Negative thermal expansion in the Ruddlesden-Popper calcium titanates

Materials exhibiting negative thermal expansion (NTE) are important for the fabrication and operation of microelectronic devices and optical systems. As an important group of Ruddlesden-Popper (RP) perovskites, calcium titanates ${\mathrm{Ca}}_{n+1}{\mathrm{Ti}}_{n}{\mathrm{O}}_{3n+1}$ [(CTO), $n=1,2,...,\ensuremath{\infty}$] have layered structures and may exhibit quasi-two-dimensional (quasi-2D) NTE within their three-dimensional structural architectures. In this paper, combining density-functional-theory calculations and the self-consistent quasiharmonic approximation method, we investigate the variation of the quasi-2D character of the phonon spectra and thermal expansion in the ${\mathrm{Ca}}_{n+1}{\mathrm{Ti}}_{n}{\mathrm{O}}_{3n+1}$ family ($n=1--3$, and $\ensuremath{\infty}$) with respect to $n$. We find that a quasi-2D NTE mechanism is active in the RP-CTOs at $n$ of 1--3, whereas a quasi-rigid-unit mode mechanism is active at $n=\ensuremath{\infty}$ (i.e., the perovskite phase). We find a NTE trend with layer number for the orthorhombic materials comprising the RP series, but the monoclinic polymorph is an outlier. For the orthorhombic members, we find the critical pressure for NTE increases with increasing $n$, but the NTE critical temperature decreases (when materials are compared at the same pressure). Additionally, the elastic moduli can be used as effective descriptors for this layer-dependent behavior of the NTE, i.e., the stiffer the RP-CTO then the lower its NTE. We also propose the integrated NTE capacity to capture the correlation between the quasi-2D NTE and $n$, and it monotonically decreases with increasing $n$.