Proximity-induced superconductivity in ferromagnetic Gd layers on Nb from a first-principles LDA+ U study

We implement the on-site Coulomb repulsion $U$ and exchange coupling $J$ within the local density approximation (LDA) into the recently developed Dirac-Bogoliubov--de Gennes (DBdG) solver [G. Csire et al., Phys. Rev. B 97, 024514 (2018)] for superconducting heterostructures, by using the screened Korringa-Kohn-Rostoker (SKKR) Green's function method. We apply this implementation to ferromagnetic (FM) Gd layers on a superconducting Nb substrate, where the $U$ and $J$ terms are considered only for the $4f$ orbitals of the Gd layers, and investigate the proximity-induced superconducting properties of the Gd layers by using the implemented $\mathrm{DBdG}+ U$ solver. Our first-principles calculations reveal that with the $U$ and $J$ terms, the density of states at the Fermi level has small contributions from $4f$ orbitals, while without the $U$ and $J$ terms, the contribution of the $4f$ orbitals somewhat increases. For the calculated quasiparticle density of states (DOS), with the $U$ and $J$ terms, there are several secondary satellite gaps, plateau-like regions, and central small V-shaped in-gap states within the bulk superconducting Nb gap, while without the $U$ and $J$ terms, the central V-shaped in-gap states appear within a much wider energy window. The in-gap states are identified to the Yu-Shiba-Rusinov states arising from the individual Gd layers with large magnetic moments rather than due to small magnetization induced in the Nb layers. We find that the normal-state DOS of the FM overlayers at the Fermi level is as important as the magnetic moment of the FM overlayers to the quasiparticle DOS. We also compute the superconducting order parameter as a function of the vertical $z$ coordinate for 10 Gd layers on a Nb substrate. The order parameter abruptly decreases in the proximity to the interface and it oscillates as a function of the $z$ coordinate in the Gd layers. This feature of quasiparticle DOS is qualitatively consistent with the previous studies. The implemented DBdG $+ U$ solver can be used to perform first-principles studies of other strongly correlated superconducting heterostructures as well as bulk superconductors.