Linear combination of bulk bands method for investigating the low-dimensional electron gas in nanostructured devices

This paper concerns the determination of the band structure of physical systems with reduced dimensionality with the method of the linear combination of bulk band (LCBB), according to the full-band energy dispersion of the underlying crystal. The derivation of the eigenvalue equation is reconsidered in detail for quasi-two-dimensional (2D) and quasi-one-dimensional (1D) systems and we demonstrate how the choice of the volume expansion in the three-dimensional reciprocal lattice space is important in order to obtain a separated eigenvalue problem for each wave vector in the unconstrained plane (for 2D systems) or in the unconstrained direction (for 1D systems). The clarification of the expansion volume naturally leads to identification of the 2D and 1D first Brillouin zone (BZ) for any quantization direction. We then apply the LCBB approach to the silicon and germanium inversion layers and illustrate the main features of the energy dispersion and the 2D first BZ for the [001], [110], and [111] quantization directions. We further compare the LCBB energy dispersion with the one obtained with the conventional effective mass approximation (EMA) in the case of (001) silicon inversion layers. As an interesting result, we show that the LCBB method reveals a valley at the edge of the 2D first BZ which is not considered by the EMA model and that gives a significant contribution to the 2D density of states.