Magnetic properties of hematite revealed by an ab initio parameterized spin model

Hematite is a canted antiferromagnetic insulator, promising for applications in spintronics. Here we present ab initio calculations of the tensorial exchange interactions of hematite and use them to understand its magnetic properties by parametrizing a semiclassical Heisenberg spin model. Using atomistic spin dynamics simulations, we calculate the equilibrium properties and phase transitions of hematite, most notably the Morin transition. The computed isotropic and Dzyaloshinskii–Moriya interactions result in a Néel temperature and weak ferromagnetic canting angle that are in good agreement with experimental measurements. Our simulations show how dipole-dipole interactions act in a delicate balance with first and higher-order on-site anisotropies to determine the material's magnetic phase. Comparison with spin-Hall magnetoresistance measurements on a hematite single crystal reveals deviations of the critical behavior at low temperatures. Based on a mean-field model, we argue that these differences result from the quantum nature of the fluctuations that drive the phase transitions.

Figure 1

(a) The crystal structure of hematite with the conventional hexagonal (gray) and the primitive rhombohedral (yellow) unit cell. The primitive basis consists of four iron atoms (blue) and six oxygen atoms (red). (b) Orientation of the spin vectors in the AFM (left) and WF (right) state. The four Fe sublattices in the primitive rhombohedral unit cell are labeled by $A, B, C$, and $D$.

Figure 2

Energy as a function of weak ferromagnetic distortion angle $\kappa$ as obtained from magnetic force theorem calculations. Inset: Magnetization vectors ${\mathit{m}}_1$ and ${\mathit{m}}_2$ of the magnetic double layers, net magnetization $\mathit{m}$, and Néel vector $\mathit{n}$. The canting angle with respect to the collinear configuration is denoted by $\kappa$.

Figure 3

Ab initio calculated interaction energy per atom, split into (a) the isotropic exchange energy, (b) the out-of-plane component of the DM vector, and (c) the two-ion anisotropy expressed as the energy difference between out-of-plane and in-plane alignment of the magnetic moments. The dashed vertical line marks the cutoff radius of $6.8\AA$ used for the spin dynamics simulations.

Figure 4

Orientation of the ab initio computed Dzyaloshinskii–Moriya vectors. Shown are all neighbors belonging to the first four shells, relative to an atom of sublattice $A$ (shown in black at the center). The DMI between atoms belonging to sublattices $A$ and $D$ is zero because there are inversion centers between the interaction partners. Depicted are orthographic projections from the (a) $\left[001\right]$ and (b) $\left[100\right]$ directions as well as a perspective projection (c).

Figure 5

Effective anisotropy due to dipole-dipole interactions. Shown is the energy difference per spin between the in-plane and out-of-plane state for both antiferromagnetic and ferromagnetic alignment of the sublattices, in dependence of the interaction range $|{\mathit{r}}_{ij}|$. The total bulk values given in the graph have been calculated from the cumulative dipole-dipole energies up to a distance of $1\mu \mathrm{m}$. The negative sign of both values indicates that the dipole-dipole interaction favors the in-plane orientation of both the Néel vector and the magnetization. The dashed vertical line marks the cutoff radius of $6.2\AA$ used for the spin dynamics simulations. The shaded areas indicate the uncertainty interval resulting from a $1\%$ relative uncertainty of the lattice parameters and a $10\%$ relative uncertainty of the atomic magnetic moment.

Figure 6

Contributions to the internal energy difference between the WF and AFM state (without external fields). Both the dipole-dipole interaction $d_{\text{dd}}$ and the two-ion anisotropy $d_{\text{tia}}$ favor an in-plane orientation of the magnetic moments while the second- and fourth-order on-site anisotropies lead to a preference of the out-of-plane state in total. The isotropic exchange and DMI energy are also affected by the transition from the AFM to the WF state but their effect is much smaller than that of the anisotropy terms.

Figure 7

Magnetization curve of the hematite spin model. The plot shows data from two simulations: heating up from the ground state (solid triangles pointing right) and cooling down from the paramagnetic state (white triangles pointing left). The Néel vector length $\left|\mathit{n}\right|$ displays the Néel transition at $T_{\text{N}}=989\mathrm{K}$. The in-plane component $|n_{\perp }|$ and the out-of-plane component $|n_z|$ of the Néel vector reveal the Morin transition at a temperature of $T_{\text{M}}=240\left(12\right)\mathrm{K}$.

Figure 8

Spin-flop field $B_{\text{sf}}$ in dependence of the angle $\vartheta$ between the magnetic field and the crystal's symmetry axis. Shown are experimental measurements at various low temperatures and at $200\mathrm{K}$ as well as simulation results at $200\mathrm{K}$. The experimental data at $200\mathrm{K}$ were taken from earlier measurements published in Ref. [10]. For the simulation data, the darker points represent the original data and the lighter points are copies of those data points based on symmetry considerations.

Figure 9

Comparison of the temperature-dependent spin-flop fields measured and simulated at $\vartheta =0{}^{\circ }$ with the rescaled theoretical predictions from a classical (cl.) and quantum (qm.) mean-field model. The experimental data were taken from earlier measurements published in Ref. [10].

Figure 10

Simulations of the spin-flop transition at $T=200\mathrm{K}$ at a magnetic field angle of (a) $\vartheta =0{}^{\circ }$ and (b) $\vartheta =90{}^{\circ }$. Plotted are the squared in-plane (ip) and out-of-plane (oop) components of the Néel vector. The black lines mark the determined spin-flop field $B_{\text{sf}}$.