Abstract.A model for the flow of calcium on the scale of one heart cell is given by a system of time-dependent reaction-diffusion equations coupled by nonlinear reaction terms. Calcium ions enter into the cell at release units distributed throughout the cell and then diffuse. At each release unit, the probability for calcium to be released increases along with the concentration of calcium, thus creating a feedback loop of waves regenerating themselves repeatedly. The validation of this model requires simulations on the time scale of several repeated waves and on the spatial scale of the entire cell. This requires long-time studies on spatial meshes that need to have a high resolution to resolve the positions of the calcium release units throughout the entire cell. We detail the development of a special-purpose numerical method and parallel implementation for this problem. Parallel performance studies demonstrate the scalability of the implementation on a distributed-memory cluster with low-latency interconnect. Convergence studies verify convergence to analytical expectations and confirm the appropriateness of all numerical parameters. Application studies on the desired time and length scales confirm that the model exhibits the desired feedback mechanism for calcium currents through the release units at suitable high levels, but the long-time studies demonstrate also that the current model with its present parameters leads to excessive calcium concentrations over time. This phenomenon could only be observed using a computational method able to reach laboratory scale final times for a domain on the scale of a complete cell.
An integrated simulator for chemical vapor deposition is introduced. In addition to a reactor scale and feature scale simulators, it consists of a \mesoscopic" scale simulator with the typical length scale of a die. It is shown that the \three-scale" integrated simulator used is a proper extension of \two-scale" deposition simulators that consist of reactor scale and feature scale simulation models. Moreover, it is demonstrated that information is provided on a new length scale, for which no information is available from the \two-scale" approach, as well as important corrections to the simulation results on the reactor scale. This enables, for instance, studies of microloading. For these demonstrations, thermally induced deposition of silicon dioxide from tetraethyloxysilane (TEOS) is chosen as the application example, which is modeled by six gaseous reacting species involved in four gas-phase and eight surface reactions.
A model designed to deal with pattern dependences of deposition processes is discussed. It is a mesoscopic scale model in the sense that it deals with spatial scales on the order of lO to 10-2rn which is intermediate between reactor scale and feature scale. This model accounts for the effects of the microscopic surface structure via suitable averages obtained by a homogenization technique from asymptotic analysis. Two studies on the low pressure chemical vapor deposition of silicon dioxide from tetraethoxysilane are presented to demonstrate the mesoscopic scale model. The first study shows the effects of microloading in regions of higher feature density. The second study shows the effects of varying operating conditions on loading and introduces a generalized Damkoehler number, which includes information about the surface patterns, for quantifying the degree of transport limitations. Some thoughts on how this model can be used to bridge reactor scale and feature scale models are presented. InfroductionThe trend toward single-wafer reactors (SWR) for deposition and etch processes in the microelectronics industry is expected to continue as wafer size continues to increase. To make SWRs more economically attractive, deposition and etch processes are being run at high rates to maintain reasonable throughputs. High rates can lead to nonunif ormities on the wafer scale because of depletion of reactants and transport limitations. Reactor scale models (RSIVI) for flow, heat transfer, and chemical reactions are well devel-
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