First-principles study of the minimal model of magnetic interactions in Fe-based superconductors

Using noncollinear first-principles calculations, we perform a systematic study of the magnetic order in several families of ferropnictides. We find a fairly universal energy dependence on the magnetization order in all cases. Our results confirm that a simple Heisenberg model fails to account for the energy dependence of the magnetization in a couple of ways: first, a biquadratic term is present in all cases and, second, the magnetic moment softens depending on the orientation. We also find that hole doping substantially reduces the biquadratic contribution, although the antiferromagnetic stripe state remains stable within the whole range of doping concentrations, and thus the reported lack of the orthorhombicity in Na-doped ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ is probably due to factors other than a sign reversal of the biquadratic term. Finally, we discover that even with the biquadratic term, there is a limit to the accuracy of mapping the density functional theory energetics onto Heisenberg-type models, independent of the range of the model.

Figure 1

Schematic view of the two-dimensional Fe planes in the Fe-based superconductors. The first- ($J_1$) and second- ($J_2$) nearest-neighbor exchange interactions are shown in solid and dashed lines. The moments on the two sublattices form an antiferromagnetic checkerboard pattern and, in our calculations, the moments on one sublattice are rotated by an angle $\theta$ relative to the other. The antiferromagnetic stripe patterns correspond to $\theta =0^{\circ }$ and $\theta ={180}^{\circ }$.

Figure 2

The energy $E\left(\theta \right)-E\left(0\right)$ and the normalized moment $M\left(\theta \right)/M\left(0\right)$ as a function of the relative angle $\theta$ of the two magnetic subsystems. Note that in all instances, the lines are a guide to the eye and do not represent a fit. (a) The angular dependence of the energy for compounds belonging to the 11, 111, 122, and 1111 families of superconductors. (b) The angular dependence of the normalized moments belonging to the 11, 111, 122, and 1111 families of superconductors. (c) The angular dependence of the energy for different doping levels of the 122 compound BaFe${}_2$As${}_2$. (d) The angular dependence of the normalized moments for different doping levels of the 122 compound BaFe${}_2$As${}_2$.

Figure 3

The two magnetic configurations, in a $4\times 4$ two-dimensional cell, which are degenerate for any pairwise Hamiltonian of arbitrary range. The closed circles correspond to spin-up moments and the crosses correspond to spin-down moments. (a) The double stripe configuration, which is the ground state of FeTe. (b) The square configuration.