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Copyright:
1996, 1999, 2006
J.A. Sethian

Applications to Materials Sciences
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The application of level set methods and Fast Marching Methods to materials sciences problems is a natural fit. As before, the central idea is to use view material boundaries as zero level sets of higher dimensional functions, and to track the evolving interfaces. Typical examples include solid-liquid boundaries, boundaries in metal layers in etching and deposition, photolithography development simulations, and solid-gas boundaries.

  • Refs. 1,2 and 3 are the examples of applying level set methods to crystal growth. They study interface stability in crystal growth and dendritic solidification, and includes the effects of heat release, Gibbs-Thomson boundary conditions, curvature-depedent heat terms, and anisotropic effects. Ref. 1 transforms the problem to a boundary integral formulation, and then extends an appropriate velocity field off of the interface to allow the update of neighboring level sets. The results show a wide variety of physical effects, including tip-splitting, side-branching, unstable growth and effects of container boundary conditions. Ref. 3 takes a different approach; rather than use a boundary integral formulation, it works directly with the diffusion equation in both regimes.

  • Refs. 4,5,6,7 and 8 discuss the application of interface techniques to problems in etching and deposition in semiconductor manufacturing . These are some of the most involved applications of level set and Fast Marching Methods to date. They study the effects of material-dependent etch rates, non-linear flux laws, visibility calculations, surface diffusion, and re-deposition/re-emission on the evolving surface profiles.

  • Ref. 9 studies the application of level set methods to optimal structural design, showing how to build schemes which construct efficient design of weight-bearing structures.

  • Refs. 10, 11 and 12 study the metallization failure, electromigration, and the growth of voids due to vacancy fluxes and transport. Previous work using this approach includes the work of Zhao. et. al. and the work of Averbuch, Israeli and Ravve. These techniques combines immersed interface methods for solving the irregular Poisson problems around the voids, level set methods to track the interface, Fast Marching Methods to extend the velocity and update the neighboring level sets. The results show the growth of voids, merger, and the effects of the various terms.


New Book and Resource on Level Set and Fast Marching Methods


References:

  1. Crystal Growth and Dendritic Solidification : Sethian, J.A., and Strain, J.D., Journal of Computational Physics, 98, pp. 231-253, 1992.
    Abstract

    We present a numerical method which computes the motion of complex solid/liquid boundaries in crystal growth. The model we solve includes physical effects such as crystalline anisotropy, surface tension, molecular kinetics and undercooling. The method is based on two ideas. First, the equations of motion are recast as a single history-dependent boundary integral equation on the solid/liquid boundary. A fast algorithm is used to solve the integral equation efficiently. Second, the boundary is moved by solving a "Hamilton-Jacobi"-type equation (on a fixed domain) formulated by Osher and Sethian for a function in which the boundary is a particular level set. This equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws. The two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary, in a way suggested by the physics of the problem. Our numerical experiments show the evolution of complex crystalline shapes, development of large spikes and corners, dendrite formation and side-branching, and pieces of solid merging and breaking off freely.


  2. Algorithms for Computing Crystal Growth and Dendritic Solidification : Sethian, J.A. and Strain, J.D., Institute of Mathematics and its Applications, University of Minnesota, pp. 107-126 (1992).
    Abstract

    We report on a numerical method for computing the motion of complex solid/liquid boundaries in crystal growth. The model we solve includes physical effects such as crystalline anisotropy, surface tension, molecular kinetics and undercooling. The method is based on a single single history-dependent boundary integral equation on the solid/liquid boundary, which is solved by means f a fast algorithm coupled to a level set approach for tracking the evolving boundary. Numerical experiments show the evolution of complex crystalline shapes, development of large spikes and corners, dendrite formation and side-branching, and pieces of solid merging and breaking off freely.

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  3. A Simple Level Set Method for Solving Stefan Problem : Chen, S., Merriman, B., Osher, S., and Smereka, P., Journal of Computational Physics, 138, pp. 8-29, 1997.


  4. A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography I: Algorithms and Two-Dimensional Simulations : Adalsteinsson, D,. and Sethian, J.A., Journal of Computational Physics, 120, 1, pp. 128--144, 1995.
    Abstract

    We apply a level set formulation to the problem of surface advancement in a two-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. The development of this model and algorithm naturally extends to three dimensions in a straightforward manner, and is described in Part II of this paper.

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  5. A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography II: Three-Dimensional Simulations : Adalsteinsson, D., and Sethian, J.A., Journal of Computational Physics, 122, 2, pp. 348--366, 1995.
    Abstract

    We apply a level set formulation to the problem of surface advancement in three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. In Part I of this paper, the basic equations and algorithms for two dimensional simulations were developed. In this paper, the extension to three dimensions is presented. We show a large collection of simulations, including three-dimensional etching and deposition into cavities under the effects of visibility, directional and source flux functions, evolution of lithographic profiles, discontinuous etch rates In Part III of this paper, effects of reflection, re-emission, surface diffusion, and multiple materials will be presented.

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  6. A Level Set Approach to a Unified Model for Etching, Deposition, and Lithography III: Re-Deposition, Re-Emission, Surface Diffusion, and Complex Simulations : Adalsteinsson, D., and Sethian, J.A., Journal of Computational Physics, 138, 1, pp. 193-223, 1997.
    Abstract

    Previously, Adalsteinsson and Sethian have applied the level set formulation to the problem of surface advancement in two and three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. Part I presented the basic equations and algorithms for two dimensional simulations, including the effects of isotropic and uni-directional deposition and etching, visibility, reflection, and material dependent etch/deposition rates. Part II focused on the extension to three dimensions. This paper completes the series, and add the effects of re-deposition, re-emission, and surface diffusion. This requires the solution of the transport equations for arbitrary geometries, and leads to simulations that contain multiple simultaneous competing effects of visibility, directional and source flux coefficients for the re-emission and re-deposition functions, multi-layered fronts and thin film layers.

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  7. Fast Marching Level Set Methods for Three Dimensional Photolithography : Sethian, J.A., Proceedings, SPIE 1996 International Symposium on Microlithography, Santa Clara, California, March, 1996.
    Abstract

    We present detailed timings of a fast marching level set method introduced by Sethian for surface advancement in photoresist development. The method merges fast narrow band level set methods entropy-satisfying schemes for the Eikonal equation, and fast heap sort algorithms. The resulting method can perform the development stage of the three-dimensional photoresist process in 80 seconds on a Sparc10 for a 200x200x200 grid.

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  8. An Overview of Level Set Methods for Etching, Deposition, and Lithography : Sethian, J.A., and Adalsteinsson, D., IEEE Transactions on Semiconductor Devices, 1996. 10, 1, pp.167-184, 1997.
    Abstract

    The range of surface evolution problems in etching, deposition, and lithography development offers significant challenge for numerical methods in front tracking. Level set methods for evolving interfaces are specifically designed for profiles which can develop sharp corners, change topology, and undergo orders of magnitude changes in speed. They are based on solving a Hamilton-Jacobi type equation for a level set function, using techniques borrowed from hyperbolic conservation laws. Over the past few years, a body of level set methods have been developed with application to microfabrication problems.

    In this paper, we give an overview of these techniques, describe the implementation in etching, deposition, and lithography simulations, and present a collection of fast level set methods, each aimed at a particular application. In the case of photoresist development and isotropic etching/deposition, the fast marching level set method}}, introduced by Sethian, can track the three-dimensional photoresist process through a $200 \times 200 \times 200$ rate function grid in under 55 seconds on a Sparc10. In the case of more complex etching and deposition, the Narrow Band level set method, introduced in by Adalsteinsson and Sethian, can be used to handle problems in which the speed of the interface delicately depends on the orientation of the interface vs. an incoming beam, the effects of visibility, surface tension, reflection and re-emission, and complex three-dimensional effects. Our applications include photoresist development, etching/deposition problems under the effects of masking, visibility, complex flux integrations over sources, non-convex sputter deposition problems, and simultaneous deposition and etch phenomena.

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  9. Structural Boundary Design via Level Set and Immersed Interfaces Methods : Sethian, J.A., and Wiegmann, A., J. Comp. Phys., 163, 2, Sep 2000, pp. 489-528
    Abstract

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  10. A numerical model of stress driven grain boundary diffusion : Wilkening, J., and Sethian, J.A., J. Comp. Phys., 193, 1, pp. 275-305, 2003
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  11. Analysis of Stress Driven Grain Boundary Diffusion, Part I : Wilkening, J., Borucki, L., and Sethian, J.A., SIAM J. Appl. Math., 64, 6, 1839-1863, 2004.
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  12. Analysis of Stress Driven Grain Boundary Diffusion, Part II : Wilkening, J., Borucki, L., and Sethian, J.A., SIAM J. Appl. Math., 64, 6, 1864-1886, 2004.
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Additional References:

  1. A Simple Level Set Method for Solving Stefan Problem : Chen, S., Merriman, B., Osher, S., and Smereka, P., Journal of Computational Physics, 138, pp. 8-29, 1997.