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The precise crystal symmetry and hence the emergence of the electricpolarization still remains an open question in the multiferroic materials$R$Mn$_2$O$_5$ ($R$ = rare-earth, Bi, Y). While previous diffraction studieshave indicated that $R$Mn$_2$O$_5$ possesses the centro-symmetric space groupP$bam$, an atomic displacement allowing for the electric polarization wouldrequire a non-centrosymmetric crystal symmetry. Our single crystal neutrondiffraction experiments on TbMn$_2$O$_5$ provide direct evidence of a reducedcrystallographic symmetry already above the magnetic and ferroelectric phasetransitions and a change in magnetic order upon entering the ferroelectricphase. This is indicated through the presence of additional nuclear Braggreflections that are otherwise forbidden for the space group P$bam$ but are ingood agreement with the polar space group P$12_11$. It implies that theexchange-striction, which arises from a symmetric $S_i \cdot S_j$ spincoupling, is the dominating mechanism for the generation of the electricpolarization in the commensurate magnetic phase of TbMn$_2$O$_5$. Furthermore,the commensurate magnetic reflections are in accordance with a quartile stepspin-spiral along the $c$-axis. Therefore, the antisymmetric $S_i \times S_j$exchange via the inverse Dzyaloshinskii-Moriya interaction contributes as welland becomes the leading term in the low temperature incommensurate spin-spiralmagnetic phase. These new findings provide important information for theunderstanding of the complex interplay between the magnetic and the structuralorder throughout the $R$Mn$_2$O$_5$ series of type-II multiferroics.

Reduced crystal symmetry as origin of the ferroelectric polarization within the incommensurate magnetic phase of TbMn2O5