New method of topography simulation in photolithography

Abstract

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 polarisationdependent cylindrical wave, the inclination factor of which is denoted as diffraction coefficient in GTD. This GTDcoefficients 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 corner-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 hig...

Rigorous three-dimensional time-domain finite-difference electromagnetic simulation for photolithographic applications

“…oftheresulting system of equations. Bischoffetal 2. formulated amethod based onthe principles ofKeller's geometrical theory of diffraction and their uniform extension (the locality principle) in order to smdy latent images in photoresists.Yeung et al106 investigated an approximate method combining physical optics with the integral equation method.…”

Improvement of the physical-optics approximation for topography simulation in optical lithography

“…In the method of Bischoff et al 6, the above discontinuities of the geometrical-optics reflected field at the shadow boundaries were removed by the addition of an Wedge diffracted field to the reflected field in accordance with the rules of the Geometrical Theory of Diffraction. In our method, however, the above problem was avoided by eliminating the geometrical-optics reflected field in favor of the physical-optics diffracted fields within each surface.…”