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We investigate the crystal structure in multiferroic tetragonal perovskiteSr$_{1/2}$Ba$_{1/2}$MnO$_3$ with high accuracy of the order of 10$^{-3}$Angstrom for an atomic displacement. The large atomic displacement for Mn ionfrom the centerosymmetric position, comparable with the off-centeringdistortion in the tetragonal ferroelectric BaTiO$_3$, is observed in theferroelectric phase ($T_\mathrm{N}$ $\leq$ $T$ $\leq$ $T_\mathrm{C}$). In starkcontrast, in the multiferroic phase ($T$ $\leq$ $T_\mathrm{N}$), the atomicdisplacement for Mn ion is suppressed, but those for O ions are enlarged. Theatomic displacements in the polar crystal structures are also analyzed in termsof the ferroelectric modes. In the ferroelectric phase, the atomicdisplacements are decomposed into dominant positive Slater, negative Last, andsmall positive Axe modes. The suppression of Slater and Last modes, the signchange of Last mode, and the enlargement of Axe mode are found in themultiferroic phase. The ferroelectric distortion is well reproduced by afirst-principles calculation based on Berry phase method, providing anadditional information on competing mechanisms to induce the polarization,electronic $p$-$d$ hybridization vs. magnetic exchange-striction.

Ferroelectric atomic displacement in multiferroic tetragonal perovskite Sr$_{1/2}$Ba$_{1/2}$MnO$_3$