Change in charge density wave order beyond the Lifshitz transition in 2H-Tatextsubscript{1pmδ}Stextsubscript{2}

We investigate electronic instabilities in 2H-TaS\textsubscript{2} and a self-intercalated variant, 2H$^\dagger$-Ta\textsubscript{1+$\delta$}S\textsubscript{2}. In conventional samples, which we determine to be slightly hole-doped, spectral gaps and backfolded features are found as fingerprints of the $3\times3$ charge density wave (CDW). Notably, the backfolded features emerge only at a temperatures below $T\approx$~65~K, substantially lower than the established CDW temperature of 78~K, suggesting an incommensurate-commensurate lock-in transition analogous to the phenomenology of the 2H-TaSe\textsubscript{2}. In contrast, the self-intercalated 2H$^\dagger$ sample exhibits substantial electron doping and signatures of a novel \tworootthree CDW. Using \textit{ab initio} calculations of the phonon spectrum, we demonstrate that the \threebythree instability ($\mathbf{q}=\sfrac{2}{3}\mathbf{\Gamma M}$) is highly sensitive to band filling. Furthermore, with increased interlayer spacing, a competing soft phonon mode emerges near $\mathbf{q}=\sfrac{1}{2}\mathbf{\Gamma K}$, corresponding to the superstructure observed in the 2H$^\dagger$ phase, although in our calculations this instability arises under hole doping rather than the electron doping inferred experimentally. These results establish band filling and interlayer spacing as key control parameters for CDW ordering vectors in 2H-TaS\textsubscript{2}, and highlight a route to engineering electronic instabilities in a prototypical layered material.