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A Wavelet Method for the Density-Gradient Equation

H. Morris and A. Limon
San Jose State University, US

quantum tunneling, density-gradient equation, unstructured grids, wavelets, boundary layer.

As MOSFET device lengths have shrunk to submicron level, so too has the oxide thickness steadily reduced. At around 4-5nm thicknesses quantum effects start to become noticeable as electrons are able to tunnel through the oxide layer. The Density-Gradient equation is a means of calculating approximate quantum corrections to existing formulae without solving the full Poisson-Schodinger system. The problem of solvingdensity gradient equationsthe gate region of a MOSFET is considered. These equations comprise a singularly perturbed system of ordinary differential equations with boundary layer type solutions. In order to treat the solution in the boundary layer correctly, special numerical techniques are needed. Several methods have been proposed in literature and include nonlinear discretization schemes which appear to be sensitive to boundary conditions. We therefore propose a new way to solve the equations using a wavelet method similar to methods used in chemical physics. This wavelet method allows us to combine the best features of each of the exiting approaches.

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