Two-dimensional nanoscale correlations in the strong negative thermal expansion material ${\mathrm{ScF}}_3$

We present diffuse x-ray scattering data on the strong negative thermal expansion (NTE) material ${\mathrm{ScF}}_3$ and find that two-dimensional nanoscale correlations exist at momentum-space regions associated with possibly rigid rotations of the perovskite octahedra. We address the extent to which rigid octahedral motion describes the dynamical fluctuations behind NTE by generalizing a simple model supporting a single floppy mode that is often used to heuristically describe instances of NTE. We find this model has tendencies toward dynamic inhomogeneities and its application to recent and existing experimental data suggest an intricate link between the nanometer correlation length scale, the energy scale for octahedral tilt fluctuations, and the coefficient of thermal expansion in ${\mathrm{ScF}}_3$. We then investigate the breakdown of the rigid limit and propose a resolution to an outstanding debate concerning the role of molecular rigidity in strong NTE materials.

Figure 1

(a) Inset: Experimental geometry for the diffuse scattering experiments in single-crystal ${\mathrm{ScF}}_3$. The main panel shows a $H=0.5$ cross section of the Ewald sphere. Blue and orange arcs indicate the extrema of the crystal rocking angle. (b) Diffuse scattering intensity collected using the experimental geometry in (a). Midzone intensity at $M$ points are indicated by dashed circles and are found in almost every BZ. Dashed lines indicate cuts through the intensity patterns shown in (c). Each cut in (c) is taken at an equivalent $K$ value but corresponds to a different $L$ value. The negligible change in widths is strong evidence of 2D spatial correlations.

Figure 2

(a) A $5\times 5\times 5$ crystallite of ${\mathrm{ScF}}_3$ in the average structure (125 octahedra). (b) The hollow cube shows the simple cubic BZ with shaded regions indicating the positions of observed scattering rods from the data in Fig. 1. Breakouts show possible real-space staggered rotation patterns which preserve internal octahedral dimensions at the indicated high-symmetry momentum-space points. Red and blue shading in the diamonds indicate equal but opposite magnitudes of rotation. (c) A 2D CLM which supports a single floppy mode (FM). As a result of the constraints, a staggered rotation by an angle $\theta$ in (d) causes a shortening of the vector locating each diamond by a factor $cos\theta$.

Figure 3

(a)–(d) show classical solutions for $\theta \left(t\right)$ which follow from Eq. (3). These are plotted for different values of $k{\theta }_0$, which uniquely quantifies the anharmonic behavior. The time axes in (a) and (c) are scaled by the FM period.

Figure 4

(a) Lattice parameter of ${\mathrm{ScF}}_3$ below $T=300$ K from Ref. [12]. (b) Transverse fluorine thermal parameter determined from x-ray pair density function analysis [18]. Superimposed on (a) and (b) are the corresponding quantities from (2) using $N_x=5.5$ for varying values of $\hslash {\omega }_p$. (c) Velocity field superimposed on a homogeneously excited FM, with phase and kinetic energy below. (d) shows the significant kinetic energy lowering when a $\pi /2$ defect is introduced.