In this report, a comprehensive neutron diffraction study as a function of temperature and magnetic field up to 5 T has been carried out to evaluate the complex magnetic structures of the multiferroic material ${\mathrm{BaHoFeO}}_4$. The compound is stabilized in an orthorhombic crystal structure with Pnma space group and consists of corner-sharing Fe(1)${\mathrm{O}}_5$ and Fe(2)${\mathrm{O}}_6$ polyhedral rings formed along the $b$ axis, exhibiting complex magnetic orderings. For $T<T_{N1}$, a collinear antiferromagnetic spin-density-wave (AFM-SDW) structure of ${\mathrm{Fe}}^{3+}$ moments with ${\mathbf{k}}_{\mathbf{1}}=(0,0,0.366)$ has been observed. For $T<T_{N2}$, a commensurate magnetic ordering with ${\mathbf{k}}_{\mathbf{2}}=(0.5,0,0.5)$ arises from both the ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Ho}}^{3+}$ sites, forming interchain AFM and intrachain AFM correlations, respectively. Further, the magnetic structure at $T<T_{N3}$ is attributed to a multi-$\mathbf{k}$ noncoplanar noncollinear antiferromagnet with ${\mathbf{k}}_{\mathbf{2}}(0.5,0,0.5)+{\mathbf{k}}_{\mathbf{3}}(0,0,0.5)$. The analysis of magnetic-field-dependent neutron powder diffraction (NPD) data reveals a significant influence of the magnetic field on the lattice parameters and the magnetic spin structures of ${\mathrm{BaHoFeO}}_4$. A complex magnetic phase diagram in the H-T plane consisting of six ordered magnetic states has been determined. The corroboration of metamagnetic transitions to the microscopic magnetic structures at respective magnetic critical fields is established. The study demonstrates a strong coupling between lattice parameters and magnetic spin orientations, positioning ${\mathrm{BaHoFeO}}_4$ as a promising temperature- and field-induced spin-lattice-coupled magnetodielectric material.

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• Dielectric properties
• Magnetic order
• Magnetic phase transitions
• Magneto-dielectric effect
• Structural properties
• Neutron diffraction