J Comput Electron (2006) 5:459–462 DOI 10.1007/s10825-006-0040-7 Meshless solution of the 3-D semiconductor Poisson equation Zlatan Aksamija · Umberto Ravaioli Published online: 18 January 2007 C  Springer Science + Business Media, LLC 2007 Abstract Finding the scalar potential from the Poisson equa- tion is a common, yet challenging problem in semiconductor modeling. One of the central problems in traditional mesh- based methods is the assignment of charge to the regular mesh imposed for the discretisation. In order to avoid this problem, we create a mesh-free algorithm which starts by assigning each mesh point to each particle present in the prob- lem. This algorithm is based on a Fourier series expansion coupled with point matching. We show that this approach is accurate and capable of solving the Poisson equation on any point distribution. Keywords Pseudospectral . Poisson equation . Meshless . Coulomb interaction 1 Introduction The Poisson equation arises frequently from problems in ap- plied physics, fluid dynamics and electrical engineering. We solve the Poisson equation in order to obtain the electric po- tential in a semiconductor device and find the electric field for electronic simulation. Finding a suitable solution mesh is complicated by the fact that there is usually a large region at the bottom of the device which has few particles and is of little interest. On the other hand, the channel region at the top of the device is small but the detailed solution in it is Z. Aksamija () . U. Ravaioli Beckman Institute, University of Illinois, 405 N. Mathews, 61801 Urbana IL., U.S.A. e-mail: aksamija@uiuc.edu e-mail: ravaioli@uiuc.edu important. There are also numerous problems arising from assigning charge to a regular mesh [1]. Therefore a regular rectangular mesh is the least desirable solution. An alterna- tive approach is to treat each individual particle as a mesh point. Meshless methods have been proposed for the semi- conductor Poisson equation in the past [3]. This work starts with a series expansion and aims at reaching a solution which is continuous and infinitely differentiable, and optimal in the least squares sense. 2 Numerical approach If we choose from the start to make each particle a mesh point, we are dealing with an arbitrary mesh. In order to seek a solution, we expand the potential into sinusoidal compo- nents. Since we have fixed boundary conditions (the voltage at the edges/contacts is given), we know all solutions must be a super-position of sinusoidal harmonics, so we write the solution as follows: V (x , y , z ) = N l  l =1 N m  m=1 N n  n=1 (l , m, n) exp  i π l L x x  × exp  i π m L y y  exp  i π n L z z  (1) where L x , L y and L z are the lengths of the solution domain in each respective direction. The Laplace operator in Cartesian co-ordinates is written as: ∇ 2 =  ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2  Springer