Incommensurate helimagnetic structure of Ba(Fe1−xScx)12O19

Sc-substituted hexagonal ferrite $\mathrm{Ba}{({\mathrm{Fe}}_{1\text{\ensuremath{-}}x}{\mathrm{Sc}}_{x})}_{12}{\mathrm{O}}_{19}$ has an incommensurate helimagnetic structure. However, the incommensurate helimagnetic structure has not been sufficiently determined, and the mechanism via which the helimagnetism develops has not been studied. Time-of-flight-Laue single-crystal neutron diffraction measurements were performed at 4--6 K on single-crystal samples of $\mathrm{Ba}{({\mathrm{Fe}}_{1\text{\ensuremath{-}}x}{\mathrm{Sc}}_{x})}_{12}{\mathrm{O}}_{19}$ having Sc concentrations of $x=0$, 0.128, 0.153, and 0.193. The incommensurate helimagnetic structure was determined by analyzing the magnitude and direction of the ${\mathrm{Fe}}^{3+}$ magnetic moments at the five Fe1--Fe5 sites ($2a, 4e, 4{f}_{1}, 4{f}_{2}$, and $12k$). The development mechanism of the helimagnetic structure was discussed from the standpoint of superexchange interaction. The crystal structure analyses revealed that the Sc substitution enabled two significant events, namely, the coordination changes of ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Sc}}^{3+}$ at the $\mathrm{Fe}2(4e)$ site, resulting in the shortened bond distances of Fe2-O1 and the preferred substitution of ${\mathrm{Fe}}^{3+}$ at the $\mathrm{Fe}4(4{f}_{2})$ site with ${\mathrm{Sc}}^{3+}$. The magnetic structure analyses revealed that the helimagnetic structure of $\mathrm{Ba}{({\mathrm{Fe}}_{1\text{\ensuremath{-}}x}{\mathrm{Sc}}_{x})}_{12}{\mathrm{O}}_{19}$ is a cone type whose base is in the $ab$ plane and height is in the $c\text{\ensuremath{-}}\mathrm{axis}$ direction. The magnetic moments of $\mathrm{Fe}2(4e), \mathrm{Fe}4(4{f}_{2})$, and $\mathrm{Fe}5(12k)$ are aligned with distinct angles, leading to the helimagnetic structure of $\mathrm{Ba}{({\mathrm{Fe}}_{1\text{\ensuremath{-}}x}{\mathrm{Sc}}_{x})}_{12}{\mathrm{O}}_{19}$, and can be explained by assuming that there is an antiferromagnetic superexchange interaction ${J}_{5}$ between $\mathrm{Fe}2(4e)$ and $\mathrm{Fe}5(12k)$ that is negligible in the $x=0$ crystal. The shorter bond distance of Fe2-O1 in Sc-rich crystals revealed by the crystal structure analyses enhances the superexchange interaction ${J}_{5}$. However, the ${\mathrm{Sc}}^{3+}$ preference for $\mathrm{Fe}4(4{f}_{2})$ weakens ${J}_{3}$ and ${J}_{4}$. The relatively strong ${J}_{5}$ leads to competition among the three superexchange interactions, ${J}_{3}, {J}_{4}$, and ${J}_{5}$, resulting in the development of the helimagnetic structure. These findings facilitate the understanding of the essentials of the basic magnetic material $\mathrm{Ba}{\mathrm{Fe}}_{12}{\mathrm{O}}_{19}$.