Dzyaloshinskii-Moriya interaction in $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ measured by magnetic circular dichroism in resonant inelastic soft x-ray scattering

Fe $L_{2,3}$-edge x-ray absorption spectra (XAS) and magnetic circular dichroism (MCD) in resonant inelastic x-ray scattering (RIXS) of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ were measured to identify the electronic structure responsible for its weak ferromagnetism caused by the Dzyaloshinskii-Moriya interaction (DMI) at room temperature. In contrast to negligible MCD in XAS, MCD in RIXS (RIXS-MCD) was clearly observed in the $dd$ excitation at 1.8 eV via excitation to charge-transfer states. Furthermore, RIXS-MCD showed a crystal orientation dependence, indicating that the observed RIXS-MCD originated from DMI. The observed RIXS-MCD is well described by ab initio charge-transfer multiplet calculations, revealing that the RIXS-MCD derives from spin flip excitations at delocalized $e_{\mathrm{g}}$ orbitals. By the combination of the experiments and calculations, RIXS-MCD has unraveled that the origin of DMI in $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ is the $e_{\mathrm{g}}$ orbitals, which are strongly hybridized with the $2p$ orbitals of oxygen atoms. The results demonstrate the importance of RIXS-MCD for identifying the electronic structure related to DMI.

Figure 1

(a) Crystal and magnetic structures of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ with rhombohedral unit cell. Black arrows on the Fe atoms indicate the direction of the magnetic moment below and above the Morin temperature, $T_{\mathrm{M}}\approx$ 250 K. (b) Projection of the crystal structure of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ onto the (111) plane. Black arrows in (b) schematically denote canted magnetic moments in the weak ferromagnetic phase. The canting angle is $\sim 0.{065}^{\circ }$ [8]. The $\langle 11\overline{2}\rangle$ and $\langle 1\overline{1}0\rangle$ directions on the (111) plane are indicated. The figures were created using vesta [11].

Figure 2

Experimental configuration for the present XMCD and RIXS-MCD measurements. The incident angle from the sample surface was ${10}^{\circ }$. Yellow arrows in the sample represent the directions of the canted magnetic moments on the (111) plane.

Figure 3

Fe $L_{2,3}$-edge XAS and RIXS results of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ single crystal. (a) Fe $L_{2,3}$-edge XAS and XMCD spectra of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$ measured in IPFY mode. (b) and (c) Excitation energy dependence of RIXS spectra measured by $\pi$-polarized x rays without a magnetic field plotted against emission energy and energy loss, respectively. Note that RIXS spectra in (b) were normalized to the intensity of Fe $L_3$ fluorescence emission for emphasizing the similar spectral shape of Fe L emission peaks, while RIXS spectra in (c) were normalized to the photon flux. Numbers in (b) and (c) indicate the excitation energies shown in (a). Constant energy-loss peaks are marked by vertical gray lines in (c).

Figure 4

Comparison of the experimental and calculated XAS and RIXS-MCD spectra of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$. (a) Experimental and calculated Fe $L_{2,3}$-edge XAS spectra. The transition energy in the calculated XAS is shifted by $-14$ eV to align the main peaks of the $L_3$ edge with the experimental ones. The peak around 719 eV indicated by the arrow in the theoretical spectrum is an artifact. (b) and (c) Experimental RIXS-MCD spectra in the configurations of ${\mathit{k}}_{\mathrm{in}}\parallel \langle 11\overline{2}\rangle$ and ${\mathit{k}}_{\mathrm{in}}\parallel \langle 1\overline{1}0\rangle$. (d) Calculated RIXS-MCD spectra in the configuration of ${\mathit{k}}_{\mathrm{in}}\parallel \langle 1\overline{1}0\rangle$. The energy loss in the calculation is shifted by $-1.35$ eV. To clearly see the RIXS-MCD, RIXS spectra ($I^{\uparrow \uparrow }$ and $I^{\uparrow \downarrow }$) are multiplied by the factor indicated on each spectra, and the difference spectra ($I^{\uparrow \uparrow }-I^{\uparrow \downarrow }$) are further multiplied by a factor of 2. Numbers labeled on the RIXS spectra indicate the excitation energies shown in (a).

Figure 5

RIXS-MCD results of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$(111) single crystal measured at $h\nu =$ 713.25 eV [No. 7 in Fig. 4] in the configuration of ${\mathit{k}}_{\mathrm{in}}\parallel \langle 1\overline{1}0\rangle$. RIXS spectra measured by LCP and RCP x rays (blue and red curves) and the dichroic difference spectra (green curves) are displayed in the same graph. The difference spectra are multiplied by a factor of 2. The schematic representations on the left top of the RIXS spectra depict the experimental configurations. The black arrows stand for the incident and scattered x rays, and the yellow arrows on the sample schematically illustrate the directions of the magnetic moments on the (111) plane. The opposite direction of the magnetic field M was used for (a) and (b). The RIXS and difference spectra in the loss energy region from $-0.8$ to 2.3 eV are enlarged on the right side to clearly show the $dd$ excitations near the elastic peak.

Figure 6

Compositions of electronic configurations in the final states of Fe $L_{2,3}$-edge XAS. (a) Calculated Fe $L_{2,3}$-edge XAS of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$. (b) and (c) Compositions of ${\left({\varphi }_{2\mathit{p}}\right)}^5{\left({\varphi }_{3\mathit{d}}\right)}^6$ configurations at $L_3$ and $L_2$ edges, respectively. (d) and (e) Compositions of the CT configurations, ${\left({\varphi }_{2\mathit{p}}\right)}^5{\left({\varphi }_{3\mathit{d}}\right)}^7{\left({\varphi }_{\text{O-2}p}\right)}^{-1}$ at $L_3$ and $L_2$ edges, respectively.

Figure 7

Possible transition channels for the $L_3$-edge RIXS of $\alpha \text{−}{\mathrm{Fe}}_2{\mathrm{O}}_3$. (a) Possible electronic spin configurations for the initial, intermediate, and final states at the $L_3$ edge. States m1, m2, and m3 stand for the intermediate states of the $2p^{5}3d^6$ configurations, and states f1, f2, and f3 for the three different final states of the $3d^5$ configurations in Table 1. (b) and (c) Schematic representation of the transition intensities for each channel in RIXS peaks ${\mathrm{A}}^{\prime }$ and ${\mathrm{B}}^{\prime }$ at three excitation energies (Nos. $3^{\prime }$, $5^{\prime }$, and $7^{\prime }$). The widths of the arrows indicate the transition intensities for each channel and can be directly compared with each other. The black arrows labeled by the Roman numerals correspond to the decay channels in (a). The transition responsible for the RIXS-MCD is highlighted by orange arrows. (d) Experimental and calculated Fe $L_3$-edge XAS spectra.