hardened photoresist. Devices are fabricated by incorporating dopant impurities either by direct diffusion or ion implantation and annealing. During subsequent processing, further diffusion takes place.

Many of the steps described above have been modeled to various degrees of accuracy by partial differential equations that describe the relevant physics. Modeling of diffusion of impurities in a nondeforming single crystal is done today with considerable sophistication that can take into account point defect interactions (Cole et al., 1990; Rorris et al., 1991). The theory of diffusion in a polycrystal, where diffusion occurs mostly along the grain boundaries (Tseng et al., 1992), and in the presence of external stresses, is not, however, adequately developed. Stresses and elastic deformations have long been investigated by applied mathematicians and mechanical engineers. There is a need to combine expertise and develop methods for the cross-coupled problems of diffusion and deformation in complex materials.

Modeling oxidation of silicon (Chin et al., 1983; Peng et al., 1991) poses problems of moving boundaries and the generation of plastic flow by stresses. Stresses are inherent in the process because oxidation yields an oxide that would normally occupy more than twice the volume of the silicon that is consumed. This volume expansion leads to flow and a shape change. In turn, the stresses affect the rate of oxidation, the diffusivity of oxygen, and so on. Self-consistent solutions are needed. Under high stresses and high temperatures, the oxide behaves like an incompressible fluid. To date, there is no completely satisfactory mathematical model for calculating the pressure during incompressible fluid flow. There are, however, promising mathematical developments for simpler models (Tayler and King, 1990; King, 1993).

Other areas of semiconductor processing have been modeled to different degrees of sophistication, but major problems remain to be solved. Examples include the modeling of etching (Kuiken, 1990), of the exposure of photoresist (Gerber, 1988), and of warping caused by titanium silicide growth (Willemsen et al., 1988; Friedman and Hu, 1992).

Overall, modeling of processing steps in microelectronics currently has substantial gaps, especially in the areas of deposition, etching, and lithography. The works referenced above constitute a starting point. Considerable amounts of information are generated by atomic-scale theories, but this knowledge is yet to be folded into a theoretical description at the appropriate mesoscopic length scales of process modeling.

There are various physical processes, not well understood, that contribute to variability of the performance of semiconductors. These include dislocations that develop during doping, diffusion of dopants in polycrystals, and elastic strain created during the various steps of depositions. One highly developed theory that seems relevant to semiconductor performance variability is the theory of dislocations (Hirth and Lothe, 1982; Eshelby, 1975). The analysis of one dislocation loop has been studied numerically (Borucki, 1993) and mathematically (Friedman et al., 1992b). However, nucleation of dislocations and the dynamics of the aggregate of dislocations remain open problems that mathematical sciences research could help to solve.

There has been a surge of mathematical analysis in the modeling of semiconductor devices, with increasing attention to small devices. A few representative recent papers are Poupaud (1993), Schmeiser and Unterreiter (1993), and Ward et al. (1993).



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