Application of finite element methods to the simulation of semiconductor devices

Abstract

In this paper a survey is presented of the use of finite element methods for the simulation of the behaviour of semiconductor devices. Both ordinary and 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. The drift-diffusion and hydrodynamic models are discussed in more detail. Some mathematical properties of the resulting nonlinear systems of partial differential equations are identified, and general considerations regarding their numerical approximations are discussed. Ordinary finite element methods of standard and non-standard type are introduced by means of one-dimensional illustrative examples. Both types of finite element method are then extended to two-dimensional problems and some practical issues regarding the corresponding discrete linear systems are discussed. The possibility of using special non-uniform fitted meshes is noted. Mixed finite element methods of standard and non-standard type are described for both one-and two-dimensional problems. The coefficient matrices of the linear systems corresponding to some methods of non-standard type are monotone. Ordinary and mixed finite element methods of both types are applied to the equations of the stationary drift-diffusion model in two dimensions. Some promising directions for future research are described.

Simulation of visible and ultra-violet group-III nitride light emitting diodes

“…The drift-diffusion equations (2)-(4), being a singular perturbed problem, can exhibit both boundary and interior layers with a rapid variation of the carrier concentrations and/or electric field [31,32]. Moreover, the stiffness of the DD equations increases nonlinearly with the bandgap of the material.…”

“…The latter choice is attractive mathematically since the carrier continuity equations are self-adjoint and linear in u and v [32], which simplifies the analysis of the DD model in the case of the Boltzmann statistics. Another advantage of the Slotboom variables is the symmetry and positive definiteness of the matrix of the linearized discrete equations.…”

“…On the other hand, one is often no less interested in the detailed distribution of the electric field and partial current densities across the heterostructure. Thus, the mixed FE method (FEM) [32] is employed in our study. The one-dimensional DD equations are re-written as a system of the first-order equations in the electrostatic potential u, z-component of the electrostatic induction vector D z , the electron and hole quasi-Fermi levels u n , u p , and the corresponding carrier fluxes along the z-coordinate J nz , J pz (the subscript ''z'' is omitted in the following equations):…”

“…A derivation of discrete FE equations proceeds in the standard way (see for example [44]): the differential equations are transformed into a weak form via multiplying them by test functions and integrating the product over the computational domain. The basis functions of the mixed FEM for the generic system (11) should belong to proper spaces (to the space L 2 of the square integrable functions for f and to the space H(div) of the functions with the square integrable derivatives for G) [32]. Thus, f (u, u n , u p ) can be approximated by a set of the piecewise constant functions Defining X = (z i , z i + 1 ), X ¼ ½z i ; z iþ1 , the basis functions M i (z) and N i (z) can be evidently written as…”

The Application of the Object-Oriented Approach to the Development of Monte Carlo Electron Devices Simulation Software

“…These approximations allow simplifying the equation. The examples of such approximations are hydrodynamic and qasihydrodynamic ones [3]. However using the hydrodynamic or qasihydrodynamic models it is necessary to implement the raw of assumption it reduces the field of application of the mentioned techniques dramatically.…”