Cycloidal spiral magnetic structures in the spin-chain compounds BaRFeO4 ( R=Yb and Tm): Ordered Yb versus partly ordered Tm

$\mathrm{Ba}R\mathrm{Fe}{\mathrm{O}}₄$ compounds containing magnetic rare-earth ions ($R=\mathrm{Yb}$, Tm) were prepared by a conventional solid-state method in air. We present detailed measurements of magnetic properties (specific heat, magnetic susceptibility) to demonstrate the presence of three successive magnetic phase transitions (at ${T}_{\mathrm{N}1}$, ${T}_{\mathrm{N}2}$, and ${T}₃$) in both compounds and employed neutron diffraction to determine the magnetic structures. Both compounds are isostructural (orthorhombic space group $Pnma$) and magnetic ions form rings and chains along the $b$ direction. All magnetic structures are incommensurate with a propagation vector $\mathbf{k}=(0,\phantom{\rule{0.28em}{0ex}}0,\phantom{\rule{0.28em}{0ex}}{k}_z)$. Magnetic ordering of ${\mathrm{Fe}}³⁺$ ions occurs at ${T}_{\mathrm{N}1}$ and ${T}_{\mathrm{N}2}$, where two different irreducible representations (irreps) order. Below ${T}_{\mathrm{N}1}$, there is a collinear spin-density wave with ordered ${\mathrm{Fe}}³⁺$ moments along the chain direction ($b$ axis). Below ${T}_{\mathrm{N}2}$, this component remains stable and an additional component inside the $ac$ plane appears. Each Fe chain adopts a collinear antiferromagnetic structure with a constant magnetic phase. The two Fe rings in the unit cell have a different chirality and a noncollinear coupling. The mixing of two irreps leads to a cycloidal spiral magnetic structure that allows spin-induced ferroelectric polarization at ${T}_{\mathrm{N}2}$. With the presence of modulated components both perpendicular and along the propagation vector $\mathbf{k}$, the magnetic structure can be viewed as a sum of a helix and a cycloid structure. For ${\mathrm{BaTmFeO}}₄$, the magnetic structure has a larger cycloidal contribution and the dielectric constant \ensuremath{\epsilon} exhibits a small peak at ${T}_{\mathrm{N}2}$. ${\mathrm{Yb}}³⁺$ moments order at ${T}₃$ with each Yb chain having a constant magnetic phase and a collinear antiferromagnetic structure stabilized by $4f\ensuremath{-}4f$ electron-exchange interactions. In contrast, no constant magnetic phase is observed at the Tm chains. Below ${T}₃$, magnetic order of Tm2 ions is induced by $3d\ensuremath{-}4f$ electron-exchange interactions and Tm1 ions remain disordered down to the lowest measured temperature $T=1.6\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ due to frustration of magnetic exchange interactions. The obtained magnetic structures are compared with those of ${\mathrm{BaYFeO}}₄$.