Lattice dynamics of the Ta$_2$Ni(Se$_{1-x}$S$_x$)$_5$ excitonic insulator candidates

We report the results of polarization-resolved Raman-scattering for the family of excitonic insulator candidate materials Ta$_2$Ni(Se$_{1-x}$S$_x$)$_5$ ($x = 0$, 0.25, 0.67, 1), focusing on the properties of the even-symmetry optical phonons. We observe all of the expected 11 Raman-active optical phonon modes, identify their symmetry, and obtain the temperature dependence for their spectral parameters, including the frequency, the relaxation rate and the integrated intensity. Additionally, the change in selection rules allows us to determine the orthorhombic-to-monoclinic structural transition temperature $T_c$ for all compositions and we find it to decrease monotonically from 328 K for Ta$_2$NiSe$_5$ to 120 K for Ta$_2$NiS$_5$. For Ta$_2$Ni(Se$_{1-x}$S$_x$)$_5$ with $x < 0.7$, we find the two lowest-energy $B_{2g}$ phonon modes to show strongly asymmetric lineshapes in the high-temperature phase due to Fano interference with an excitonic continuum; we develop a quantitative description of their lineshape, allowing us to obtain the phonon parameters also above $T_c$. For Ta$_2$NiSe$_5$ we also observe signatures of acoustic modes scattered by the structural domains formed below $T_c$. For Ta$_2$NiS$_5$, all $B_{2g}$ phonon modes preserve symmetric lineshape, however, their energy anomalously decreases on cooling. The temperature dependence of the spectral parameters of most modes is found to be consistent with anharmonic effects; for the alloy compositions $x=0.25,0.67$ they also display two-mode behavior. Finally, we demonstrate that the frequencies of the modes are generally in agreement with first principles calculations and the latter also allow to explain the differences between the coupling strengths of the $B_{2g}$ phonons with the excitonic continuum.