Self-consistent calculation of ionized-donor distribution in a nanostructured heterojunction with corrugated gate

We calculate self-consistently the spatial distribution of ionized silicon donors in the barrier of a nanostructured GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As heterostructure with a corrugated top gate, as well as the electron density of the corresponding modulated two-dimensional electron gas (2DEG). The geometry of the periodically corrugated Schottky gate gives rise to an inhomogeneous occupation of deep donor levels (DX centers) at room temperature, which freezes in when the device is cooled down. This charge pattern can nearly compensate the desired field effect near the GaAs-${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As interface. On the other hand, moderate doping of the barrier is predicted to lead to an improvement of the confinement potential, which affects the electrons in the 2DEG. Our numerical treatment of the laterally modulated electron density combines a Fang-Howard variational approach describing quantum confinement in the growth direction and a Thomas Fermi approximation modeling the lateral inhomogeneities, and covers the gate-voltage-induced crossover from a weakly modulated 2DEG to isolated quantum dots.