4.4.2 Chemical-Mechanical Planarization

If a geometry is represented as LS, Boolean operations can easily be calculated. They can be applied to simulate various semiconductor processes in a simplified manner. For example, the simplest model for chemical-mechanical planarization (CMP) assumes that everything is cut away above a certain height. Hence, the new structure ${\mathcal{M}}'$ is obtained by

$${\mathcal{M}}'={\mathcal{M}}\cap\lbrace{\vec{x}}:{\vec{x}}\cdot{\vec{n}}\leq {c}\rbrace,$$ (4.14)

where ${\vec{n}}$ is the normal vector and ${c}$ the distance to the origin of the cutting plane ${\mathcal{P}}_{\text{cmp}}=\lbrace{\vec{x}}:{\vec{x}}\cdot{\vec{n}}={c}\rbrace$ . By setting up a LS function ${\Phi}_{\text{cmp}}$ , whose zero LS represents ${\mathcal{P}}_{\text{cmp}}$ (4.14) can be written as

$${\Phi}'=\max({\Phi},{\Phi}_{\text{cmp}}).$$ (4.15)

Here ${\Phi}$ and ${\Phi}'$ are the LS functions describing ${\mathcal{M}}$ and ${\mathcal{M}}'$ , respectively.

Figure 4.5 shows an example, for which CMP is used to flatten the geometry after an isotropic deposition process. Boolean operations are applied to both, the LS function representing the initial structure and the LS function representing the final structure after the deposition process.

Figure 4.5: After an isotropic deposition process the structure is exposed to CMP. This process is realized using Boolean operations applied on the LS representations of the initial surface (blue) and the surface after the deposition (yellow).