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.
Highlights
The dramatic advances in microelectronics during the past two decades are largely a result of "shrinking" the technology
For given voltage bias and operating conditions, as device size shrinks the local field strength inside the device increases and the interior layers in the solution fields become more abrupt. Other physical effects such as quantum tunneling in the inversion layer become significant and several numerical difficulties commonly arise. These numerical difficulties are typically associated with the following issues: (1) an inadequate physical model that ignores physics that was negligible at the previous scale; (2) numerical effects associated with the high local gradients in the solution that adversely impact the convergence of the nonlinear iterative solver; (3) other numerical effects such as oscillations in the approximate solutions that are intrinsically tied to the resolution of the underlying grid and the stability of the chosen discretization scheme
Part of the difficulty has to do with the scale and the reliability of a deterministic mathematical model such as those based on augmented drift-diffusion or hydrodynamic PDE systems
Summary
The dramatic advances in microelectronics during the past two decades are largely a result of "shrinking" the technology. For given voltage bias and operating conditions, as device size shrinks the local field strength inside the device increases and the interior layers in the solution fields become more abrupt Other physical effects such as quantum tunneling in the inversion layer become significant and several numerical difficulties commonly arise. Since the issue of multiscale capability is a topical research subject in a number of modeling applications areas, we suggest that this is a good framework for interpreting the above problems-that is, different microscale (here quantum level) and macroscale (carrier transport by drift diffusion) effects need to be accommodated This concept has not yet been explored in sufficient detail to develop alternative practical simulation models and strategies and remains an open research opportunity
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