Magnetic phase diagram and multiferroicity of Ba$_{3}$MnNb$_{2}$O$_{9}$: A spin- $\frac{5}{2}$ triangular lattice antiferromagnet with weak easy-axis anisotropy

We have performed magnetic, electric, thermal, and neutron powder diffraction (NPD) experiments as well as density functional theory (DFT) calculations on Ba$_{3}$MnNb$_{2}$O$_{9}$. All results suggest that Ba$_{3}$MnNb$_{2}$O$_{9}$ is a spin-$\frac{5}{2}$ triangular lattice antiferromagnet (TLAF) with weak easy-axis anisotropy. At zero field, we observed a narrow two-step transition at $T_{\mathrm{N1}}$ = 3.4 K and $T_{\mathrm{N2}}$ = 3.0 K. The neutron diffraction measurement and the DFT calculation indicate a 120$^{\circ}$ spin structure in the $ab$ plane with out-of-plane canting at low temperatures. With increasing magnetic field, the 120$^{\circ}$ spin structure evolves into up-up-down ($uud$) and oblique phase showing successive magnetic phase transitions, which fits well to the theoretical prediction for the 2D Heisenberg TLAF with classical spins. Multiferroicity is observed when the spins are not collinear but suppressed in the $uud$ and the oblique phase. We discuss the results in comparison with our previous works on its sister compounds with small spins, Ba$_{3}$NiNb$_{2}$O$_{9}$ (S = 1) (J. Hwang $et$ $al$., Phys. Rev. Lett. {\bf 109}, 257205 (2012) and Ba$_{3}$CoNb$_{2}$O$_{9}$ (S = $\frac{1}{2}$) (M. Lee $et$ $al$., Phys. Rev. B {\bf 89}, 104420 (2014)).