Empty perovskites as Coulomb floppy networks: Entropic elasticity and negative thermal expansion

Floppy networks (FNs) provide valuable insight into the origin of anomalous mechanical and thermal properties in soft matter systems, from polymers, rubber, and biomolecules to glasses and granular materials. Here, we use the same FN concept to construct a quantitative microscopic theory of empty perovskites, a family of crystals with ${\mathrm{ReO}}_3$ structure, which exhibit a number of unusual properties. One remarkable example is ${\mathrm{ScF}}_3$, which shows a near-zero-temperature structural instability and large negative thermal expansion (NTE). We trace these effects to an FN-like crystalline architecture formed by strong nearest-neighbor bonds, which is stabilized by net electrostatic repulsion that plays a role similar to osmotic pressure in polymeric gels. NTE in these crystalline solids, which we conceptualize as Coulomb floppy networks, emerges from the tension effect of Coulomb repulsion combined with the FN's entropic elasticity and has the same physical origin as in gels and rubber. Our theory provides an accurate, quantitative description of phonons, thermal expansion, compressibility, and structural phase diagram, all in excellent agreement with experiments. The entropic stabilization of critical soft modes, which play only a secondary role in NTE, explains the observed phase diagram. Significant entropic elasticity resolves the puzzle of a marked, $\approx 50\%$ discrepancy between the experimentally observed bulk modulus and ab initio calculations. The Coulomb FN approach is potentially applicable to other important materials with markedly covalent bonds, from perovskite oxides to iron chalcogenides, whose anomalous vibrational and structural properties are still poorly understood.

Figure 1

Crystal structure of ${\mathrm{ScF}}_3$ and schematics of RUM and Coulomb floppy network models. (a) Coordination polyhedra rendering of the ${\mathrm{ReO}}_3$-type cubic empty perovskite structure. (b)–(d) The RUM-based mechanistic models. Assuming rigid ${\mathrm{ScF}}_6$ octahedra, or squares in the two-dimensional version (b), the structure is fully constrained, with no floppy modes. (c) A generalized model of Refs. [13, 14] has one floppy mode, and (d) the one-dimensional chain model has two [20]. In the absence of tension at equilibrium, $\theta ={\theta }_0$, the classical RUM models are unstable and are phenomenologically stabilized by postulating the bending rigidity of flexible joints with potential energy $U\sim {(\theta -{\theta }_0)}^2$. [(f), (g)] A freely jointed floppy network structure in ${\mathrm{ScF}}_3$ is stabilized by net Coulomb repulsion. At equilibrium, the network is under tension (negative pressure). Thermal motion of F ions (green spheres) transverse to rigid Sc-F bonds pulls Sc ions closer together and acts against Coulomb tension, leading to NTE. (h) In a polymer gel, a disordered floppy network is stabilized by internal osmotic pressure [1, 2, 3, 4].

Figure 2

Transverse phonons of Coulomb floppy network in ${\mathrm{ScF}}_3$. Curves are results of our theory, Eq. (4), and symbols are the experimental data points for ${\mathrm{ScF}}_3$ from Ref. [30]. (a) The spectrum of the six transverse F phonon modes at $T=0$, $P=0$, calculated using point charge approximation ($\epsilon =0.3$, dashed lines) and with the account for the ionic charge distribution ($\epsilon =0.006$, solid lines). (b) The phonon spectrum at room temperature ($T=300$ K, $P=0$, $\epsilon =0.047$, solid line) and at a critical point ($P=P_c\left(T\right)$, $\epsilon =0$, dashed line). (c) Schematics of the F atom displacements for phonon eigenmodes described by Eqs. (6): soft rotating breathing mode (RBM) usually associated with quasi-RUMs, hard ferroelectric (FE) mode, and their counterparts, anti-RBM and anti-FE modes. (d) The phonon density of states (DOS) for the six modes in (b), calculated for ${\alpha }_F=1.0$ and ${\alpha }_F=1.3$. Dashed line shows linear interpolation.

Figure 3

Phase diagram and negative thermal expansion in ${\mathrm{ScF}}_3$. (a) Dependence of the critical temperature, $T_c$, on half of the lattice spacing, $r$. Symbols represent the experimental data for ${\mathrm{Sc}}_{1-x}{\mathrm{Ti}}_x{\mathrm{F}}_3$ (circles) and ${\mathrm{Sc}}_{1-x}{\mathrm{Al}}_x{\mathrm{F}}_3$ (squares) from Refs. [22, 23]. Solid lines are theoretical results for ${\alpha }_F=1$ and Sc-F bond dipole parameter $\delta ={\delta }_0+{\delta }_1\mathrm{\Delta }r/r_0$ (see Appendix pp2) and include correction for the reduction of the confinement parameter with $r$, assuming $\mathrm{\Delta }{u}_0=\mathrm{\Delta }r$. ${\delta }_0$ and ${\delta }_1$ were used as fitting parameters, resulting in ${\delta }_0=0.128$ and ${\delta }_1=0.35$ for ${\mathrm{Sc}}_{1-x}{\mathrm{Ti}}_x{\mathrm{F}}_3$ and ${\delta }_1=0.20$ for ${\mathrm{Sc}}_{1-x}{\mathrm{al}}_x{\mathrm{F}}_3$. The inset shows theoretical $P_c\left(T\right)$ phase diagram given by Eq. (9) with data points representing the experimental results for ${\mathrm{ScF}}_3$ from Ref. [29]. (b) Solid lines show theoretical dependencies of the lattice NTE (blue), and Sc-F bond length (green). Symbols represent the experimental data from Refs. [29, 33]. Dashed line is the result for infinitely rigid Sc-F bond, Eq. (11).