Evaluating the piezoelectric response from the elastic modulus of unpoled ceramics around the ferroelectric transition

The measurement of the piezoelectric coefficients of ceramic materials is affected by several sources of error, first of all the uncertainty on the degree of poling, and may be hindered by several factors, including excessive electrical conductivity or coercive field for the available poling field. Recently, we have proposed a method for evaluating the intrinsic piezoelectric response that unpoled ceramics would have if fully poled. It does not directly provide the piezoelectric coefficients, but has the considerable advantage of not requiring poling, and therefore is insensitive to leakage currents and to the magnitude of the coercive field. The method is based on the measurement of the softening of the elastic compliance s that occurs at the paraelectric-to-ferroelectric transition, and which can be written as [1] Ds_piezo = d+.(1/e).d(1) where s is the compliance, d is the piezoelectric coefficient and e the dielectric permittivity tensors. This formula does not take into account the depoling field and is therefore particularly useful, after performing an angular average, for describing unpoled ceramics with negligible internal depoling fields, conditions valid for not too large untextured grains and measuring frequencies below the MHz range. Thus, elastic and dielectric measurements through the ferroelectric transition of an unpoled ceramic provide an angular average of d, whose magnitude is proportional to the square root of the piezoelectric softening. This is the softening in excess with respect to the background compliance of the paraelectric phase, taking into account the linear anharmonic stiffening with cooling. Figure 1a) shows the case of a PbTiO3 based ceramic, where there is only one ferroelectric transition causing a constant softening, as predicted by the Landau theory in its simplest form. Figure 1b) shows the case of BaTiO3, with its series of ferroelectric transitions, and accordingly more complicated piezoelectric softening. Yet, the magnitude of the step at the first cubic/tetragonal transition has been shown [2] to be in good agreement with the angular average of Eq. (1), using the literature values of the dielectric and piezoelectric tensors of BaTiO3. If porosity affected in the same manner ?, d and s, then the magnitude of the piezoelectric softening would automatically provide a measure of the effective piezoelectric response, but porosities that are very high or with particular geometries may be impossible to be taken into account. Due to the angular averaging of all the piezoelectric and dielectric constants in Eq. (1), measuring the softening of only one elastic modulus of unpoled ceramics is not sufficient for obtaining the crystal or effective ceramic dij constants, but would be very useful, for example, in evaluating the effect of doping or process conditions on the piezoelectric response of materials with pure ferroelectric transitions, without the need of poling and without caring of leakage currents. In this respect, Fig. 1b) shows how easily with this method it can be shown that ferroelectricity persists in BaTiO3-? even in the metallic state induced by doping with O vacancies. Indeed, doping electrons in this manner only shifts to lower temperature the various transitions but does not depress the magnitude of the corresponding softenings [3]. This demonstrates that the measurement of the elastic response can be a valid tool in evaluating the piezoelectric and ferroelectric activity of materials that are also conducting by design or accident, as for thermoelectrics and multiferroics. A limitation of the analysis of the piezoelectric softening is that it provides reliable information on the ferroelectric/piezoelectric state only for proper ferroelectric transitions. If other modes, such as octahedral rotations in perovskites, concur to the ferroelectric transition, then softening arises also from these non-polar modes. References [1]Cordero, F., Craciun, F., Trequattrini, F., Galassi, C. Piezoelectric softening in ferroelectrics: Ferroelectric versus antiferroelectric PbZr1-xTixO3, Phys. Rev. B, 93 (2016) 174111. [2]Cordero, F. Quantitative evaluation of the piezoelectric response of unpoled ferroelectric ceramics from elastic and dielectric measurements: Tetragonal BaTiO3, J. Appl. Phys. 123 (2018) 094103. [3]Cordero, F., Trequattrini, F., Craciun, F., Langhammer, H. T., Quiroga, D. A. B. and Silva, Jr., P. S. Probing ferroelectricity in highly conducting materials through their elastic response: Persistence of ferroelectricity in metallic BaTiO3-? Phys. Rev. B 99 (2019) 064106