Microscopic nature of correlations in multiorbital $A{\text{Fe}}_2{\text{As}}_2$ $(A=\text{K},\text{Rb},\text{Cs})$: Hund's coupling versus Coulomb repulsion

We investigate via LDA+DMFT (local density approximation combined with dynamical mean field theory) the manifestation of correlation effects in a wide range of binding energies in the hole-doped family of Fe pnictides $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ $(A=\mathrm{K},\text{Rb},\text{Cs})$ as well as the fictitious ${\text{FrFe}}_2{\text{As}}_2$ and $a$-axis stretched ${\text{CsFe}}_2{\text{As}}_2$. This choice of systems allows for a systematic analysis of the interplay of Hund's coupling $J_H$ and on-site Coulomb repulsion $U$ in multiorbital Fe pnictides under negative pressure. With increasing ionic size of the alkali metal, we observe a nontrivial change in the iron $3d$ hoppings, an increase of orbitally-selective correlations, and the presence of incoherent weight at high binding energies that do not show the typical lower Hubbard-band behavior but rather characteristic features of a Hund's metal. This is especially prominent in $ab$-stretched ${\text{CsFe}}_2{\text{As}}_2$. We also find that the coherent/incoherent electronic behavior of the systems is, apart from temperature, strongly dependent on $J_H$, and we provide estimates of the coherence scale $T^{*}$. We discuss these results in the framework of reported experimental observations.

Figure 1

Momentum-resolved spectral function of ${\text{KFe}}_2{\text{As}}_2$ (top) and ${\text{CsFe}}_2{\text{As}}_2$ (bottom). Strong correlations in these materials introduce renormalization effects as well as broadening in the spectral function due to finite quasiparticle lifetimes compared to DFT(LDA). We observe a notable broadening and suppression of spectral features especially in the energy range $[-2.0,-0.5]$ eV.

Figure 2

The density of states of the Fe $3d$ orbitals at the Fermi level as obtained from (a) LDA+DMFT and as comparison with (b) DFT. The electronic correlations induce a marked deviation from the DFT results: The contribution of the Fe $3d_{z^2}$ orbital is strongly enhanced, while a more pronounced decrease of the Fe $3d_{xy}$ orbital is found towards the end system of stretched ${\text{CsFe}}_2{\text{As}}_2$. Also, the trend in the Fe $3d_{xz/yz}$ orbital in DFT is completely evened out in LDA+DMFT.

Figure 3

Filling of the Fe $3d$ orbitals as obtained from (a) LDA+DMFT and (b) DFT(LDA). Correlations reduce the overall Fe $3d$ filling, with the most correlated $3d_{xy}$ and $3d_{xz/yz}$ orbitals being the closest to half filling in contrast to the DFT result. Along the alkali metal series, the occupation of Fe $3d_{z^2}$ and $3d_{x^2-y^2}$ orbitals reduces, while that of the $3d_{xy}$ and $3d_{xz/yz}$ orbitals increases after a local minimum in ${\text{RbFe}}_2{\text{As}}_2$, which has the smallest $3d_{xy}$ occupation of all systems studied.

Figure 4

The density of states for (a) the Fe $3d_{z^2}$ orbital and (b) the Fe $3d_{xy}$ orbital as obtained from LDA+DMFT for the three compounds ${\text{KFe}}_2{\text{As}}_2,{\text{CsFe}}_2{\text{As}}_2$, and an $a$-axis stretched ${\text{CsFe}}_2{\text{As}}_2$. Note the emergence of Hubbard-like peaks around $-1.2$ eV and $+1$ eV for $3d_{z^2}$ and around $-1$ eV and $+1$ eV for $3d_{xy}$. With arrows we mark the trend of the changes in the spectral function along the series.

Figure 5

(a) Quasiparticle lifetimes given by ${\tau }_m=-{\left(Z_m\mathrm{Im}{\mathrm{\Sigma }}_m\left(i{0}^{+}\right)\right)}^{-1}$ of the Fe $3d$ orbitals along the $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ series and (b) the corresponding effective masses $\frac{m^{*}}{m_{\mathrm{LDA}}}$.

Figure 6

Temperature dependence of the (a) scattering rates $-Z_m\mathrm{Im}{\mathrm{\Sigma }}_m\left(i{0}^{+}\right)$ of the Fe $3d$ orbitals of ${\text{KFe}}_2{\text{As}}_2$. The shaded area indicates the coherent domain. The coherence temperature estimate $T^{*}$ as deduced from the scattering rate is quite low, located around 50 K. (b) The corresponding effective masses.

Figure 7

Sketch of some of the most likely Fe atomic orbital configurations in hole-doped 122 iron pnictide systems. The electronic filling is indicated by $n$ and the total spin by $S$.

Figure 8

The histogram of the Fe $3d$ atomic state for ${\text{KFe}}_2{\text{As}}_2$ at $T=145$ K. The probability corresponds to the fraction of time the Fe $3d$ orbitals spend in one of the $2^{10}=1024$ possible states. Within the interval of constant electron number $N$ the states are sorted by increasing energy, i.e., the leftmost states within an interval correspond to high-spin states, while the rightmost states correspond to low-spin states.

Figure 9

(a) Probabilities of the most likely atomic states of the Fe $3d$ atomic orbitals for the $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ series. The probability corresponds to the fraction of time the atom in the DMFT calculation spends in a specific state. (b) The summed probability of all atomic states with a given total spin $S$. High-spin states become more likely for increasing lattice parameter $a$, while the probability of low-spin states is reduced.

Figure 10

The density of states for the Fe $3d_{z^2}$ orbital of the stretched ${\text{CsFe}}_2{\text{As}}_2$ compound as a function of the on-site Coulomb repulsion $U$ and Hund's coupling $J_H$. (a) An increase only in $U$ leads to an increase in renormalization, i.e., effective masses and a pronounced Hubbard-like peak at $-1.5$ eV but no other qualitative changes are observed. (b) The Hund's coupling $J_H$ greatly increases the decoherence of the electronic states at the Fermi level and leads to a significant shift of spectral weight down to lower energies. (c) The combined effect of $U$ and $J_H$ is qualitatively very similar to an increase in $J_H$ alone.

Figure 11

The density of states for the Fe $3d_{xy}$ orbital of the stretched ${\text{CsFe}}_2{\text{As}}_2$ compound as a function of the on-site Coulomb repulsion $U$ and Hund's coupling $J_H$. Similar to the Fe $3d_{z^2}$ orbital, we see that the shift of spectral weight to lower energies is almost exclusively dependent on the Hund's coupling $J_H$. Being the most correlated orbital, the DOS of the Fe $3d_{xy}$ orbital at the Fermi level is almost gapped for $U=6$ eV and $J_H=1.2$ eV.

Figure 12

The structural parameters for the considered materials ${\text{KFe}}_2{\text{As}}_2$ (K), ${\text{RbFe}}_2{\text{As}}_2$ (Rb), ${\text{CsFe}}_2{\text{As}}_2$ (Cs) (from Ref. [45]) and the fictitious systems ${\text{FrFe}}_2{\text{As}}_2$ (Fr) and stretched ${\text{CsFe}}_2{\text{As}}_2$ $\left({\mathrm{Cs}}_{\text{stretch}}\right)$. (a) The $a$ parameter and absolute height of the As atom above the Fe plane. (b) The $c$-lattice parameter.

Figure 13

Eight largest effective Fe-Fe hopping parameters $t$ obtained from a 10-band tight binding fit. Also shown are some second nearest neighbor parameters indicated by the label NN2. The nontrivial evolution of the hopping values with increasing lattice parameter and reduced ${\mathrm{As}}_z$ height leads to different degrees of localization and effects of correlation in the Fe $3d$ orbitals along the series ${\text{KFe}}_2{\text{As}}_2,{\text{RbFe}}_2{\text{As}}_2,{\text{CsFe}}_2{\text{As}}_2$, and ${\text{FrFe}}_2{\text{As}}_2$.

Figure 14

The first six largest hopping parameters $t$ for the Fe-Fe hopping obtained from a 16-band tight binding fit. ${\mathrm{Fe}}^i$ denotes the $i\text{th}$ atom out of the two equivalent iron atoms in the irreducible Brillouin zone.

Figure 15

The density of states for (a) the Fe $3d_{xy}$ orbital and (b) the Fe $3d_{x^2-y^2}$ orbital for $U=4$ eV and $J_H=0.8$ eV as obtained from LDA+DMFT for the three compounds ${\text{KFe}}_2{\text{As}}_2,{\text{CsFe}}_2{\text{As}}_2$, and an expanded structure of ${\text{CsFe}}_2{\text{As}}_2$.

Figure 16

The density of states for the Fe $3d_{xz/yz}$ and $3d_{x^2-y^2}$ orbital of the stretched ${\text{CsFe}}_2{\text{As}}_2$ compound as a function of the on-site Coulomb repulsion $U$ and Hund's coupling $J_H$. (a),(d) An increase only in $U$ leads to a slightly better pronounced lower Hubbard bandlike feature but otherwise no qualitative changes to the high energy features. (b),(e) The Hund's coupling $J_H$ greatly increases the decoherence of the electronic states at the Fermi level and leads to a significant shift of spectral weight down to negative energies. (c),(f) The combined effect of $U$ and $J_H$ is qualitatively very similar to an increase in $J_H$ alone.