Ginzburg-Landau (GL) theory is a mathematical theory used to describe superconductivity. Ginzburg-Landau (GL) theory is used to explain the difference between Type-I and Type-II superconductors and enables the calculation of two critical magnetic fields H_c1 and H_c2 [10] . Ginzburg-Landau theory was derived from the BCS microscopic theory by Lev Gorkov, showing that it also appears in some limit of microscopic theory and applying microscopic interpretation of all its parameters.

The basic postulate of GL is that if ψ is small and varies slowly in space, the free-energy density can be expanded in a series of the form:

where α and β are phenomenological parameters, (β is positive and the sign of α is temperature dependent), is an effective mass, is the charge of an electron, is the magnetic vector potential and is the magnetic field [5] [10] .

If, Equation (2) reduces to the free energy of the normal state;.

Now, by minimizing the free energy with respect to fluctuations in the order parameter and the vector potential, one arrives at the Ginzburg-Landau Equations given by:

The current density from the Hamilton’s energy of particles is given by;

where

where

where is supercurrent density.

Equation (3) determines the order parameter ψ based on the applied magnetic field and Equation (5) yields the superconducting current density. The Ginzburg-Landau Equation provides complete information about the superconducting state that gives the spatial distribution of the Cooper pair density taking into account a possible variation in their concentration, where as describes the local distribution of the magnetic field in the superconductor.

In the absence of external magnetic field (at free surface), there will not be superconducting current(current flow) and the Equation for ψ becomes:

This Equation has a trivial solution ψ = 0 and it corresponds to normal state of. Below superconducting transition temperature (T_c), Equation (6) is expected to have a non-trivial solution (i.e.) and the Equation can be rearranged into:

If the second part of Equation (3) is positive, then there is a non zero solution for ψ and this can be achieved

by assuming the temperature dependence of α such that with and. For, the expression is positive and the second part of Equation (3) is negative and only ψ = 0

solves the Ginzburg-Landau Equation. For, the second part of Equation (3) is positive and there is a non-trivial solution for ψ. Thus Equation (7) can be expressed as:

Equation (8) yields Ginzburg Landau order parameter [5] [11] .

The Ginzbrug-Landau coherence length (ξ_GL) is a measure of the distance in the superconducting electron concentration that can not change drastically in a spatially-varying magnetic field. The Ginzbrug-Landau coherence length (ξ_GL) is a temperature-dependent as well as a material dependent quantity. In the case of absence of the magnetic vector potential, Equation (3) reduces to:

Now, let us consider a wave function that varies only in the z-direction with zero applied magnetic field. In this case, the first GL Equation is one dimensional.

i.e,

Assuming ψ is real and neglecting the term in comparison with α, Equation (10), becomes:

For the plane wave function, the solution of Equation (11) is in the form of,

Substituting the value of plane wave function into Equation (11) (in terms of), we get,

This implies that,

Solving for ξ_GL at superconducting state that means, where α is negative yields,

where, and

Equation (14) yields the GL coherence length [2] . Since α depends on temperature as, then we can conclude that coherence length is temperature dependent.

Now let us consider the cases:

Case (I), For superconducting state (T < T_c),

and

Case (II), For normal state (T > T_c),

Case (III), at T = T_c, the GL theory is not valid

Where the length ξ_GL(0) is known as the zero temperature GL coherence length.

The surface current flows in a very thin layer of thickness (λ_GL) which is called the Ginzburg-Landau penetration depth [2] . The temperature and magnetic field dependence of the penetration depth appear quite naturally in Ginzburg-Landau (GL) theory. Like the London model, the GL model is independent of the underlying mechanism for superconductivity. Ginzburg-Landau theory is strictly valid only in superconducting phase boundary and is thus not generally applicable at low temperatures [2] . In the Ginzburg-Landau theory, a complex order parameter(ψ) is a function of temperature, magnetic field and the spatial coordinates [5] [10] . The total free energy per unit volume of the superconducting state in the presence of a magnetic field is minimizing this expression with respect to the first GL Equation and with respect to the current density(GL-II) Equation;

where and.

Using Equation (17), we get the expression for current density, as follows

since

Neglecting and Equation (20), becomes;

Using Maxwell’s Equation:

Taking the curl on both sides of Equation (22), we get

where, and

From Equation (22) and Equation (24), we get

Since and, we get

Therefore,

where and.

Therefore,

For

from Equation (30) [9] it follows that, the Ginzburg-Landua penetration depth (λ_GL(T)) varies as a function of temperature as:

where λ_L(0) is the London penetration at absolute zero temperature [9] .

The upper critical magnetic field (UCMF) is the magnetic field which completely suppresses superconductivity in type-II superconductors. More properly, the UCMF is a function of temperature (and pressure) and if not specified absolute zero and standard pressure are implied. Supercondcting region nucleates spontaneously within a normal conductor when the applied magnetic field is decreased below a value denoted by H_c2 [8] . At the onset of superconductivity, is small and we linearize the GL Equations as follows;

Since, , and.

The upper critical magnetic field (H_c2) can be calculated by linearizing Equation (34) and substituting the value of as:

The magnetic field in a superconducting region at the onset of superconductivity is just the applied field, so that and Equation (35) becomes

where and where the eigne value of mo-

mentum crystal is.

Therefore,

Since the expression of the Hamiltonian’s energy given in Equation (37) does not depend on coordinates (y and z) the corresponding momentum components (,) are conserved.

The largest value of the magnetic field (B) for which the solution of Equation (40) of the lowest eigenvalue is given by

Let us take the smallest eigenvalues n = 0 and K_z = 0 corresponding to the highest field in which superconductivity can nucleate in the interior of a bulk sample which occurs with the upper critical magnetic field in the coefficients change of sign. From Equation (41), we have;

where is the cyclotron frequency and is given by

since, solving for, we get:

From the relation that means, and.

We obtain the expression of the temperature dependent upper critical magnetic field (),

where or and

Now, consider anisotropy in mass, by introducing the effective mass tensor to the kinetic energy term of the GL Equation (3), i.e.

where is an effective mass tensor which is given by,

Since the coherence length depends on the effective mass as, the resulting Equation is

formally identical with the Schrödinger Equation of a particle with charge, an isotropic mass tensor in a uniform magnetic field H and the energy levels that have the harmonic oscillator are given by;

Let us consider Newton’s law of motion under the influence of lorentz force

i.e.

where v is the velocity.

The upper critical magnetic field can be expressed by using cyclotron frequency with the lowest free energy;

The solution of upper critical magnetic field by applying elliptical orbits traversed with cyclotron frequency is given by,

The solution of the lowest free energy corresponds to n = 0 and by using Equation (49), we get

where θ is the angle of the magnetic field that makes with the z-axis

.

From the general expression of coherence length Equation (45) we have,

Using the expression for the flux quantization, and Equation (45), can be expressed as,

For fields parallel and perpendicular to the symmetry plane we can write Equation (53) as:

and

Equations (54) and (55) are the mathematical expressions of the upper critical magnetic field (H_c2) for fields parallel and perpendicular to the symmetry axis [4] .