Strong magnetoelastic effect on the magnetoelectric phenomena of TbMn${}_2$O${}_5$

Comparative studies of magnetoelectric susceptibility ($\alpha$), magnetization ($M$), and magnetostriction ($u$) in TbMn${}_2$O${}_5$ reveal that the increment of $M$ owing to the field-induced Tb${}^{3+}$ spin alignment coins a field-asymmetric line shape in the $\alpha (H)$ curve, being conspicuous in a low temperature incommensurate phase but persistently subsisting in the entire ferroelectric phase. Correlations among electric polarization, $u$, and $M^2$ variation represent linear relationships, unambiguously showing the significant role of Tb magnetoelastic effects on the low field magnetoelectric phenomena of TbMn${}_2$O${}_5$. An effective free energy capturing the observed experimental features is also suggested.