Adiabatic approximations of coupled dipole-proton model for hydrogen-bonded ferroelectrics

In several hydrogen-bonded ferroelectrics, such as ${\mathrm{KH}}_2{\mathrm{PO}}_4$-type ferroelectrics, the protons are strongly coupled to the dipoles produced in the heavy-ion sublattice. It is not clear whether such a coupled dipole-proton system can be described by two traditional simple models, the model of interacting tunneling protons and the model of interacting dipoles, and which one is more appropriate. We show that both models follow from the coupled dipole-proton model under the opposite adiabatic approximations, and that the model of interacting dipoles seems to correspond better with the experiments.

Figure 1

(a) KDP structure and the coupled dipole-proton model. A proton in a double-well hydrogen-bond potential is a two-level system with the tunneling-splitting energy $\Delta$. The ${\mathrm{PO}}_4$ dipole ${\mu }_i$ induces the term $-K{\mu }_i{x}_k$ in the hydrogen-bond potential of four neighboring protons $(k=1,2,3,4)$, where $K$ is the dipole-proton coupling constant and $x_k$ is the proton position along the hydrogen bond, relative to its center. (b) The calculated energy splitting between the two lowest-energy states of the coupled dipole-proton pair (E, exact Hamiltonian; AAD, adiabatic approximation for dipole; AAP, adiabatic approximation for proton).

Figure 2

MFA results for the regions in the $\epsilon$, $\eta$ space (a), (b) and the dependencies of $t_c$, $t_0$, and $1.5t_s$ on $\epsilon$ along the $\eta ={\eta }_1∕\epsilon$ curves (c), (d) of the coupled dipole-proton system in the AAD region $\epsilon <1$ (a), (c) and in the AAP region $\epsilon >1$ (b), (d).