Thermal expansion and phase stability of B F 3 ( B = Sc , Y, La, Al, Ga, In) from first principles

${\mathrm{ScF}}₃$ has attracted much attention because of its simple crystal structure and negative thermal expansion (NTE) over a wide temperature range. No other binary trifluoride has the same crystal structure and exhibits NTE in as wide a temperature range as ${\mathrm{ScF}}₃$. There has hitherto been no sufficient explanation for this fact. Herein, to discuss the phase stability and the NTE behavior of ${\mathrm{ScF}}₃$, the six binary trifluorides $B{\mathrm{F}}₃$ ($B$ = Sc, Y, La, Al, Ga, In) are thoroughly compared by using $ab\phantom{\rule{3.33333pt}{0ex}}initio$ molecular-dynamics calculations with the on-the-fly machine-learning technique, first-principles lattice-dynamics calculations, and electronic-structure analyses based on group-representation theory and the band-unfolding method. The present study reveals that (i) the ${a}^{\ensuremath{-}}{a}^{\ensuremath{-}}{a}^{\ensuremath{-}}$ octahedral rotational distortion (ORD) in ${\mathrm{ScF}}₃$ increases the frequency of the crucial phonons for realizing NTE, leading to a weakening of the NTE behavior under hydrostatic pressure; (ii) the group-representation theory predicts the emergence of ORDs triggered by the second-order Jahn-Teller effect in cubic ${\mathrm{ScF}}₃, {\mathrm{YF}}₃, {\mathrm{AlF}}₃, {\mathrm{GaF}}₃$, and ${\mathrm{InF}}₃$; (iii) cubic ${\mathrm{ScF}}₃$ and ${\mathrm{YF}}₃$ do not possess ORD without pressure due to their slight orbital overlap between the cations and anions; (iv) ${\mathrm{ScF}}₃$ has a delicate balance between the repulsive force term and the energy stabilization term, induced by the octahedral rotational phonon modes; and (v) metastable cubic ${\mathrm{YF}}₃$ and ${\mathrm{LaF}}₃$ exhibit NTE behaviors. Our findings would provide an understanding of a material-design principle for realizing NTE.