Crystalline and magnetic anisotropy of the 3$d$-transition metal monoxides MnO, FeO, CoO, and NiO

The magnetic-ordering and orbital-occupancy induced distortions of the rocksalt structure below the Néel temperature are computed for antiferromagnetic MnO, FeO, CoO, and NiO by means of spin-polarized density functional theory including generalized-gradient corrections and an on-site Coulomb repulsion $U$. The important role of the occupation of the $t_{2g}$ minority-spin states is studied in detail for the occurring rhombohedral and monoclinic distortions. The magnetic anisotropy energy is calculated to determine the orientation of the local magnetic moments in the antiferromagnetic crystals. We take into account both the influence of spin-orbit coupling and the transverse electron interaction. The spin-orbit interaction drives the magnetic anisotropy in CoO and FeO due to the partially filled $t_{2g}$ subshell while transverse electron interaction plays an important role for the magnetic anisotropy in MnO and NiO due to the completely empty or filled $t_{2g}$ subshell. The results for the structural and magnetic anisotropies are discussed in the light of the available experimental data.

Figure 1

Illustration of the AFM II antiferromagnetic ordering and the lattice parameters of the TMOs. TM ions with opposite spins are indicated by large blue or green spheres while the oxygen ions are given by small gray spheres. The arrows serve as a guide to the eye to denote opposite magnetization directions and not the actual orientation of the magnetization vector in the unit cell. In the left image the pseudocubic angle $\alpha$ and the rhombohedral distortion angle $\Delta \alpha$ are defined. The right image shows the monoclinic unit cell together with the definitions of the monoclinic angle $\beta$, the monoclinic distortion angle $\Delta \beta$, and the pseudo-orthorhombic lattice constants $a^{\prime }$, $b^{\prime }$, and $c^{\prime }$.

Figure 2

Spin density distribution for (a) FeO and (b) CoO in the magnetic unit cell. The spin densities shown here correspond to the minority-spin densities of the TM atoms and clearly show the partial occupancy of the minority-spin $t_{2g}$ states. The red spheres indicate the positions of the oxygen ions in the distorted octahedron.

Figure 3

Level splittings and occupation of the $3d$ states due to the octahedral crystal field with the dominating orthorhombic distortions (schematically). Only the minority-spin channel is described. (a) Fe${}^{2+}$, (b) Co${}^{2+}$.

Figure 4

Pressure dependence of the distortion angles. (a) The rhombohedral distortion angle $\Delta \alpha$ for MnO (black solid line and circles) and NiO (red solid line and circles). (b) The monoclinic distortion angle $\Delta \beta$ (solid lines and circles), the orthorhombic distortions $b^{\prime }/a^{\prime }-1$ (dotted lines and diamonds), and $c^{\prime }/a^{\prime }-1$ (dashed lines and squares) are depicted for FeO (green) and CoO (blue).

Figure 5

Magnetic anisotropy energy $\Delta E(\theta ,\phi )$ for FeO (a) and CoO (b). Results are shown for two angles $\phi =0^{\circ }$ (red circles) and $\phi ={90}^{\circ }$ (black squares). The values obtained by ab initio calculations are fitted to Eq. (5) (red solid and black dashed lines).

Figure 6

Orbital contribution ${\mu }_l(\theta ,\phi )$ to the total magnetic moment for FeO (green), CoO (blue), and NiO (red). Circles (squares) indicate $\phi =0^{\circ }$ ($\phi ={90}^{\circ }$).

Figure 7

Isosurface of the magnetic anisotropy energy (5) for (a) $K_d>0$ (yielding an easy axis) and (b) $K_d<0$ (yielding an easy plane). (c) Visualization of the magnetic anisotropy energy obtained from SO coupling for FeO, CoO, and NiO together with the magnetic anisotropy obtained from the magnetic dipole interaction. The anisotropy of the energy $\Delta E(\theta ,\phi )$ is shown in a polar plot where the distance of each point to the origin of the coordinate system indicates the magnitude of $\Delta E(\theta ,\phi )$ in that direction. The blue unit spheres are normalized to the maximum anisotropy energy ($\left|K\right|+|K^{\prime }|=1$ and $|K_d|=1$, respectively).