Weak and Strong Correlations in Fe Superconductors

Abstract

In this chapter I will discuss the strength of electronic correlations in the normal phase of Fe-superconductors. It will be shown that the agreement between a wealth of experiments and DFT+DMFT or similar approaches supports a scenario in which strongly correlated and weakly correlated electrons coexist in the conduction bands of these materials. I will then reverse-engineer the realistic calculations and justify this scenario in terms of simpler behaviors easily interpreted through model results. All pieces come together to show that Hund’s coupling, besides being responsible for the electronic correlations even in the absence of a strong Coulomb repulsion is also the origin of a subtle emergent behavior: orbital decoupling. Indeed Hund’s exchange decouples the charge excitations in the different Iron orbitals involved in the conduction bands thus causing an independent tuning of the degree of electronic correlation in each one of them. The latter becomes sensitive almost only to the offset of the orbital population from half-filling, where a Mott insulating state is invariably realized at these interaction strengths. Depending on the difference in orbital population a different “Mottness” affects each orbital, and thus reflects in the conduction bands and in the Fermi surfaces depending on the orbital content.

Appendix: The Slope of the Linear Z α (n α ) in the Orbital-Decoupling Regime

The direct proportionality of the quasiparticle weight to the individual orbital population when approaching the Mott insulator is the main evidence of the orbital decoupling mechanism induced by Hund’s coupling in the models for FeSC. However it can be noticed from Fig. 11.14 that the slope of the linear behavior is not universal.

In order to have an indication on how the bandstructure determines the slopes of the linear Z(n) for each orbital, I have performed a simplified analysis on a Hubbard model with two bands of equal half-bandwidth D (with hopping integrals as appropriate for a doublet in a cubic environment, see, e.g., [88]), slightly hybridized (by an interorbital hopping V = 0. 05D) and split by a crystal field Δ. The Kanamori density–density of the interaction is used, with U = 3D and $J = U/4$ and the model is solved within slave-spin mean-field.

A Mott insulator is found at half-filling and strong orbital differentiation in the mass enhancements for a large region of doping around it: as expected, for electron-doping (hole-doping being the same, for particle-hole symmetry), for a small crystal-field splitting the upper orbital is closer to individual half-filling and more correlated than the lower one. This situation becomes extreme very close to half-filling and an orbital-selective Mott transition takes place eventually (albeit retarded by the hybridization—unlike the five-orbital case in the regime relevant for FeSC, where the hybridization actually prevents the OSMT from happening). This happens when the upper orbital reaches individual half-filling and has a Mott gap at the chemical potential while the lower band remains metallic until global half-filling, where it becomes Mott insulating too.

Fig. 11.16

Orbitally resolved quasiparticle weight as a function of the respective orbital population in a two band Hubbard model with U = 3D and $J/U = 0.25$ and a light hybridization V = 0. 05D, for several values of the crystal-field splitting Δ between the two orbital energies. The linear behavior typical of the orbital-decoupling near the Mott insulating state is indeed found in both orbitals. Inset: the slope of the steeper curve (corresponding to the orbital higher in energy) in the electron-doped case scales with the bare crystal-field splitting

The curves Z _α (n _α ) are indeed linear (see Fig. 11.16) and the slope is steeper for the upper band. Thus at a given orbital population, the mass enhancement is heightened by the presence of other more correlated (possibly even insulating) orbitals. It is found that the slopes of the curves for the upper orbital (the most correlated and closest to half-filling, that can be viewed as mimicking the t2g orbitals in the realistic calculations) scale exactly (see inset in Fig. 11.1d) with the bare-crystal field splitting (it is worth recalling here that the crystal field renormalized by the interactions changes with the filling, instead).

The physics described in this simplified model is quite similar to the one we have investigated in the ab-initio calculations for iron superconductors. Indeed this scaling seems to apply to the t _2g orbitals in the three compounds studied in [30]: while for the studied iron pnictides (LaFeAsO and BaFe₂As₂) the xy orbital has a steeper slope than the xz∕yz (which have in turn a slope steeper than the eg orbitals), in FeSe there is an inversion, and the curve for the xy is less steep than for the xz∕yz. Now, this seems to reflect the bare crystal fields here too, since as reported in table in Fig. 11.3 the bare energy for the xy is above the one for the xz∕yz in LaFeAsO and BaFe₂As₂, while it is below in the chalcogenides.

Thus it seems that albeit the renormalized crystal-field makes the xy closer to half-filling and hence more correlated than the xz/yz, the footprint of the values of the crystal-field in the non-interacting system remains in the slopes of the Z _α (n _α ). Further work is however needed in order to clarify if this is always true, and what is the mechanism behind.