Magnetoelectric phase diagrams of the rare-earth (R) Mn perovskites RMnO_3are theoretically studied byfocusing on crucial roles of the symmetric magnetostriction or the Peierls-type spin-phonon coupling throughextending our previous work M. Mochizuki et al., Phys. Rev. Lett. 105, 037205 (2010). We first constructa microscopic classical Heisenberg model for RMnO_3including the frustrated spin exchanges, single-ionanisotropy, and Dzyaloshinskii-Moriya interaction. We also incorporate the lattice degree of freedom coupled tothe Mn spins via the Peierls-type magnetostriction. By analyzing this model using the replica-exchange MonteCarlo technique, we reproduce the entire phase diagram of RMnO_3in the plane of temperature and magnitude ofthe orthorhombic lattice distortion. Surprisingly it is found that in the ab-plane spiral spin phase, the (S ·S)-typemagnetostriction plays an important role for the ferroelectric order with polarization P awhose contribution iscomparable to or larger than the contribution from the (S x S)-type magnetostriction, whereas in the bc-planespiral phase, the ferroelectric order with P 11c is purely of (S x S) origin. This explains much larger P in theab-plane spiral phase than the bc-plane spiral phase as observed experimentally and gives a clue how to enhancethe magnetoelectric coupling in the spin-spiral-based multiferroics. We also predict a noncollinear deformationof the E-type spin structure resulting in the finite (S x S) contribution to the ferroelectric order with Pa,and a wide coexisting regime of the commensurate E and incommensurate spiral states, which resolve severalexperimental puzzles.
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