Magnetic and ferroelectric orderings in multiferroicα-NaFeO2

The triangular based antiferromagnet $\ensuremath{\alpha}$-${\mathrm{NaFeO}}_{2}$ has been studied by magnetization, dielectric, and neutron diffraction measurements as a function of temperature and magnetic field. The appropriate $(H\ensuremath{-}T)$ phase diagram was constructed revealing a complex behavior due to a competition between several magnetic phases. In zero field, the system undergoes a sequence of magnetostructural transitions; initially from paramagnetic $R\overline{3}m{1}^{\ensuremath{'}}$ phase to the incommensurate spin density wave (ICM1) at ${T}_{\mathrm{N}1}=11$ K with the nonpolar (3+1) magnetic superspace group $C2/m{1}^{\ensuremath{'}}(0,\ensuremath{\beta},\frac{1}{2})s0s$, then, below ${T}_{\mathrm{N}2}=7.5$ K, the ICM1 phase coexists with the polar cycloidal ordering (ICM2) possessing the $Cm{1}^{\ensuremath{'}}(0,\ensuremath{\beta},\frac{1}{2})0s$ superspace symmetry and finally the commensurate collinear ordering (CM) with the nonpolar magnetic space group ${P}_{a}{2}_{1}/m$ develops below ${T}_{\mathrm{N}3}=5.5$ K as the ground state of the system. A small amount of ICM2 coexists with the ICM1 and CM phases resulting in a nonzero measured polarization below ${T}_{\mathrm{N}2}$. Magnetic field destabilizes the collinear ground state and promotes the polar ICM2 phase resulting in a drastic increase of the polarization. The symmetry of the zero field cycloidal structure allows the two orthogonal components ${\mathbit{p}}_{1}\ensuremath{\propto}{\mathbit{r}}_{ij}\ifmmode\times\else\texttimes\fi{}({\mathbit{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\mathbit{S}}_{j})$ and ${\mathbit{p}}_{2}\ensuremath{\propto}{\mathbit{S}}_{i}\ifmmode\times\else\texttimes\fi{}{\mathbit{S}}_{j}$ to contribute to the macroscopic polarization through the inverse DM effect. The applied magnetic field reduces the symmetry of the ICM2 phase down to the triclinic $P1(\ensuremath{\alpha},\ensuremath{\beta},\ensuremath{\gamma})0$, resulting in admixture of another cycloidal and helical components both generating magnetic field switchable polarization ${\mathbit{p}}_{3}$ perpendicular to ${\mathbit{p}}_{1}$ and ${\mathbit{p}}_{2}$.