Negative Thermal Expansion in the Homologous Series of Ruddlesden-Popper Calcium Titanate

Recently, it was predicted that the hybrid-improper ferroelectric compound Ca$_{3}$Ti$_{2}$O$_{7}$ ($n=2$) with the layered Ruddlesden-Popper (RP) structure ($A_{n+1}B_n$O$_{3n+1}$) exhibits a pressure-tunable negative thermal expansion (NTE) [Huang \emph{et al}., Phys.\ Rev.\ Lett.\ \textbf{117}, 115901 (2016)]. The NTE is proposed to arise from the quasi-two-dimensional nature of an acoustic mode and strong Ti--O bonding. Here, the effect of layer thickness \textit{n} on the NTE and thermodynamic properties of the homologous titanates is investigated using the self-consistent quasi-harmonic approximation (SC-QHA) method implemented within density functional theory. We show a significant dependence of the lattice dynamical modes with \emph{n}, and compare our results and associated microscopic underpinnings of the NTE responses to that proposed for Ca$_{3}$Ti$_{2}$O$_{7}$ ($n=2$). We conclude by describing a model for the anharmonic lattice properties in terms of layer dimensionality.