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Magnetically induced electric polarization reversal in multiferroic TbMn$_2$O$_5$: Terbium spin reorientation studied by resonant x-ray diffraction

In multiferroic TbMn$_2$O$_5$, the behavior of the terbium ions forms a crucial part of the magneto-electric coupling. The result is a magnetically induced reversal of the electric polarization at 2 T. In this article we present the first direct measurement of the terbium magnetic structure under applied magnetic fields. Contrary to the current interpretation of the magnetic properties of \tmo, we show that upon the electric polarization reversal the terbium sub-lattice adopts a canted antiferromagnetic structure with a large component of magnetic moment parallel to the \textit{a}-axis. Furthermore, we provide evidence for a coupling between the manganese 3d magnetic structure and the terbium $4f$ magnetism, which is of great significance in the elusive magneto-electric mechanisms at play.