Discretization Scheme for the Density-Gradient Equation and Effect of Boundary Conditions

In recent years, the density gradient theory (DG) (M.G. Ancona and H.F. Tiersten 1987, Phys. Rev. B. 35(15): 7959–7965, M.G. Ancona 1990, Phys. Rev. B. 42: 1222) has been established as a viable alternative to the solution of the Schrödinger equation for solving problems such as charge density distribution in MOS inversion layers and MOS tunneling (M.G. Ancona 1998, J. Tech. CAD(11), M.G. Ancona et al. 2000, IEEE Trans. Electron Devices 47: 1449). Primary advantages of the DG method over the Schrödinger method are flexibility in extending to multi-dimension and easiness in incorporating into the conventional drift-diffusion or hydrodynamic solver (C.S. Rafferty et al. 1998, Proc. SISPAD, p. 137, A. Wettstein et al. 2001, IEEE Trans. Elec. Dev. 48: 279). However, the DG term that represents the quantum effects is a singular perturbation term and requires special care for discretization (X. Wang 2001, Master's thesis, University of Massachusetts, Amherst). In this work, we examine the validity of the linear vs. the nonlinear discretization scheme and the effect of boundary conditions on the scheme used.