Fig. 5: AFD modulation triggered AFE order in R-NaNbO3.
From: Improper antiferroelectricity in NaNbO3-based perovskites driven by antiferrodistortive modulation
a Phonon dispersions of cubic NaNbO3 along the M-T-R path exhibiting a flat soft phonon band (in thick) involving AFD modulation (irrep. T4). b Structure model of 2 × 4 × 2 superstructure. c Local AFD mode amplitudes resolved along [010]p. d Cation displacements of AFE Δ5 mode. Experimental value derived from Fig. 1d is shown in dashed lines. e Electric polarization resulted from the AFE Δ5 mode. AFD amplitude and polarization are determined based on A-site-centered perovskite cells. f Energy profiles in order-parameter (eigenmode) space (M3+,R4+,T4,Δ5,Z1) along selected paths bridging (M3+,R4+,T4,Δ5,Z1) = (1, 1, 1, 0, 0) and (M3+,R4+,T4, Δ5,Z1) = (1,1,1,1,1) states. The amplitudes of the order parameters (horizontal axis) are normalized to 1. The vertical coordinate indicates the energy levels when gradually incorporating Δ5 and Z1 modes into the reference structure with only the M3+, R4+, and T4 modes [namely, (M3+,R4+,T4,Δ5,Z1) = (1,1,1,0,0)] along different paths.
