Relativistic framework for microscopic theories of superconductivity. I. The Dirac equation for superconductors

Abstract

We present a unified treatment of relativistic effects in superconductors. The relativistically correct ͑Dirac-type͒ single-particle Hamiltonian describing the quasiparticle spectrum of superconductors is deduced from symmetry considerations and the requirement of the correct nonrelativistic limit. We provide a complete list of all order parameters consistent with the requirement of Lorentz covariance. This list contains the relativistic generalizations of the BCS and the triplet order parameters, among others. Furthermore, we present a symmetry classification of the order parameters according to their behavior under the Lorentz group, generalizing previous treatments that were based on the Galilei group. The considerations in this paper are based only on the concepts of pairing and Lorentz covariance. They can therefore be applied to all situations in which pairing takes place. This includes BCS-type superconductors, as well as the heavy-fermion compounds, hightemperature superconductors, pairing of neutrons and protons in neutron stars, and superfluid helium 3.

Premature switching in graphene Josephson transistors

“…Moreover, as a surface two dimensional (2D) electron system, graphene can be easily coupled to special electrodes such as superconductors with controllable interfaces, while its charge density and type are continuously tunable by an electrostatic gate. Thus, coupling graphene to superconducting electrodes provides an ideal platform for investigating the interplay between superconductivity and relativistic quantum electrodynamics, which will be useful for understanding of high temperature and heavy-element superconductors [8]. Superconducting order is also predicted to emerge in pure and doped graphene [9].…”

Relativistic spin-polarized KKR theory for superconducting heterostructures: Oscillating order parameter in the Au layer of Nb/Au/Fe trilayers

“…Capelle and Gross [20] showed that in the relativistic case, 16 different order parameters exist with different symmetry properties (these order parameters are 4 × 4 matrices). However, a symmetry analysis with respect to the Lorentz group results in only five different types of order parameters that are consistent with the requirement of covariance (in the nonrelativistic case there are different types of order parameters, which describe singlet and triplet superconductivity).…”

“…The microscopic theory of inhomogeneous superconductors is based on the Bogoliubov-de Gennes (BdG) equations [19]. The relativistic generalization of this theory-called DiracBogoliubov-de Gennes (DBdG) equations-was established by Capelle and Gross [20,21]. To be able to treat arbitrary geometries including semi-infinite geometries without the use of a giant supercell, in this paper we develop a relativistic spinpolarized screened Korringa-Kohn-Rostoker (KKR) method [22][23][24][25][26][27][28][29][30][31] for the solution of the DBdG equations.…”

Non-Abelian Anyon Superconductivity

“…Pairing of fermions with a relativistic spectrum has been explored in two [32] and three [33] spatial dimensions. We can decouple the quartic interaction term in (7) via a Hubbard-Stratonovich transformation and look for a selfconsistent pairing ij :…”

“…In the presence of pairing, the mean-field Hamiltonian is quadratic and is diagonalized by fermionic operators a jk that are linear combinations of i and y j [33]:…”