# Re-emission and Re-deposition

## Physical Effects/Overview of Mechanism:

In a variety of problems, etching and/or deposition occurs with a non-unity sticking coefficient, and hence the flux at any point depends not only on the source but also on the reflected flux from other parts of the surface. Thus, in addition to the need to calculate visibility from all parts of the surface to all other parts, one must solve an integral equation which contains the interaction matrix.

## Level Set Methodology:

The level set methodology advances the surface by solving the appropriate initial value partial differential equation, using the correct viscous limit of the Hamilton-Jacobi equation. We construct the interaction matrix by explicitly locating the front, and then placing points along the front as nodes in the creation of the interaction matrix. Visibility, which is often time consuming, may be easily calculated using the level set function. A direct matrix solver, which is often used to solve the integral equation, may be replaced by an iterative approach which provides an explicit, calculatable bound on the number of terms required (that is, iterations) as a function of the sticking coefficient.

## Results and Sample Simulations:

In the below examples, we assume luminiscent re-emission; thus, re-emitted and redeposited particles are ejected with uniform angle probability.

• ### Effect of Re-deposition on Uni-directional Etching: Various Sticking Coefficients

Under directional etching directly from the vertical, the two figures contrast complete etching (sticking coefficient 1.0) with etching plus deposition, in which the total amount of mass removed is again re-deposited.
 Full Etching Etching plus Re-deposition

• ### Effect of Sticking Coefficient on Deposition:

Next, we vary the sticking coefficient for a pure deposition process. A uni-directional deposition beam enters from the vertical, and the sticking coefficient varies from 1.0 down to .2.
 Full Deposition: Sticking Coefficient 1.0 Full Deposition: Sticking Coefficient .5
 Full Deposition: Sticking Coefficient .2

• ### Three-Dimensional Simulations: Deposition into Via

Finally, we vary the sticking coefficient for a pure deposition process in 3D. The example is fully three-dimensional; no two-dimensional simplications are used whatsoever. A full source deposition beam comes from a plate above, and the sticking coefficient varies from 1.0 down to .001.
 Full Deposition: Sticking Coefficient 1.0 Full Deposition: Sticking Coefficient .1
 Full Deposition: Sticking Coefficient .001