Magnetic structure of the magnetoelectric material Ba$_2$MnGe$_2$O$_7$

A detailed investigation of Ba$_2$MnGe$_2$O$_7$ was performed in its low-temperature magnetoelectric state combining neutron diffraction with magnetization measurements on single crystals. In the paramagnetic state at 10 K, polarized neutron diffraction was applied to reveal the components of the susceptibility tensor. The crystal and magnetic structures below the antiferromagnetic transition temperature of $T_N \approx 4$ K were determined using unpolarized neutron diffraction. This data implies no structural phase transition from 10 K down to 2.5 K and is well described within the tetragonal space group $P\bar{4}2_1m$. We found that in zero magnetic field the magnetic space group is either $C_cmc2_1$ or $P_c2_12_12_1$ with antiferromagnetic order along the [110] or [100] directions, respectively, while neighboring spins along the [001] axis are ordered antiferromagnetically. A non-collinear spin arrangement due to small canting within the $ab$ plane is allowed by symmetry and observed experimentally. The ordered moment is found to be 3.24(3) $\mu_B$/Mn$^{2+}$ at 2.5 K and the temperature-field dependent magnetic phase diagram is mapped out by macroscopic magnetization. Distinct differences between the magnetic structure of Ba$_2$MnGe$_2$O$_7$ as compared to those of Ba$_2$CoGe$_2$O$_7$ and Ca$_2$CoSi$_2$O$_7$ are discussed.