Abstract
Classical and quantum solutions of the planar accumulation layer problem for a degenerate semiconductor within the parabolic effective-mass approximation are presented. The Hartree approximation and the potential theory equation are used to describe the electron-electron and electron-positive charge interactions. Details of the exact analytical solution of the classical Thomas-Fermi equation are presented. The straightforward self-consistent quantum approach is developed. It results in the nonlinear equations similar to that in the quantum field or disordered solid state theories. The nonlinear quantum equations are solved numerically. It has been established that the nanosize accumulation layers containing two two-dimensional-confined bands can be regarded as the most stable states of the inhomogeneous system of such a kind. In the case of inversion layers, the single-band states can be expected to arise. Useful general relations between crystal parameters and confinement energies have been found. The applicability of the quantum solution is demonstrated by using recent literature data obtained by the high-resolution angle-resolved photoemission spectroscopy. © 2007 The American Physical Society.