SciPost Thesis Link
| Title: | On the scope of McMillan's formula | |
| Author: | Jan Berges | |
| As Contributor: | Jan Berges | |
| Type: | Master's | |
| Field: | Physics | |
| Specialties: |
|
|
| Approaches: | Theoretical, Computational | |
| URL: | https://janberges.de/theses/Master_Jan_Berges.pdf | |
| Degree granting institution: | Universität Bremen | |
| Supervisor(s): | Tim Wehling, Gerd Czycholl | |
| Defense date: | 2016-11-01 |
Abstract:
Ever since McMillan's formula has been published in 1968, it has been widely used to obtain estimates of the critical temperature of superconductors as a function of three effective material parameters, namely an average phonon frequency ⟨ω⟩, the electron-phonon coupling strength λ and the Coulomb pseudo-potential μ*, which can be extracted from experiment or first-principles calculations. It constitutes an approximation to the more general Eliashberg theory of superconductivity from which it was derived by fitting analytic approximations of the underlying equations to exact numerical results. Although for the latter the special phononic density of states of niobium has been assumed, which was simply at hand at that time, the validity of the resulting formula turned out to be much more general. The aim of the present work is to trace the steps that lead from the theory of the fundamental interactions between electrons and phonons to the handy formula for the critical temperature and to perform further tests on its scope, many of them, supposedly, have already been carried out somewhere in its past of almost half a century and fallen into oblivion or, more probably, just overlooked this time. Special attention is paid to potential discrepancies emerging from exceptional densities of electronic states and the question whether and possibly how the multi-band case with non-scalar coupling strengths can be brought into accordance. Notwithstanding that in the course of the investigations no references to specific materials are made but rather simple models applied, it is intended that the results be of use for the understanding of novel, especially two-dimensional materials. For this purpose, an appropriate software is developed which may be used not only to obtain electronic self-energies on the imaginary or real frequency axis as solutions of the multi-band Eliashberg equations or analytically continued by means of Padé approximants, respectively, but also to solve the linearized critical-state equations for a parameter of choice, which may be either the critical temperature itself, the phonon frequency or any element of the matrices defining the coupling strengths, for the respective other quantities fixed.