Drift-diffusion Limit Research Articles

Molecular dynamics simulations of the laser ablation of silicon with the thermal spike model

The purpose of this work is to model laser ablation of silicon on an atomistic scale in combination with a mesoscale model for the description of the electron-phonon interaction and an electron-temperature dependent interaction potential. The well-known continuum two-temperature model (TTM) for solids with highly excited electrons is extended from metals to silicon by explicitly taking charge carrier transport effects into account (nTTM). This is accomplished by the drift-diffusion limit of the Boltzmann-transport equation leading to the so called thermal-spike model (TSM). The model is further enhanced by extending the static modified Tersoff potential to a dynamical carrier excitation dependent interaction potential. We compare the TSM and nTTM with regard to physical correctness, numerical stability and applicability in the context of large-scale massive parallel high performance computing.

Optimal transport and colossal ionic mechano-conductance in graphene crown ethers.

Biological ion channels balance electrostatic and dehydration effects to yield large ion selectivity alongside high transport rates. These macromolecular systems are often interrogated through point mutations of their pore domain, limiting the scope of mechanistic studies. In contrast, we demonstrate that graphene crown ether pores afford a simple platform to directly investigate optimal ion transport conditions, i.e., maximum current densities and selectivity. Crown ethers are known for selective ion adsorption. When embedded in graphene, however, transport rates lie below the drift-diffusion limit. We show that small pore strains (1%) give rise to a colossal (100%) change in conductance. This process is electromechanically tunable, with optimal transport in a primarily diffusive regime, tending toward barrierless transport, as opposed to a knock-on mechanism. These observations suggest a novel setup for nanofluidic devices while giving insight into the physical foundation of evolutionarily optimized ion transport in biological pores.

An asymptotic-preserving scheme for the semiconductor Boltzmann equation toward the energy-transport limit

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and energy as mean free path goes to zero. As opposed to the classical drift-diffusion limit where the stiff collisions are all in one scale, new difficulties arise in the two-scale stiff collision terms because the simple BGK penalization [15] fails to drive the solution to the correct limit. We propose to set up a spatially dependent threshold on the penalization of the stiffer collision operator such that the evolution of the solution resembles a Hilbert expansion at the continuous level. Formal asymptotic analysis and numerical results confirm the efficiency and accuracy of our scheme.

Drift–diffusion limit of a model for the dynamics of epithelial and mesenchymal cell monolayers

This paper is devoted to deriving formally the drift–diffusion limit for a kinetic-like model describing the dynamics of a monolayer sample of epithelial and mesenchymal cells, which move via chemotaxis on a flat surface, proliferate, and interact among themselves. The aim is to verify if the macroscopic equations resulting from the underlying model are able to mimic a biologically consistent scenario, where epithelial cells tend to adhere to one another while mesenchymal cells diffuse through the sample.

Critical mass phenomenon for a chemotaxis kinetic model with spherically symmetric initial data

The goal of this paper is to exhibit a critical mass phenomenon occurring in a model for cell self-organization via chemotaxis. The very well-known dichotomy arising in the behavior of the macroscopic Keller–Segel system is derived at the kinetic level, being closer to microscopic features. Indeed, under the assumption of spherical symmetry, we prove that solutions with initial data of large mass blow-up in finite time, whereas solutions with initial data of small mass do not. Blow-up is the consequence of a momentum computation and the existence part is derived from a comparison argument. Spherical symmetry is crucial within the two approaches. We also briefly investigate the drift-diffusion limit of such a kinetic model. We recover partially at the limit the Keller–Segel criterion for blow-up, thus arguing in favour of a global link between the two models.

M5 mesoscopic and macroscopic models for mesenchymal motion

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.

Drift-diffusion limits of kinetic models for chemotaxis: A generalization

We study a kinetic model for chemotaxis introduced by Othmer, Dunbar, and Alt [23], which was motivated by earlier results of Alt, presented in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser, it was rigorously shown that, in three dimensions, this kinetic model leads to the classical Keller-Segel model as its drift-diffusion limit when the equation of the chemo-attractant is of elliptic type [4], [5]. As an extension of these works we prove that such kinetic models have a macroscopic diffusion limit in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type, which is the original version of the chemotaxis model.

Global Solutions of Nonlinear Transport Equations for Chemosensitive Movement

A widespread phenomenon in moving microorganisms and cells is their ability to reorient themselves depending on changes of concentrations of certain chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which also takes into account evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species. The basic type of model was discussed by Alt [J. Math. Biol., 9 (1980), pp. 147--177], [J. Reine Angew. Math., 322 (1981), pp. 15--41] and by Othmer, Dunbar, and Alt [J. Math. Biol., 26 (1988), pp. 263--298]. Chalub et al. rigorously proved that, in three dimensions, these kinds of kinetic models lead to the classical Keller--Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [Monatsh. Math.}, 142 (2004), pp. 123--141], [On the Derivation of Drift-Diffusion Model for Chemotaxis from Kinetic Equations, ANUM preprint 14/02, Vienna Technical University, 2002]. In [H. Hwang, K. Kang, and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B., to appear] it was proved that the macroscopic diffusion limit exists in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type. So far in the rigorous derivations, only the density of the chemo-attractant was supposed to influence the motion of the chemosensitive species. Here we show that in the macroscopic limit some types of evaluations of gradient fields of the chemical stimulus result in a change of the classical parabolic Keller--Segel model for chemotaxis. Under suitable structure conditions, global solutions for the kinetic models can be shown.

Discretization of the Multiscale Semiconductor Boltzmann Equation by Diffusive Relaxation Schemes

In this paper we derive diffusive relaxation schemes for the linear semiconductor Boltzmann equation that work in both the kinetic and diffusive regimes. Similar to our earlier approach for multiscale transport equations, we use the even- and odd-parity formulation of the kinetic equation, and then reformulate it into the diffusive relaxation system (DRS). In order to handle the implicit anisotropic collision term efficiently, we utilize a suitable power series expansion based on the Wild sum, which yields a time discretization uniformly stable with any desired order of accuracy, yet is explicitly solvable with the correct drift-diffusion limit. The velocity discretization is done with the Gauss–Hermite quadrature rule equivalent to a moment expansion method. Asymptotic analysis and numerical experiments show that the schemes have the usual advantages of a diffusive relaxation scheme for multiscale transport equations and are asymptotic-preserving.

Analytical and Computational Advances for Hydrodynamic Models of Classical and Quantum Charge Transport

In recent years, substantial advances have been made in understanding hydrodynamic models, both from the standpoint of analytical infrastructure, as well as the parameters which play a decisive effect in the behavior of such models. Both classical and quantum hydrodynamic models have been studied in depth. In this survey paper, we describe several results of this type. We include, for example, well-posedness for both classical and quantum reduced models, and the relaxation drift–diffusion limit as examples of analytical results. As examples of computational results, we include some discussion of effective algorithms, but most importantly, some information gleaned from extensive simulation. In particular, we present our findings of the prominent role played by the mobilities in the classical models, and the role of hysteresis in the quantum models. All models are self-consistent. Included is discussion of recent analytical results on the use of Maxwell’s equations. Benchmark devices are utilized: the MESFET transistor and the n+/n/n+ diode for classical transport, and the resonant tunneling diode for quantum transport. Some comparison with the linear Boltzmann transport equation is included.

Extended moment method for electrons in semiconductors

The semiclassical Boltzmann equation for electrons in semiconductors with the Kane dispersion law or the parabolic band approximation is considered and systems of moment equations with an arbitrary number of moments are derived. First, the paper deals with spherical harmonics in the formalism of symmetric trace-free tensors. The collision frequencies are carefully studied for the physical properties of silicon. Then, for the parabolic band approximation, the hierarchy of equations for full moments of the phase density and the corresponding closure problem is discussed. In particular, a set of 2 R scalar and vectorial moments is considered. To answer the question which number R one has to chose in order to retain the physical contents of the Boltzmann equation, the moment equations are examined in the drift-diffusion limit and in an infinite crystal in a homogeneous electric field (transient and stationary cases) for increasing number of moments R. The number R must be considered to be sufficient, if its further increase does not change the result considerably and the appropriate numbers for the processes are given.

The Drift–Diffusion Limit for Electron–Phonon Interaction in Semiconductors

A scattering operator modeling the interaction of electrons with phonons of given constant energy in semiconductors is analyzed by means of an expansion in terms of spherical harmonics. An application is the derivation of the drift–diffusion limit of the corresponding Boltzmann equation for the transport of electrons in a semiconductor crystal.

An analytical expression for base transit time in an exponentially doped base bipolar transistor

An analytical expression of base transit time for exponentially doped base, which is applicable to both homojunction and heterojunction bipolar transistors, is developed. The present treatment includes dopant dependent mobility variation, bandgap narrowing and finite velocity saturation effects in the calculation of base transit time and thus is a generalization of the results reported recently by Rosenfield and Alterovitz[6]. These effects degrade base transit time significantly and must be incorporated in the calculation. the finite velocity saturation effect will progressively play a pivotal role as the base width is scaled down and for double heterostructure bipolar transistor. The transition of base transit time from the ballistic and the drift-diffusion limit will occur at a longer base width if the base is exponentially doped with a sufficiently high doping index.

Simulation study of InP-based PNP heterojunction bipolar transistors and incorporation of nonclassical effects

This paper describes a numerical approach to the modeling of PNP HBTs in the InP-based materials systems (InP/InGaAs and InAlAs/InGaAs). Initial device analysis was achieved in the drift-diffusion limit by self-consistent numerical solution of the Poisson, carrier continuity and conductor equations subject to the device's geometry and boundary conditions imposed by the device's biasing. Simulation results are compared with the available experimental results and good agreement is found. For the InP/InGaAs and InAlAs/InGaAs heterojunctions, the valence band discontinuities are larger than for the AlGaAs/GaAs system so grading of the emitter-base junction and tunneling effects are important. For completeness, nonclassical effects were also considered. For the emitter-base junctions, hole tunneling was considered, particularly at low forward bias. The inverse dependence of the hole tunneling on effective mass was found to lead to significantly more light hole than heavy hole tunneling in calculating the emitter injection current. In addition, since very narrow base regions (25-35 nm) can be employed while keeping the base spreading resistance low due to the electron's higher mobility, ballistic hole transport should also be considered.