Role of the orbital moment in a series of isostructural weak ferromagnets

The orbital contribution to the magnetic moment of the transition-metal ion in the isostructural weak ferromagnets $A{\mathrm{CO}}_3$ ($A=$ Mn,Co,Ni) and ${\mathrm{FeBO}}_3$ was investigated by a combination of first-principles calculations, nonresonant x-ray magnetic scattering, and x-ray magnetic circular dichroism. A nontrivial evolution of the orbital moment as a function of the $3d$ orbitals filling is revealed, with a particularly large value found in the Co member of the family. Here, the coupling between magnetic and lattice degrees of freedom produced by the spin-orbit interaction results in a large single-ion anisotropy and a peculiar magnetic-moment-induced electron cloud distortion, evidenced by the appearance of a subtle scattering amplitude at space-group-forbidden reflections and significant magnetostrictive effects. Our results, which complement a previous investigation on the sign of the Dzyaloshinskii-Moriya interaction across the series, highlight the importance of spin-orbit coupling in the physics of weak ferromagnets and prove the ability of modern first-principles calculations to predict the properties of materials where the Dzyaloshinskii-Moriya interaction is a fundamental ingredient of the magnetic Hamiltonian.

Figure 1

Trigonal crystal structure (space group $R\overline{3}c$, No. 167) of the weak ferromagnets $A{\mathrm{CO}}_3$ ($A=$ Mn,Co,Ni) and ${\mathrm{FeBO}}_3$ in the hexagonal axes description. Large blue spheres: transition-metal (TM) atoms (Mn, Fe, Co, Ni); medium-size yellow spheres: C/B; small red spheres: O. The arrows represent the magnetic moments of the TM atoms in the AFM phase.

Figure 2

Representative magnetic reflections dependence on the magnetic field direction for two different polarization states of the diffracted x-ray beam in ${\mathrm{CoCO}}_3$. The data points represent the diffracted intensity integrated over a rocking scan while the solid curves correspond to the best fit to Eq. (4). The data were collected at $T=5–6$ K and $\psi ={83}^{\circ },0^{\circ },{108}^{\circ },{30}^{\circ }$ for the (003), (107), $\left(\overline{1}17\right)$, and (009) reflections, respectively. A small constant background originating from residual multiple scattering has been removed from all the data sets. For each reflection, the intensity is normalized to the maximum value across both polarization channels. A schematic drawing of the vertical scattering geometry, along with the definition of the ${\mathbf{u}}_1{\mathbf{u}}_2{\mathbf{u}}_3$ reference frame used to express the cross sections of Eq. (1) and the magnetic field angle $\eta$, is reported in the top panel.

Figure 3

Orbital-to-spin angular momenta ratio for the compounds of the series $A$(C,$\mathrm{B}){\mathrm{O}}_3$ ($A=$ Mn,Fe,Co,Ni) as a function of the momentum transfer. For each compound, the $l\left(Q\right)/s\left(Q\right)$ value for each magnetic reflection (and the corresponding error bar) was calculated combining the measurements at different temperatures and azimuth values [31]. The dashed line represents a fit to Eq. (3) of the ${\mathrm{CoCO}}_3$ data considering an isotropic approximation of the magnetic form factors.

Figure 4

Absorption spectra measured with the external magnetic field parallel (${\mathrm{XAS}}^{-}$) and antiparallel (${\mathrm{XAS}}^{+}$) to the helicity of the incident circularly polarized light and corresponding circular dichroism (XMCD). The data were collected at $T=3$ K using a magnetic field ${\mu }_{0}H=0.4$ T applied in the $\mathbf{ab}$ plane of the crystal. The dashed gray line represents the integrated XMCD signal used for the application of the sum rules: the $l/s$ value refers to corresponding orbital-to-spin angular momenta ratio. The XAS data are normalized such that the post-edge spectral weight is equal to unity.

Figure 5

Direction of the weak net magnetization $\mathbf{M}$ as a function of the external magnetic field $\mathbf{H}$ direction in the basal plane of the crystal for different values of the anisotropy parameter $h=\frac{{\mu }_{0}HM}{K}$ discussed in the text. Each curve $\alpha \left(\beta \right)$ corresponds to a solution of Eq. (7) for a different value of $h$. The inset shows the definitions of the magnetization ($\alpha$) and external magnetic field ($\beta$) angles in the $\mathbf{ab}$ basal plane of the $R\overline{3}c$ crystal structure.

Figure 6

Calculated dependence of the (009) magnetic reflection in the $\sigma \text{−}{\pi }^{\prime }$ polarization channel on the external magnetic field direction for different values of the anisotropy parameter $h=\frac{{\mu }_{0}HM}{K}$ discussed in the text. The diffracted intensity was calculated using analogous solutions of Eq. (7) to the ones shown in Fig. 5.

Figure 7

Dependence of the ${\mathrm{CoCO}}_3\left(009\right)$ reflection on the magnetic field direction in $\sigma \text{−}{\pi }^{\prime }$ at different temperatures across the Néel transition at $T_N=16.7\left(5\right)$ K [31].

Figure 8

Fit detail of two representative data sets of Fig. 7 measured (a) well below and (b) close to the Néel transition. The solid lines represent the best fit to the calculated intensity (see Fig. 6) leaving the anisotropy constant $K$ as a free parameter. The results of the fit of the data sets at all temperatures are shown in Fig. 9.

Figure 9

Temperature dependence of the in-plane anisotropy constant $K$ in ${\mathrm{CoCO}}_3$. The data points were obtained through analogous fits to the ones shown in Fig. 8. The solid line represents the best fit to the quadratic law $\propto {(T_N-T)}^2$ proposed by Kaczer [38].

Figure 10

(a), (b) $\left(\overline{2}07\right)$ intensity magnetic field direction dependence as a function of the sample azimuth $\psi$ for two different polarization states of the diffracted x-ray beam. For each azimuth value and polarization channel, the measured intensity (represented through the color scale) as a function of the field direction is normalized to its maximum value. A detail of the $\psi ={65}^{\circ }$ data sets (dashed horizontal lines) is shown in Fig. 11. (c), (d) Global fit of the $\left(\overline{2}07\right)$ magnetic field dependence shown in (a) and (b). The $\sigma \text{−}{\sigma }^{\prime }$ and $\sigma \text{−}{\pi }^{\prime }$ data at each azimuth value have been fitted to Eq. (8) and the second of Eq. (4), respectively, as shown in Fig. 11 for $\psi ={65}^{\circ }$. The ratio between the magnetic and forbidden charge amplitudes in $\sigma \text{−}{\sigma }^{\prime }$ was kept constant across all $\psi$ values, while the multiple scattering amplitude was left free to vary: the resulting values are plotted in Fig. 14.

Figure 11

$\left(\overline{2}07\right)$ intensity magnetic field dependence in (a) $\sigma \text{−}{\sigma }^{\prime }$ and (b) $\sigma \text{−}{\pi }^{\prime }$ at $\psi ={65}^{\circ }$ [dashed horizontal line of Figs. 10 and 10]. The symbols represent the measured diffracted intensity, while the solid line refers to the global fit as explained in the text. In (a) the dashed green line and the dashed-dotted light blue line correspond to the charge and magnetic contribution to the global fit, respectively. The data were collected at $T=4$ K and are normalized to the peak intensity of the $\sigma \text{−}{\sigma }^{\prime }$ channel. A constant background originating from multiple scattering has been removed from the $\sigma \text{−}{\pi }^{\prime }$ data set.

Figure 12

Magnetic field dependence of the forbidden charge scattering amplitude normalized to its peak value. The drawing illustrates the empirical model for the Co electron cloud distortion discussed in the text. The Co ion of one of the two ferromagnetic sublattices ($A$) is shown together with the two negative charges used to model the elongation along the magnetic moment ${\mathbf{\mu }}_A$ direction. The definition of the field angle $\alpha$ is analogous to the one given in the inset of Fig. 5 in the case of negligible magnetocrystalline anisotropy ($\mathbf{H}\parallel \mathbf{M}$).

Figure 13

Temperature dependence of the $\left(\overline{2}07\right)$ magnetic and charge amplitudes for $\psi ={65}^{\circ }$. The data points were obtained through fits analogous to the one of Fig. 11. The solid and dashed lines refer to the best fit to a power law for the magnetic and charge amplitudes with critical exponents $\beta$ and $\gamma$, respectively.

Figure 14

Multiple scattering (a) $\sigma \text{−}{\sigma }^{\prime }$ amplitude and (b) $\sigma \text{−}{\pi }^{\prime }$ intensity azimuthal dependence. The data points are the results of the fit of the data shown in Figs. 10 and 10, while the solid line represents calculations performed using a mixed kinematical/dynamical approach where the standard kinematical structure factors of the secondary and tertiary reflection are weighted by terms calculated in a dynamical framework.

Figure 15

Charge density of the ${\mathrm{Co}}^{2+}3d$ valence electrons for an external field direction (a) $0^{\circ }$, (b) ${60}^{\circ }$, and (c) ${90}^{\circ }$ away from the crystallographic $\mathbf{a}$ axis as derived from the multiplet calculations discussed in the text. The plots refer to one of the two ${\mathrm{Co}}^{2+}$ clusters used for the calculations (cluster two). The colors correspond to different spin directions with red and blue for down and up character, respectively.

Figure 16

Angular shift of the (119) Bragg peak as a function of the magnetic field direction at $T=5$ K. The solid line represents a fit to the shift calculated using the unit-cell deformation of Eq. (9). The deformation of the in-plane lattice parameters resulting from the fit is shown in Fig. 17.

Figure 17

Relative deformation of the in-plane lattice parameters of the $R\overline{3}c$ hexagonal unit cell at $T=5$ K as obtained from the fit of the (119) angular shift of Fig. 16.