  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.
