1.1.1 Lithography

  Optical lithography has enabled the continuous shrinking of the device geometries for the last decade. Optical reduction steppers decreased the feature size by increasing the numerical aperture and reducing the wavelength. Nowadays, the mainstream technology for is based on 248 nm exposures [vdH94]. As the feature size of modern semiconductor devices reaches the submicron range, optical methods are approaching their limits due to effects like light interference and scattering. To simulate optical lithography three steps have to be performed: (1) imaging, (2) resist-bleaching, and (3) resist-development. Imaging calculates the incident light passing the optical aperture through the wafer surface by means of Fourier optics. As the result of imaging the aerial image which refers to the electromagnetic field at the wafer surface, is given. Resist-bleaching is concerned which the chemical bleaching reaction depending on the electromagnetic field distribution within the resist. Therefore, the Maxwell equations have to be solved in the nonplanar resist, including backscattering effects caused by the underlying substrate material. Solving the Maxwell equations with rigorous frequency-domain methods requires the solution of a system with millions of simultaneous equations. Due to the fact that each solution variable depends on each other, the system matrix is not positive definite and fully occupied. Iterative solution methods like the conjugate gradient method are not successful [Won95]. The final resist development process is more or less a selective etching process. Several two-dimensional lithography simulators were released in recent years that are able to handle both bleaching and development processes [Old79] [Mac85] [Luc92]. However two-dimensional simulations are not sufficient to predict topologies of the mask patterns. Three-dimensional lithography simulators are required and have been developed for photo, electron and X-ray lithography [Jon81] [Bar89] [Toh91]. One of the difficulties in photo lithography simulation is the resist development algorithm. The so-called string algorithm is quite accurate but can cause looping effects of strings [Jew75] [Toh91]. Therefore, the data structures and the resist manipulations are complicated and slow. On the contrary, the cell removal algorithm is absolutely stable but the resolution of the simulation domain, especially at curved surfaces, is limited by memory requirements. Within process simulators the resist development is often done by etching modules with the algorithms described above.