Identification of soft modes across the commensurate-to-incommensurate charge density wave transition in 1$T$-TaSe$_2$
1$T$-TaSe$_2$ is a prototypical charge density wave (CDW) material for which electron-phonon coupling and associated lattice distortion play an important role in driving and stabilizing the CDW phase. Here, we investigate the lattice dynamics of bulk 1$T$-TaSe$_2$ using angle-resolved ultralow wavenumber Raman spectroscopy down to 10 cm$^{-1}$. Our high-resolution spectra allow us to identify at least 27 Raman-active modes in the commensurate (CCDW) phase. Contrary to other layered materials, we do not find evidence of interlayer breathing or shear modes, suggestive of $AA$ stacking in the bulk, or sufficiently weak interlayer coupling. Polarization dependence of the mode intensities allows the assignment of their symmetry, which is supported by first-principles calculations of the phonons for the bulk structure using density functional theory. A detailed temperature dependence in the range $T$ = 80 - 500 K allows us to identify soft modes associated with the CDW superlattice. Upon entering the incommensurate (ICCDW) phase above 473 K, we observe a dramatic loss of resolution of all modes, and significant linewidth broadening associated with a reduced phonon lifetime as the charge-order becomes incommensurate with the lattice.