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

Abstract

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

On the validity of the effective mass approximation and the Luttinger k.p model in fully depleted SOI MOSFETs

“…Its value depends on the number N of trigonometric functions / n used in the calculation. As discussed in detail in [15] the expansion of Eq. (3) should be performed within an appropriate expansion volume such as nP=L A 6 2P=a, 8n 6 N (which can be satisfied with L A ¼ Na=2 [23]).…”

“…This constraint is imposed in order to establish a unique relationship between the envelope-function theory and the microscopic k.p theory from which it was derived [20,15]. For that reason, the straightforward transformation in real space k z Ài d dz ( leading to a set of local coupled differential equations [17]) gives rise to nonvanishing envelope-functions Fourier components F i ðkÞ outside the first BZ, and generates an error in the kinetic energy [20].…”

“…For that reason, the physics and electronic device community is actively developing far more computationally efficient semi-empirical approaches, that can work out the electronic structure of strained semiconductor devices. In this context, the 'full-band' semi-empirical computational methods fall into two general categories: The atomistic approach such as the tight-binding (TB) [5,6] and atomistic linear combination of bulk bands (LCBB) [7] and alternatively, models that use a continuous description of the matter such as a 'full-zone'-k.p model [8,9] within the ''envelope-function approximation" (EFA) [10][11][12][13][14] and empirical pseudo-potential within the k-space LCBB [15,16].…”

Modeling of modern MOSFETs with strain

“…An approach based on the full band structure computed with the empirical pseudo-potential method (EPM) [44] is promising and has been recently generalized to include strain and spin-orbit interaction [45]. Although it uses a realistic band structure, the method is computationally demanding and needs to be improved to include the selfconsistent solution of Poisson equation.…”

Semi-classical transport modelling of CMOS transistors with arbitrary crystal orientations and strain engineering

“…Recently the Linear Combination of Bulk Bands (LCBB) method [66][67][68] has been used jointly with a non-local pseudopotential solver to calculate the band-structure in electron and hole inversion layers by accounting at the same time for both the strain and the quantization effects [69]. The overall procedure is computationally demanding, but the results can be used to test the accuracy of the simpler models, such as the EMA or the k · p approach.…”