Negative thermal expansion near two structural quantum phase transitions

Recent experimental work has revealed that the unusually strong, isotropic structural negative thermal expansion in cubic perovskite ionic insulator ${\mathrm{ScF}}_3$ occurs in excited states above a ground state tuned very near a structural quantum phase transition, posing a question of fundamental interest as to whether this special circumstance is related to the anomalous behavior. To test this hypothesis, we report an elastic and inelastic x-ray scattering study of a second system ${\mathrm{Hg}}_2{\mathrm{I}}_2$ also tuned near a structural quantum phase transition while retaining stoichiometric composition and high crystallinity. We find similar behavior and significant negative thermal expansion below 100 K for dimensions along the body-centered-tetragonal $c$ axis, bolstering the connection between negative thermal expansion and zero-temperature structural transitions. We identify the common traits between these systems and propose a set of materials design principles that can guide discovery of new materials exhibiting negative thermal expansion.

Figure 1

Structural phase diagrams of (a) $3d$ transition-metal trifluorides ${\mathrm{BF}}_3$ and (b) mercurous halides ${\mathrm{Hg}}_2{X}_2$. Open circles represent solid solutions ${\mathrm{Sc}}_{1-x}{\mathrm{Ti}}_x{\mathrm{F}}_3$ [18] and ${\mathrm{Hg}}_2{\left({\mathrm{Br}}_{1-x}{\mathrm{I}}_x\right)}_2$ [19], respectively. Insets show the basic volume-defining polyhedral units: (a) the ${\mathrm{BF}}_6$ octahedron and (b) the elongated square dipyramid (ESD). (c)–(f) Schematic structures of the (c) cubic trifluoride, (d) rhombohedral trifluoride, (e) body-centered-tetragonal mercury halide, and (f) orthorhombic mercury halide. The lower panels in (e) and (f) show views down the 001 axis, showing the shift pattern of the Hg dimer across the structural transition, and gray lines show the bct unit cell and the black diamond in (f) shows the orthorhombic unit cell.

Figure 2

Temperature-dependent lattice parameters of [(a),(c)] ${\mathrm{ScF}}_3$ and ${\mathrm{TiF}}_3$ from Refs. [10, 12, 21], and [(b),(d),(f)] ${\mathrm{Hg}}_2{\mathrm{I}}_2$ and ${\mathrm{Hg}}_2{\mathrm{Br}}_2$. The mercurous halide $c$ axis parameters were collected from the 004 and 008 reflections and the $a,b$ parameters are derived from the 300 and 030 twin reflections. The panel (e) shows a schematic free-energy landscape of high, low, and transitioning systems near the SQPT. (f) Shows an intermolecular potential appropriate for bond-stretch coordinates. The mean separation is indicated by a dashed curve, illustrating the positive thermal expansion effect of this type of excitation. Horizontal arrows in (e) and (f) show the fluctuation domain in each case. (g) Planar lattice parameters for ${\mathrm{Hg}}_2{\mathrm{Br}}_2$ and ${\mathrm{Hg}}_2{\mathrm{I}}_2$.

Figure 3

(a) Dynamical structure factor for ${\mathrm{ScF}}_3$ collected using inelastic x-ray scattering at the (2.5,3.5,0.5) reciprocal vector, corresponding to the $R$ point of the simple cubic Brillouin zone, shown as an inset. (b) Dynamical structure factor for ${\mathrm{Hg}}_2{\mathrm{I}}_2$ collected using inelastic x-ray scattering at the (2.5,3.5,0) reciprocal vector, corresponding to the $X$ point of the body-centered-tetragonal Brillouin zone, shown as an inset. (c) Soft-mode frequency squared determined from fits to the data in (a), showing the incipient ferroelastic state underlies the SNTE effect in ${\mathrm{ScF}}_3$. (d) Soft-mode frequency squared determined from fits to the data in (b), showing the incipient ferroelastic state also underlies the SNTE effect in ${\mathrm{Hg}}_2{\mathrm{I}}_2$.

Figure 4

(a) Development of the structures of the transition-metal trifluorides, with purple spheres representing ${\mathrm{F}}^{-}$ and silver spheres representing ${\mathrm{B}}^{+3}$ ions. (b) Development of the structures of the mercurous halides with purple spheres representing $X^{-}$ and silver spheres representing ${\mathrm{Hg}}^{-}$ ions. ${\mathrm{B}}^{+3}$ or $X^{-}$ ionic radius increases from left to right in (a) and (b), respectively. Overlaid lines show the ESD of Fig. 1 inset, where solid lines lie in the plane of the page and dashed lines show bonds that protrude out of the plane. (c),(d) Comparison of the observed bond distances and expectations based on hard-sphere packing (dashed lines) for (c) transition-metal trifluorides in the cubic phase [52] and (d) mercurous halides in the bct phase. Symbols in (d) are from corresponding references: triangles [39], squares [40], pentagons [38], hexagons [37], septagons [53], and circles [36].