Dirty displacive ferroelectrics

Abstract

We define a dirty displacive ferroelectric material as a displacive ferroelectric in which each unit cell is different from every other unit cell yet there is an average translational symmetry. The experimental results of the temperature dependence of the optic index of refraction $n\left(T\right)\mathrm{}$ in two dirty displacive ferroelectrics with the tungsten-bronze crystal structure are presented. In this one crystal system these data show that as the amount of disorder increases the behavior of $n\left(T\right)\mathrm{}$ becomes qualitively different from what is expected in normal ferroelectrics. We argue that this phenomenon is related to the following observation which we have previously reported: In all displacive ferroelectrics, in the ferroelectric phase, the Lyddane-Sachs-Teller (LST) relationship does not correctly predict the temperature dependence of the clamped low-frequency dielectric constant $\epsilon \mathrm{}\left(0\right)\mathrm{}$. A more complete derivation of a simple model that explains this result is given. This model considers the mechanism by which localized impurities or deviations from stoichiometry are coupled to the optic modes and shows that this type of impurity contribution to $\epsilon \mathrm{}\left(0\right)\mathrm{}$ can be considerably enhanced by this coupling. Thus, in BaTi${\mathrm{O}}_3$ an impurity concentration of as little as 3 × ${10}^{17}$ ${\mathrm{cm}}^{-3\mathrm{}}$ can explain the very large disagreement (a factor 5) in the ratio of the value of $\epsilon \mathrm{}\left(0\right)\mathrm{}$ measured by standard and capacitance techniques to the LST calculations of $\epsilon \mathrm{}\left(0\right)\mathrm{}$. In dirty displacive ferroelectrics there is no recognizable soft optic vibrational mode. Thus, to describe the existence of large peaks in $\epsilon \mathrm{}\left(0\right)\mathrm{}$ and the qualitatively unexpected behavior in $n\left(T\right)\mathrm{}$ we extend the idea of the model, used to explain the BaTi${\mathrm{O}}_3$ data, to encompass a distribution of localized charges. The contribution to $\epsilon \mathrm{}\left(0\right)\mathrm{}$ from one frequency range of a localized impurity oscillator can be enhanced by not only the optic modes but also by those impurities which have a higher frequency of oscillation. Since in dirty displacive ferroelectrics one has essentially ${10}^{23}$ ${\mathrm{cm}}^{-3\mathrm{}}$ impurities, such effects can be very important. Further, localized regions with very high dielectric constant will tend to polarize staticly or dynamically far above the transition temperature $T_c$. These localized regions of polarization can explain the $n\left(T\right)\mathrm{}$ data. In fact one may invert the process and, using these data, obtain a temperature-dependent polarization $P\left(T\right)\mathrm{}$, or more exactly, $\mathrm{}\left|P\right(T\left)\right|$. This $P\left(T\right)\mathrm{}$, which involves no adjustable parameters, compares well with the reversable spontaneous polarization $P_s$ in the ferroelectric phase. We believe that this model, which does not necessarily involve a soft optic mode, explains the behavior of dirty displacive ferroelectrics.