Structural distortion and collinear-to-helical magnetism transition in rutile-type $\mathrm{Fe}{\mathrm{O}}_2$

The recent discovery of $\mathrm{Fe}{\mathrm{O}}_2$ under high pressures has aroused great interest. With the fully anisotropic density-functional theory+U method, we present a predictive study of structural and magnetic transitions of rutile-type $\mathrm{Fe}{\mathrm{O}}_2$ after reproducing the experimental spiral wave vector in isostructural $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$. A second-order structural distortion (tetragonal to orthorhombic) involving octahedral rotation occurs at a critical pressure of $3\sim 4\mathrm{GPa}$. From a global search in the Brillouin zone, the ground-state spin order of rutile-type $\mathrm{Fe}{\mathrm{O}}_2$ is found to be collinear below 22 GPa and transforms to a helix at higher pressures. The phases remain insulating throughout the whole pressure range, with a change from an indirect gap in the high-spin state to a direct gap in the low-spin state. Our work extends the fundamental understanding of iron oxides and provides schemes that treat strongly correlated magnetic systems in a proper and effective way.

Figure 1

(a)–(f) Magnetic supercells for different spiral states of $\beta \text{−}\mathrm{Mn}{\mathrm{O}}_2$, uniquely determined by q and α. Arrows on atoms represent local magnetic moments. Collinear ferromagnetic [FM, (a)] and antiferromagnetic [AFM, (b)] states correspond to $\mathbf{q}=\left(0,0,0\right)$ (Г point) with $\alpha =0^{\circ }$ and 180° respectively. At $\mathbf{q}=\left(0,0,0.5\right)$ (Z point), the collinear state double-AFM [DAFM, (c)] is characterized by $\alpha ={180}^{\circ }$ or 0°, the noncollinear spin helix (d) by $\alpha ={90}^{\circ }$. The experimental helical magnetism $\left[\mathbf{q}=\left(0,0,2/7\right),\alpha ={129}^{\circ }\right]$ is demonstrated in the primitive cell (e) and its plan view (f). (g) Computed spiral dispersions along the Г-Z path with ${\alpha }_0=0^{\circ }$ and 90° separately, where the energy is scaled to a formula unit with respect to the AFM type.

Figure 2

Spiral dispersions along the rutile high-symmetry $k$ path calculated with (a) full relaxation, corresponding to 0 GPa, or with the volume fixed at (b) $27.5{\AA }^3/\mathrm{f}.\mathrm{u}.$, (c) $25{\AA }^3/\mathrm{f}.\mathrm{u}.$, and (d) $24{\AA }^3/\mathrm{f}.\mathrm{u}.$, representing increasing pressures. (a) and (b) are computed in a HS state whereas (c) and (d) in a LS state. The spin-spiral search starts with ${\alpha }_0=0^{\circ }$, 90°, and ${180}^{\circ }$ separately for the ambient-pressure tetragonal (T) phase and high-pressure orthorhombic (O) phase. The global ground states (red arrow) together with some other special states (black arrow) are indicated. For comparison, dispersions of the less stable tetragonal phase at high pressures are also given with ${\alpha }_0={180}^{\circ }$. Figures 2 and 2 are magnified Г-Z segments of the orthorhombic $({\alpha }_0={180}^{\circ })$ dispersions in Figs. 2 and 2, respectively, with increased sampling density. Energies are scaled to a formula unit with respect to the tetragonal AFM (Г point, $\alpha ={\alpha }_0={180}^{\circ }$) phases.

Figure 3

(a) Total energy (square) and average magnitude of the magnetic moments of Fe (triangle), as a function of volume. The energy-volume relations are fitted with the Birch-Murnaghan EOS, and then converted to enthalpy-pressure curves (inset) to determine the critical pressure of the magnetic phase transition ($p_{c2}$). (b) The spontaneous strain and octahedral rotation angles in the HS orthorhombic phase as a function of pressure, fitted by linear and quadratic functions, respectively. The critical pressure of the orthorhombic distortion ($p_{c1}$) is obtained by extrapolation. (c) Structural evolution across the two phase transitions, represented by structures at 0, 20, and 31 GPa.

Figure 4

Total, Fe-$d$ projected, O-$p$ projected DOS, and corresponding band dispersions of (a) tetragonal HS-DAFM phase at 0 GPa, (b) orthorhombic HS-DAFM phase at 20 GPa, and (c) orthorhombic LS-helix phase at 31 GPa. Only one spinor is shown in (a) and (b) because the other one has the same energy dispersion. (d) DOS projected to O-$p$ orbitals and the $t_{2g}$, $d_{xy}$, $d_{{\mathrm{z}}^2}$ orbitals of a spin-up Fe for the phase at 0 GPa, with respect to the CFT coordinate system (inset) [50]. With fixed lattice parameters, two imaginary states are simulated for comparison, where (e) Hubbard U is reduced to 2.5 eV or (f) the oxygen atoms are deliberately displaced to suppress the Jahn-Teller distortion.

Figure 5

Phonon dispersions of (a) tetragonal HS-DAFM phase at 0 GPa, (b) orthorhombic HS-DAFM phase at 20 GPa, (c) orthorhombic LS-helix phase at 31 GPa.