Green's-function theory of phase transitions in the hydrogen-bonded PbHPO4-type ferroelectrics with pseudospin model

In this paper the pseudo-spin-lattice-coupled-mode (PLCM) model has been extended to study the static and the dynamic properties of PbHP${\mathrm{O}}_4$ and its deuterated form PbDP${\mathrm{O}}_4$. Using a double-time temperature-dependent Green's-function technique, we have derived the expressions for the transition temperature ($T_C$), Curie-Weiss constant ($C$), spontaneous polarization ($P_s$), dielectric constant ($\epsilon$), and the renormalized pseudospin-lattice-coupled energy spectrum and fitted them to the corresponding experimental data. A single set of the Blinc—de Gennes parameters, being different for different crystals, has been found to fit the experimental values of $T_C$, $C$, $P_s$, and $\epsilon$ very well both for PbHP${\mathrm{O}}_4$ and PbDP${\mathrm{O}}_4$ salts. In contrast to the case of K${\mathrm{H}}_2$P${\mathrm{O}}_4$, very small values of the tunneling frequency (≅2.5 ${\mathrm{cm}}^{-1\mathrm{}}$) and the frequency of the soft proton mode (≅4 ${\mathrm{cm}}^{-1\mathrm{}}$) are observed from our calculations, which agree with the results of recent far-infrared, dielectric, and Raman-scattering experiments. This is a first attempt to explain the simultaneous occurrence of a low soft-mode frequency (corresponding to a low value of the tunneling integral) and a large shift in $T_C$ in some hydrogen bonded ferroelectrics like PbHP${\mathrm{O}}_4$. We need not consider the additional coupling term $B_{\mathrm{ij}}{S}_i^x{S}_j^x$ along with the transverse Ising model for the explanation of the above results as suggested by Blinc et al. [Phys. Rev. B 20, 1991 (1979)]. Another interesting feature clearly observed from our calculations verifying the experimental data of neutron diffraction is that the interchain coupling between the protons of the neighboring chain is much weaker than the intrachain coupling between the two O—H——O dipoles in a chain. The former interaction is found to be about half of the latter one. Finally, comparing the results of our calculations on different hydrogen bonded ferroelectrics with the PLCM model, we suggest the possibility of a unified theory of phase transitions in these crystals having the characteristics of both order-disorder and displacive mechanisms.