Solution of Self-Consistent Kohn–Sham and Poisson Equations for Quasi Two-Dimensional Electron Gas in the Accumulation Layer of Semiconductor with Nonparabolic Conduction Band

Abstract

We generalize the method proposed previously for a self-consistent solution of the system of the Kohn–Sham and Poisson equations to the case of high density of electrons localized near the surface of a semiconductor due to size quantization of the electron spectrum in a potential well in the band banding region. We obtain a completely self-consistent iterative solution for a quasi two-dimensional electron gas in the accumulation layer at the surface of a semiconductor with a degenerate electron gas in the bulk. Both parabolic and non-parabolic energy spectra of conduction electrons are considered. For a parabolic conduction band, we obtain the spatial distributions of the electron density and electrostatic potential, as well as the energies of size-quantized levels and their dependence on the depth of the near-surface potential well. A significant decrease in the density of three-dimensional electrons was revealed in the region where the quasi two-dimensional electron gas occupied the accumulation layer. We calculated the dependencies of the excess electron surface density and the capacitance of the structure on the surface potential. The results are presented in dimensionless form that can be used to semi-quantitatively evaluate the parameters of accumulation layers by the known level of doping and the value of band bending. In the case of the nonparabolic conduction band, due to the finite bandgap width, the spectrum of quasi two-dimensional electron gas is obtained in the two-band Franz–Kane approximation by solving the eigenvalue problem for the new single-band equation of the effective-mass method. The density of states and the effective mass of electrons in the 2D subbands is found to increase linearly with energy. We have derived the expression that analytically approximates the calculated energy spectrum. The position of levels, the dispersion of energy spectrum, and the dependence of effective mass on energy, which were found for the size-quantized sub-bands, are in agreement with the published results for characteristics of quasi two-dimensional electrons at n-InAs surface that were directly measured by angle resolved photoelectron spectroscopy and magnetotunneling spectroscopy. Conditions are considered when the nonparabolicity of the conduction band enables quasi two-dimensional electrons to absorb the normally incident radiation due to intersubband transitions.