Analysis of Dual Gate Structures Using Double-Well and WKB Quantization Rules

Abstract

An alternative method for direct calculation of energy levels for symmetric dual gate structures with body widths of 3–10 nm is presented in the current paper, using double well analysis and Wentzel-Kramers-Brillouin (WKB) quantization rules. The double well analysis explains the lower energy level splitting, as well as the twin shape of the corresponding wave function with odd–even symmetry. The potential is approximated using a parabolic potential. The perturbation analysis, which already gives good accuracy for higher energies and lower electron effective mass, is compared with the current approach. The combination of the double well and WKB method gives improved accuracy for energies below barrier maximum potential, over perturbation potential, while a simple analytic second-order WKB correction term is needed for energies just above the potential peak. Using the WKB representation of wave function, the effect of finite barrier height on wave function penetration and on energy levels is analytically treated, for all energy levels, with barriers as low as 0.5 eV. Theoretical models are verified using numerical eigenvalue/eigenvector solvers, while device modeling results are compared with self-consistent simulator.