Multiferroicity of RMn$_{1-x}$Ga$_{x}$O$_{3}$ (R = Ho, Y) and Ho$_{1-x}$Y$_{x}$MnO$_{3}$

RMn$_{1-x}$Ga$_{x}$O$_{3}$ (R = Ho or Y) and Ho$_{1-x}$Y$_{x}$MnO$_{3}$ single-crystals have been prepared. The experimental results revealed that the $c$ axis decreases with increasing temperature with a larger $\left| {\mbox{d}c/\mbox{d}T} \right|$ above $T_{C}$ than below it. This shows that the cooperative MnO$_{5}$ site rotations responsible for the ferrielectricity expend energy to induce the ferroic R$^{3+}$-ion displacements along the $c$ axis. Ga doping raises the ferrielectric Curie temperature $T_{C}$ and the Mn-spin reorientation temperature $T_{SR}$ while lowering $T_{N}$ of the Mn spins and the Ho magnetic ordering temperature $T_{2}$. The data show (i) an important coupling between the Mn$^{3+}$-ion and Ho$^{3+}$-ion spins; (ii) a $T_{SR}$ that is driven by the cooperative MnO$_{5}$ site rotation and R$^{3+}$-ion displacements that control the $c$ lattice parameter. Y doping favors the formation of $P6'_{3}$\textit{cm'} magnetic phase below $T_{N}$, and enhances the temperature region of $P6'_{3}$\textit{cm'} phase. Therefore, $T_{SR}$ for the transition from $P6'_{3}$\textit{c'm to P6'}$_{3}$\textit{cm'} phase increases with increasing $x$, but $T_{SR}$ disappears for $x \quad >$ 0.8 samples because the $P6'_{3}$\textit{cm'} phase already occupies the whole temperature region below $T_{N}$. The thermal conductivity data also support an enhanced spin-lattice interaction above $T_{N}$ in the geometrically frustrated (GF) Mn-spin system.