Noncollinear Ordering of the Orbital Magnetic Moments in Magnetite

The magnitude of the orbital magnetic moment and its role as a trigger of the Verwey transition in the prototypical Mott insulator, magnetite, remain contentious. Using $1s2p$ resonant inelastic x-ray scattering angle distribution (RIXS-AD), we prove the existence of noncollinear orbital magnetic ordering and infer the presence of dynamical distortion creating a polaronic precursor for the metal to insulator transition. These conclusions are based on a subtle angular shift of the RIXS-AD spectral intensity as a function of the magnetic field orientation. Theoretical simulations show that these results are only consistent with noncollinear magnetic orbital ordering. To further support these claims we perform Fe $K$-edge x-ray magnetic circular dichroism in order to quantify the Fe average orbital magnetic moment.

Figure 1

(a) The cubic unit cell of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ and the magnetic coupling between the Fe sites. Octahedral ($O_h$) ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Fe}}^{2+}$ ions are antiferromagnetically coupled to the tetrahedral ($T_d$) ${\mathrm{Fe}}^{3+}$ ions. (b) Fe $K$-edge measurements in ${\mathrm{Fe}}_3{\mathrm{O}}_4$ single crystal performed at Beam line ID12 of the European Synchrotron Facility (ESRF) [38]. The top panel shows XAS results as a function of the sample azimuthal angle $\theta$. The bottom panel shows the corresponding XMCD experimental (dotted) and theoretical (solid) Fe $K$ pre-edge signals. Two model calculations are presented: (i) Calc1 is the optimized calculation where a partially quenched orbital magnetic moment of $0.26{\mu }_{\mathrm{B}}$ per unit formula of ${\mathrm{Fe}}_3{\mathrm{O}}_4$ was concluded, and (ii) Calc2 is the theoretically expected XMCD signal for a fully quenched orbital magnetic moment scenario.

Figure 2

Fe $1s2p$ RIXS-AD measurements performed at Beam line ID26 of the ESRF [46, 47]. (a) A sketch of the scattering geometry employed. The magnetic field (${\mathit{B}}_{\mathbf{\text{Field}}}$) is aligned nearly parallel to the incident wave vector (${\mathit{k}}_{\mathbf{in}}$) which corresponds to the [110] direction. (b) Experimental and theoretical dichroism RIXS planes computed as the difference between the RIXS plane at $\varphi =90°$ and at $\varphi =0°$. The full experimental (dotted) and theoretical (solid) 360° RIXS-AD signals of the features labeled 1, 2, and 3 in the RIXS dichroic maps are shown in (c), respectively. The angular dependence signal is normalized as follows: $\mathrm{RIXS}\text{−}\mathrm{AD}=(\mathrm{RIXS}(\varphi )-\mathrm{Min}[\mathrm{RIXS}(\varphi )])/(\mathrm{Max}[\mathrm{RIXS}(\varphi )-\mathrm{Min}[\mathrm{RIXS}(\varphi )]])$.

Figure 3

Fe $1s2p$ RIXS-AD measurements. (a) A sketch of the scattering geometry employed. The magnetic field (${\mathit{B}}_{\mathbf{Field}}$) is aligned to the $(\{[-\cos (40°)]/\sqrt{2}\},\{[\cos (40°)]/\sqrt{2}\},\sin (40°))$ direction. The angular dependence is measured by rotating the sample about the [110] direction ($\varphi$ rotation). $\varphi =0°$ is defined when the incident polarization vector (${\mathit{\epsilon }}_{\mathbf{in}}$) is aligned to the [$-110$] direction. The experimental (dotted) and calculated (solid) angular dependence of the three main features (labeled 1, 2, and 3) are shown in panels (b) and (c).

Figure 4

The angular dependence of the orbital magnetic moment (${\mu }_L$) of the four ${\mathrm{Fe}}^{2+}$ ions as a function of the rotation of the magnetic field (${\mathit{B}}_{\mathbf{\text{Field}}}$) about the [110] orientation. (a) Sketch of the rotation geometry. The angular dependence of the orbital magnetic moment projected along the direction of ${\mathit{B}}_{\mathbf{\text{Field}}}$ and two perpendicular noncollinear contributions are shown in panels (b), (c), (d), and (e). The average orbital magnetic moment of the four ${\mathrm{Fe}}^{2+}$ ions are shown in (f). Note that the two non-collinear contributions in (f) overlap.