1. Theoretical Introduction

Challenging world of high temperature superconductors has proven that microscopic theories such as BCS, which undeniably pushed the science of superconducting states a huge step further and proven to describe a vast majority of type I superconductors, proceeded to fail to clearly develop a prediction and model explanation of them. There were many attempts but rather no commonly accepted theory has emerged for many years so. After that, many thought to improve the BCS itself because of the formulation of the Green functions used. There were a lot of quasi-classical methods which have worked, despite they weren’t event meant to do so. Nevertheless, the main problem in creating a microscopic theory is that we shall deal with non-equilibrium response to applied fields and not only nonstationary behaviour of the quantum states.

For that so, in many purposes, the quasi-classical Ginzburg-Landau theory with a time dependent dynamics is a powerful tool to describe kinetics of superconductors within a certain limits. Mainly because it leaves a microscopic theory behind and leaves us with properties of the system as a whole, which is a lot easier to describe.

The fact is that the dynamics of vortices controls almost all of the magnetic properties in type II SC, especially the high temperature ones. The most important parameter in our theory would be a order parameter, which is proportional to the wave function of superconducting electrons (as BCS theory has also proven). As the electrons couple via many complex interactions in the superconductor, especially a repulsive Coulomb force, they do form a Bose condensate. The parameter is of course the normalised, complex function with the same phase for all particles in the condensate in equilibrium state, without current. The phase only changes if the symmetry for the condensate does change, clearly when the supercurrent is present. Thus, there exists a coherent phase for all of the electrons in the condensate.

As in all of physical states, we shall also consider a single particle excitation with a gap:

so all excitations have energies above |Δ|.

The Ginzburg-Landau theory(G&L 1950) deals with second order phase transitions. We start with a phenomenological theory with fundamental assumptions:

a) There exists a complex, scalar order parameter 𝝍 (𝚫, as we stated above)

b) There exists a homogeneous free energy(F) density, which can be expanded in a power series of an order parameter

c) The coefficients of the expansion are functions of a absolute temperature

with an assumption that the equation is valid near critical temperature.

𝛼 ~ (𝑇−𝑇𝑐)𝛼_0, 𝛽(𝑇) ~ 𝛽_0 = 𝑐𝑜𝑛𝑠𝑡

The order parameter has an absolute minimum −𝛼/𝛽≠0 when 𝑇<𝑇𝑐 and 0 when it is above. It poorly describes the effects rather distant from Tc and in the Tc itself. The free energy expansion is supplemented with Maxwell equations for magnetic field. It predicts in momentum that the Cooper pair would have a double charge for both electrons. When we include magnetic potential.

We can easily provide study of Gibbs energy:

After the standard procedure of energy minimalization we have corresponding, stationary Landau equations:

With corresponding boundary conditions(notice that the H is only the outside magnetic field and ∇𝑥𝐴 doesn’t imply it being equal H ):

Which is inconvenient and therefore , we make a variation of Gibbs energy putting ∇𝑥𝐻 to the equation, which gives us real boundary condition of ∇𝑥𝐴=𝐻.

We then put the remark that the vortices( very briefly, rapid changes of order parameter) tend to meet the boundary of the sample perpendicularly. Thus, as the circulating supercurrents form the vortex, It generates magnetic field tangential of the core, and perpendicular to the sample boundary.

Then we put a gauge transformation in the presence of magnetic field. We have two particular solutions.

a) 𝜓 = 0 and standard rotation, when H(applied field) is large. b) and A=0 when H=0. This is pure superconducting state without field applied.

We define coherence length which represents the length scale on which normalised order parameter varies from 0 to 1. We also define penetration depth which is equivalent to London theory of superconductivity

This was the case for time independent GL theory. Now we proceed to introduce TDGL, especially for type II SC.

Below a certain field, we still have a perfect diamagnet and no long range order for the electrons. In fields above 𝐻𝑐1 the magnetic flux penetrates the material. The flux is carried by the vortices

with parameters described above. Vortices can move during saturation as a result of interaction with other vortices and a transport current. It gives rise to an electric field, therefore resistivity. Many studies base on making the vortices immobile. In type II SC, where the vortex degree of freedom is not negligible, the normal state is separated by those 𝐻𝑐1 𝑎𝑛𝑑 𝐻𝑐2 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒𝑦 𝑑𝑖𝑣𝑖𝑑𝑒 𝑡𝑜 ~ 𝜅2. Typical solution for us is between those two field barriers. Magnetic fields constrained to vortices overlap, giving us nearly homogeneous superposition. The behaviour of so called Abrikosov vortices is described by the figure below.

The equations for TDGL theory generalised by Schmidt, which give us the dynamics of vortex in SC read as:

Where 𝜙 is a scalar potential, and 𝛼,𝛽,𝐷 are material parameters. The most general boundary conditions consist of:

After that, as before, we can put a simple gauge transformation to a function. The theory consists of more information, but for the case of this paper those won’t be mentioned. The theory itself is further improved by considering topological symmetry in the superconductors and its’ impact on vortices creation.