Specific heat analyses on optical-phonon-derived uniaxial negative thermal expansion system $Tr$Zr₂ ($Tr$ = Fe and Co₁₋ₓNiₓ)

Recently, huge uniaxial negative thermal expansion (NTE) along a c-axis has been observed in transition-metal ($Tr$) zirconides $Tr$Zr₂ with a tetragonal CuAl₂-type structure. In a recent study on FeZr₂ [M. Xu et al., Nat. Commun. 14, 4439 (2023)], the importance of optical phonons to the emergence of the c-axis NTE in FeZr₂ has been proposed. In this study, the physical properties of $Tr$Zr₂ ($Tr$ = Fe and Co₁₋ₓNiₓ) have been studied by specific heat, sound velocity measurements, and theoretical phonon calculations to discuss the importance of optical phonons to the emergence of the c-axis NTE in CoZr₂ and FeZr₂. From analyses of lattice specific heat, we found that Ni substitution results in a systematic decrease in oscillator strength for the Einstein modes with 8.74 meV (CoZr₂). From phonon calculations, the low-energy optical phonon branches at the Γ point were observed for CoZr₂ and FeZr₂ with c-axis NTE, but not in NiZr₂ with positive thermal expansion. The enhancement of phonon density of states near the above-mentioned optical phonon energy in CoZr₂ and FeZr₂ is consistent with the specific heat analyses. We propose the importance of the low-energy optical phonons to the emergence of the c-axis NTE in TrZr₂.