Thermal expansion mechanisms, heat transport, and band-gap renormalization in strongly anharmonic Rb$_2$SO$_4$, Rb$_2$Ca$_2$(SO$_4$)$_3$, and Ba$_3$Lu(BO$_3$)$_3$ from machine-learned interatomic potentials

Abstract

Rb$_2$SO$_4$, Rb$_2$Ca$_2$(SO$_4$)$_3$, and Ba$_3$Lu(BO$_3$)$_3$ are structurally complex materials whose thermal expansion has been interpreted, on the basis of high-temperature x-ray diffraction, in terms of specific polyhedral rearrangements. We study their lattice dynamics and its consequences for thermal and electronic properties using molecular dynamics with actively learned moment tensor potentials trained on density-functional theory data. The anharmonicity measure $σ^A$ (compares the anharmonic contribution to the interatomic forces with the total forces acting on atoms) reaches $\approx$0.3-0.5 for $β$-Rb$_2$SO$_4$, Rb$_2$Ca$_2$(SO$_4$)$_3$, and Ba$_3$Lu(BO$_3$)$_3$ and $\approx$0.75 for orientationally disordered $α$-Rb$_2$SO$_4$, and the imaginary harmonic modes of Rb$_2$Ca$_2$(SO$_4$)$_3$ and $α$-Rb$_2$SO$_4$ are stabilized at finite temperature. The simulations reproduce the measured thermal expansion and support the mechanisms proposed from diffraction: expansion along $c$ in $β$-Rb$_2$SO$_4$ arises from both elongation and straightening of corrugated Rb chains, BO$_3$ reorientation accompanies the anisotropic expansion of Ba$_3$Lu(BO$_3$)$_3$, and bending of polyhedral linkages limits expansion along the weakly expanding directions and in cubic Rb$_2$Ca$_2$(SO$_4$)$_3$. Green-Kubo lattice thermal conductivities differ by more than an order of magnitude between compounds with comparable $σ^A$, showing that the anharmonicity measure alone does not determine heat transport. Phonon-induced band-gap renormalization, including lattice thermal expansion, reaches 2.7 and 1.7 eV at 1250 K in Rb$_2$Ca$_2$(SO$_4$)$_3$ and Ba$_3$Lu(BO$_3$)$_3$ and 1.4 eV at 650 K in $β$-Rb$_2$SO$_4$, with zero-point renormalizations of 0.64, 0.70, and 0.31 eV for $β$-Rb$_2$SO$_4$, Rb$_2$Ca$_2$(SO$_4$)$_3$, and Ba$_3$Lu(BO$_3$)$_3$, respectively.