Electronic structure and magnetic anisotropies of antiferromagnetic transition-metal difluorides

We compare calculations based on density functional theory (DFT) with available experimental data and analyze the origin of magnetic anisotropies in ${\mathrm{MnF}}_2, {\mathrm{FeF}}_2, {\mathrm{CoF}}_2$, and ${\mathrm{NiF}}_2$. We confirm that the magnetic anisotropy of ${\mathrm{MnF}}_2$ stems almost completely from the dipolar interaction, while magnetocrystalline anisotropy energy (originating in spin-orbit interaction) plays a dominant role in the other three compounds, and discuss how it depends on the details of band structure. The latter is critically compared to available optical measurements. The case of ${\mathrm{CoF}}_2$, where magnetocrystalline anisotropy energy strongly depends on $U$ (the Hubbard parameter in $\mathrm{DFT}+\mathrm{U}$), is put into contrast with ${\mathrm{FeF}}_2$ where theoretical predictions of magnetic anisotropies are nearly quantitative.

Figure 1

Schematic band structure of rutile-type ${\mathrm{MnF}}_2, {\mathrm{FeF}}_2, {\mathrm{CoF}}_2$, and ${\mathrm{NiF}}_2$ in their AFM state. Spin up and down bands are degenerate. Note that for ${\mathrm{MnF}}_2$, all five upper $d$-bands are in group B (group A is an empty set).

Figure 2

Overview of all four fluorides. Left to right: ${\mathrm{MnF}}_2$ without $U, {\mathrm{FeF}}_2$ with $U=0.2$ Ry, ${\mathrm{CoF}}_2$ with $U=0.1$ Ry, and ${\mathrm{NiF}}_2$ again without $U$.

Figure 3

Nominal gap in ${\mathrm{FeF}}_2$ as a function of $U$. Note that the apparent optical gap is larger because some optical transitions may be suppressed.

Figure 4

On the left, optical spectra of ${\mathrm{FeF}}_2$ (relative permittivity) calculated for $U=0.1$ Ry. Agreement with experimental data on the right (based on Fig. 4 of Pištora et al. [25]) is striking, despite the fact that these measurements were taken above $T_N$.

Figure 5

Calculated imaginary part of permittivity, which is proportional to absorption, for ${\mathrm{CoF}}_2$ ($U=0.1$ Ry). The inset shows a feature below 2 eV, which could be related to the observed narrow band [27].

Figure 6

Calculated MCA for ${\mathrm{FeF}}_2$ and ${\mathrm{CoF}}_2$ as a function of the Hubbard parameter $U$.

Figure 7

Crystal structure (rutile) applies to all four difluorides under study. Magnetic structure on the left corresponds to orientation along the easy axis (except for ${\mathrm{NiF}}_2$) and we denote its energy by $E_{\parallel }$. Magnetic structure on the right is defined to have energy $E_{\perp }$. The essence of spin flop is that upon applying magnetic field $B=B_{\mathrm{sf}}$ parallel to $c$ in the left magnetic structure the moments switch to the structure on the right (here, if $B_{\mathrm{sf}}>0$, the moments are slightly canted towards $c$).

Figure 8

Band structure of ${\mathrm{MnF}}_2$ calculated with large $U$ corresponding to Ref. [38] (0.43 Ry).