Self-consistent mapping: Effect of local environment on formation of magnetic moment in $\alpha -{\mathrm{FeSi}}_2$

The Hohenberg-Kohn theorem establishes a basis for mapping the exact energy functional to a model one provided that their charge densities coincide. We suggest to use a mapping in a similar spirit, but here the parameters of the formulated multiorbital model should minimize the difference between the self-consistent charge and spin densities. The analysis of the model allows for detailed understanding of the role played by different parameters of the model in the physics of interest. After finding the areas of interest in the phase diagram of the model, we return to the ab initio calculations and check if the effects discovered are confirmed or not. Because of the last controlling step, we call this approach hybrid self-consistent mapping approach (HSCMA). As an example of the approach we present a study of the effect of silicon atoms substitution by the iron atoms and vice versa on the magnetic properties in the iron silicide $\alpha -{\mathrm{FeSi}}_2$. We find that while the stoichiometric $\alpha -{\mathrm{FeSi}}_2$ is nonmagnetic, the substitutions generate different magnetic structures depending on the type of local environment of the substitutional Fe atoms. Besides, contrary to the commonly accepted statement that the destruction of the magnetic moment is controlled only by the number of Fe-Si nearest neighbors, we find that actually it is controlled by the Fe-Fe next-nearest-neighbor hopping parameter. This finding led us to the counterintuitive conclusion: an increase of Si concentration in ${\mathrm{Fe}}_{1-x}{\text{Si}}_{2+x}$ ordered alloys may lead to ferromagnetism. The calculation within GGA-to-DFT confirms this conclusion.

Figure 1

(Left) Structure of $\alpha -{\mathrm{FeSi}}_2$; Si atoms are shown by blue balls, Fe atoms shown by grey balls. (Right) Partial density of electronic states (pDOS) of Fe atoms; black line shows $t_{2g}$ states, red line shows $e_g$ states. The zero on the energy axis is chosen at the Fermi energy.

Figure 2

pDOS for host ${\mathrm{Fe}}_0$ (left) and substitutional ${\mathrm{Fe}}_{\mathrm{I}}$ (right) in alloy A. Black line shows $t_{2g}$ states and red line shows $e_g$ states. The zero on the energy axis is the Fermi energy.

Figure 3

(Top) Nearest and next-nearest neighbors of ${\mathrm{Fe}}_0$ with corresponding hopping integrals (Fe and Si atoms are shown by grey and blue balls, correspondingly) and the $t_2-t_3$ map of magnetic moments; the blue lines show the values of hopping integrals $t_2$and $t_3$ from Table 4. (Bottom) Model pDOS for hopping integral $t_3=0.0$ (left) and $t_3=-0.65$ (right). Hopping integral $t_2=1.0$.

Figure 4

Alloys C and D. (Top) NN and NNN environment of iron atoms. Si atoms are shown by blue balls, grey and black balls stand for ${\mathrm{Fe}}_0$ and substitutional ${\mathrm{Fe}}_{\mathrm{I}}$ atoms, correspondingly. (Middle) Dependence of the MMs on hopping integrals $t_1$ and $t_2$ (hopping integrals $t_3$ and $t_4$ are switch on). (Bottom) Dependence of MMs on hopping integrals $t_1$ and $t_2$ (hopping integrals $t_3$ and $t_4$ are switch off). Blue lines show the values of hopping integrals $t_1=0.9$ and $t_2=1.0$ for alloy C and $t_1=0.85$ and $t_2=0.95$ for alloy D (Table 3); these values provide the best fitting to the ab initio charge density.

Figure 5

Alloy C: model pDOS (left panel) and the map of the magnetic moments (right panel) for the different values of hopping integral $t_3\left(\text{Fe-Fe}\right)$. The blue lines at the last map show the values of hopping integrals $t_1=0.9$ and $t_2=1.0$ (Table 3), which provide the best fitting to the ab initio charge density.

Figure 6

The comparison of the ab initio (blue lines) and the model (black lines) pDOS of ${\mathrm{Fe}}_{\mathrm{I}} d$ electrons (left) and ${\mathrm{Fe}}_{\mathrm{II}} d$ electrons (right) in alloy B. (Top) $t_{2g}$ electrons and (bottom) $e_g$ electrons.

Figure 7

Alloy B. (Top, left) NN and NNN environment of ${\mathrm{Fe}}_0$ atoms; color encodings: Si are blue, ${\mathrm{Fe}}_0$ are grey, substitutional ${\mathrm{Fe}}_{\mathrm{I}}$ and ${\mathrm{Fe}}_{\mathrm{II}}$ are black and green balls, correspondingly. (Top, center) The dependence of ${\mathrm{Fe}}_0$ MMs on hopping integrals $t_1$ and $t_2$ (hopping integrals $t_3$ and $t_4$ are switch on). (Top, right) The dependence of MM on the ${\mathrm{Fe}}_0$ atom on the hopping integrals $t_1$ and $t_2$ (hopping integrals $t_3$ and $t_4$ are switch off). (Middle and bottom) Same for ${\mathrm{Fe}}_{\mathrm{I}}$ and ${\mathrm{Fe}}_{\mathrm{II}}$ atoms, correspondingly. Blue lines on the maps show the values of hopping integrals $t_1=0.9$ and $t_2=0.95$ (Table 3), which provide the best fitting to the ab initio charge density.

Figure 8

The dependence of MMs on the host ${\mathrm{Fe}}_0$ and the substitutional ${\mathrm{Fe}}_{\mathrm{I}}$ and ${\mathrm{Fe}}_{\mathrm{II}}$ atoms on hopping $t_3$ in $\alpha -{\mathrm{FeSi}}_2$, alloys C and B (from left to right). The solid lines are the dependencies, obtained in model calculations, the dots show the MMs from ab initio calculations at a distance $R$ between NNN Fe-Fe, according to (6). The scale for the distances (in angstroms) is given on the top of the figures.

Figure 9

Different environments for Fe ions lead to the formation of local MMs when part of the Fe ions are replaced by Si ions (the host Si atoms are not shown). The optimized lattice parameters and the calculated MMs of the Fe atoms are given under the structures.