Unified model for the transverse electric susceptibility in K${\mathrm{H}}_2$P${\mathrm{O}}_4$ -and N${\mathrm{H}}_4$${\mathrm{H}}_2$ ${\mathrm{PO}}_4$-type crystals

The dielectric properties of N${\mathrm{H}}_4$ ${\mathrm{H}}_2$ P${\mathrm{O}}_4$-type crystals are investigated using an antiferroelectric version of an extended pseudospin model previously applied to ferroelectric K${\mathrm{H}}_2$ P${\mathrm{O}}_4$-type crystals. The model takes into account the transverse polarization induced by the proton displacements along the hydrogen bonds. The Hamiltonian is based on a four-sublattice version of the Slater-Takagi protonic configurations around the P${\mathrm{O}}_4$ groups and includes also a long-range antiferroelectric interaction energy. The temperature dependence of the susceptibilities along the $a$ and $c$ directions is calculated by applying a four-cluster approximation to the model. The theoretical results are compared to the available data for the dielectric constants along the $a$ and $c$ directions of N${\mathrm{H}}_4$ ${\mathrm{H}}_2$ P${\mathrm{O}}_4$, ${\mathrm{ND}}_4$ ${\mathrm{D}}_2$ P${\mathrm{O}}_4$, and N${\mathrm{H}}_4$ ${\mathrm{H}}_2$ As${\mathrm{O}}_4$. An excellent fit to the data for both the $a$ and $c$ directions is achieved by taking a single set of energy parameters for each crystal. It is found that both the long-range antiferroelectric interaction and the Slater short-range energy play an important role in the antiferroelectric ordering. We also present a detailed comparison of our model for antiferroelectric N${\mathrm{H}}_4$ ${\mathrm{H}}_2$ P${\mathrm{O}}_4$-type crystals with that of Ishibashi et al. and we find that our results for the transverse susceptibility differ from theirs. This difference arises from our inclusion of the direct contribution of the proton displacements to the transverse polarization.