Spatial inhomogeneity and temporal dynamics of a 2D electron gas in interaction with a 2D adatom gas

Fundamental interest for 2D electron gas (2DEG) systems has been recently renewed with the advent of 2D materials and their potential high-impact applications in optoelectronics. Here, we investigate a 2DEG created by the electron transfer from a Ag adatom gas deposited on a Si(111)$\sqrt{3}\times\sqrt{3}$-Ag surface to an electronic surface state. Using low-energy electron microscopy (LEEM), we measure the Ag adatom gas concentration and the 2DEG-induced charge transfer. We demonstrate a linear dependence of the surface work function change on the Ag adatom gas concentration. A breakdown of the linear relationship is induced by the occurrence of the Ag adatom gas superstructure identified as Si(111)$\sqrt{21}\times\sqrt{21}$-Ag only observed below room temperature. We evidence below room temperature a confinement of the 2DEG on atomic terraces characterised by spatial inhomogeneities of the 2DEG-induced charge transfer along with temporal fluctuations. These variations mirror the Ag adatom gas concentration changes induced by the growth of 3D Ag islands and the occurrence of an Ehrlich-Schwoebel diffusion barrier of 155$\pm$10meV.