Sparse field level set method for non-convex Hamiltonians in 3D plasma etching profile simulations

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Abstract

Level set method [S. Osher, J. Sethian, J. Comput. Phys. 79 (1988) 12] is a highly robust and accurate computational technique for tracking moving interfaces in various application domains. It originates from the idea to view the moving front as a particular level set of a higher dimensional function, so the topological merging and breaking, sharp gradients and cusps can form naturally, and the effects of curvature can be easily incorporated. The resulting equations, describing interface surface evolution, are of Hamilton–Jacobi type and they are solved using techniques developed for hyperbolic equations. In this paper we describe an extension of the sparse field method for solving level set equations in the case of non-convex Hamiltonians, which are common in the simulations of the profile surface evolution during plasma etching and deposition processes. Sparse field method itself, developed by Whitaker [R. Whitaker, Internat. J. Comput. Vision 29 (3) (1998) 203] and broadly used in image processing community, is an alternative to the usual combination of narrow band and fast marching procedures for the computationally effective solving of level set equations. The developed procedure is applied to the simulations of 3D feature profile surface evolution during plasma etching process, that include the effects of ion enhanced chemical etching and physical sputtering, which are the primary causes of the Hamiltonian non-convexity.

Introduction

In various fields of science and engineering often arise phenomena in which different materials (or phases) can coexist without mixing. A surface bounding two materials is called an interface, a phase boundary, or front depending upon situation. In some processes a boundary is moving by external driving forces and its velocity does not depend on its geometrical properties. Since the evolution of a boundary is unknown and it should be determined as a part of solutions, the problems including such a boundary are called in general a free boundary problems. The motion of a phase boundary between ice and water is a typical example, and it has been well studied (the Stefan problem).

Another important class of problems are those where the evolution of an interface does not depend on the physical situation outside the boundary, but only on its geometry. There are several examples of such a behavior in material sciences and they are sometimes called the interface controlled problems. Examples are not limited only to material sciences. Some of those comes from geometry, crystal growth problems and image processing.

Level set method, introduced by Osher and Sethian [1], is a powerful technique for analyzing and computing moving fronts in a variety of different settings. Some references to earlier works with similar ideas, as well as deeper analytical results concerning foundations of this method can be found in [3]. The level sets are used in image processing, computer vision, computational fluid dynamics, material science, and many other fields. Detailed exposition of the theoretical and numerical aspects of the method, and applications to different areas can be found in books [4] and [5], and recent review articles [6], [7], [8]. Ref. [9] is a popular and lucid introduction to the subject.

The profile surface evolution in plasma etching, deposition and lithography development is a significant challenge for implementation of numerical methods in front tracking. The level set methods for evolving interfaces are specially designed for profiles which can develop sharp corners, change of topology and undergo orders of magnitude changes in speed. They are based on Hamilton–Jacobi type equation for the level set function using techniques developed for solving hyperbolic partial differential equations. During last several years several variants of the level set methods have been developed with application to micro fabrication problems. In this paper we describe shortly the level set method as well as sparse field method for solving the level set equations. The sparse-field method itself, developed by Whitaker [2], and broadly used in image processing community, is an alternative to the usual combination of narrow band and fast marching procedures for the computationally effective solving of the level set equations [4], [5]. After that, we analyze the case of non-convex Hamiltonians in more details. This type of problem is of special interest in studying the evolution of the profile surface during the etching process, especially if we treat it as an interface controlled problem.

Our primary goal is to develop an accurate, stable and efficient 3D code for tracking of the etching profile evolution that includes different physical effects such as anisotropy, visibility conditions and material-dependent propagation rates, yet being computationally effective to run on desktop PCs. This work is one of the preparation steps for accomplishing it.

Section snippets

Level set method

The basic idea behind the level set method is to represent the surface in question at a certain time t as the zero level set (with respect to the space variables) of a certain function ϕ(t,x), the so-called level set function. The initial surface is given by {x|ϕ(0,x)=0}. The evolution of the surface in time is caused by forces or fluxes of particles reaching the surface in the case of the etching process. The velocity of the point on the surface, normal to the surface, will be denoted by V(t,x)

Sparse field method for non-convex Hamiltonians

Several approaches for solving the level set equations exist which increase the accuracy and decrease the computational effort. They are all based on using some sort of adaptive schemes. The most important is the narrow band level set method [4], [5], widely used in the etching process modeling tools (for a detailed review see [13]), and recently developed the sparse-filed method [2], [11], [12], implemented in ITK medical image processing library [14], as well as in the general purpose image

Results

The details about the code design and implemented algorithms will be published elsewhere. Our implementation is based on ITK library [14]. The classes describing the level set function and the level set filter are reimplemented according to the procedures for treating non-convex Hamiltonians described in the previous section. Here we will present some results of calculations illustrating our approach to dealing with non-convex Hamiltonians in the etching profile simulations. We did not try to

Conclusions

In this paper we have presented an extension of the sparse field method for solving the level set equations in the case of non-convex Hamiltonians, suitable for the application in the 3D simulations of the profile surface evolution during the plasma etching and deposition processes. The obtained results show that it is possible to use the Lax–Friedrichs scheme in conjunction with the sparse field method, and to preserve sharp interfaces and corners by optimizing the amount of smoothing in it.

Acknowledgement

This work was supported by the Lam Research Corporation and Tera-level Nano Devices Project, 21c Frontier R&D Program of Korea Ministry of Science and Technology.

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