Wavelets in Nonlinear Semiconductor Devices Semiconductor device behavior can be described by a system of coupled partial differential equations (PDEs) with associated boundary conditions, requiring the conservation of charge and energy. In physics one is more interested in the quantities of charge concentration, average velocity, and mean energy, for example. From an engineering standpoint, potential, fields, current, and I -V curves are the desired parameters. In this chapter we will study the drift-diffusion (DD) model, which is the simplest version of the Boltzmann transport equation (BTE) coupled with Poisson’s equation. The DD model has handled most engineering problems to date reasonably. | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER TEN Wavelets in Nonlinear Semiconductor Devices Semiconductor device behavior can be described by a system of coupled partial differential equations PDEs with associated boundary conditions requiring the conservation of charge and energy. In physics one is more interested in the quantities of charge concentration average velocity and mean energy for example. From an engineering standpoint potential fields current and I - V curves are the desired parameters. In this chapter we will study the drift-diffusion DD model which is the simplest version of the Boltzmann transport equation BTE coupled with Poisson s equation. The DD model has handled most engineering problems to date reasonably well. Having studied the DD model we will use spherical expansion and Galerkin s method to solve the 1D BTE obtaining more advanced information of hot carrier effects and ballistic transport for deep-submicron SMOS devices or high-frequency compound semiconductor devices. Interpolating wavelets will be employed to derive the sparse point representation SPR that reduces the computation burden in nonlinear modeling. Multiwavelets are used for the first time to replace the ad hoc upwind algorithms. PHYSICAL MODELS AND COMPUTATIONAL EFFORTS The Boltzmann transport equations BTE and Maxwell s equations establish a relationship between charge distribution and electric potential. Under most operating conditions the quasi-static approximation holds for the electric field inside semiconductor devices and it is appropriate to use Poisson s equation instead of Maxwell s equations. The electron distribution f is governed by the BTE yg k Vrf r k - q r Vkf r k f 5 k k f r k d3k V tJc J - f r k S k k dV 474 PHYSICAL MODELS AND COMPUTATIONAL EFFORTS 475 where the involved quantities are as follows vg group velocity e electric field S k k differential electron .
