Advanced Numerical Methods and Software Approachesfor Semiconductor Device Simulation

Abstract

In this article we concisely present several modern strategies that are applicable to driftdominated carrier transport in higher-order deterministic models such as the driftdiffusion, hydrodynamic, and quantum hydrodynamic systems. The approaches include extensions of “upwind” and artificial dissipation schemes, generalization of the traditional Scharfetter – Gummel approach, Petrov – Galerkin and streamline-upwind Petrov Galerkin (SUPG), “entropy” variables, transformations, least-squares mixed methods and other stabilized Galerkin schemes such as Galerkin least squares and discontinuous Galerkin schemes. The treatment is representative rather than an exhaustive review and several schemes are mentioned only briefly with appropriate reference to the literature. Some of the methods have been applied to the semiconductor device problem while others are still in the early stages of development for this class of applications. We have included numerical examples from our recent research tests with some of the methods. A second aspect of the work deals with algorithms that employ unstructured grids in conjunction with adaptive refinement strategies. The full benefits of such approaches have not yet been developed in this application area and we emphasize the need for further work on analysis, data structures and software to support adaptivity. Finally, we briefly consider some aspects of software frameworks. These include dial-an-operator approaches such as that used in the industrial simulator PROPHET, and object-oriented software support such as those in the SANDIA National Laboratory framework SIERRA.

Enriched residual free bubbles for semiconductor device simulation

“…The results for potential, electron concentration and hole concentration after the first iteration of this procedure are illustrated in Figs. 6,7,8,9,10,11 for each of the numerical methods listed above.…”

“…Three different finite element simulations were run: Bubnov-Galerkin, the bubble enrichment procedure outlined in Sect. 3.2 and SUPG (as outlined by Carey et al [8]) where four meshes with increasing element distortion were used to test their performance. In the first study, a comparison was made between a regular mesh with no refinement (Fig.…”

“…Of course, one strategy to overcome oscillations is to adaptively refine elements towards the positions of high gradient, and this is exactly what is shown in [8]. But this strategy suffers from the obvious drawback of high computational expense and we seek other more efficient strategies.…”

Performance of a Petrov–Galerkin algebraic multilevel preconditioner for finite element modeling of the semiconductor device drift‐diffusion equations

“…The terms in the weak form include the standard Galerkin term (first term in (4)- (6)) and an SUPG-type term (second term in (5) and (6)). This methodology is based on a variation of the streamline upwind Petrov-Galerkin (SUPG)-type FE formulations of Hughes et al [25,26] and Shakib [27] for convection-diffusion systems and has similarities to the flux upwind Petrov-Galerkin method for the drift-diffusion equations of Carey and coworkers [28,29]. The stabilized FE method allows solution of convection-diffusion-type systems by decreasing numerical oscillations due to convection effects.…”

An accelerated monotone iterative method for the quantum-corrected energy transport model

“…[3][4][5]6,[9][10][11][12][13][14][15]17,18,21,23,25,26,28] and references therein. Among them, a class of macroscopic quantum mechanical models based on the density-gradient (DG) theory of Ancona and Tiersten [1] have been shown to accurately simulate multi-dimensional MOSFET devices with gate lengths ranging from 50 nm down to 6 nm [3,5,[9][10][11][12][13]18,21,23,25,28].…”