Threshold effects in the energy spectrum of quasi-two-dimensional electrons of the accumulation layer

We consider the problem of determining the threshold values of the electric field $F$ at which new size-quantized subbands appear in the accumulation layer on the surface of an $n$-type semiconductor. The difficulties of such determination in experimental and computational works presented in the literature are discussed. An explanation of the available facts is proposed as a manifestation of the quadratic dependence of the energy $E$ of the shallow level on the depth of the potential well near the appearance threshold. Formulae of threshold dependence $E(F)$ for the case of parabolic conduction band in the bulk of semiconductor are obtained. The possibility of application of the threshold approximation not only to the main, but also to the excited subbands is shown. In the case of non-parabolic conduction band, the threshold character in dependence of the size-quantized level energy on quasimomentum along the surface is considered. Numerical calculations of two-dimensional spectra under the conditions of the appearance of the main subband, the first and the second excited ones have been performed with $n$-InAs parameters and analyzed by means of the derived expressions for the threshold behavior. A method to determine the threshold of a subband appearance from available data in the region above the threshold is proposed. An instability of the self-consistent solution of the system from the Poisson equation and the effective mass equation in the case of the second excited subband is observed and investigated. Arguments are presented in favor of interpreting this instability as evidence of the formation of two-dimensional valence-type subbands with negative mass in the accumulation layer. We discuss a possible connection between the appearance of such a spectrum in the potential well, the deep of which is comparable to the bandgap, and L.V. Keldysh's assumption about the origin of the amphotericity of impurities that create deep levels in the semiconductor bandgap.