Crossover of Equilibrium and Nonequilibrium Carrier Density in Germanium Two-Dimensional Hole Gases

We performed a detailed study on the enhancement-mode two-dimensional hole gases (2DHGs) in the Ge/Ge$_{1-x}$Si$_{x}$ (0.15 \textless x \textless 0.4) heterostructures. High mobility of 200,000 cm$^{2}$/V-s was achieved with clear quantum Hall plateaus observed. By varying the depth of 2DHG layer, a cross-over of equilibrium and non-equilibrium 2DHG density is reported in any two-dimensional system for the first time. Furthermore, an unexpectedly low effective capacitance in the shallow-channel devices was also observed. Shubnikov-de Haas oscillations showed beatings in a very shallow 2DHG (\textasciitilde 9 nm) due to second subband population.