Resolving the local distortions of Ising-like moments in magnetoelectric Ho-doped langasite

The magnetic properties of Ho-doped langasites ${(\mathrm{La}:\mathrm{Ho})}_{3}\mathrm{G}{\mathrm{a}}_{5}\mathrm{Si}{\mathrm{O}}_{14}$ are dominated by the Ising-like magnetic moments of the $\mathrm{H}{\mathrm{o}}^{3+}$ ions. In their saturated regime, the induced magnetic state breaks both time and space inversion symmetries, leading to a linear magnetoelectric effect. However, due to distortions induced by a shared Ga/Si occupancy of the $2d$ sites, resolving the microscopic nature of the magnetic configuration remains a difficult task. Here, we combine polarized neutron diffraction and angular-dependent magnetization experiments to determine the local distortions of the $\mathrm{H}{\mathrm{o}}^{3+}$ magnetic moments in doped langasites ${(\mathrm{L}{\mathrm{a}}_{1\ensuremath{-}x}\mathrm{H}{\mathrm{o}}_{x})}_{3}\mathrm{G}{\mathrm{a}}_{5}\mathrm{Si}{\mathrm{O}}_{14}$ with $x\ensuremath{\approx}0.015$ and 0.045. We propose a model for a field-induced magnetic configuration with arbitrary orientations of the local Ising axis of $\mathrm{H}{\mathrm{o}}^{3+}$ in distorted positions. The operations of broken local ${C}_{2}$ symmetry and rotations around the trigonal ${C}_{3}$ axis connect different sites, restoring the global $P321$ symmetry of the crystal and simplifying the description of the magnetic properties. The superposition of two distorted $\mathrm{H}{\mathrm{o}}^{3+}$ positions connected by ${C}_{2}$ symmetry determines the local magnetic susceptibility tensor, which no longer appears Ising-like at low fields.