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Page 6 of 14 Fujii et al. Microstructures 2023;3:2023045 https://dx.doi.org/10.20517/microstructures.2023.43

like to non-Rayleigh-like behavior decreases. As a result, Figure 4 shows marked departures from Rayleigh-
like behavior at ac field magnitudes under 1 kV/cm in the tetragonal phase field for the BaTiO . Above these
3
fields, the permittivity first rises very quickly as new domains are nucleated and then goes through a
maximum before decreasing as the field amplitude is further increased. This is in contrast to the data in
Figures 3 and 5 for finer-grained ceramics, where no marked maximum in the relative permittivity is
observed up to 4 kV/cm over the same temperature range.

Figure 5 shows the ac field dependence of the dielectric properties of the undoped BaTiO ceramics with a
3
grain size of 1.2 µm measured at various temperatures. It was observed that the ac field dependence was
suppressed compared to the large-grained sample over the whole temperature range, as has been reported
[23]
elsewhere . Grain boundaries are regions where the crystallography of the material is disrupted, and in
many cases, these grain boundaries act as pinning centers for the domain walls [25,30-33] .

The dielectric data for samples with different grain sizes were quantified by a pseudo-Rayleigh analysis that
replaced ε ' with ε' - the dielectric constant measured at a small field amplitude j. α' was replaced with α' -
i
j
init
the slope for the dielectric constants measured at the field amplitudes j and i (i > j). In this case, pseudo
refers to the fact that the permittivity does not increase truly linearly as a function of ac field. Figure 6 shows
the temperature dependence of the pseudo-Rayleigh parameters of the BaTiO ceramics with a grain size of
3
1.2 µm and 76 µm. The field i was chosen to ~0.5 kV/cm or ~2 kV/cm to quantify domain wall
contributions at subcoercive fields. The temperature dependence of the ε' 0.03kV/cm was similar to that of the
small signal dielectric constant, except that no dielectric peak was observed at 125 °C for the large-grained
[25]
ceramics due to its higher T of 128 °C . The observation that the ε' 0.03kV/cm of the 1.2 µm grain size sample is
c
larger than that of the 76 µm grain size sample below T is due to the so-called grain size effect of BaTiO .
[7]
3
c
For the 76 µm grain size sample, the α' values were larger than those of the 1.2 µm grain sample due to
i
reduced domain wall pinning at grain boundaries. The α' values were increased at the rhombohedral to
i
orthorhombic phase transition and orthorhombic to tetragonal phase transition temperatures. In addition,
they were increased at 80 °C for the 76 µm grain size sample. This is probably due to a combination of
increased domain wall mobility and decreased polarization with increasing temperature. For the 1.2 µm
grain size sample, the increases at 80 °C were not observed due to suppressed domain wall mobility. For the
76 µm grain size sample, α' 2kV/cm is larger than α' 0.5kV/cm above -70 °C, indicating a superlinear ac field
dependence. This, in turn, suggests a sublinear response below this temperature, which would be consistent
with a restriction in long-range domain wall motion and/or a reduction in the nucleation of new domains.

Given that domain wall motion is a function of the concentration of pinning sites in the sample, it is also
interesting to consider the effect of point defects as potential pinning points. Here, the oxygen vacancy
concentration was varied by sintering undoped BaTiO ceramics at different oxygen partial pressures, as
3
described elsewhere . Figure 7 shows the ac field dependence of the dielectric properties of BaTiO
[24]
3
ceramics sintered at 10 atm pO with a 1.2 µm grain size measured at various temperatures. It was found
-9
2
that the response was similar to that of the sample sintered at 10 atm pO , except that the larger increase in
-2
2
the dielectric properties at moderate ac fields than that of the sample sintered at 10 atm pO below T and
-2
2
c
larger loss above T .
c
The effect of the pO on the dielectric nonlinearity was quantified at various temperatures, as shown in
2
Figure 8. Neither sample showed a true Rayleigh regime, presumably due to a combination of defect dipoles
associated with oxygen vacancies, residual stresses , and pinning from the grain boundaries. It was found
[34]
that the temperature dependence of the pseudo-Rayleigh parameters was similar, except that the α' of
i
-2
10 atm pO sample were larger than those of the 10 atm pO sample, which is attributable to the de-
-9
2
2