Soft-mode dynamics of partially deuteratedKH2PO4-type crystals

The dynamics of isotopically disordered ferroelectric crystals of $\mathrm{K}{\left({\mathrm{H}}_{1-x}{\mathrm{D}}_x\right)}_2\mathrm{P}{\mathrm{O}}_4$ type is discussed on the basis of the coupled proton-deuteron tunneling-lattice optic-mode model. It is shown that one of the collective modes of the disordered system exhibits a soft-mode behavior at the ferroelectric transition temperature. The collective mode frequencies and their pressure derivatives are calculated by a modified random-phase approximation as a function of deuteron concentration and compared with recent low-temperature Raman data of Peercy. The theory allows an estimate of the various parameters of the model and provides strong support for the proton tunneling picture of hydrogen-bonded ferroelectrics.