Ginsburg-Landau theory

The Ginsburg-Landau theory , also called the GLAG theory (after the first letters of the inventors Witali Ginsburg , Lew Landau , Alexei Abrikossow , Lew Gorkow ), is a theory to describe superconductivity . Ginsburg and Abrikosov received for 2003 together with Leggett the Nobel Prize in Physics .

In contrast to the BCS theory , which seeks an explanation on a microscopic basis, it investigates the macroscopic properties of superconductors with the help of general thermodynamic arguments. It is a phenomenological theory, which was correct at the time of its formation in 1950, except that initially the charge of the place of Cooper pairs from the general charge parameter was selected. In 1959, Gorkow was able to derive the Ginsburg-Landau theory from the BCS theory, whereby the identification in particular was recognized. ${\displaystyle -2e}$ ${\displaystyle q}$ ${\displaystyle q\,\rightarrow \,-2e}$

The Ginsburg-Landau theory is a calibration theory . The theory specially formulated for superconductors is a special case of the more general Landau theory of phase transitions.

Mathematical formulation

Building on Landau's theory of second order phase transitions , Landau and Ginsburg argued that the free energy of a superconductor near the phase transition can be expressed by a complex order parameter . This describes to what extent the system is in the superconducting state; corresponds to the normal state without superconductivity. ${\displaystyle F}$ ${\displaystyle \psi }$ ${\displaystyle \psi =0}$

The free energy is then:

${\displaystyle F=F_{n}+\alpha |\psi |^{2}+{\frac {\beta }{2}}|\psi |^{4}+{\frac {1}{2m^{*}}}\left|\left(-\mathrm {i} \hbar {\vec {\nabla }}+2e{\vec {A}}\right)\psi \right|^{2}+{\frac {|{\vec {B}}|^{2}}{2\mu _{0}}}}$,

With

• ${\displaystyle F_{n}}$: the free energy in the normal state,
• ${\displaystyle \alpha }$and : phenomenological parameters, ${\displaystyle \beta }$
• ${\displaystyle m^{*}}$: effective mass (later identified with the mass of Cooper pairs)
• ${\displaystyle {\vec {A}}}$: the vector potential and
• ${\displaystyle {\vec {B}}}$: the magnetic induction that is related to about the relationship . ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {B}}=\operatorname {rot} {\vec {A}}}$

The identification of the charge with that of Cooper pairs ( ) was used in the term for the minimal coupling . ${\displaystyle -2e}$

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

${\displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\frac {1}{2m^{*}}}\left(-\mathrm {i} \hbar {\vec {\nabla }}+2e{\vec {A}}\right)^{2}\psi =0}$ and

${\displaystyle {\vec {j}}=-{\frac {2e}{m^{*}}}\operatorname {Re} \left\{\psi ^{*}\left(-\mathrm {i} \hbar {\vec {\nabla }}+2e{\vec {A}}\right)\psi \right\}}$.

It denotes the electrical current density and the real part . ${\displaystyle {\vec {j}}}$ ${\displaystyle \operatorname {Re} }$

Fabrice Béthuel , Frédéric Hélein , Haïm Brezis and Sylvia Serfaty made significant advances in the mathematical treatment of the Ginsburg-Landau model . They showed, among other things, that the vortex for large values ​​of the order parameter is determined by the values ​​of a renormalized energy.

Interpretation of a special case

If you consider a homogeneous superconductor without an external magnetic field, the first Ginsburg-Landau equation simplifies to:

${\displaystyle \alpha \,\psi +\beta |\psi |^{2}\psi =0}$,

The trivial solution of this equation corresponds to the normal state of the metal (non-superconducting state), which is present at temperatures above the transition temperature . ${\displaystyle \psi \,=0}$ ${\displaystyle T_{\text{c}}}$

A non-trivial solution is expected below the transition temperature . With this assumption, the above equation can be transformed into: ${\displaystyle \psi \not =0}$

${\displaystyle |\psi |^{2}=-{\frac {\alpha }{\beta }}}$ .

The magnitude of the complex number on the left side of the equation is nonnegative ; H. , with it its square and with it the right side of the equation. For the non-trivial solution of , the term on the right-hand side must be positive , i.e. H. . This can be achieved by assuming the following temperature dependence for : ${\displaystyle \geq 0}$ ${\displaystyle \psi }$ ${\displaystyle >0}$ ${\displaystyle \alpha }$

${\displaystyle \alpha (T)\,=\alpha _{\text{0}}(T-T_{\text{c}})}$, with . ${\displaystyle {\frac {\alpha _{\text{0}}}{\beta }}>0}$

• Below the transition temperature ( ) the expression is negative, the right side of the above equation is positive and there is a non-trivial solution for . The following also applies in this case: ${\displaystyle T<T_{\text{c}}}$ ${\displaystyle \alpha (T)/\beta }$ ${\displaystyle \psi }$

${\displaystyle \,|\psi |^{2}=-{\frac {\alpha _{0}(T-T_{c})}{\beta }}}$, d. H. approaches zero when the temperature tends towards the transition temperature from below . Such behavior is typical of a second order phase transition. ${\displaystyle \psi }$ ${\displaystyle T}$ ${\displaystyle T_{c}}$

• Above the transition temperature ( ) the expression is positive and the right side of the above equation is negative. In this case only the Ginsburg – Landau equation solves . ${\displaystyle T>T_{\text{c}}}$ ${\displaystyle \alpha (T)/\beta }$ ${\displaystyle \psi \,=0}$

In the Ginsburg – Landau theory it is assumed that those electrons that contribute to superconductivity are condensed into a superfluid . According to this describes precisely this proportion of electrons. ${\displaystyle \,|\psi |^{2}}$

Relationships with other theories

To the Schrödinger equation

The first Ginsburg – Landau equation shows interesting similarities to the time-independent Schrödinger equation ; Note, however, that here is not a probability amplitude, as in quantum mechanics , but has the quasi-classical meaning given ( is the density of the superconductor carriers, the Cooper pairs). Mathematically, it is a time-independent Gross-Pitaevskii equation , which is a nonlinear generalization of the Schrödinger equation. The first equation thus determines the order parameter as a function of the applied magnetic field. ${\displaystyle \psi }$ ${\displaystyle |\psi |^{2}}$ ${\displaystyle \psi }$

To the London equation

The second Ginsburg – Landau equation gives the supercurrent and corresponds to the London equation .

About the Higgs mechanism

Formally, there is a great similarity between the phenomenological description of superconductivity by Ginsburg and Landau and the Higgs kibble mechanism in high-energy physics . The Meißner-Ochsenfeld effect of superconductivity is described with the aid of a finite penetration depth of magnetic induction. But at the same time this corresponds to a mass term in the electromagnetic calibration fields of high energy physics, if one uses the usual translation ( is Planck's quantum of action , divided by , and the speed of light ). The penetration depth is interpreted as the Compton wavelength of the mass . ${\displaystyle \lambda }$ ${\displaystyle A^{\alpha }}$ ${\displaystyle \lambda =\hbar /(M\cdot c)}$ ${\displaystyle \hbar }$ ${\displaystyle 2\pi }$ ${\displaystyle c}$ ${\displaystyle M}$

Derivation from the theory

Many interesting results can be derived from the Ginsburg-Landau equations. Probably the most significant is the existence of two characteristic lengths in superconductors.

Coherence length

The first is the coherence length ξ ,

${\displaystyle \xi ={\sqrt {\frac {\hbar ^{2}}{2m^{*}|\alpha |}}}}$.

which describes the magnitude of the thermodynamic fluctuations in the superconducting phase.

Penetration depth

The second is the depth of penetration , ${\displaystyle \lambda }$

${\displaystyle \lambda ={\sqrt {\frac {m^{*}}{4\mu _{0}e^{2}\psi _{0}^{2}}}}}$

where denotes the order parameter in equilibrium, without an electromagnetic field. The penetration depth reflects the depth to which an external magnetic field can penetrate the superconductor. ${\displaystyle \psi _{0}}$

Note: SI units were used here. The cgs units often used in the literature result in:

${\displaystyle \lambda ={\sqrt {\frac {m^{*}\cdot c^{2}}{4\pi e^{2}\psi _{0}^{2}}}}}$

Ginsburg-Landau parameters

The ratio of these two characteristic lengths is called the Ginsburg-Landau parameter . Depending on their size, superconductors can be divided into two classes with different physical properties (according to Abrikossow 1957): ${\displaystyle \kappa ={\frac {\lambda }{\xi }}}$

• Type I superconductors are those with . ${\displaystyle \kappa <1/{\sqrt {2}}=0,707\dots }$
• Type II superconductors are those with . They retain their superconducting properties even under the influence of strong magnetic fields (for certain alloys up to 25 Tesla ). ${\displaystyle \kappa >1/{\sqrt {2}}=0,707\dots }$

It is a second order phase transition.

Flow hoses

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

credentials

1. ↑ VL Ginzburg: On superconductivity and superfluidity (what I have and have not managed to do), as well as on the 'physical minimum' at the beginning of the 21st century . In: Chemphyschem. tape 5 , no. 7 , July 2004, p. 930-945 , doi : 10.1002 / cphc.200400182 , PMID 15298379 .
2. ^ For example Tinkham, Introduction to superconductivity, McGraw Hill 1996, p. 19, De Gennes, Superconductivity of metals and alloys, Westview 1999, p. 24
3. ↑ Abrikossow, Sov. Phys. JETP, 5, 1957, p. 1174, see also Tinkham Introduction to Superconductivity, McGraw Hill 1996, p. 11

Selected publications

• VL Ginzburg, LD Landau: To the Theory of Superconductivity. In: Zh. Eksp. Teor. Fiz. 20 (1950), p. 1064. English translation in: LD Landau: Collected papers. Pergamon Press, Oxford 1965, p. 546.
• AA Abrikossow: On the Magnetic Properties of Superconductors of the Second Group. In: Zh. Eksp. Teor. Fiz. 32, 1957, pp. 1442-1452.
• LP Gor'kov: Microscopic Derivation of the Ginzburg-Landau Equations in the Theory of Superconductivity. In: Sov. Phys. JETP. 36, 1364 (1959)

Books

• D. Saint-James, G. Sarma, EJ Thomas: Type II superconductivity. Pergamon Press, 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, ISBN 9971-5-0210-0 . (online at: physik.fu-berlin.de )
• Pierre-Gilles de Gennes , Superconductivity of Metals and Alloys. WA Benjamin, 1964. (Perseus Books, Reading, Mass. 1999, ISBN 0-7382-0101-4 )

Web links

• Nobel Lecture by Ginzburg