Lattice dynamics of layered transition-metal dihalides

We present an extended-shell-model (ESM) calculation of the dispersion relations and of the phonon density of crystals with either Cd${\mathrm{I}}_2$ or Cd${\mathrm{Cl}}_2$ structure, namely, of V, Mn, Fe, Co, and Ni dichlorides and dibromides. The dynamical matrix, including the Coulomb and short-range contributions originating from the anion static dipoles, is constructed from model interionic potentials. While anion-anion Born-Mayer repulsion and van der Waals interactions are taken as in alkali halides, the parameters of metal-halogen short-range potential are obtained from equilibrium nearest-neighbor distance and cohesive energy. To this purpose we have derived an expression for the cohesive energy suitable to crystals with reduced ionicity and open $d$-shell configurations. A transverse metal-metal force constant and a correction to the transverse anion-anion interlayer force constant are added in order to fulfill the rotational invariance conditions. The three electrostatic parameters, net charge $Z$, anion shell charge $Y$, and anion shell-core displacement $w_0$, are adjustable and are best-fitted to the experimental Raman and far-infrared data. A reliable scale of ionicities ranging from $Z=0.71\mathrm{}$ of Ni${\mathrm{Br}}_2$ to 0.91 of Mn${\mathrm{Cl}}_2$ is obtained, and the theoretical dynamical charges and elastic constants are reported. The calculated dispersion curves are in good agreement with the few available neutron data. For Mn${\mathrm{Cl}}_2$, where a complete comparison with neutron data is possible, we show how the various ESM contributions influence the dynamical anisotropy and the lattice stability. Finally, we discuss the role of central metal-metal interaction, three-body forces, monopolar and quadrupolar metal-ion breathing (all neglected in the calculations) in order to explain the residual discrepancies.