Entropy-Based Moment Closures in Semiconductor Models

Abstract

We investigate aspects of entropy-based moment closures which are used to simplify kinetic models of particle systems. Closures of this type use variational principles to formally generate balance laws for velocity moments of a kinetic density. These balance laws form a symmetric hyperbolic system of partial differential equations that satisfies an analog of Boltzmann's famous H-Theorem. However, in spite of this elegant structure, practical implementation of entropy-based closures requires that several analytical and computational issues be settled.

Our presentation is devoted to the development of electron transport models in semiconductor devices. In this context, balance laws for velocity moments are generally referred to as hydrodynamic models. Such models provide a reasonable alternative to kinetic and Monte Carlo approaches, which are usually expensive, and the well-known drift-diffusion model, which is much simpler but a has a limited range of validity.

We first analyze the minimization problem that defines the entropy closure. It is known that there are physically relevant cases for which this problem is ill-posed. Using a dual formulation, we find so-called complementary slackness conditions which give a geometric interpretation of ill-posed cases in terms of the Lagrange multipliers of the minimization problem. Under reasonable assumptions, we show that these cases are rare in a very precise sense.

We also develop pertubations of well-posed entropy-based closures, thereby making them useful for modeling systems with heat flux and anisotropic stress. Heat flux has long been known to be an important component of electron transport in semiconductors. However, we also observe that anisotropy in the stress tensor also plays an important role in regions of high electric field. This conclusion is made based on our simulations of two different devices.

Finally, we devise a new split scheme for hydrodynamic models. The splitting is based on the balance of forces in the hydrodynamic model that recovers the drift-diffusion equation in the asymptotic limit of small mean-free-path. This scheme removes numerically stiffness and excessive dissipation typically associated with standard shock-capturing schemes in the drift-diffusion limit. In addition, it significantly reduces numerical current oscillations near material junctions.