Continuing advances in the miniaturization of integrated circuits have imposed new challenges to designers of semiconductor devices, as traditional circuit analysis tools are no longer applicable. In this study, we focus on the gate region of MOSFET devices with an oxide thickness of order 4-6 nanometers. For oxide layers of this width, quantum effects start to become noticeable and the standard equations of semiconductor physics require quantum corrections. The Density-Gradient equation [1], [2] is a means of calculating approximate quantum corrections to existing formulae without solving the full Poisson-Schodinger system. We use a one-dimensional approximation to reduce the D-G equations to a set of singularly perturbed ODE~Rs [3]. These equations have interior layer solutions, which compromise the numerical treatment of the problem if care is not taken to resolve the large gradients in the solution within the layer correctly. Several approaches have been proposed to resolve the boundary layer region including nonlinear discretization schemes [4], [5]. However, these methods have difficulty resolving interior layers [6]; therefore, we propose a multilevel adaptive scheme to solve the model equations. The method is akin to multigrid, but utilizes high-order inter-grid operators, which preserve a nested space structure throughout the various resolution levels, similar to the methods described by Goedecker [7] and Warming & Beam [8]. This multilevel method has several advantages over the previous discretization schemes, the most important being that it adapts dynamically to interior layers without a priori knowledge of the location or geometry of the layer.

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[2] Ancona M.G. 1998. Simulation of quantum confinement effects in Ultra-Thin-oxide MOS structures. J. Tech. CAD (11).

[3] Ward, J. F., Odeh M., and Cohen D. S., Asymptotic methods for metal oxide semiconductor field affect transistor modeling. SIAM J. Appl. Math 50 1099-1125 (1990).

[4] Ancona M.G. and Biegel B.A. 2000. Nonlinear discretization scheme for the density-gradient equations. In: Proc. SISPAD~R00, p. 196 (2000).

[5] Wettstein A., Schenk A., and Fichtner W. 2001. Quantum device-simulation with the density-gradient model on unstructured grids. IEEE Trans. Elec. Dev. 48: 279.

[6] Wang X. and Tang T.-W. 2002. Discretization scheme for Density-Gradient Equation. Journal of Computational Electronics 1, 389-392.

[7] Goedecker S., Ivanov O., ~SSolution of Partial Differential Equations Using Wavelets~T, Comp. In Phys, 12, 548 (1998) .

[8] Warming R., Beam R, ~SDiscrete Multiresolution Analysis Using Hermite Interpolating: Biorthogonal Multiwavelets,~T SIAM J. Sci. Comput. 22, 4, 1269-1377 (2000).