Large isotropic negative thermal expansion above a structural quantum phase transition

Perovskite structured materials contain myriad tunable ordered phases of electronic and magnetic origin with proven technological importance and strong promise for a variety of energy solutions. An always-contributing influence beneath these cooperative and competing interactions is the lattice, whose physics may be obscured in complex perovskites by the many coupled degrees of freedom, which makes these systems interesting. Here, we report signatures of an approach to a quantum phase transition very near the ground state of the nonmagnetic, ionic insulating, simple cubic perovskite material ${\mathrm{ScF}}_3$, and show that its physical properties are strongly effected as much as 100 K above the putative transition. Spatial and temporal correlations in the high-symmetry cubic phase determined using energy- and momentum-resolved inelastic x-ray scattering as well as x-ray diffraction reveal that soft mode, central peak, and thermal expansion phenomena are all strongly influenced by the transition.

Figure 1

Background material: (a) Structural phase diagram showing the $c\text{−}r$ phase boundary vs the $B$-site mean radius in $3d$ transition metal trifluorides and in the solid solutions ${\mathrm{Sc}}_{1-x}{\mathrm{Al}}_x{\mathrm{F}}_3$ and ${\mathrm{Sc}}_{1-x}{\mathrm{Y}}_x{\mathrm{F}}_3$. Data taken from Refs. [5, 8, 9, 10, 11]. (b) Simple cubic structure of ${\mathrm{ReO}}_3$, ${\mathrm{ScF}}_3$, and the high-temperature phases of transition metal trifluorides. ${\mathrm{Sc}}^{+3}$ ions sit at the center of regular corner-linked octahedral cages formed by ${\mathrm{F}}^{-}$ ions (red ellipsoids). (c), (d) 100 planar section of the octahedra, illustrating the displacement pattern of modes attributed to NTE. The area of the box in (d) is reduced by ${cos}^2\theta$ of the area in (c). (e) The first Brillouin zone of the simple cubic structure, showing the zone center $\mathrm{\Gamma }$ $(0,0,0)$, zone face center $X$ $(\pi ,0,0)$, zone edge center $M$ $(\pi ,\pi ,0)$, and zone corner $R$ $(\pi ,\pi ,\pi )$ points. Soft collective rigid unit modes (RUMs) which nearly preserve internal octahedral bonds are permitted on the zone boundaries, as indicated by shadowing. The lowest vibrational modes at the (f) $M$ and (g) $R$ points involve coordinated staggered rotations of octahedra in the patterns shown. The rhombohedral phase can be described as a static tilt according to the $R_4^{+}$ pattern.

Figure 2

(a) Lattice volume as a function of temperature determined from tracking the Bragg peak $(3,0,0)$ in single-crystalline ${\mathrm{ScF}}_3$. Inset: The lattice volume over a larger thermal window determined on powder samples [7]. (b) Overview of the lattice mode dispersion at select momenta and energies. Strong branch softening along the $M\text{−}R$ cut is shown by arrows. The longitudinal (LA) and transverse (TA) acoustic branches are shown near $\mathrm{\Gamma }$. Black solid lines are a guide to the eye. Red circles in the inset mark the momenta $R+\delta$ and $M+\epsilon$.

Figure 3

Color maps of the IXS signal at the (a) $M$, (b) $R$, (c) $M+\epsilon$, and (d) $R+\delta$ points in reciprocal space. Resolution-convoluted fitting of the IXS data in (a)–(d) results in squared mode energies (e) and elastic peak intensity (f). Dashed lines in (e) show the extrapolation of the mode frequencies from classical mean-field theory. Dotted lines show the expected effect of a small 74 MPa pressure on the $R$ point mode energy, suggesting this small pressure is necessary to pass through the QPT.

Figure 4

Disorder phase diagram for $\mathrm{ScF}{}_3$. Compositional disorder is quantified via the $B$-site variance ${\sigma }_B^2=\langle r_B-\langle r_B\rangle {\rangle }^2$, which has the effect of increasing the stability of the lower-symmetry, rhombohedral structural phase. Red and orange symbols show the effect of Y and Al substitution, respectively. Dotted lines indicate a linear fit to the $T_c\left({\sigma }_B^2\right)$, not including the pure sample $\mathrm{ScF}{}_3$. The vertical axis intercept reveals a temperature scale $\simeq 80\text{−}100$ K, where central peak elastic scattering and thermal expansion saturation are also observed in pure $\mathrm{ScF}{}_3$.