Solving the Drift Diffusion Equations

The Drift Diffusion equations constitute the simplest and most commonly used model for simulating semiconductor devices. This paper contains a comparitive study of the performance and stability of several algorithms that solve these coupled equations without decoupling them. The considered techniques include Successive Over-Relaxation schemes, Nonlinear Conjugate Gradients, and Damped Inexact Newton methods using various linear solvers such as GMRES and Multigrid. Numerical simulation of semiconductor devices is of course of considerable technological importance; in addition, it also presents a very interesting and challenging problem from the perspective of numerical partial differential equation solution algorithms. The extreme electric field and temperature sensitivity of electron and hole distribution functions in the semiconductor bands, which makes semiconductors useful for electronic devices, causes instabilities in the partial differential equations describing semiconductor devices, which need to be addressed when designing solvers for these equations. The Drift Diffusion model is one of the simplest models that describes the classical transport of charge carriers in a semiconductor. These equations take the form of coupled nonlinear Poisson-like equations. Many solvers decouple these equations using the Gummel method (Section 2.2). This paper focusses on algorithms that do not involve decoupling these equations. Specifically, 2D models discretized over a uniform Cartesian grid are considered. A few Relaxation schemes, Nonlinear Conjugate Gradients and Damped Inexact Newton methods have been studied. The powerful Multigrid algorithm has been considered as a candidate linear solver for the Inexact Newton method. The Drift Diffusion Model is introduced in Section 1. Discretization and the formulation of the problem as a sparse system of nonlinear equations is described in Section 2. The algorithms that have been implemented and tested are described in Section 3, and the experimental results are presented in Section 4. 1 The Drift Diffusion Model The classical transport of electrons and holes in a semiconductor can be described by the evolution of their probability distribution functions in space and time. This is captured by the Boltzmann equation: df dt = ∂f ∂t + v K . ∂f ∂x K + qEK . ∂p K (1) where f(x K , p K , t) is the distribution function (in space, momentum and time) for the electrons/holes. Note that for the steady state (no t dependence) simulation of a 2D system, this equation would require a 4D PDE solution. This is usually an impractical equation to solve. For most devices, the distribution function usually remians quite close to the equilibrium Fermi-Dirac distribution f(p K ) = (1 + exp( kBT ) )−1 and we can integrate out the p K dependence, and work only with the x K dependence (now a 2D problem). If this integration is done accounting for only the first order moments of the distribution function in p K , the Drift Diffusion model is obtained. It turns out that this level of approximation amounts to having exactly the Fermi-Dirac distribution for electrons and holes, except that the chemical potential μ is assumed to be different for the electrons and holes (ie. they are no longer in mutual equilibrium). Thus the distribution is determined by the temperature T , and two spatially dependent potentials φn and φp called the quasiFermi potentials for the electrons and holes respectively. The Drift diffusion equations consist of the Poisson equation for electrostatics, and the continuity equation for electrons and holes. It can be shown that the electron and hole currents are: j Ke = μen∇K φn j Kh = μhp∇K φp (2) where μe, μh are called the mobility of the electrons and holes respectively, while n and p are the number densities of the mobile electrons in the conduction band and holes in the valence band respectively. Thus the equations to be solved take the form: ∇K .(κ∇K ψ) = − ρ/ǫ0 ∇K .(μen∇K φn) = (G−R)/e ∇K .(μhp∇K φp) = (G−R)/e (3)