the equation is not negative, that is, Thus also be square, and thus the right side of the equation. For the non- trivial solution ψ of the term on the right-hand side must be positive, ie. This can be achieved by adopting the following temperature dependence for α:

• Below the critical temperature (), the expression is negative, the right-hand side of the above equation is positive and there is a non - trivial solution for ψ. Also in this case:

• Above the critical temperature ( ) of the expression is positive, and the right side of the above equation is negative. In this case, only the Ginzburg -Landau solves equation.

In Ginsburg Landau theory it is believed that those electrons which contribute to the superconductivity, are condensed to form a super liquid. After that just describes this proportion of electrons.

Relations with other theories

To the Schrödinger equation

The first Ginzburg -Landau equation has interesting similarities to the time-independent Schrödinger equation; However, note that ψ as in quantum mechanics is here not a probability amplitude, but the specified quasi- classical meaning ( is the density of the carriers of superconductivity, the Cooper pairs ). Mathematically, it is a time-independent Gross- Pitaevskii equation which is a nonlinear generalization of the Schrödinger equation. Thus, the first equation determines the order parameter ψ as a function of applied magnetic field.

The London equation

The second Ginzburg -Landau equation gives the supercurrent and corresponds to the London equation.

The Higgs mechanism

Formally, there is a great similarity between the phenomenological description of superconductivity by Ginzburg and Landau and the Higgs - Kibble mechanism in high energy physics: the Meissner effect of superconductivity, which amounts to the generation of a finite penetration depth of the magnetic induction, corresponding to the mass production the gauge fields of high energy physics, if one uses the usual translation ( is Planck's constant divided by, and the speed of light). That is, the penetration depth is identified as the Compton wavelength of a mass m.

Derivations from the theory

From the Ginzburg -Landau equations can be derived many interesting results. Probably the most important is the existence of two characteristic lengths in superconductors.

Coherence length

The first is the coherence length ξ,

Which describes the size of the fluctuations in the thermodynamic of the superconducting phase.

Penetration depth

The second is the depth λ,

ψ0 wherein the order parameter is in equilibrium, without the electromagnetic field, respectively. The penetration depth is the depth of back up to the external magnetic field to penetrate into the superconductor.

Ginsburg -Landau parameter

The ratio of these two characteristic lengths is called the Ginzburg -Landau parameter. Depending on its size can be superconductors in two classes with different physical properties:

• Type I superconductors are such.
• Type II superconductors are with such. They retain their superconducting properties even under the influence of strong magnetic fields (for certain alloys up to 25 Tesla).

This result can be derived by means of a dual Ginzburg - Landau theory for superconductors. It is in both cases a phase transition of second order.

Flux tubes

Another important result of the Ginzburg - Landau theory was found in 1957 by Alexei Alexeyevich Abrikossow. In a type II superconductor in a high magnetic field in the form of channels with quantized flux penetrates. These so-called flux tubes or vortices form a - often hexagonal - Abrikossow lattice.

Credentials

Selected Publications

• V.L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950). English Translation in: LD Landau, Collected Papers (Oxford: Pergamon Press, 1965) p. 546
• A. A. Abrikossow, Zh. Eksp. Teor. Fiz. 32, 1442 (1957 )
• L.P. Gor'kov, Sov. Phys. JETP 36, 1364 (1959 )

• D. Saint- James, G. Sarma and EJ Thomas, Type II Superconductivity Pergamon (Oxford 1969)
• M. Tinkham, Introduction to Superconductivity, McGraw- Hill (New York 1996)
• Hagen Kleinert, Gauge Fields in Condensed Matter, Vol I, World Scientific (Singapore, 1989); Paperback ISBN 9971-5-0210-0 (also available online here )
• Pierre -Gilles de Gennes, Superconductivity of Metals and Alloys, WA Benjamin, 1964 [ ISBN 0-7382-0101-4 ]

• Superconductivity
• Thermodynamics