Silent soft mode in hexagonal barium titanate observed by hyper-Raman scattering.

In a recent paper' Inoue et al. discussed a soft optical phonon that drives the second-order phase transition from hexagonal BaTi03, space group P63/mme (D6t, ), to a lower-symmetry structure at T, =222 K. The new phase is not ferroelectric and its structure is not known. The soft mode is Raman and infrared inactive and therefore is a so-called "silent" mode. Using hyper-Raman scattering, Inoue et al. ' observed that the zone-center E2„mode was underdamped and became soft as the temperature T, was approached from above. Since it remained underdamped the phonon frequency could be followed near the transition and gave a mean-field Landau-Cochran exponent y = 1. On the basis of the E2„ driving mode, they indicated that the lower-symmetry phase should be of symmetry 222(D2) or mm 2(Cz, ). In this Comment, we use space-group theory to discuss results which follow from knowing that the zone-center E2„mode is driving the transition. We obtain the lowersymmetry space group from the order parameter, we give the free energy for the structure including its elastic coupling, and we show the possible optical distortions allowed by the soft mode. The E2„mode is labeled as the I 5 representation in Table I of Stokes and Hatch. The onset of the order parameter of this representation brings about the symmetry reduction to the space groups C222, (D2), Cmc2, (C2, ), and P2, (C2) depending upon the direction of the order parameter. The subgroup C222, (D2) is improper ferroelastic and the transition from the paraphase P63lmmc(D6h ) is allowed to be a continuous. The subgroup Cmc2, (C2, ) is improper ferroelastic and improper ferroelectric. This transition from the paraphase is also allowed to be continuous. The subgroup P2, (Cz) .is improper ferroelastic and improper ferroelectric but the transition must be discontinuous. Since the observed lower-symmetry phase is nonferroelectric, the phase C222t(Dz) is most likely the lower-symmetry phase of hexagonal barium titanate. It is of course possible that, even though the improper ferroelectric properties are allowed by symmetry for the other two phases, they are not significant because of weak coupling. However, it is likely that the ferroelectric properties would be observed if allowed so we expect the structure below the transition to be C222&(Dz). From Tables IX and X in Stokes and Hatch we can obtain the free energy of the order parameter. For the I 5 representation, the free energy to fourth order takes the form bF = —,a, (ri, +g2)+ —,b, (g, +g2) 1 2 2 1 (l)