12.3 The Transport of Particles above the Wafer

In an integrated simulation of transport phenomena and surface evolution the transport phenomena above the wafer surface determine the deposition and etching rates. They can broadly be divided into two classes according to the mean free path length, although this distinction is only a rough classification and the suitable model in each case may depend on other considerations as well:

• If the mean free path length is much larger than the diameter of the simulation domain, the collision of single particles can be neglected and the transport can be simulated using the radiosity approach.
• If, on the other hand, the mean free path length is much smaller than the simulation domain, the collisions between single particles play a major role and their concentration is determined by the diffusion equation.

In the first case the simulation methods are similar to those in computer graphics, where the reflections of light rays must be simulated. When particles hit the surface, they either stick to the surface or they continue their way after luminescent or specular reflection. Luminescent reflection must be used for low energy particles and specular reflection for particles of high energy, i.e., ions.

The deposition rate depends on factors such as visibility between the source and the position of the trench, the angle dependent distribution of the flux of source particles, and the angle of incidence of particles hitting the surface. Particles don't always stick once they hit the surface, but are re-emitted with a certain probability. The sticking coefficient determines which fraction of particles actually sticks and hence is between zero and unity.

In the second case transport is governed by the diffusion equation

$${\partial c \over \partial t} = \nabla\cdot (D \nabla c),$$

where $c$ is the concentration and $D$ the diffusion constant. The boundary conditions (cf. Figure 12.8) are usually as follows: at the top of the simulation domain a Dirichlet boundary condition is assumed, i.e., a constant concentration is supplied by the reactor; on the left and right hand side a Neumann boundary concentration is assumed, i.e., the fluxes are zero; and finally the fluxes on the wafer surface are determined by the deposition rates.