Chemically Induced Surface Evolution with Level Sets (ChISELS) is a level-set based computer code in development at Sandia National Laboratories to model at the feature scale the evolution of surfaces of micro devices during fabrication on patterned wafers by surface micromachining (SMM) processes (cf. Figure 1). The initial focus of the models in ChISELS is on low pressure chemical vapor deposition (LPCVD), one of the processes employed in Sandia National Laboratories' SUMMiT V technology, so the models employed are ballistic transport of chemical species from the reactor to and between surfaces and surface chemistry only; all of which are designed to function in a massively parallel computational framework.

BTRM

All gas-phase transport in our model is assumed to occur in the free-molecular flow regime (i.e., particle to particle collisions are negligible). This is a good approximation for the low-pressure conditions of interest here and yields equations similar to those for the more familiar problem of radiation heat transfer (See Figure 2). In ChISELS we adopt the ballistic transport and reaction model (BTRM) developed and described by Cale and coworkers [1] [3]. Details of this model and it's additional assumptions can be found in the cited references. An important aspect of this method is the need to calculate view-factors from each point on the evolving surface to all other surface points in the simulation domain. For this purpose, standard mathematical and numerical techniques as described and implemented in Chaparral [4] are applied.

Deposition or etching occurs through the chemical reaction of gas phase species with bulk and surface species at a surface. The thermodynamics and heterogeneous surface chemistry of these reactions are modeled in ChISELS by coupling with Surface Chemkin [5]. This requires the specification of a chemical reaction mechanism for each surface reaction to be modeled in the simulation.

The equation that models the flux of a species, k, to a surface, i , is

where is the flux from the bulk of the reactor, is the view factor between surface i and k, is the flux to surface j of species K, is the reaction rate of species j on surface k, and repeated indices in the product indicate summation.

Equation 1 is a nonlinear relationship (because of the chemical reaction terms) that requires careful consideration in order to solve efficiently. For the case of low-reactive probabilities this equation is solved by an iterative scheme as described by Walker http://www.t12.lanl.gov/home/toposim.

Level-Set Method

Feature scale micro system fabrication modelers such as ChISELS are, at heart, topology modelers, i.e. they model the evolution of a free boundary according to the physics that cause it to move. ChISELS uses an implicit surface-tracking approach called the level set method [2]. In the level-set method, a domain-spanning function, is defined; the zero-value contour, or level set, of which conforms to the feature surface. The surface is evolved by solving the scalar partial-differential equation,

over the volume and integrating through time. The velocity, v , is called the extension velocity and is defined over the entire domain. The extension velocity must be chosen so that the
level set evolves in such a way that it mimics the evolution of the feature surface; i.e. it is chosen based on the velocity of the surface---the deposition or etch rate. The level set method avoids the debilitations of the explicit methods because the mesh which is used to solve Equation 2 does not deform, so distortion effects are avoided. Likewise, because a volume-defined function is evolved, merging surfaces do not create problems in the method.

The level-set partial differential equation is solved by the so-called semi-Lagrangian method---an augmented method of characteristics for wave equations with a non-constant wave velocity [6]. This is a method of the predictor-corrector type, and thus each time step has two parts, or stages. This method was chosen over finite element or finite difference techniques with implicit time integration because there is no matrix associated with the solution of the level-set equation to invert. Thus it requires very little memory and inter-processor communication, so it is exceptionally amenable to parallel implementation. Also, because only interpolation is required, there are less stringent requirements of the grid.

Once the extension velocity is computed at the current time step, the predicted level-set function, at the next time is computed from

References

[1] T. S. Cale, T. P. Merchant, L. J. Borucki and A. H. Labun, Thin Solid Films, 356, 152-175, 2000.

[2] J. A. Sethian, "Level Set Methods", Cambridge Univ. Press, Cambridge, 1996.

[3] T. S. Cale and V. Mahadev, in Thin Films: Modeling of Film Deposition for Microelectronic Applications, Vol. 22, Ed. S. Rossnagel, Academic Press, 176-277, 1996.

[4] M. W. Glass,"{CHAPARRAL}: A library for solving enclosure radiation heat transfer problems", Sandia National Laboratories, Albuquerque, NM, 2001.

[5] SURFACE CHEMKIN: A Software Package for the Analysis of Heterogeneous Chemical Kinetics at a Solid-Surface -- Gas-Phase Interface Interface, Reaction Design Inc. 2001 (See also: http://www.reactiondesign.com).

[6]J. Strain, J. Comput. Phys., 161, 512-528, 2000.