Engineering-oriented simulations of quantum mechanical tunneling are often based on density-gradient (DG) theory. This paper presents an analytical solution to the DG equation for quantum tunneling through an ultra-thin oxide in a MOS capacitor with an n+ poly-silicon gate obtained using the method of matched asymptotic expansions. Tunneling boundary conditions extend the approximation into the entire region of the poly-silicon gate, oxide barrier, and substrate. An analytical solution in the form of an asymptotic series is obtained in each region by treating each part of the domain as a separate singular perturbation problem. The solutions are then combined through ‘matching’ to obtain an approximate solution for the whole domain. Analytical formulae are given for the electrostatic potential and the electron density profiles. The results capture the features of the quantum effects which are quite different from classical physics predictions. The analytical results compare well with exact numerical solutions over a broad range of voltages and different oxide thicknesses. The analytical results predict the enhancement of the quantum tunneling effect as the oxide thickness is reduced.

Transmission lines (T-Lines) are widely used in millimeter wave applications on silicon-based complementary metal-oxide semiconductor (CMOS) technology. Accurate modeling of T-lines to capture the related electrical effects has, therefore, become increasingly important. This paper describes a method to model the capacitance and conductance of T-Lines on CMOS multilayer, lossy substrates based on conformal mapping, and region subdivision. Tests show that the line parameters (per unit length) obtained by the method are frequency dependent and very accurate. The method is also suitable for parallel multiconductor interconnect modeling for high frequency circuits.