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    • Introduction Lead based perovskites find numerous applicatio

    2018-11-02

    Introduction Lead-based perovskites find numerous applications in technology. They are used in high-performance piezoelectric ceramics [1], electrooptical materials [2] and non-volatile random access memory [3]. Antiferroelectric materials attract particular interest because their lattice dynamics has not been completely understood up to the present. The classical antiferroelectrics PbZrO3 and PbHfO3 exhibit a critical increase in the dielectric permittivity upon approaching the phase transition [4]. This indicates the presence of a soft ferroelectric phonon mode. On the other hand, the phase transition leads to the formation of the antiferroelectric phase [5] instead of the ferroelectric one. The physical reason for the formation of the antipolar order in these materials is currently not completely clear. Under small doping by ferroelectric additions, classical antiferroelectric PbZrO3 undergoes phase transitions to incommensurate phases [6], which have an even more complex form of structural modulation than antiferroelectricity. Until recently, there was no information on the phonon dispersion of lead-based antiferroelectrics. It has been shown that PbZrO3 single crystals have transverse acoustic (TA) phonon branch, which is highly flattened at finite wavevectors in the [11 0] pseudocubic direction [7]. In the following study on PbHfO3 we have shown an anti-crossing between the transverse acoustic and low raltegravir potassium optic branches [8]. In that paper we used the simplest rigid ion model for describing the dispersion curves. Here we consider a more elaborate and commonly used shell model [9] which accounts for the electronic polarizability of ions. The results of the description within these two models are compared to each other in terms of the predicted phonon energies and the polarization eigenvectors.
    Analysis method The rigid-ion model was initially introduced by Kellermann for describing the dispersion curves in sodium chloride [10]. Later on it was adapted by Cowley for modeling the lattice dynamics in perovskites [9]. This model is the simplest one to account for the effects of electric fields created when charged ions are displaced from their equilibrium positions. In the case of perovskite structure the rigid-ion model has eight independent parameters: six constants describing the central short-range forces between the nearest neighbor ions and two constants defining the charges of the ions. The charge of the third ion is determined by the condition of electroneutrality of the unit cell. To fit the model we took the experimental data (wavevectors, energies and energy determination uncertainities), corresponding to T=773K from Ref. [8]. We used all the available experimental points for transverse and longitudinal phonons in [1 0 0] and [1 1 0] directions simultaneously. The lattice constant was taken as a=4.15Å. The least-squares fits were accomplished using Matlab software. The shell model (SM) was first introduced by Woods et al. [11] for describing phonons in alkali halides and adapted for use with perovskite structure by Cowley [9]. In the present work, we used the SM slightly modified to account for anisotropy of oxygen polarizability, as it was done by Fontana et al. [12] for KNbO3. In addition to the experimental data determined by inelastic X-ray scattering, we used with the SM an additional experimental point, which is the value of the dielectric constant of the material (we took ε as about 575±50 [4]). The model dielectric constant was calculated by the method discussed by Cowley [9]. In order to improve the conditionality of the fitting problem we have fixed the electronic polarizabilities of Pb and Hf to their expected values. For Pb we used α=4.9Å3[13] and for Hf we used the value known for the chemically similar Zr (α=0.47Å3[13]).
    Results and discussion Fig. 1 shows the experimentally determined phonon energies together with the results of the fits by the two models. The result of the fit by the rigid-ion model is shown by dashed lines, while the SM fit is shown by solid lines. Both models provide an adequate description of the characteristic flattened transverse acoustic branch in the (q, q, 0) direction and the avoided crossing between the acoustic and optic branches. Both models reproduce the change in the group velocity for the optic branch in (q, q, 0) direction at q=0.35. The models give notably different frequencies for the small-wavevector optical branches, where there were no experimental data to directly fit to. The largest difference is observed for the longitudinal optical branch. The zone-center frequency of the transverse optical (ferroelectric) mode is notably lower for the SM.