Influence of Impurities on the Curie Temperature of Ferroelectric Crystals

Abstract

The method of Green's functions is used to derive an expression for the susceptibility of an ionic crystal with defects in a homogeneous electric field in a harmonic approximation. For the sake of simplicity, the polarizability of the ions is not considered and the general expression is specialized for the case of isovalent substitutional impurities. Using the temperature dependence of an anomalously low frequency [${\ensuremath{\omega}}^{2}=\ensuremath{\gamma}(T\ensuremath{-}{T}_{C})$] of one transverse optic mode of vibration of long wavelength in ferroelectric crystals, it is shown that the influence of such defects on the static dielectric constant is seen as the shift of the Curie temperature, as is known experimentally. The Curie-temperature shift is expressed by means of the force constants between ions, and in the crystal model in question it depends linearly on the concentration of impurities in a broad range of concentrations. The Curie-temperature shift is used to estimate the magnitude of the relative change in frequency of the anomalous optic ("ferroelectric") mode in the (Ba,Sr)Ti${\mathrm{O}}_{3}$ crystal, which could be determined by the inelastic scattering of neutrons.