Theory of superconductivity : phenomenology, mean field and fluctuations

Abstract

More than one century has passed since Heike Kamerlingh Onnes discovered the phenomena of superconductivity but this field is still full of potential. This work will consider how we can understand the phenomena of superconductivity using quantum mechanics and how we can go beyond the standard superconducting model developed in the middle of the 20th century. We will start our consideration by discussing the basic experimental facts of a superconductor and the work of the London brothers that explains the Meissner effect by using the two-fluid theory where some of the electrons would condense into a super-fluid. The physical explanation of how this happens done by Leon Cooper who explained that at sufficiently low temperatures electrons can form stable pairs and condensate as bosons is discussed. This, though, is not enough to explain the phenomena of superconductivity. The first attempt at an explanation was the phenomenological theory by Ginzburg and Landau (GL) which we derive by considering the necessity of an order-parameter, a quantity which is small near the critical temperature, allowing us to write the free-energy as a expansion on this small parameter. By using the variational principle we are able get the two GL equations. However, there is still the need for a theory based on microscopic arguments. We will calculate the BCS Hamiltonian which lies at the heart of the theory developed by Bardeen, Cooper and Schrieffer by using mean-field theory and quasiparticles. This theory explains the behavior of a clean s-wave superconductor and naturally arrive at the conclusion that the spectrum of the excitation of the quasiparticles has a gap. To complete our analysis of basic theory we will discuss how we can link GL theory with BCS theory by the Green function formalism first develop by Gor’kov. We are going to clearly see that the GL equations which were obtained by phenomenological arguments can be derived from microscopic arguments. Following this discussion the application of this formalism is done by considering a system with just spin-magnetic interaction and a system with more than one electronic band, broadening therefore the scope of the BCS theory. For this two situations, the values for the critical temperature will change from the mean-field result. And, as a last topic we will tackle fluctuations. The mean-field theory is an approximation and has its limits of application. Going beyond we can use the fluctuation theory. How this theory is obtained and the so-called "Fluctuation driven shift of the critical temperature" are presented at the end of this dissertation.