Magnetic interactions in iron superconductors studied with a five-orbital model within the Hartree-Fock and Heisenberg approximations

We analyze the magnetic interactions of a five-orbital model for iron superconductors treated within the Hartree-Fock and Heisenberg approximations. We find that the exchange constants depend nontrivially on the Fe-As-Fe angle and on the charge and orbital filling. Within the localized picture, columnar ordering is found for intermediate Hund's coupling $J_H$. At smaller $J_H$, an unusual orbital reorganization stabilizes checkerboard ordering. Ferromagnetism appears at large $J_H$. Ferromagnetic correlations are enhanced with electron doping whereas large hole doping stabilizes checkerboard antiferromagnetism, explaining the change in magnetic interactions upon replacement of Fe by Co or Mn. For intermediate and large values of $U$, the Hartree-Fock approximation shows results similar to those for strong coupling though with a double-stripe phase instead of ferromagnetism. Itinerancy enhances the stability of the columnar ordering. A comparison of the two approaches reveals a metallic region of the phase diagram where strong-coupling physics is determinant.

Figure 1

(a) Sketch of the two different orbital configurations ($\left\{\lambda \right\}=x^2-y^2$ and $3z^2-r^2$) of the $S=2$ state for $n=6$. (b) Calculated exchange constants for $n=6$, $U=5$ eV, and $\alpha =35.3^{\circ }$ as a function of Hund's coupling. (c) Illustration of the virtual transitions involving filled orbitals responsible for the FM exchanges for $n=6$ at large $J_H/U$. (d) Exchange constants as a function of $J_H/U$ for $n=5$ and $U=5$ eV assuming a $S=5/2$ state. A $(\pi ,\pi )$ ground state is always favored. (e) Energies of the magnetic ground states as functions of $J_H/U$ for the same parameters as in (b). The double-stripe DS order, although never the ground state, is included for comparison. Vertical lines separate regions with different ground states [$(\pi ,\pi ),(\pi ,0)$, and FM].

Figure 2

Hartree-Fock magnetic phase diagram calculated at fixed density $n=6$ as a function of $U$ and $J_H/U$. Gray stands for nonmagnetic (NM), blue for the AF $(\pi ,0)$ high-moment (HM) state satisfying Hund's rule, red for the AF $(\pi ,0)$ low-moment (LM) state that violates Hund's rule, orange for AF $(\pi ,\pi )$, and green for the double-stripe DS state with charge modulation (see the text). Shaded areas mark insulating phases. Superposed dashed lines on the magnetic regions show the phase transition lines between the $(\pi ,\pi )$ and HM $(\pi ,0)$ states and between the HM $(\pi ,0)$ and the FM phase predicted by the $S=2$ Heisenberg model.

Figure 3

Mean-field magnetic phase diagram in the grand canonical ensemble (a) as a function of $U$ and $J_H/U$ and for $n=6$ and (b) as a function of $U$ and $n$ for $J_H/U=0.22$. The same color code as in Fig. 2 applies. White regions are unstable towards phase separation (PS).

Figure 4

(a) First- and second-nearest-neighbor exchange constants $J_1^{\left\{\lambda \right\}}$ and $J_2^{\left\{\lambda \right\}}$ as a function of $\alpha$, the angle formed by the Fe-As bond and the Fe plane, for $n=6$, $U=5$ eV, and $J_H/U=0.22$. (b) Energies of the most competitive magnetic orderings as a function of $\alpha$ for the parameters in (a). The sequence of magnetic orderings with increasing $\alpha$ is $(\pi ,0)$, $(\pi ,\pi )$, and FM. Vertical lines delimit the experimentally relevant values of $\alpha$.

Figure 5

Dependence of $E_0^{x^2-y^2}$ and $E_0^{3z^2-r^2}$ for $n=6$ and $U=5$ eV as a function of the Fe-As-Fe angle $\alpha$ for (a) $J_H/U=0.22$ and (b) Hund's coupling for $\alpha =35.3^{\circ }$.

Figure 6

Energies of the magnetic ground states as functions of $J_H/U$ calculated within the Heisenberg picture (left) and Hartree-Fock phase diagram (right) for the same parameters as in Figs. 2 and 3, except for the crystal field splitting ${\epsilon }_{3z^2-r^2}-{\epsilon }_{x^2-y^2}$, which is 100 meV larger here. In the right panel the color code is the same as in Fig. 2. Superposed dashed lines on the magnetic regions show the phase transition lines between the $(\pi ,\pi )$ and HM $(\pi ,0)$ states and between the HM $(\pi ,0)$ and the FM phase predicted by the $S=2$ Heisenberg model.

Figure 7

Dependence of the orbital filling as a function of the intraorbital interaction for (a) $J_H/U=0.15$ and (b) $J_H/U=0.22$. Vertical lines separate regions with different ground states.