Modeling the patterned two-dimensional electron gas: Electrostatics

Abstract

We present analytical calculations of the potential in a two-dimensional electron gas (2DEG) generated by patterned polygon gates on the surface of a heterostructure. They give the bare and screened potentials and reveal the effect of different boundary conditions on the surface. The formulas for the bare electrostatic potential from patterned gates are simple enough to be plotted in spreadsheets; they give threshold voltages, estimates of the region occupied by the 2DEG, and the energies of some collective infra-red excitations. We also consider the screened potentials in linear response, where no part of the 2DEG is fully depleted, which can again be found within an electrostatic approximation. The behavior of the exposed surface between the gates affects the potential strongly. Surface states provide perfect pinning of the Fermi energy in the ‘‘equipotential’’ model, the usual assumption, but this requires charge to move to the surface from the 2DEG. The charge on the surface is held fixed in response to a gate voltage in the ‘‘frozen’’ model, which typically gives a lower cutoff voltage but stronger confinement of electrons in a split-gate wire. The difference between the two models is large, emphasizing that accurate modeling needs a thorough understanding of the surface states.