Comparative Analysis of Pseudo-Potential and Tight-Binding Band Structure Calculations with an Analytical Two-Band k·p Model: Conduction Band of Silicon

The k·p theory allows to describe the band structure analytically. After the pioneering work by Luttinger and Kohn [1] the six-band k·p method has become widely used to model the valence band in silicon. The conduction band in silicon is usually approximated by three pairs of equivalent minima located near the Xpoints of the Brillouin zone. It is commonly assumed that close to the minima the electron dispersion is well described by the effective mass approximation. A non-parabolicity parameter is introduced to describe deviations in the density of states from the purely parabolic expression, which become pronounced at higher electron energies. In ultra-thin body FETs, however, the non-parabolicity affects the subband energies substantially, and it was recently indicated that anisotropic, direction-dependent non-parabolicity could explain a peculiar mobility behavior at high carrier concentrations in a FET with (110) film orientation [2]. Therefore, and because of its inability to address properly the band structure modification under stress, a more refined description of the conduction band minima beyond the usual single-band non-parabolic approximation is needed. A recently proposed two-band k·p model for the conduction band of silicon [3] is compared with other band structure models, notably the nonlocal empirical pseudo-potential method [4] and the spds* nearest-neighbor tight-binding model [5]. The two-band k·p model is demonstrated to predict results consistent with the empirical pseudo-potential method, and to accurately describe the band structure around the valley minima, including the effective masses and the band non-parabolicity. The tight-binding model, on the other hand, overestimates the gap between the two lowest conduction bands at the valley minima, which results in an underestimation of the non-parabolicity effects. In biaxially stressed Si films the electron mobility can be nearly doubled [6]. The reason for the mobility enhancement lies in the stress-induced band structure modification. The degeneracy between the six equivalent valleys is lifted due to stress-induced valley shifts. This reduces inter-valley scattering. In case of tensile biaxial stress applied in the (100) plane the four in-plane valleys move up in energy and become depopulated. The two populated out-of-plane valleys have favorable conductivity masses, which together with reduced inter-valley scattering results in the observed mobility increase [7]. The technologically relevant uniaxial stress along [110] has received little attention in the research community and was systematically investigated experimentally just recently [8]. Inherent to [110] uniaxial stress, the shear distortion of the crystal lattice induces pronounced modifications in the conduction band. Contrary to biaxial stress, the electron mobility data for [110] stress suggest that the conductivity mass depends on stress. Any dependence of the effective masses on stress is neglected within the single-band description of the conduction band and can only be introduced phenomenologically. In order to describe the dependence of the effective mass on stress a single-band description is not sufficient, and coupling to other bands has to be taken into account. The two-band k·p model gives analytical expressions for the shear strain-dependence of the band structure parameters. Shear strain modifies substantially both the longitudinal and transversal effective masses [3,8,9]. The transversal mass determines the mobility in FETs with ultra-thin body. In these FETs the electron mobility enhancement induced by [110] tensile stress is therefore solely caused by a decrease of the conductivity mass in the stress direction. Predictions of the two-band k·p model are in good agreement with those of the pseudo-potential method [10]. This work was supported in part by the Austrian Science Fund FWF, project P9997-N14.