Type II superconductors
Superconductivity is a phenomenon in which electrons pair together into Cooper pairs creating a coherent state with such remarkable properties as zero resistivity and perfect diamagnetism. According to the way a superconducting material responds to an external magnetic field, one can divided superconductors into two different classes: Type I and Type II. In type I superconductors below the critical value magnetic flux is expelled from the bulk of the material (Messner effect). Raising the external magnetic field past the critical value , the Meissner effect is broken and the superconductivity is destroyed.
On the other hand in type II superconductor, there are three regions divided by two critical values and in magnetic phase diagram as shown in Fig. 1.
When external magnetic field is below , the Meissner effect exists in the
superconductor, and the superconductive states have no resistance. In the region
certain amount of the magnetic flux can penetrate the superconductor. The sample is divided into two states, normal areas (cores) and superconducting areas. At relatively small magnetic flux penetrates the material mostly in normal cores. The flux lines enter the material with vorticity quantized, the vortex was also called Abrikosov vortex or fluxion, each vortex carries one unit of flux . This phenomenon is called mixed state or Shubnikov phase. The superconductivity is destroyed and there are no vortices in the superconductor while the strength of the external magnetic field larger than the second critical value .
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Because of type II superconductor can endure stronger magnetic than type I superconductor, the type II superconductor is quite practicable for both academic and industrial development.
Abrikosov vortices in the mixed state, lattice and its melting
Two major characteristics of the mixed state are the coherence length ξ which is the size of the normal core and the London penetration depth λ on whose scale the
supercurrent (current associated with Cooper pair) decays. The regular arrangement of the vortices is called Abrikosov lattice similar to an atomic lattice. Usually they arrange themselves in a form of hexagon to minimize mutual repulsion. If we raise the temperature above a certain value , the Abrikosov lattice melts into the vortex liquid phase.
Fig. 1-1.
Schematic magnetic phase diagram of a type II superconductor.
(1) Meissner State (2) Mixed State (3) Normal State.
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London and Ginsburg-Landau approximations
The two basic approximations to phenomenologically describe superconductivity on microscopic scale are London and Ginsburg-Landau approximations. The difference of these two approximations is that the London approximation assumes the constant order parameter and Ginsburg-Landau approximation uses the wave function to describe the order parameter with constant external magnetic field. Microscopic theories like the BCS theory for conventional s – wave superconductors, while can explain the phenomenon with homogeneous order parameter successfully, are too complicated to treat an inhomogeneous mixed state in external magnetic field, Ginsburg-Landau theory can describe the
phenomenon of superconductor more easily.
Layered superconductors
Generally high cuprites are layered materials, which consist of the copper-oxide planes. The electrons comprising Cooper pairs move mostly within the copper-oxide planes and the properties of superconductivities become largely two dimensional. The anisotropy parameter gamma is not very large in optimally doped YBCO γ 5~7, but becomes very large for BSCCO or underdoped YBCO (of order 50 or higher). In this case, the thermal fluctuations are practically two dimensional. This is seen experimentally by the scaling of magnetization, specific heat etc. In addition many other layered materials (like organic superconductors) are also nearly two dimensional. Recently layered superconductor BSCCO became a major material for applications like the THz wave generator. In addition to its phenomenological significance, the 2D system is by far simpler to simulate compared to the 3D one (which is still rather inaccessible to the MC method).
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Present work
This thesis focuses on the properties of the two dimensional type II superconductors with the external magnetic field in the region . In this field range, the superposition of magnetic fields of vortices makes the internal field nearly
homogeneous. Moreover for | | , the magnetic field is high enough so that so called higher Landau levels excitations can be neglected. Low energy states all belong to the lowest Landau level (LLL). Therefore we only consider the order parameter ,
constrained to the LLL. In high superconductors (and in some relatively low layered materials), the thermal fluctuations on the microscopic scale are not negligible.
Strongly thermal fluctuations on the microscopic scale affect such characteristics of the high temperature superconductors as specific heat, magnetization, structure factor etc. They lead to Abrikosov lattice melting and complicate the vortex matter phase diagrams. In real materials vortices are generally pinned by impurities creating disorder in the vortex system.
Disorder determines the most important characteristic of a superconductor – critical current.
The materials keep the main property of a superconductor, its zero resistance, only when the vortices are pinned. In this case the vortex system become a “vortex glass” or a ”Bragg glass”. Hence, it is important to find the glass line of the vortex system.
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Previous simulation and theoretical results
The Monte Carlo simulations on phase transition of type II superconductor were done over the years by many researchers. Clean system in the presence of thermal fluctuations was simulated by Y. Kato and N. Nagaosa1 who used the quasi-periodic boundary condition within the LLL approximation in Landau gauge. The finite size scaling of the algorithm was estimated to be ( is defined as the degrees of freedom). J .Hu and A. H.
MacDonald9 used quasimomentum basis to speed up the simulation, so that the finite size scaling becomes . J .Hu and A. H. MacDonald, Kato and Nagaosa, found the first order phase transition from crystalline to liquid phase, by the double peak in probability of energy distribution !.
Disordered system was first simulated only recently by M. S. Li and T. Nattermann3. They also claimed the model of the disorder system with expanding the random Gaussian disorder in Hermite polynomials. They presented the results of the flux lattice melting transition and the behavior of different correlation factor. They concluded that the phase transition from the curves of reduced temperature dependence of structure factor splayed out near the melting temperature. No glass transition was found for the highest value of disorder considered the disorder parameter ζ# 0.01.
Theoretical estimates of melting temperature with and without disorder. D. Li, B.
Rosenstein and V. Vinokur13 provided a theory to determine the glass transition in a
disorder vortex system. In their theory, the behavior of the magnetization in vortex glass is different from the one in vortex liquid. I tried to find the glass line of a disorder system by analyzing this quantity.
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There are six chapters in this thesis: the model of 2D the lowest Landau level GL free energy and quasi-momentum basis are introduced in Chapter 2. The Metropolis algorithm and the Monte Carlo calculations are shown in Chapter 3. The results and discussion of clean system are studied in Chapter 4. The vortex systems with disorder are analyzed in Chapter 5 and I discussed the conclusion in Chapter 6.