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Recently, photolithography simulation above topographical substrates becomes a more and more interesting topic in submicrotechnology. Besides originating standing waves in the resist during exposure, substrate slopes cause specular reflections. Moreover one can state diffraction effects if geometrical dimensions are in the same order of magnitude like exposure wavelength. The worst case that might occur is the so-called concave mirror effect.
Different methods for field calculation are known from optical theory. Closed analytical treatments can be done merely for simple geometries. Direct solutions of Maxwells equations or Helmholtz equation as a boundary value problem by means of numerical methods (FDM or FEM/BEM) are difficult and computation time consuming. Highly sophisticated computers (especially massively parallel machines) are required to realize acceptable operation times.
We propose an alternative method, which is mainly based upon using the basic principles of Keller’s Geometrical Theory of Diffraction (GTD) and their uniform extension (UTD), namely the locality principle, the boundary diffraction wave representation and Keller’s ray conception which includes diffracted rays. The first step now consists in the separation of a given diffracting surface in such a way that analytical solutions for the separated regions are known (canonical problem). Particularly in the two-dimensional (2d) case, a topography can be approached by putting inclined (plane) faces together. This leads to the canonical problem of wedge diffraction, the exact solution of which was given at first by Sommerfeld. An asymptotic evaluation of his diffraction integral yields a partition of the total field in the geometrical-optical field and a diffracted field. In the 2d-case the latter is represented by a direction- and polarisation- dependent cylindrical wave, the inclination factor of which is denoted as diffraction coefficient in GTD. This GTD- coefficients fail both in the near of and directly on the geometrical-optical boundaries and the edge of the wedge. (We have to distinguish between the two shadow- and the two reflexion-boundaries.) In such transition regions UTD-coefficients guarantee sufficient accuracy of field calculation, as can be shown by a comparison with exact solution (convergent expansion of the diffraction integral). These extended UTD-coefficients consist of four terms, each of them is related to one shadow- or reflexion-boundary. Remaining problems (especially in the 3d-case) like curved wedges and comer- or vertex-diffraction are discussed shortly. Wedges with impedance faces (e.g. reflectivity < 1) are involved by the application of a heuristical method from microwave theory. Furthermore, geometrical-optical field calculation within the bounds of GTD is described in brief.
Based on our model explained above, a complete two-dimensional topography simulator was created. Running on a simple IBM-AT-386 the algorithm employs as much as or less time than the procedures basing on direct numerical solutions and requiring highly sophisticated machines. Resist bleaching is taken into consideration by several bleaching steps, according to Dill’s differential equation in their difference approach. Numerical results for typical simulation situations (notching, grain, trapezium, concave mirror) are presented and compared with literature. Additionally, special polarization effects could be predetermined by means of simulation and proved experimentally. Under certain conditions an essential improvement would be achieved by the application of polarized light
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