Influence of the cation distribution, atomic substitution, and atomic vacancies on the physical properties of ${CoFe}_2{O}_4$ and ${NiFe}_2{O}_4$ spinel ferrites

${\mathrm{CoFe}}_2{\mathrm{O}}_4$ and ${\mathrm{NiFe}}_2{\mathrm{O}}_4$ are well-known insulating and ferrimagnetic spinel ferrites with high Curie temperatures, an important characteristic for electronic and spintronic applications. We used first-principles calculations to investigate how their electronic and magnetic properties can be altered or tuned by the presence of structural point defects. We considered successively the effects of cation distribution in the spinel lattice for stoichiometric compounds and of atom substitutions or vacancies. Our calculations demonstrate that a deviation from the perfectly inverse distribution of cations increases the magnetization and decreases the width of the band gap at the Fermi level. In contrast to cation vacancies, oxygen vacancies are not expected to strongly affect the magnetization. We show that ${\mathrm{NiFe}}_2{\mathrm{O}}_4$ crystals with an excess of Ni cations can display a spin-polarized hole conductivity. We finally calculated the formation energy of the different defects and we give details on their gap states.

Figure 1

Total energy $E$ of the cubic conventional cell containing 8 f.u. of the perfect inverse spinels ${\mathrm{NiFe}}_2{\mathrm{O}}_4$ and ${\mathrm{CoFe}}_2{\mathrm{O}}_4$ and band-gap width $E_{\text{g}}$ at the Fermi level $E_{\text{F}}$, as a function of the space group number [45, 46]. The different space groups correspond to different cation distributions in the occupied Oh sites, which are generated by swapping one or two pair(s) of M/Fe atoms. The ground-state energy of the most stable structure is chosen as the origin of the energies and the space groups are ordered by increasing values of $E$. The band gap width $E_{\text{g}}$ corresponds to the lowest energy difference between the highest occupied band and the lowest unoccupied band for one of the two spin directions.

Figure 2

Spin- and site-projected DOS (per formula unit) calculated for the perfect bulk (a) ${\mathrm{NiFe}}_2{\mathrm{O}}_4$ and (b) ${\mathrm{CoFe}}_2{\mathrm{O}}_4$ inverse spinel structures with the space group $P{4}_{1}22$. Majority and minority spin DOS curves are respectively shown in the right and left panels of each subfigure. Blue lines next to the DOS, with/without arrows, show the theoretical orbital filling of ${\mathrm{Co}}^{2+}$ and ${\mathrm{Ni}}^{2+}$ cations.

Figure 3

(a) Variations of the total energy $\mathrm{\Delta }{E}_{\text{inv}}$ and (b) of the pseudocubic lattice parameter $a$, as a function of the inversion parameter $\lambda$. $\mathrm{\Delta }{E}_{\text{inv}}$ is given with respect to the energy of the perfect normal spinel structure, $\mathrm{\Delta }{E}_{\text{inv}}\left(\lambda \right)=E\left(\lambda \right)-E\left(0\right)$. Open circles correspond to the energies calculated by DFT for all possible structures generated from the primitive cell with space group $P{4}_{1}22$ ($\lambda =0,0.25,0.5,0.75,1$) or from the conventional cell ($\lambda =0.125,0.875$). Crosses are the ensemble average of these calculated energies, which are used for the quadratic fits represented by the dashed lines. Lattice parameters are given for the most stable structures.

Figure 4

Schematic representation of the density of states calculated (a) for NFO and (b) for CFO and for different values of the inversion degree $\lambda$. The energy range of the band gap corresponds to the white area. Minority- and majority-spin DOS are respectively shown with red and blue colors. The green dashed line corresponds to the highest occupied state.

Figure 5

Same as in Fig. 4, but for the nonstoichiometric crystals $M_{0.875}{\mathrm{Fe}}_{2.125}{\mathrm{O}}_4$ ($\alpha =0.125$) and $M_{1.125}{\mathrm{Fe}}_{1.875}{\mathrm{O}}_4$ ($\beta =0.125$).

Figure 6

Formation energies $E_{\text{f}}\left[{\text{V}}_{\text{X}}^q\right]$ of charged atom vacancies in NFO for (a) the oxygen-poor conditions ($\mathrm{\Delta }{\mu }_{\text{O}}=-2.2$ eV) and (b) for the oxygen-rich conditions ($\mathrm{\Delta }{\mu }_{\text{O}}=-0.5$ eV). Formation energies of charged atom vacancies in CFO under similar oxygen-poor and oxygen-rich conditions are respectively given in the panels (c) and (d). The formation energies are given as a function of the position of the Fermi level $\mathrm{\Delta }{E}_{\text{F}}$, which is given with respect to the VBM of the perfect bulk compounds. Only the energies corresponding to the most stable charge states $q$ of each kind of vacancies are shown and these charge states are indicated. The green lines indicate the equilibrium Fermi level $\mathrm{\Delta }{E}_{\text{F}}^{\text{eq}}(T_{\text{g}},T_{\text{R}})$ calculated according to the equations and temperatures given in Sec. 2.

Figure 7

Same as in Fig. 4, but for [(a),(b)] NFO and [(c),(d)] CFO crystals with different cations vacancies and different charge states $q$.

Figure 8

Same as in Fig. 4, but for oxygen-deficient (a) NFO and (b) CFO crystals.

Figure 9

Stability phase diagrams of (a) NFO and (b) CFO. The chemical potential ranges where CFO and NFO are more stable than competing phases are shown by shaded areas. A and B points are median values for oxygen-rich ($\mathrm{\Delta }{\mu }_{\text{O}}=-0.5$ eV) and oxygen-poor ($\mathrm{\Delta }{\mu }_{\text{O}}=-2.2$ eV) conditions respectively.