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), $n=1,2,...,\infty$] 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 $\infty$) 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 $n=\infty$ (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 $n$.

中文翻译：

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Ruddlesden-Popper钛酸钙的负热膨胀

表现出负热膨胀（NTE）的材料对于微电子设备和光学系统的制造和操作非常重要。钛酸钙是Ruddlesden-Popper（RP）钙钛矿的重要组成部分${钙}_{ñ+\mathrm{1个}}{钛}_{ñ}{\mathrm{Ø}}_{3ñ+\mathrm{1个}}$ [（CTO）， $ñ=\mathrm{1个}，\mathrm{2个}，...，\infty$它们具有分层结构，并且在其三维结构体系结构中可能表现出准二维（准2D）NTE。本文结合密度泛函理论计算和自洽准谐波近似方法，研究了声子谱的准二维特征的变化和热膨胀。${钙}_{ñ+\mathrm{1个}}{钛}_{ñ}{\mathrm{Ø}}_{3ñ+\mathrm{1个}}$ 家庭 （$ñ=\mathrm{1个}–3$， 和 $\infty$） 关于 $ñ$。我们发现准二维二维NTE机制在RP-CTO中处于活动状态$ñ$ 1–3，而准刚性单元模式机制在 $ñ=\infty$（即钙钛矿相）。我们发现包含RP系列的正交材料的NTE趋势具有层数，但单斜晶型是一个异常值。对于正交晶系成员，我们发现NTE的临界压力随着压力的增加而增加$ñ$，但NTE临界温度会降低（在相同压力下比较材料时）。另外，弹性模量可用作NTE的这种依赖于层的行为的有效描述符，即RP-CTO越硬，则其NTE越低。我们还提出了集成的NTE能力，以捕获准2D NTE与$ñ$，并且随着增加而单调减少 $ñ$。