# Christian A. RinghoferArizona State University | ASU · School of Mathematical and Statistical Sciences

Christian A. Ringhofer

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171

Publications

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## Publications

Publications (171)

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 r...

In this article, we present a simulator for modeling transport of charge carriers and electrically active defect centers in solar cells by treating them on an equal footing, which allows us to address metastability and reliability issues. The exact nonlinear differential equations set solved by our solver is presented. The formulation of such diffe...

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 sta...

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 sta...

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 kine...

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...

We have developed a 2D diffusion-reaction simula-tor 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...

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 migra...

We are interested in flows on general networks and derive a kinetic equation describing general production, social or transportation networks. Corresponding macroscopic transport equations for large time and homogenized behavior are obtained and studied numerically. This work continues a recent discussion [Averaged kinetic models for flows on unstr...

In this work, we report on development of one-dimensional reaction-diffusion simulator needed to understand the kinetics of Cu-related metastabilities observed in CdTe PV devices. Evolution of intrinsic and Cu-related defects in CdTe solar cells has been studied in time-space domain self-consistently with free carrier transport. Resulting device pe...

We review transport equations and their usage for the modeling and simulation of nanopores. First, the significance of nanopores and the experimental progress in this area are summarized. Then the starting point of all classical and semiclassical considerations is the Boltzmann transport equation as the most general transport equation. The derivati...

In this work, we report on developing 1D reaction-diffusion solver to understand the kinetics of p-type doping formation in CdTe absorbers and to shine some light on underlying causes of metastabilities observed in CdTe PV devices. Evolution of intrinsic and Cu-related defects in CdTe solar cell has been studied in time-space domain self-consistent...

It is well known that Cu plays an important role in CdTe solar cell performance as a dopant. In this work, a finite-difference method is developed and used to simulate Cu diffusion in CdTe solar cells. In the simulations, which are done on a two-dimensional (2D) domain, the CdTe is assumed to be polycrystal-line, with the individual grains separate...

We develop a model for the evolution of wealth in a non-conservative economic
environment, extending a theory developed earlier by the authors. The model
considers a system of rational agents interacting in a game theoretical
framework. This evolution drives the dynamic of the agents in both wealth and
economic configuration variables. The cost fun...

In this work, the multiscale problem of modeling fluctuations in boundary layers in stochastic elliptic partial differential equations is solved by homogenization. Homogenized equations for the covariance and variance of the solution of stochastic elliptic PDEs are derived. In addition to the homogenized equations, a scaling law for the covariance...

An impurity diffusion-reaction model is applied to Cu migration in CdTe layer of CdTe solar cells. In this simulation, the reactions between major defects, such as Cu interstitials, Cd interstitials, Cd vacancies and Cu at Cd site, in CdTe and the diffusion of each of them are calculated numerically. The simulation yields transient Cu distributions...

We introduce a new mean field kinetic model for systems of rational
agents interacting in a game-theoretical framework. This model is
inspired from non-cooperative anonymous games with a continuum of
players and Mean-Field Games. The large time behavior of the system is
given by a macroscopic closure with a Nash equilibrium serving as the
local the...

This paper is concerned with the static, one dimensional modelling of a semiconductor device (namely the pn-junction) when a bias is applied. The governing equations are the well known equations describing carrier transport in a semiconductor which consist of a system of five ordinary differential equations subject to boundary conditions imposed at...

We present and analyze a model for the evolution of the wealth distribution
within a heterogeneous economic environment. The model considers a system of
rational agents interacting in a game theoretical framework, through fairly
general assumptions on the cost function. This evolution drives the dynamic of
the agents in both wealth and economic con...

We introduce a new mean field kinetic model for systems of rational agents
interacting in a game theoretical framework. This model is inspired from
non-cooperative anonymous games with a continuum of players and Mean-Field
Games. The large time behavior of the system is given by a macroscopic closure
with a Nash equilibrium serving as the local the...

In Perdaen et al [1] we developed a Discrete Event Simulation (DES) for a re-entrant semiconductor factory with a push dispatch policy at the beginning of the line and a pull dispatch policy at the end of the line. The simulation uses a heuristic to move the transition point between both policies, the push-pull point (PPP), along the production lin...

We present an overview over recent developments of traffic flow mod-els for production networks. Particular emphasis is given to the implementation of service rules for complex systems, involving multiple product types and re -entrant loops. A rather general scheduling concept is introduced and demonstrated on some numerical experiments.

Production networks are usually defined as a set of processes utilized to efficiently integrate suppliers, manufacturers, and customers so that goods are pro-duced and distributed in the right quantities, to the right locations, and at the right time, in order to reduce costs while satisfying delivery conditions. More precisely, in our setting, we...

We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.

In state-of-the-art devices, it is well known that quantum and Coulomb effects play significant role on the device operation. In this paper, 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. We also demonstrate,...

This paper deals with the spatial discretization of partial differential equations arising from Galerkin approximations to the Boltzmann equation, which preserves the entropy properties of the original collision operator. A general condition on finite difference methods is derived, which guarantees that the discrete system satisfies the appropriate...

This paper is concerned with the spatial discretization of the energy transport model for charged particles. A finite difference method is given which dissipates the entropy of the system on the discrete level, thus exhibiting the correct long-time behavior of the solution. This method represents a generalization of the Scharfetter–Gummel exponenti...

We analyze the convergence properties of a spectral collocation method for the Wigner-Poisson equation, a nonlinear pseudodifferential equation describing quantum mechanical transport phenomena. Spectral accuracy and nonlinear stability of the momentum discretization are proven. The convergence results are verified numerically on a test sample.

As semiconductor devices are scaled into nanoscale regime, first velocity saturation starts to limit the carrier mobility due to pronounced intervalley scattering, and when the device dimensions are scaled to 100 nm and below, velocity overshoot (which is a positive effect) starts to dominate the device behavior leading to larger ON-state currents....

In memoriam Naoufel Ben Abdallah. Abstract. A system of diffusion-type equations for transport in 3d confined structures is derived from the Boltzmann transport equation for charged particles. Transport takes places in confined structures and the scaling in the derivation of the diffusion equation is chosen so that transport and scattering occur in...

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 t...

We derive semi - classical approximations to quantum,transport models in thin slabs with applications to SOI (Silicon Oxide on Insulator) - type semiconductor devices via a sub - band approach. In the regime considered the forces acting on the particles across the slab are much,larger than the forces in the lateral direction of the slab. In a semi...

A production system which produces a large number of items in many steps can be modelled as a continuous flow problem. The resulting hyperbolic partial differential equation (PDE) typically is nonlinear and nonlocal, modeling a factory whose cycle time depends nonlinearly on the work in progress. One of the few ways to influence the output of such...

The need for service rules, or policies, in supply chains arises if not all the parts processed in the chain are considered identical, but are distinguished by certain attributes. We develop and analyze a methodology to model arbitrary service rules in large supply chains based on a kinetic (traffic flow like) theory and a level set approach. The f...

Fluctuations in the biofunctionalized boundary layers of nanowire field-effect biosensors are investigated by using the stochastic linearized Poisson-Boltzmann equation. The noise and fluctuations considered here are due to the Brownian motion of the biomolecules in the boundary layer, i.e., the various orientations of the molecules with respect to...

The purpose of this paper is to develop a model which allows for the study and optimization of arbitrarily complex supply networks, including order policies and money flows. We propose a mathematical description that captures the dynamic behavior of the system by a coupled system of ordinary differential delay equations. The underlying optimization...

Biologically sensitive field‐effect transistors (BioFETs) are a promising technology for detecting pathogens, antigen‐antibody complexes, and tumor markers. A BioFET is studied for a biotin‐streptavidin complex. Biotin‐streptavidin is used in detection and purification of various biomolecules. The link between the Angstrom scale of the chemical rea...

Field-effect nanobiosensors (or Biofets, biologically sensitive field-effect transistors) have recently been demonstrated experimentally and have thus gained interest as a technology for direct, label-free, real-time, and highly sensitive detection of biomolecules. The experiments have not been accompanied by a quantitative understanding of the und...

Optimizing manufacturing systems consists in generating large-quantity outputs to fulfill cus-tomers demands. But naturally machines may fail and the production process is either slowed down or completely interrupted. In order to keep production running, we are interested in assigning repair crews to currently broken-down machines. But due to the l...

We use the stochastic linearized Poisson-Boltzmann equation to model the fluctuations in nanowire field-effect biosensors due to changes in the orientation of the biomolecules. Different orientations of the biomolecules with respect to the sensor surface due to Brownian motion have different probabilities. The probabilities of the orientations are...

We develop continuum models of re-entrant factory production systems that treat the flow of products in analogy to traffic flow. Specifically, we model the dynamics of material flow through a re-entrant factory via a parabolic conservation law describing the product density and flux in the factory. We first extract the transport coefficients, in pa...

In this work the properties of a biotin-streptavidin BioFET have been studied numerically with homogenized boundary interface conditions as the link between the oxide of the FET and the analyte which contains the bio-sample. The biotin-streptavidin reaction pair is used in purification and detection of various biomolecules; the strong streptavidin-...

This paper presents a continuum -traffic flow like -model for the flow of products through complex production networks, based on statistical information obtained from extensive observations of the system. The resulting model consists of a system of hyperbolic conservation laws, which, in a relax-ation limit, exhibit the correct diffusive properties...

In this paper a bottom-up approach for modeling field-effect biosensors (BioFETs) is developed. Starting from the given positions of charged atoms, of a given molecule, the charge and the dipole moment of a single molecule are calculated. This charge and dipole moment are used to calculate the mean surface density and mean dipole moment at the biof...

BioFETs (biologically sensitive field-effect transistors) are field-effect biosensors with semi-conducting transducers. Their device structure is similar to a MOSFET, except that the gate structure is replaced by an aqueous solution containing the analyte. The detection mechanism is the conductance modulation of the transducer due to binding of the...

In these notes, we review the recent theory of quantum hydrodynamic and diffusion models derived from the entropy minimization
principle. These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system
of equations by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantu...

BioFETs (biologically active field-effect transistors) are biosensors with a semiconductor transducer. Due to recent experiments demonstrating detection by a field effect, they have gained attention as potentially fast, reliable, and low-cost biosensors for a wide range of applications. Their advantages compared to other technologies are direct, la...

In this chapter, we review the recent theory of quantum diffusion models derived from the entropy minimization principle.
These models are obtained by taking the moments of a collisional Wigner equation and closing the resulting system of equations
by a quantum equilibrium. Such an equilibrium is defined as a minimizer of the quantum entropy subjec...

The solutions of the nonlinear Schrödinger equation are of great
importance for ab initio calculations. It can be shown that such solutions
conserve a countable number of quantities, the simplest being the local norm
square conservation law. Numerical solutions of high quality, especially for
long time intervals, must necessarily obey these conserv...

The classical Coulomb potential and force can be calculated efficiently using fast multi-pole methods. Effective quantum potentials,
however, describe the physics of electron transport in semiconductors more precisely. Such an effective quantum potential
was derived previously for the interaction of an electron with a barrier for use in particle-ba...

We derive a modification of the semiclassical Fermi Golden Rule collision operator based on quantum thermodynamic principles.
The resulting operator is nonlocal in space and acknowledges the presence of steep potential gradients and potential barriers.
The resulting quantum mechanical transport equation—the Wigner quantum Boltzmann equation—increas...

Effective quantum potentials describe the physics of quantum-mechanical electron transport in semiconductors more than the
classical Coulomb potential. An effective quantum potential was derived previously for the interaction of an electron with
a barrier for use in particle-based Monte Carlo semiconductor device simulators. The method is based on...

A BioFET (biologically active field-effect transistor) is a biosensor whose transducer consists of a semiconductor, usually silicon. The device structure of a BioFET is similar to a MOSFET whose gate structure has been replaced by a functionalized oxide surface and an aqueous solution (see Fig. 1). BioFETs can be viewed as generalizations of ISFETs...

We develop a methodology to investigate optimal dynamic policies for a large class of supply networks. The basic underlying model is a fluid dynamic model for continuous product flows in continuous time. Discrete as well as continuous controls are implemented.

We analyze the stochastic large time behavior of long supply chains via a traffic flow type random particle model. So items travel on a virtual road from one production stage to the next. Random breakdowns of the processors at each stage are modeled via a Markov process. The result is a conservation law for the expectation of the part density which...

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. In a previous work, we have derived a hyperbolic conservation law for the part density and flux in the supply chain. In the present paper, we introduce internal variables (named attributes: e.g. the time to due-date) and...

High-volume, multistage continuous production flow through a re-entrant factory is modeled through a conservation law for a continuous-density variable on a continuous-production line augmented by a state equation for the speed of the production along the production line. The resulting nonlinear, nonlocal hyperbolic conservation law allows fast and...

The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the dis...

Two quantum-kinetic models of ultrafast electron transport in quantum wires are derived from the general-ized electron-phonon Wigner equation. The various assumptions and approximations allowing one to find closed equations for the reduced electron Wigner function are discussed with an emphasis on their physical relevance. The models correspond to...

By following a strategy introduced in previous works, quantum extensions of the classical electron-phonon scattering operator are deduced from first prin-ciples. These quantum collision operators satisfy a quantum H-theorem and relax towards quantum equilibria. Then, under an assumption of dominant elastic interactions, a hierarchy of quantum Spher...

To manage the increasing dynamics within complex production networks, a decen-tralised and autonomous control of material flows is a promising approach. This aims at an easier control of logistic processes in the network, whereby the local decision rules should lead to self-organisation and good logistic performance on the global level. There are t...

A quantum kinetic equation approach is adopted in order to incorporate quantum effects such as collisional broadening due to finite lifetime of single particle states, and collisional retardation due to finite collision time. A quantum correction to the semiclassical electron distribution function is obtained using an asymptotic expansion for the q...

In this Note, we generalize the Boltzmann collision operator modeling binary particle–particle collisions to a quantum framework using nonlocal quantum entropy principles. To cite this article: P. Degond, C. Ringhofer, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

We consider a supply chain consisting of a sequence of buffer queues and processors with certain throughput times and capacities. Based on a simple rule for releasing parts, i.e. batches of product or individual product items, from the buffers into the processors we derive a hyperbolic conservation law for the part density and flux in the supply ch...