Complex magnetic incommensurability and electronic charge transfer through the ferroelectric transition in multiferroic Co$_{\mathrm{3}}$TeO$_{\mathrm{6}}$

Polarized and unpolarized neutron diffraction has been carried out to investigate the nature of the magnetic structures and transitions in multiferroic Co$_{\mathrm{3}}$TeO$_{\mathrm{6}}$. Below T$_{\mathrm{M1}}=$26 K long range order develops which is fully incommensurate in all three crystallographic directions. Below T$_{\mathrm{M2}}=$19.5 K commensurate magnetic peaks develop in the $\Gamma_{\mathrm{4}}$ irreducible representation, along with a splitting of the ICM peaks along the $h$ direction. Below T$_{\mathrm{M3}}=$18 K this additional magnetic incommensurability disappears, ferroelectricity develops, a commensurate $\Gamma_{\mathrm{3}}$ irreducible representation appears, and the $k$ component of the ICM wave vector disappears. Synchrotron x-ray data demonstrate a significant shift of the electronic charge distribution from the Te ions, which together with the unusually small electric polarization and its strong magnetic field dependence suggest this material is an antiferroelectric. Below T$_{\mathrm{M4}}=$15 K the $k$ component of the ICM structure reappears, along with second-order ICM Bragg peaks, which polarized neutron data demonstrate are magnetic in origin with a small net magnetization. See also PRB \textbf{85}, 094431 (2012); PRB \textbf{88}, 184427 (2013).