(Color online) Schematic of the phase transition induced by an electrical field applied along  axis above for single crystal. In paraelectric phase the polarization varies with field below , while in ferroelectric phase the polarization sits in the interval above .
(Color online) Field dependence of the dielectric permittivity for single crystal at . The open squares represent the measured values, and the dotted line (blue) is to guide the eyes. Remarkably different are obtained for either phase when fitting with Eqs. (4) and (5), as listed in Table I. The dashed (red) and solid (green) lines are calculated with the coefficients derived from the paraelectric and ferroelectric data, respectively. The ferroelectric coefficients have a serious problem in describing the dielectric properties of the paraelectric phase, and vice versa. The inset shows the much better fit quality (black dashed-dotted line) of an eighth-order expansion for both phases.
(Color online) The quartic coefficient determined with sixth- and eighth-power expansions in the vicinity of . Phase dependent values are derived when using sixth-power expansion, whereas phase independent can be obtained with the eighth-power expansion. The values below are estimated by fitting the dielectric nonlinearity with constant and . However, this is less convincing since the temperature dependence of and have been arbitrarily omitted.
(Color online) Temperature dependence of the quartic coefficients (a) and (b). The values of are determined by combining the dielectric nonlinearity measurements along , , and  orientations, and the values and temperature derivatives calculated from three combinations agree well with each other.
(Color online) Divergency between the quantity and in the ferroelectric tetragonal phase. The values of dielectric permittivity and spontaneous polarization are extracted from literature (Ref. 15).
(Color online) Inverse dielectric permittivity along the spontaneous polarization in the vicinity of , a comparison between the calculations and the experimental data. For convenience the phase transition temperatures are normalized to . The scattered dark marks are experimentally measured by different researchers on different crystals: by Drougard (Ref. 19), 엯 by Merz (Ref. 15), and ◻ in this work. The dashed (blue), dash-dotted (green), and solid (red) lines are calculated with Li et al., Bell, and the proposed potential in this work, respectively. Note that despite the great difference in the source sample, the permittivity exhibits tremendous consistence on departing from . This potential has a much better quality in fitting the experimental data than the two existing potentials do.
(Color online) Spontaneous polarization along pseudo cubic  of , a comparison between the calculations and the experimental data. The scattered marks represent the experimental data: ◻, ▵, and ◆ are extracted from literature (Refs. 15, 25, and 26), respectively. The dashed (dark), dash-dotted (red), and solid (blue) lines are calculated with Li et al., Bell, and the proposed potential in this work, respectively. The polarization values of the rhombohedral phase are exactly reproduced by the improved thermodynamic potential.
Estimated values of the polarization dependent terms in Eq. (6) and typical values of the corresponding coefficients. The value of is taken from Ref. 4.
Anharmonic coefficients determined from the field dependence of dielectric permittivity shown in Fig. 2. The values proposed by Li et al. are also listed for comparison.
Literature available values of the anharmonic coefficients for determined with the sixth-power expansion in the vicinity of .
Analytical calculated and experimentally determined values of and for single crystals. The anharmonic coefficients and their temperature derivatives are taken from literature (Refs. 4–6). The spontaneous polarization are taken from literature (Ref. 15).
Coefficients of Landau thermodynamic potential in Eq. (22), where is temperature in K.
Article metrics loading...
Full text loading...