5.1 Modeling of Resist Development

The resist profile evolving during development is simulated with the cell-based topography algorithm introduced in Section 2.2. The surface advancement is carried out by applying structuring elements at the exposed surface, which successively remove cells of the underlying cellular geometry by switching the material index of the cells from resist (R) to vacuum (0).

**Figure 5.1:** Structuring element model for resist development, considered as isotropic wet etching process with locally varying etching rates.
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Since development corresponds to an isotropic etching process, a sphere is used as structuring element. In contrast to a common wet etching step, where each material has a constant etch rate which depends on the selected etchant, the development rate depends on the grade of polymerization or dissolution of the photo-active compound (PAC) of the resist accumulated during exposure. Therefore the development rate is inhomogeneous and modeled with spheres of varying size. Their radius is determined by the local development rate resulting from lithography simulation multiplied with the time-step (see Fig. 5.1).

Alternatively to the rate obtained from a precedent exposure simulation the values for the development rate of the single cells were assigned with mathematical expressions. The equations were used for the characterization of typical benchmark examples which directly address specific requirements to be fulfilled for a correct propagation of the surface.

Either way, for resist development after lithography simulation as well as for the mathematically formulated test cases a sufficiently high number of cells has to be chosen in order to be able to resolve the strong variations of the rate originating from standing waves or notching effects during photo-resist exposure. Such standing wave effects which result from reflective interference are indicated on the right hand side of Fig. 5.1. We know that the distance between maxima and minima of the intensity which determines the development rate is given by $\lambda$ /4. For a deep ultraviolet (DUV) illumination wavelength of 248nm and a refractive index of 1.65 for the photo-resist, this distance is 37nm. For reliable simulations this distance has to be resolved by at least 10 cells, which leads to a cell density of 300cells/ $\mu\mathrm m$ . The lithography examples from Section 5.3 were simulated with this resolution. The benchmark examples (Section 5.2) have been scaled to a lower resolution of 200cells/ $\mu\mathrm m$ .