We consider generalizations of the classical inverse problem to Bayesien type estimators, where the result is not one optimal parameter but an optimal probability distribution in parameter space. The practical computational tool to compute these distributions is the Metropolis Monte Carlo algorithm. We derive kinetic theories for the Metropolis Monte Carlo method in different scaling regimes. The derived equations yield a different point of view on the classical algorithm. It further inspired modifications to exploit the difference scalings shown on a simulation example of the Lorenz system.

The retina is a part of the central nervous system that is accessible, well documented, and studied by researchers spanning the clinical, experimental, and theoretical sciences. Here, we mathematically model the subcircuits of the outer plexiform layer of the retina on two spatial scales: that of an individual synapse and that of the scale of the receptive field (hundreds to thousands of synapses). To this end we formulate a continuum spine model (a partial differential equation system) that incorporates the horizontal cell syncytium and its numerous processes (spines) within cone pedicles. With this multiscale modeling approach, detailed biophysical mechanisms at the synaptic level are retained while scaling up to the receptive field level. As an example of its utility, the model is applied to study background-induced flicker enhancement in which the onset of a dim background enhances the center flicker response of horizontal cells. Simulation results, in comparison with flicker enhancement data for square, slit, and disk test regions, suggest that feedback mechanisms that are voltage-axis modulators of cone calcium channels (for example, ephaptic and/or pH feedback) are robust in capturing the temporal dynamics of background-induced flicker enhancement. The value and potential of this continuum spine approach is that it provides a framework for mathematically modeling the input-output properties of the entire receptive field of the outer retina while implementing the latest models for transmission mechanisms at the synaptic level.

NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract Session 3647 Interdisciplinary Research on Modeling and Scheduling of Semiconductor Manufacturing Operations Donald Collins, Ph.D., Manufacturing Engineering Technology, Forouzan Golshani, Ph.D., Computer Science, Frank Hoppensteadt, Ph.D., Math and Electrical Engineering, Christian Ringhofer, Ph.D., Math, Jennie SI, Ph.D., Electrical Engineering, Kostas Tsakalis, Ph.D., Electrical Engineering Arizona State University Abstract This paper will describe the ongoing interdisciplinary research work of a group of faculty and students working within the Systems Science and Engineering Research Center at Arizona State University (SSERC) in collaboration with INTEL and Motorola. One of the strongest driving forces in the economy of most developed countries is manufacturing. In the United States, one of the most important components of this driving force is the manufacturing of semiconductors in fabrication facilities (FAB). Not surprisingly, tremendous efforts have been expended to reduce the art of semiconductor manufacturing to a science. While the improvements in scale and yield of semiconductor manufacturing has been spectacular and are well known, the improvement in operational methods of process flow and process scheduling have not been as impressive. The most obvious reason for this is the inherent physical/chemical complexity of these processes. Semiconductor manufacturing processes are hundreds of steps long, and factories typically operate multiple processes with each one running many product devices to serve a dynamic marketplace. Process flows are re-entrant and the work in process can be subject to rework. Finite equipment and human resources exhibit a wide range of variability of availability and performance. There are batching or set-up considerations on machines that process by wafer, by lot, or by batch. One practical consequence of all of this complexity is that decisions must be made on very different time-scales. The smallest time-scale being every few minutes during floor operations, usually about resource allocation - which step/lot combination to run next, which down machine to repair next, and so on. The medium time-scale of every few days or weeks, responding to market changes. To the largest time-scale of every few months concerning process changes, design improvements, etc. The integration of all of these operational decisions defines the performance of the factory. The key difficulty in making these decisions lies in the intricacy by which they affect the long- and short-term factory performance, as well as each other.

We are interested in the derivation of optimality conditions for controlled interacting agent systems. We establish the relation between mean field optimality conditions and the optimality condition of the mean field control problem. This link is important for many recently published articles on control strategies for agent based systems since it establishes the precise relation between multipliers for the individual agents and the probability density distribution of the multipliers in the mean field limit. The relation to different notions of differentiability are also shown.

In state of the art devices, it is well known that quantum and Coulomb effects play significant role on the device operation. In this book chapter we demonstrate that a novel effective potential approach in conjunction with a Monte Carlo device simulation scheme can accurately capture the quantum-mechanical size quantization effects. Inclusion of tunneling within semi-classical simulation schemes is discussed in details. We also demonstrate, via proper treatment of the short-range Coulomb interactions, that there will be significant variation in device design parameters for devices fabricated on the same chip due to the presence of unintentional dopant atoms at random locations within the channel of alternative technology devices.

The main goal of this work was to eliminate the ambiguity between the observed performance changes under stress and their physical root cause by enabling a depth of modeling that takes account of diffusion and drift at the atomistic level coupled to the electronic subsystem responsible for a PV device’s function. Development of advanced physical models describing capture/emission/ recombination phenomena relevant to multivalent dopants, amphoteric centers, and donor-acceptor pairs was also accomplished. These models, incorporating grain bulk, grain boundaries, and hetero-interface properties, were implemented in a self-contained simulation tool that drastically reduces interpretation ambiguity and, for the first time, allows for predictive design of thin-film PV devices.

Thin-film modules of all technologies often suffer from performance degradation over time. Some of the performance changes are reversible and some are not, which makes deployment, testing, and energy-yield prediction more challenging. Manufacturers devote significant empirical efforts to study these phenomena and to improve semiconductor device stability. Still, understanding the underlying reasons of these instabilities remains clouded due to the lack of ability to characterize materials at atomistic levels and the lack of interpretation from the most fundamental material science. The most commonly alleged causes of metastability in CdTe device, such as “migration of Cu,” have been investigated rigorously over the past fifteen years. Still, the discussion often ended prematurely with stating observed correlations between stress conditions and changes in atomic profiles of impurities or CV doping concentration. Multiple hypotheses suggesting degradation of CdTe solar cell devices due to interaction and evolution of point defects and complexes were proposed, and none of them received strong theoretical or experimental confirmation. It should be noted that atomic impurity profiles in CdTe provide very little intelligence on active doping concentrations. The same elements could form different energy states, which could be either donors or acceptors, depending on their position in crystalline lattice. Defects interact with other extrinsic and intrinsic defects; for example, changing the state of an impurity from an interstitial donor to a substitutional acceptor often is accompanied by generation of a compensating intrinsic interstitial donor defect. Moreover, all defects, intrinsic and extrinsic, interact with the electrical potential and free carriers so that charged defects may drift in the electric field and the local electrical potential affects the formation energy of the point defects. Such complexity of interactions in CdTe makes understanding of temporal changes in device performance even more challenging and a closed solution that can treat the entire system and its interactions is required. In this book chapter we first present validation of the tool that is used to analyze Cu migration in single crystal (sx) CdTe bulk. Since the usual diffusion analysis has limited validity, our simulation approach presented here provides more accurate concentration profiles of different Cu defects that lead to better understanding of the limited incorporation and self-compensation mechanisms of Cu in CdTe. Finally, simulations are presented that study Cu ion’s role in light soaking experiments of CdTe solar cells under zero-bias and forward-bias stress conditions.

We present a 2D Unified Solver for modeling metastability and reliability in solar cells. The current version of the solver is adopted to model carrier and defect transport in thin film photovoltaic devices. A unique feature of the solver is that defects transport, carrier generation and recombination mechanisms and grain boundary descriptions are treated on an equal footing. We propose and implement the standard implicit Euler method with Newton iteration for solving the reactions, which leads to improved accuracy and numerical stability. We use Gummel's method to solve a coupled set of reaction-diffusionPoisson equations which has been discretized using the Scharfetter-Gummel scheme. We verify the solver on the test case of 2D chlorine anneal. We illustrate that process temperature profiles can be simulated as well.

Thin-film modules of all technologies often suffer from performance degradation over time. Some of the performance changes are reversible and some are not, which makes deployment, testing, and energy-yield prediction more challenging. Manufacturers devote significant empirical efforts to study these phenomena and to improve semiconductor device stability. Still, understanding the underlying reasons of these instabilities remains clouded due to the lack of ability to characterize materials at atomistic levels and the lack of interpretation from the most fundamental material science. The most commonly alleged causes of metastability in CdTe devices, such as ‘migration of Cu’, have been investigated rigorously over the past fifteen years. Still, the discussion often ended prematurely with stating observed correlations between stress conditions and changes in atomic profiles of impurities or CV doping concentration. Multiple hypotheses suggesting degradation of CdTe solar cell devices due to interaction and evolution of point defects and complexes were proposed, and none of them received strong theoretical or experimental confirmation. It should be noted that atomic impurity profiles in CdTe provide very little intelligence on active doping concentrations. The same elements could form different energy states, which could be either donors or acceptors, depending on their position in crystalline lattice. Defects interact with other extrinsic and intrinsic defects; for example, changing the state of an impurity from an interstitial donor to a substitutional acceptor often is accompanied by generation of a compensating intrinsic interstitial donor defect. Moreover, all defects, intrinsic and extrinsic, interact with the electrical potential and free carriers so that charged defects may drift in the electric field and the local electrical potential affects the formation energy of the point defects. Such complexity of interactions in CdTe makes understanding of temporal changes in device performance even more challenging and a closed solution that can treat the entire system and its interactions is required.

Kinetic exchange models of markets utilize microscopic binary descriptions of wealth transfers to derive a Boltzmann-like equation describing the evolution of the corresponding wealth distribution. We develop such a model to describe a binary form of welfare called need-based transfer (NBT), inspired by the gift-giving of cattle practiced among the Maasai of East Africa. Variants of such welfare schemes can be attributed to other human and animal communities. Specifically, we consider NBTs relative to a given welfare threshold such that individuals with surplus give to individuals with need in order to preserve the recipient’s continued viable participation in the economy. Our NBT kinetic model considers redistribution rules parameterized to vary between regressive and progressive redistribution.

The dynamics of insurance plans have been under the microscope in recent years due to the controversy surrounding the implementation of the Affordable Care Act (Obamacare) in the United States. In this paper, we introduce a game between an insurance company and an ensemble of customers choosing between several insurance plans. We then derive a kinetic model for the strategies of the insurer and the decisions of the customers and establish the conditions for which a Nash equilibrium exists for some specific customer distributions. Finally, we give some agent-based numerical results for how the plan enrollment evolves over time which show qualitative agreement to "experimental'' results in the literature from two plans in the state of Massachusetts.

We have developed a 2D diffusion-reaction simulator suitable for modeling CdTe solar cells that contain active defects. It utilizes a self-consistent numerical scheme in which the device is mapped onto a finite element mesh. To demonstrate its versatility, we apply the simulator to the problems of defect migration during the annealing process and the extraction of current-voltage characteristics and quantum efficiencies. Our simulator also includes grain boundaries which are known to play an important role in CdTe devices as they strongly affect the migration of defects such as chlorine (Cl).

Continuum models of re-entrant production systems are developed that treat the flow of products in analogy to traffic flow. Specifically, the dynamics of material flow through a re-entrant factory via a parabolic conservation law is modeled describing the product density and flux in the factory. The basic idea underlying the approach is to obtain transport coefficients for fluid dynamic models in a multi-scale setting simultaneously from Monte Carlo simulations and actual observations of the physical system, i.e. the factory. Since partial differential equation (PDE) conservation laws are successfully used for modeling the dynamical behavior of product flow in manufacturing systems, a re-entrant manufacturing system is modeled using a diffusive PDE. The specifics of the production process enter into the velocity and diffusion coefficients of the conservation law. The resulting nonlinear parabolic conservation law model allows fast and accurate simulations. With the traffic flow-like PDE model, the transient behavior of the discrete event simulation (DES) model according to the averaged influx, which is obtained out of discrete event experiments, is predicted. The work brings out an almost universally applicable tool to provide rough estimates of the behavior of complex production systems in non-equilibrium regimes.

An improved model of copper p-type doping in CdTe absorbers is proposed that accounts for the mechanisms related to tightly bound Cu(i)-Cu(Cd) and Cd(i)-Cu(Cd) complexes that both limit diffusion and cause self-compensation of Cu species. The new model explains apparent discrepancy between DFT-calculated and fitted diffusion parameters of Cu reported in our previous work, and allows for better understanding of performance and metastabilities in CdTe PV devices.

Absorber material with high and stable p-type doping that does not impede free carrier lifetime is the key component enabling efficiency and stability improvements in thin-film CdTe technology. To better understand the compensation mechanism and the metastable effects related to Cu acceptors, the most common p-type dopant in CdTe, i.e., a detailed kinetic model describing the behavior of intrinsic and Cu-related defects in this material, has been developed and applied for the first time. Migration and reactions of these point defects in single crystal CdTe have been investigated by solving diffusion-reaction equations in the time-space domain self-consistently with free carrier transport. The simulation results supported by reasonable match to experimental data have shed light on the nature of limited Cu incorporation (also known as the Cu solubility limits) and Cu self-compensation during the annealing and cooling processes.

In this work, we report on development of one-dimensional (1D) finite-difference and two-dimensional (2D) finite-element diffusion-reaction simulators to investigate mechanisms behind Cu-related metastabilities observed in CdTe solar cells [1]. The evolution of CdTe solar cells performance has been studied as a function of stress time in response to the evolution of associated acceptor and donor states. To achieve such capability, the simulators solve reaction-diffusion equations for the defect states in time-space domain self-consistently with the free carrier transport. Results of 1-D and 2-D simulations have been compared to verify the accuracy of solutions.

We demonstrate a self-consistent numerical scheme for simulating an electronic device which contains active defects. As a specific case, we consider copper defects in cadmium telluride solar cells. The presence of copper has been shown experimentally to play a crucial role in predicting device performance. The primary source of this copper is migration away from the back contact during annealing, which likely occurs predominantly along grain boundaries. We introduce a mathematical scheme for simulating this effect in 2D and explain the numerical implementation of the system. Finally, we will give numerical results comparing our results to known 1D simulations to demonstrate the accuracy of the solver and then show results unique to the 2D case.

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