Physics-based analytical model of quantum-mechanical electron wave function penetration into thin dielectric films for capacitance evaluation

Recently, quantum-mechanical (QM) effects in MOSFETs have been studied extensively to overcome predicted performance limitation as stated in G. Timp et al. (1999) and A. Pacelli et al. (1999). In MOS structures with a thin gate SiO₂ film, the dark-space near the Si/SiO₂ interface influences capacitance-voltage (C-V) characteristics as presented in A. Shimizu et al. (2001). In addition, it is reported that the electron penetration influences C-V characteristics in case of high impurity density based in S. Mudanai et al. (2001). So, simulations should consider several QM effects. Unfortunately, generally speaking, recent numerical calculations including quantum effects spend a long time to get results. In this paper, we derive exact analytical equations or more precise approximation for electron distribution functions. We examine applicability of proposed expressions to C-V analysis.