Physically-Based Models of Reactive Ion Etching
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The process of microelectronic fabrication relies on transferring the design pattern to a semiconductor wafer with incredible accuracy and precision. The precursor to this pattern transfer is a lithography step in which a pattern is tranferred to a photosensitive polymer on the wafer's surface, but the actual transfer of this pattern to the wafer is accomplished by an etching process.
Reactive ion etching (RIE) is the industry standard method for etching through openings created in the polymer mask. RIE is a synergistic combination of two opposing etch technologies, chemical etching and physical etching. Whereas chemical etching involves bathing the substrate to be etched in a corrosive chemical which chemically attacks the surface through mask openings, and physical etching involves firing high energy particles at the surface through the mask opening to blast away substrate material, RIE is an elegant combination of both these processes, holding onto the advantages of each.
Like chemical etching, RIE is highly selective, etching only materials with the target composition; like physical etching, RIE is highly anisotrophic, etching in a single direction from the mask opening. The mechanism by which this is accomplished is based upon using energetic particles to activate a reaction of a chemical species with the surface. As shown in Figure 1, ions are generated in a plasma and accelerated toward the surface and the mask opening. The plasma also creates highly reactive neutral species that are not accelerated by the electric field and thus arrive at the surface without a preferential direction. Where both ions and neutrals are present, i.e., on parts of the surface oriented horizontally such as the bottom of the pit being etched, a highly selective reaction is activated and the target material is removed.
To model such a system, we approach the problem on the scale of a single feature being etched. On this scale, which is significantly smaller than the mean free path of any of the gas or plasma species, all species can be considered to move in straight lines, interacting only with the surface. This 'ballistic approximation' allows us to compute the fluxes of species arriving at any point on the surface from the source plasma and from emissions by other points on the surface. If we know or can guess how reactions at the surface depend on incoming fluxes, we then have enough information to compute the distribution of fluxes everywhere on the surface and the resulting etch rates everywhere on the surface.
Feature scale simulations that use the above ballistic transport and reaction model (BTRM)1 must solve a set of integral equations representing the integrated flow of species to and from control volumes along the evolving surface. As with many problems with integral equations at their centers, solving this problem involves a large amount of bookkeeping! To keep track of all of the reaction, transport, and discretization tasks involved, we have created a trio of codes, DQR.
- D - a level-set based, 3D surface discretization and evolution code
- Q - a deterministic 3D fractional transport computation code
- R - a psuedosteady reaction state computation code
This trio of codes works together to represent and track a model of the evolving system as it is etched, accounting for transport between the source and the surface and surface and other parts of the surface, and the complex resulting psuedo-equilibrium. DQR is part of the CCNI Commons, a collection of intellectual property developed in conjunction with the CCNI facilities, allowing sponsors and developers to appropriately share in rights to research advancements.
Q computes the view factor matrix Q, or the set of differential transmission probability between pairs of oriented triangular elements in a surface mesh, accounting for shadowing and occlusion by the rest of the surface. This view factor contains all the information about the geometry of the surface required to compute the transport of material throughout the structure.
Q is able to do this quickly using an octree-based preprocessor to efficiently find potential occlusions while ignoring elements away from the line-of-sight. The figure below shows an example of the complicated path that a ray might take in a space with reflective boundary condition. As the ray proceeds through the space, cells in the octree hashing table that it passes through are shown.
As a result of this and other accelerating data structures, Q is extremely quick in determining view factors between pairs of surface elements, finding all the view factors in a 50k element structure in under a minute on a modern desktop computer.
R computes the other half of the BTRM, the reaction component. Because the time scale of etching is on the order of minutes and the transmission of material from surface to surface can be on nanosecond time scales, the reaction rates along the surface can be thought of as in a psuedo-equilibrium. The reaction rates as a function of position along the surface must be known to compute both the distribution of fluxes transmitted from surface to surface and the resulting etch rates as a function of position. It is this psuedo-equilibrium which R calculates, using the view factor matrix Q computed by Q, and expressions for reaction rates as functions of incoming fluxes.
An example of such a rate calculation is shown in Figure 4, in which a generic RIE chemistry with one ion and one neutral species is considered in an L-shaped trench. On the left of Figure 4 is the trench shown from the top to reveal the plane representing the plasma and the open ends of the trench at which reflective boundary conditions are affixed. On the right of the figure is a view through the back of the trench revealing the local etch rates along the trench. Note the accelerated etch rate at the junction of the L, due to the increased ionic flux due to the decreased shadowing at that point.
1. Cale and Raupp, J. Vacuum Sci. and Technol. A, 1990.