Phonon dynamics in the layered negative thermal expansion compounds Cu$_x$Ni$_{2-x}$(CN)$_4$

This study explores the relationship between phonon dynamics and negative thermal expansion (NTE) in Cu$_x$Ni$_{2-x}$(CN)$_4$. The partial replacement of nickel (II) by copper (II) in Ni(CN)$_2$ leads to a line phase, CuNi(CN)$_4$ (x = 1), and a solid solution, Cu$_x$Ni$_{2-x}$(CN)$_4$ (0 $\leq$ x $\leq$ 0.5). CuNi(CN)$_4$ adopts a layered structure related to that of Ni(CN)$_2$ (x = 0), and interestingly exhibits 2D NTE which is about 1.5 times larger. Inelastic neutron scattering (INS) measurements combined with first principles lattice dynamical calculations provide insights into the effect of Cu$^{2+}$ on the underlying mechanisms behind the anomalous thermal behavior in all the Cu$_x$Ni$_{2-x}$(CN)$_4$ compounds. The solid solutions are presently reported to also show 2D NTE. The INS results highlight that as the Cu$^{2+}$ content increases in Cu$_x$Ni$_{2-x}$(CN)$_4$, large shifts to lower energies are observed in modes consisting of localized in- and out-of-plane librational motions of the CN ligand, which contribute to the NTE in CuNi(CN)$_4$. Mode Gr\"uneisen parameters calculated for CuNi(CN)$_4$ show that acoustic and low-energy optic modes contribute the most to the NTE, as previously shown in Ni(CN)$_2$. However, mode eigenvectors reveal a large deformation of the [CuN4] units compared to the [NiC4] units, resulting in phonon modes not found in Ni(CN)$_2$, whose NTE-driving phonons consist predominately of rigid-unit modes. The deformations in CuNi(CN)$_4$ arise because the d$^9$ square-planar center is easier to deform than the d$^8$ one, resulting in a greater range of out-of-plane motions for the adjoining ligands.