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    J.A. Sethian
  • Semiconductor Manufacturing

    ( A special web page on semiconductor manufacturing may be found here )

    Part of the process of manufacturing a computer chip consists of digging holes in silicon wafers and coating them with materials. At various stages, level set methods can be used to track the evolution of the surface profile as it is shaped and formed.

    One of the first steps is photolithography, which looks a lot like silkscreening. Imagine looking down on a pattern:

    In photolithography, a mask is made of the above pattern, and the wafer is exposed to light which weakens the ability of the exposed material to resist an etching substance. During the photoresist development stage, the material is etched away, leaving deep holes that form initial paths in the silicon.

    A next step consists of etching and development; which is like a combination of sculpting and painting. Thin layers of material are both etched away and deposited, with the goal of constructing the desired shape at the end.

    The actual shape that comes out of this process depends on many factors, including the placement of the mask pattern, the length of exposure to light, the orientation of the etching and deposition beams, and the effectiveness of these beams.

    Fast Marching Methods and Photolithography Development

    Numerical simulations of photolithography development consist of several stages. First, Maxwell's equations are solved to see how the light beam changes the resist properties of the exposed material for a given pattern mask. Next, the material is "eaten away" during photoresist development; one way to think of this process is like an acid eating away at a substance; in some places, the acid can penetrate quickly, in other places, not so fast.

    We can use the Fast Marching Method to compute the evolution of the surface as it digs out the trench. The result of the solving Maxwell's equation gives us the speed of a front at each point in the material. Then we let a surface propagate with that speed, and etch out the material.
    (72K) (286K)
    Evolution of single contact hole Evolution under mask pattern
    Model Gaussian rate function Rate calculated using TMA's Depict

    Level Set Methods and Etching and Deposition

    Under etching and deposition, additional layers of materials are stripped away and added. The etching/deposition can occur
    • Isotropically: Material is uniformly added/subtracted, for example, a deposition layer condense from a gas which is everywhere above the surface.
    • Directionally: Material is directionally added/subtracted; for example, etching might occur under bombardment of particles coming from a particular direction.
    • From many angles: Material emanating from a source is added/subtracted; for example, etching particles might come from a source located above the profile.
    The effect of this etching/deposition depends on many factors, such as
    • Material properties: it's easier to tunnel through jello than concrete
    • Surface Visibility: if you want to get a suntan, don't hide in a dark corner
    • Beam orientation and strength: you'll go blind if you stare at an eclipse
    These and other combined effects are shown in the movies below:

    Acid enters from open sides: front merges many times: (569K)

    Isotropic Etching into a well: (129K)

    Deposition into a well: Visibility (shadowing) is important: (101K)

    Cutaway view: as neck pinches, inside is shadowed: (244K)

    Isotropic etching: thin curtain vanishes first: (385K)

    Directional etching from behind causes shadows: (253K)


    Calculations were performed using both the level set method and the Fast Marching Method. In the photolithography simulation, the fast marching method, which exploits a heap sort algorithm and the stationary time formulation was used; in the deposition example, a non-convex Lax-Friedrichs algorithm for non-convex sputter yield laws was used to update a narrow band level set function. Additional movies


    Adalsteinsson, D., and Sethian, J.A., A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography, I: Two-Dimensional Simulations, J. Comp, Phys., Vol. 120, No. 1, pp. 128-144, 1995. Download publications
    Adalsteinsson, D., and Sethian, J.A., A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography, II: Three-dimensional Simulations, J. Comp, Phys., Vol. 122, No. 2, pp. 348-366, 1995. Download publications
    Adalsteinsson, D., and Sethian, J.A., A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography, III: Re-Deposition, Re-Emission, Surface Diffusion, and Complex Simulations, J. Comp, Phys., J. Comp. Phys., 138, 1, pp. 193-223, 1997. Download publications
    Sethian, J.A., A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Nat. Acad. of Sciences, 1995. Download publications