Fig. 2: Potential energy surfaces (PESs) from soft phonon modes.

PESs of a the ferroelectric instability mode at the Γ point, b the in-phase tilting mode at the M point, and c the out-of-phase tilting mode at the R point, respectively for KNbO3. The corresponding PESs for NaNbO3 is shown for d the ferroelectric instability mode at the Γ point, e the in-phase tilting mode at the M point, and f the out-of-phase tilting mode at the R point, respectively. Solid lines represent the fourth-order (quartic) polynomial fit, while the dashed lines represent the second-order (harmonic) polynomial fit. To investigate the volume dependence, the unit cell volume is scaled from 0.96 to 1.04 with an interval of 0.01 (from dark to lighter shades) based on the DFT optimized volume.