Structural Relationship between Negative Thermal Expansion and Quartic Anharmonicity of Cubic ${\mathrm{ScF}}_3$

Cubic scandium trifluoride (${\mathrm{ScF}}_3$) has a large negative thermal expansion over a wide range of temperatures. Inelastic neutron scattering experiments were performed to study the temperature dependence of the lattice dynamics of ${\mathrm{ScF}}_3$ from 7 to 750 K. The measured phonon densities of states show a large anharmonic contribution with a thermal stiffening of modes around 25 meV. Phonon calculations with first-principles methods identified the individual modes in the densities of states, and frozen phonon calculations showed that some of the modes with motions of F atoms transverse to their bond direction behave as quantum quartic oscillators. The quartic potential originates from harmonic interatomic forces in the ${\mathrm{DO}}_9$ structure of ${\mathrm{ScF}}_3$, and accounts for phonon stiffening with the temperature and a significant part of the negative thermal expansion.

Figure 1

(a) ${\mathrm{DO}}_9$ structure of ${\mathrm{ScF}}_3$. (b) Geometry and variables for the mechanical model of Sc-F bonds.

Figure 2

(a) Neutron-weighted ${\mathrm{ScF}}_3$ phonon DOS from incident energies of 118.7 (black lines), 79.5 (green dashed lines), and 30.0 meV (red or grey lines), scaled to conserve spectral areas and offset for clarity. Five vertical lines are aligned to peak centers at 7 K and labeled by numbers. Errors at the top are from counting statistics, and are similar at all temperatures. (b) Shifts of phonon peak centers relative to 7 K data. The solid lines are linear fits. For each point, the spectrum with the best resolution was used. Error bars are mean differences between the spectra of different incident energies at all temperatures.

Figure 3

(a) Calculated phonon dispersions along high symmetry directions of ${\mathrm{ScF}}_3$ at 0 K. $X=(1,0,0)\pi /a$, $M=(1,1,0)\pi /a$,and $R=(1,1,1)\pi /a$. (b) Total and partial phonon DOS curves at 0 K from first-principles calculation, neutron-weighted phonon DOS with instrument broadening at 120 meV added, and experimental neutron-weighted phonon DOS at 7 K. (c) Grüneisen parameters ($\gamma$) calculated with the QHA for modes along high symmetry directions. Colors or greyscale correspond to the phonon dispersions in (a).

Figure 4

Experimental [15] and calculated linear thermal expansion coefficients.

Figure 5

Phonon mode $R4+$, its frozen phonon potential, and the quadratic (harmonic) and quartic fits to the frozen phonon potential. The range of the quadratic fit is from $-0.1$ to 0.1 Å for the transverse displacements of F atoms.