Evolution of Superatomic-Charge-density-wave and Superconductivity under Pressure in AuTe$_2$Se$_{4/3}$

Superatomic crystal is a class of hierarchical materials composed of atomically precise clusters assembled via van der Waals or covalent-like interactions. AuTe$_2$Se$_{4/3}$, an all-inorganic superatomic superconductor exhibiting superatomic-charge-density-wave (S-CDW), provides a first platform to study the response of their collectively quantum phenomenon to the external pressure in superatomic crystals. We reveal a competition between S-CDW and superconductivity using cutting-edge measurements on thin flakes at low pressures. Prominently, the pressure modulation of S-CDW ordering is 1$\sim$2 order of magnitudes (0.1 GPa) lower than that of conventional atomic superconductors. As pressure increases to 2.5 GPa, the $T_{\mathrm{CDW}}$ is suppressed and the superconducting transition temperature ($T_{\mathrm{c}}$) is firstly enhanced, and reaches the maximum then quenches with increasing pressure. Above 7.3 GPa, a second superconducting phase emerges, and then a three-fold enhancement in the transition temperature ($T_{\mathrm{c}}$) happens. Analyses of the crystal structure and theoretical calculations suggest a pressure-mediated switch of the conduction channel from the $a$- to the $b$-axis occur, followed by a dimensional crossover of conductivity and the Fermi surface from 2D to 3D.