Charge Orders in Fully Intercalated Bilayer TaSe$_2$: Dependence on Interlayer Stacking and Intercalation Sites

Recent advancements have established self-intercalation as a powerful technique for manipulating quantum material properties, with precisely controllable intercalation concentrations. Given the inherently rich phase diagrams of transition metal dichalcogenides (TMDCs), studying the self-intercalated TMDCs can offer promising candidates for investigating the interplay between various orderings. This work focuses on fully intercalated bilayer TaSe$_2$ (Ta$_3$Se$_4$), which has recently been fabricated experimentally. By performing first-principles calculations, we demonstrate the suppression of an intrinsic $3\times3$ charge density wave (CDW) in parent TaSe$_2$ layers, and the emergence of $2\times 2$, $\sqrt{3} \times \sqrt{3}$, or the absence of a CDW in the intercalated layers, depending on the interlayer stacking orders and intercalation sites being occupied. Particularly, the $2\times 2$ CDW shows an increase in electronic states at the Fermi level compared to its non-CDW phase. This unusual behavior contrasts with that of typical CDW materials in TMDCs. Furthermore, superconductivity is preserved in these Ta$_3$Se$_4$ structures, with superconducting transition temperatures comparable to or substantially smaller than those of TaSe$_2$. Spin-orbit coupling is found to enhance the density of states at Fermi levels while simultaneously reducing the electron-phonon coupling matrix elements. These two competing effects result in varying impacts on superconductivity across different Ta$_3$Se$_4$ structures. Moreover, our calculations indicate that magnetic order is absent. Our study deepens the understanding of underlying physics in Ta$_3$Se$_4$, and provides experimentally feasible candidates for studying CDW, superconductivity, and their interplay.