Analytical Solution to the Density-Gradient Equation for MOS Quantum Tunneling

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.