Top-gate density tuning ($\Delta n_s=ϵ\Delta V_g/(ed)$)

A gate separated by dielectric thickness $d$ has geometric capacitance per area $C/A=ϵ/d$ in the parallel-plate approximation. An electron-density change is $\Delta n_s=(ϵ/ed)\Delta V_g$ with the stated polarity/reference convention. Fringing, trapped charge, depletion and finite quantum capacitance can modify the relation. An absolute gate voltage additionally requires the zero-density offset or another calibration.