Dopant diffusion in phases with a moving interface

We consider the diffusion of a dopant through a moving interface in the suicide film-Si system during silicide layer growth. The dopant concentration distribution is derived in analytical form by the integral Fourier transform method with subsequent reduction of the dopant redistribution problem to numerical solution of two integral equations. The results are presented in the form of curves plotting the time dependence of dopant concentration on both sides of the interface for various values of diffusion coefficients and interface velocity. The effect of physical parameters on the variation of dopant concentration near the interface is demonstrated.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 93–97, 1987.     

Nonlinear diffusion of impurities in semiconductors

We study the concentration-dependent diffusion of dopant impurities into semiconductors. In particular, we examine the two-dimensional diffusion in the vicinity of a mask. Numerical solutions are obtained for dopant diffusion with fixed-total-concentration and with constant-surface-concentration. For the fixed-total-concentration case, we also obtain approximate power series solutions. Our numerical and approximate results are compared with analytical and numerical results obtained by other investigators.     

Single dopant diffusion in semiconductor technology

The paper deals with the analysis of pair diffusion models in semiconductor technology. The underlying model contains reaction‐drift‐diffusion equations for the mobile point defects and dopant‐defect pairs as well as reaction equations for immobile dopants which are coupled with a non‐linear Poisson equation for the chemical potential of the electrons. For homogeneous structures we present an existence and uniqueness result for strong solutions. Starting with energy estimates we derive further a priori estimates such that fixed point arguments due to Leray–Schauder guarantee the solvability of the model equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Diffusion of Dopant in Crystalline Silicon: An Asymptotic Analysis

An existing model of the diffusion of dopant in crystallinesilicon is presented. The model is used in predicting semiconductordevice properties. For small concentrations of dopant, the diffusionis linear. However, at large concentrations, enhanced diffusionoccurs. This may be modelled by a concentration-dependent diffusioncoefficient that is approximately constant at small concentrationsbut increases linearly at higher concentrations. The initialdistribution of implanted dopant is confined to a small regionof the crystal, and has a high peak concentration. This initialdistribution develops into a high-concentration region withadvancing steep fronts and decaying tail regions. In the enhancedregion, the dominant balance is given by the filtration equation,and the tail regions are dominated by quasi-steady convection—diffusion.These two regions are analysed and matched together, using singularperturbation theory.     

The stress-strain state for heterodiffusive saturation of a sphere

We study the mechanico-diffusive phenomena of saturation of a sphere for the case of two routes of dopant migration. We establish the basic qualitative and quantitative regularities. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 51–59.     

Propagation and conditional propagation of chaos for pressureless gas equations

We study the existence and uniqueness of a weak solution of a viscous d-dimensional system of pressureless gas equations. We construct a nonlinear diffusion by using the propagation and conditional propagation of chaos. The latter diffusion is associated with the above pressureless gas equations.Mathematics Subject Classification (2000):60H15, 35R60, 60H30     

Boundary conditions at a thin membrane for normal diffusion,classical subdiffusion,and slow subdiffusion processes

We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one-dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system.     

Diffusion approximation of the Wright-Fisher model of population genetics: Single-locus two alleles

We investigate an autoregressive diffusion approximation method applied to the Wright-Fisher model in population genetics by considering a Markov chain with Bernoulli distributed independent variables. The use of an autoregressive diffusion method and an averaged allelic frequency process lead to an Orn-stein-Uhlenbeck diffusion process with discrete time. The normalized averaged frequency process possesses independent allele frequency indicators with constant conditional variance at equilibrium. In a monoecious diploid population of size N with r generations, we consider the time to equilibrium of averaged allele frequency in a single-locus two allele pure sampling model.     

On the Uniform Convergence of the Empirical Density of an Ergodic Diffusion

We investigate the uniform convergence of the density of the empirical measure of an ergodic diffusion. It is known that under certain conditions on the drift and diffusion coefficients of the diffusion, the empirical density f _t converges in probability to the invariant density f, uniformly on the entire real line. We show that under the same conditions, uniform convergence of f _t to f on compact intervals takes place almost surely. Moreover, we prove that under much milder conditions (the usual linear growth condition on the drift and diffusion coefficients and a finite second moment of the invariant measure suffice), we have the uniform convergence of f _t to f on compacta in probability. This revised version was published online in June 2006 with corrections to the Cover Date.     

Time-dependent diffusion operators on <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We study the Cauchy problem for time-dependent diffusion operators with singular coefficients on L¹-spaces induced by infinitesimal invariant measures. We give sufficient conditions on the coefficients such that the Cauchy-Problem is well-posed. We construct associated diffusion processes with the help of the theory of generalized Dirichlet forms. We apply our results in particular to construct a large class of Nelson-diffusions that could not been constructed before.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

Miscible displacement in a Hele-Shaw cell

We formulated a theory of simple mixtures of incompressible miscible liquids in terms of the mass averaged velocityu and the solenoidal volume averaged velocityW, We derived simplified equations for miscible displacement in a Hele-Shaw cell. We obtained a steady solution of these equations corresponding to displacement under gravity with prescribed values of concentration and mean normal stress at the inlet and exit of the cell. We studied the stability of this steady flow. This differs from previous works which treat the stability of unsteady miscible displacement using a quasi-static assumption and classical equations based on divu=0. In our problem, replacingu withW gives rise to a difference in the mean normal stress, which alters the pressure drop across the cell and changes the velocity of free fall. We found that the stability equations are the same in the two formulations, but the boundary conditions are slightly different; however the difference will be small if diffusion is slow or the thickness of the cell is small. The results show that steady miscible displacement in a Hele-Shaw cell is stable to long and short waves. Within certain ranges of parameters, the displacement of glycerin into water can be unstable. This instability is basically of a Rayleigh-Taylor type, regularized by diffusion. As the diffusion parameterS becomes smaller, the waves of disturbances become finer and are confined to an increasingly thin diffusion layer. Water displacing glycerin is always stable. This is due to the fact that the steady equilibrium profile is not steep enough to create a fingering instability.Dedicated to Prof. Klaus Kirchgässner on the occasion of his sixtieth birthday     

Metropolis–Hastings Algorithms with acceptance ratios of nearly 1

We develop the results on polynomial ergodicity of Markov chains and apply to the Metropolis–Hastings algorithms based on a Langevin diffusion. When a prescribed distribution p has heavy tails, the Metropolis–Hastings algorithms based on a Langevin diffusion do not converge to p at any geometric rate. However, those Langevin based algorithms behave like the diffusion itself in the tail area, and using this fact, we provide sufficient conditions of a polynomial rate convergence. By the feature in the tail area, our results can be applied to a large class of distributions to which p belongs. Then, we show that the convergence rate can be improved by a transformation. We also prove central limit theorems for those algorithms.     

On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion

We prove that the Hamilton–Jacobi equation for an arbitrary Hamiltonian H (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical C1,α solutions. The proof is achieved using a new Hölder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.     

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L_∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization. AMS subject classification 65L10, 65L12, 65L15     

On the diffusion equation and diffusion wavelets on flat cylinders and the n‐torus

In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case. Copyright © 2010 John Wiley & Sons, Ltd.     

Extremal Behavior of Diffusion Models in Finance

We investigate the extremal behavior of a diffusion X _t given by the SDE , where W is standard Brownian motion, μ is the drift term and σ is the diffusion coefficient. Under some appropriate conditions on X _t we prove that the point process of ε -upcrossings converges in distribution to a homogeneous Poisson process. As examples we study the extremal behavior of term structure models or asset price processes such as the Vasicek model, the Cox–Ingersoll–Ross model and the generalized hyperbolic diffusion. We also show how to construct a diffusion with pre-determined stationary density which captures any extremal behavior. As an example we introduce a new model, the generalized inverse Gaussian diffusion. This revised version was published online in July 2006 with corrections to the Cover Date.     

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

     

Logarithmic Estimates for the Density of Hypoelliptic Two-Parameter Diffusions

We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)[0, 1]² for the L^p-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of2² log p_s, t(x, y), as ↓0, where x is the initial condition of the diffusion, = , and p_s, t(x, y) is the density of the hypoelliptic two-parameter diffusion.