In physics, Ginzburg–Landau theory, often called Landau–Ginzburg theory, named after Vitaly Ginzburg and Lev Landau, is a mathematical physical theory used to describe superconductivity. In its initial form, it was postulated as a phenomenological model which could describe type-I superconductors without examining their microscopic properties. Alexei Abrikosov later used the theory to describe type-II superconductors as well. One GL-type superconductor is YBCO, and generally all cuprates.
Later, a version of Ginzburg–Landau theory was derived from the Bardeen–Cooper–Schrieffer microscopic theory by Lev Gor'kov, thus showing that it also appears in a limit of the microscopic theory and giving it a microscopic interpretation of all its parameters. The theory can also be given a general geometric setting, placing it in the context of Riemannian geometry, where in many cases exact solutions can be given. This general setting then extends to quantum field theory and string theory, again owing to its solvability, and its close relation to other, similar systems.
Ginzburg was awarded a third of the 2003 Nobel prize in physics for his development of the theory. Abrikosov was awarded another third of this prize for his description of type-II superconductors using the theory.
Introduction Free energy Based on Landau's previously established theory of second-order phase transitions Ginzburg and Landau argued that the free energy density of a superconductor near the superconducting transition can be expressed in terms of a complex order parameter field , where the quantity is a measure of the local density of superconducting electrons analogous to a quantum mechanical wave function. In the convention used in this article, the relation is , which is the density of Cooper pairs. While is nonzero below a phase transition into a superconducting state, no direct interpretation of this parameter was given in the original paper. Assuming smallness of and smallness of its gradients, the free energy density has the form of a field theory and exhibits U(1) gauge symmetry. In Gaussian units, it is written as
where is the free energy density of the normal phase, and are phenomenological parameters that are functions of (and often written just and ). the mass of a Cooper pair, is the charge of a Cooper pair, is the magnetic vector potential, and is the magnetic field.
The total free energy is given by .
Ginzburg–Landau equations By minimizing with respect to variations in the order parameter and the vector potential , one arrives at the Ginzburg–Landau equations:
where denotes the dissipation-free electric current density, or supercurrent density. The first equation, which bears some similarities to the time-independent Schrödinger equation but is principally different due to a nonlinear term, determines the order parameter, . The second equation is instead Maxwell-like and provides the superconducting current and the magnetic field.
Additionally, the boundary condition at the surface of the superconductor can be determined to be
where is the vector normal to the surface and denotes the location of the boundary.
Close to the superconducting transition, it is assumed that is a positive constant and that
where is a positive constant and is the critical temperature.
Simple interpretation
Consider a homogeneous superconductor where is constant and there is no external magnetic field. The equation for the free energy then simplifies to
The equilibrium order parameter is found by minimizing this equation with respect to .
Above the critical temperature, is positive and the only minimum is given by . This corresponds to the absence of superconductivity.
Below the superconducting transition, is negative, and the minima are given by
The density of electrons participating in the supercurrent is given by
Thus, this corresponds to the superconducting phase.
Characteristic quantities Coherence length and penetration depth The Ginzburg–Landau equations predicts two characteristic lengths in a superconductor. The first characteristic length is termed coherence length. In one dimension at zero magnetic field, the first GL equation becomes
At an interface between a superconductor and a normal conductor, this equation is solved by
where
is the coherence length. This length scale also sets the exponential law according to which small perturbations of density of superconducting electrons recover their equilibrium value .
The second length scale is the penetration depth. Consider an external magnetic field applied to a semi-infinite superconducting slab. Assuming that is constant on the length scale we are considering, the second GL equation becomes
This is equation is solved for the magnetic field by
where
is the penetration depth. By identifying the order parameter with the density of superconducting electrons, it becomes apparent that this is equivalent to the penetration depth in the London theory.
The ratio between these two length scales, is called the Ginzburg–Landau parameter. The importance of this property is that it distinguishes between type I and type II superconductors; superconductors for which are of type I, and those for which are of type II.
Critical magnetic field To calculate the magnetic field above which superconductivity is destroyed in the bulk, is considered to be constant and in equilibrium. Using these assumptions, the free energy is
At the phase transition, the free energies of the normal and superconducting phases should be equal. This gives
Dimensionless form of the Ginzburg–Landau equations In Gaussian units, the GL equations read
with the boundary condition
By changing to a unit system which is defined by the characteristic quantities, the equations can be made dimensionless. For this, the units become
Then, the GL equations become dimensionless:
with the boundary condition
In this unit system, the equations then depend only on the Ginzburg–Landau parameter .
Applications Thin films One of the systems that can be analysed with the GL theory is a thin film. Consider a type-I () superconducting film extending infinitely far in the y- and z-directions, with a thickness . In addition, an external magnetic field is applied in the z-direction.
By symmetry, only derivatives to x can be nonzero. This means that the gauge of the vector potential can be chosen such that
Using this gauge, and choosing to be real, the first dimensionless GL equations becomes
Because of the assumption that the superconductor is of type-I, it is reasonable to assume that is approximately constant over the thickness of the thin film. The second GL equation then becomes
By using that the magnetic field must be continuous at the boundary, i.e. , the solution to for the magnetic field is
In the second-order approximation of the first GL equation, the relation between and can be found to be
In this system, the difference in the free energies of the superconducting and normal state is
Using the expression for and that at the critical magnetic field, this has to be zero. Therefore,
At the transition point, , so
Therefore, the critical magnetic field is higher for a thin film than for a bulk superconductor. Moreover, it can be derived that for a thin film, the phase transition between the normal and superconducting phase is of second order, while it is of first order in a bulk material. This change in the order of the phase transition happens for . Flux quantization The order parameter can be written in terms of the modulus and the phase as . The supercurrent is then
Now consider a hollow superconducting cylinder that encloses a non-superconducting material. In the non-superconducting region, and external magnetic field is applied. Integrating the supercurrent over a contour inside the hollow cylinder that encloses the non-superconducting region, one gets
Inside the wall, there is no current and the first integral vanishes. From electromagnetism, it is known that
where is the magnetic flux. Finally, the first integral on the left side is constrained by the fact that must be a single-valued function. Because for any integer ,
Putting it together, these constraints imply that
Thus, the flux inside a hollow cylinder can only be a multiple of the magnetic flux quantum . This phenomenon is known as flux quantization.
Fluctuations
The phase transition from the normal state is of second order for Type II superconductors, taking into account fluctuations, as demonstrated by Dasgupta and Halperin, while for Type I superconductors it is of first order, as demonstrated by Halperin, Lubensky and Ma.
Classification of superconductors
In the original paper Ginzburg and Landau observed the existence of two types of superconductors depending on the energy of the interface between the normal and superconducting states. The Meissner state breaks down when the applied magnetic field is too large. Superconductors can be divided into two classes according to how this breakdown occurs. In Type I superconductors, superconductivity is abruptly destroyed when the strength of the applied field rises above a critical value Hc. Depending on the geometry of the sample, one may obtain an intermediate state consisting of a pattern of regions of normal material carrying a magnetic field mixed with regions of superconducting material containing no field. In Type II superconductors, raising the applied field past a critical value Hc1 leads to a mixed state (also known as the vortex state) in which an increasing amount of magnetic flux penetrates the material, but there remains no resistance to the flow of electric current as long as the current is not too large. At a second critical field strength Hc2, superconductivity is destroyed. The mixed state is actually caused by vortices in the electronic superfluid, sometimes called fluxons because the flux carried by these vortices is quantized. Most pure elemental superconductors, except niobium and carbon nanotubes, are Type I, while almost all impure and compound superconductors are Type II.
The most important finding from Ginzburg–Landau theory was made by Alexei Abrikosov in 1957. He used Ginzburg–Landau theory to explain experiments on superconducting alloys and thin films. He found that in a type-II superconductor in a high magnetic field, the field penetrates in a triangular lattice of quantized tubes of flux vortices.Abrikosov, A. A. (1957). The magnetic properties of superconducting alloys. Journal of Physics and Chemistry of Solids, 2(3), 199–208.
Geometric formulation The Ginzburg–Landau functional can be formulated in the general setting of a complex vector bundle over a compact Riemannian manifold. This is the same functional as given above, transposed to the notation commonly used in Riemannian geometry. In multiple interesting cases, it can be shown to exhibit the same phenomena as the above, including Abrikosov vortices (see discussion below).
For a complex vector bundle over a Riemannian manifold with fiber , the order parameter is understood as a section of the vector bundle . The Ginzburg–Landau functional is then a Lagrangian for that section:
The notation used here is as follows. The fibers are assumed to be equipped with a Hermitian inner product so that the square of the norm is written as . The phenomenological parameters and have been absorbed so that the potential energy term is a quartic mexican hat potential; i.e., exhibiting spontaneous symmetry breaking, with a minimum at some real value . The integral is explicitly over the volume form
for an -dimensional manifold with determinant of the metric tensor .