Epitaxial strain effects in the spinel ferrites ${\text{CoFe}}_2{\text{O}}_4$ and ${\text{NiFe}}_2{\text{O}}_4$ from first principles

The inverse spinels ${\text{CoFe}}_2{\text{O}}_4$ and ${\text{NiFe}}_2{\text{O}}_4$, which have been of particular interest over the past few years as building blocks of artificial multiferroic heterostructures and as possible spin-filter materials, are investigated by means of density-functional theory calculations. We address the effect of epitaxial strain on the magnetocrystalline anisotropy and show that, in agreement with experimental observations, tensile strain favors perpendicular anisotropy whereas compressive strain favors in-plane orientation of the magnetization. Our calculated magnetostriction constants ${\lambda }_{100}$ of about $-220\text{ppm}$ for ${\text{CoFe}}_2{\text{O}}_4$ and $-45\text{ppm}$ for ${\text{NiFe}}_2{\text{O}}_4$ agree well with available experimental data. We analyze the effect of different cation arrangements used to represent the inverse spinel structure and show that both local spin-density approximation plus $U$ and generalized gradient approximation plus $U$ allow for a good quantitative description of these materials. Our results open the way for further computational investigations of spinel ferrites.

Figure 1

(Color online) Inverse spinel structure of spinel ferrites. Oxygen cations are depicted as small red spheres (corner positions), and the coordination polyhedra surrounding cation sites are shaded. (a) Random cation distribution on the octahedrally coordinated $B$ sites, i.e., all $B$ sites remain equivalent. (b) $Imma$ cation arrangement used throughout this work. Inequivalent $B$ sites are represented by different shadings of the corresponding octahedra.

Figure 2

Cation distribution within different $x\text{−}y$ planes for different values of $z$, corresponding to setting 1 (a) and setting 3 (b) (see main text for details). Only layers containing $B$-site cations are shown. Divalent cations $\left({\text{Co}}^{2+},{\text{Ni}}^{2+}\right)$ are depicted as filled black circles, trivalent cations $\left({\text{Fe}}^{3+}\right)$ as open black circles, and oxygen anions as black crosses. Note that we use the convention where the origin is located at the midpoint between two $A$ sites.

Figure 3

(Color online) Total and projected DOS per formula unit for ${\text{CoFe}}_2{\text{O}}_4$. Left (right) panels correspond to GGA $\left(\text{GGA}+U\right)$ calculations. The $d$ states of $\text{Co}\left(O_h\right)$ (upper panels), $\text{Fe}\left(O_h\right)$ (middle panels), and $\text{Fe}\left(T_d\right)$ (lower panels) are separated into $t_{2g}$ (green/dark gray) and $e_g$ (red/black) contributions for the $O_h$ sites and into $e$ (green/dark gray) and $t_2$ (red/black) contributions for the $T_d$ sites. The total DOS is shown as shaded gray area in all panels. Majority- (minority-) spin projections correspond to positive (negative) values.

Figure 4

(Color online) Total and projected DOS per formula unit for ${\text{NiFe}}_2{\text{O}}_4$. Left (right) panels correspond to GGA $\left(\text{GGA}+U\right)$ calculations. The $d$ states of $\text{Ni}\left(O_h\right)$ (upper panels), $\text{Fe}\left(O_h\right)$ (middle panels), and $\text{Fe}\left(T_d\right)$ (lower panels) are separated into $t_{2g}$ (green/dark gray) and $e_g$ (red/black) contributions for the $O_h$ sites and into $e$ (green/dark gray) and $t_2$ (red/black) contributions for the $T_d$ sites. The total DOS is shown as shaded gray area in all panels. Majority- (minority-) spin projections correspond to positive (negative) values.

Figure 5

(Color online) Calculated $c/a$ ratio of ${\text{NiFe}}_2{\text{O}}_4$ as a function of epitaxial strain ${\epsilon }_{xx}$ obtained from $\text{GGA}+U$ calculations and different cation arrangements (settings) on the $O_h$ sites.

Figure 6

(Color online) ${\text{CoFe}}_2{\text{O}}_4$ $\text{GGA}+U$ total-energy differences for orientation of the magnetization along various crystallographic in-plane directions with respect to the [001] direction (black solid lines): ▲ [100], ▼ [010], $◀$ [110], and $▶$ $\left[1\overline{1}0\right]$. Red broken lines denote crystallographic directions which include also out-of-plane components, namely, [101] (dashed-dotted line), [011] (dashed-double dotted line), and [111] (dashed line). Left (right) panels contain the results corresponding to setting 1 (setting 3).

Figure 7

(Color online) ${\text{NiFe}}_2{\text{O}}_4$ $\text{LSDA}+U$ $\left(\text{GGA}+U\right)$ total-energy differences for orientation of the magnetization along various crystallographic in-plane directions with respect to the [001] direction are shown in the upper (lower) panels (black solid lines): ▲ [100], ▼ [010], $◀$ [110], and $▶$ $\left[1\overline{1}0\right]$. Red broken lines denote crystallographic directions which include also out-of-plane components, namely, [101] (dashed-dotted line), [011] (dashed-double dotted line), and [111] (dashed line). Left (right) panels contain the results corresponding to setting 1 (setting 3).