Application of finite element methods to the simulation of semiconductor devices

In this paper a survey is presented of the use of finite element methods fo r the simulation of the behaviour of semiconductor devices. Both ordinary a nd mixed finite element methods are considered. We indicate how the various mathematical models of semiconductor device behaviour can be obtained from the Boltzmann transport equation and the appropriate closing relations. Th e drift-diffusion and hydrodynamic models are discussed in more detail. Som e mathematical properties of the resulting nonlinear systems of partial dif ferential equations are identified, and general considerations regarding th eir numerical approximations are:discussed. Ordinary finite element methods of standard and non-standard type are introduced by means of one-dimension al illustrative examples, Both types of finite element method are then exte nded to two-dimensional problems and some practical issues regarding the co rresponding discrete linear systems are discussed. The possibility of using special non-uniform fitted meshes is noted. Mixed finite element methods o f standard and non-standard type are described for both one- and two-dimens ional problems. The coefficient matrices of the linear systems correspondin g to some methods of non-standard type are monotone. Ordinary and mixed fin ite element methods of both types are applied to the equations of the stati onary drift-diffusion model in two dimensions. Some promising directions fo r future research are described.