Abstract
In this work we have implemented the Bogoliubov-de Gennes (BdG) equation in a screened Korringa-Kohn-Rostoker (KKR) method for solving, self-consistently, the spin-polarised superconducting state for 3d crystals including substitutional impurities. The generalisation to the fully relativistic Dirac-Bogoliubov-de Gennes (DBdG) equations is also implemented for 3d crystals. This method combines the full complexity of the underlying electronic structure and Fermi surface geometry with a simple phenomenological parametrisation for the superconductivity. We apply this theoretical framework to the known s-wave superconductors Nb, Pb, and MgB$_2$. In these materials multiple distinct peaks at the gap in the density of states were observed, showing significant gap anisotropy which is in good agreement with experiment. For Pb the effects of spin-orbit coupling and the surface gap anisotropy are also addressed. Qualitatively, the results can be explained in terms of the k-dependent Fermi velocities on the Fermi surface sheets exploiting concepts from Bardeen-Cooper-Schrieffer (BCS) theory.We then investigate how impurities affect the superconducting state by applying the theoretical framework to bulk Nb with \textcolor{black}{non-magnetic} impurities. Without non-magnetic impurities, Nb has an anisotropic gap structure with two distinct peaks around the Fermi level. In the presence of non-magnetic impurities those peaks are broadened due to the scattering between the two bulk superconducting gaps, however the peaks remain separated. As a second example of self-consistent real-space solutions of the BdG equations we examine superconducting clusters embedded within a non-superconducting bulk metallic host. This allows us to estimate the coherence length of the superconductor and we show that, within our framework, the coherence length of the superconductor is related to the inverse of the gap size, just as in bulk BCS theory. The resulting local density of states (LDOS) in the superconductor is non-zero at the Fermi level due to the metallic host, giving it a striking resemblance to the pseudogap phase in copper-oxide based superconductors.
Finally we investigate how magnetic impurities affect the superconducting state by embedding 3$^{rd}$ row d-block magnetic impurities into bulk and surface Pb. In the presence of magnetic impurities, there is a pair-breaking effect that results in \textcolor{black}{sub-gap} Yu-Shiba-Rusinov (YSR) states which we decompose into contributions from the individual orbital character. In bulk Pb we find that not only are there two distinct YSR resonance pairs coming from the $t_{2g}$ and $e_{g}$ orbitals, there is a significant but smaller response from the `s' component of the impurity contributing to a third pair of YSR resonances. The intensity of the peaks is governed by the LDOS at the Fermi level of the impurity in the normal state. This finding is only reinforced when investigating how magnetic impurities, as an adatom and as an embedded impurity, affect the surface electronic structure. In both cases the degeneracy of the $t_{2g}$ and $e_{g}$ is further split, however in some cases no YSR resonances associated with `d' orbitals are observed due to the majority and minority peaks being completely below or completely above the Fermi level. This highlights the important fact that multiple YSR states in the presence of 3d magnetic impurities cannot be attributed to the d-moment alone.
| Date of Award | 21 Jan 2021 |
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| Original language | English |
| Awarding Institution |
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| Supervisor | Martin Gradhand (Supervisor) & James F Annett (Supervisor) |