# Hybrid Numerical Simulation of Electrostatic Force Microscopes Considering Charge Distribution

## Dr. Uzzal Binit Bala

**Jan. 11, 2007, 2 p.m. HF 136**

For modeling and simulating micro-electro-mechanical systems (MEMS), multi physics aspects must be taken into consideration. From the numerical point of view additional problems arise since frequently we are confronted with multi-scale problems. Therefore advanced numerical methods have to be applied. At the same time the coupled mechanical and electrical behavior have to be taken into account. This can be achieved by dividing the model into an electrical part and a mechanical part. The interaction between them can conveniently be realized by using a staggered simulation approach. An interesting example of MEMS is the so-called electrostatic force microscope (EFM) which can be used for scanning samples with nearly atomic resolution.

For developing a model of the EFM different effects have to be considered. For example long distance interaction, charge distribution and non-linearity of the material properties, singularity etc. In order to take into consideration these effects the simulation region is divided into three regions. As high values of the electric field will occur at the pick of the tip, a special numerical method is needed to calculate these electric field more effectively. For this reason an augmented FEM method will be applied to region near the tip. Since charge distribution and nonlinearities of the dielectric properties may have to be considerd, a versatile numerical method such as finite element method (FEM) should be applied to the next region. As boundary element method (BEM) works well when the boundary is infinite or semi-infinite, the large distance interaction between the tip and the cantilever can be conveniently treated by using BEM in the rest of the region. Later all these three numerical methods will be coupled with each other.

In this talk a coupled simulation will be presented considering charge distribution on the measuring object. Since the scanning process of EFM is dynamic, one has to deal with a moving sample and moving boundaries. As a result the mesh has to be updated at each time step. The approach presented here for mesh updating is based on an arbitrary Lagrangian Eulerian (ALE) algorithm.