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A model of evolution of temperature and density of ions in an electrolyte

Andrzej Raczyński (2005)

Applicationes Mathematicae

We study existence and nonexistence of solutions (both stationary and evolution) for a parabolic-elliptic system describing the electrodiffusion of ions. In this model the evolution of temperature is also taken into account. For stationary states the existence of an external potential is also assumed.

A new exponentially fitted triangular finite element method for the continuity equations in the drift-diffusion model of semiconductor devices

Song Wang (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we present a novel exponentially fitted finite element method with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices. The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkin finite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded as an extension to two dimensions of the...

A numerically efficient approach to the modelling of double-Qdot channels

A. Shamloo, A.P. Sowa (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

We consider the electronic properties of a system consisting of two quantum dots in physical proximity, which we will refer to as the double-Qdot. Double-Qdots are attractive in light of their potential application to spin-based quantum computing and other electronic applications, e.g. as specialized sensors. Our main goal is to derive the essential properties of the double-Qdot from a model that is rigorous yet numerically tractable, and largely circumvents the complexities of an ab initio simulation....

Asymptotic stability of stationary solutions to the drift-diffusion model in the whole space

Ryo Kobayashi, Masakazu Yamamoto, Shuichi Kawashima (2012)

ESAIM: Control, Optimisation and Calculus of Variations

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution...

Coulomb Interaction Effects on the Spin Polarization and Currents in Quantum Wires with Spin Orbit Interaction

Anton Heidar Thorolfsson, Andrei Manolescu, D.C. Marinescu, Vidar Gudmundsson (2012)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

We analyze the charge and spin distributions induced in an interacting electron system confined inside a semiconductor quantum wire with spin orbit interaction in the presence of an external magnetic field. The wire, assumed to be infinitely long, is obtained through lateral confinement in three different materials: GaAs, InAs, and InSb. The spin-orbit coupling, linear in the electron momentum is of both Rashba and Dresselhaus type. Within the Hartree-Fock approximation the many-body Hamiltonian...

Estimate of the Hausdorff measure of the singular set of a solution for a semi-linear elliptic equation associated with superconductivity

Junichi Aramaki (2010)

Archivum Mathematicum

We study the boundedness of the Hausdorff measure of the singular set of any solution for a semi-linear elliptic equation in general dimensional Euclidean space n . In our previous paper, we have clarified the structures of the nodal set and singular set of a solution for the semi-linear elliptic equation. In particular, we showed that the singular set is ( n - 2 ) -rectifiable. In this paper, we shall show that under some additive smoothness assumptions, the ( n - 2 ) -dimensional Hausdorff measure of singular set...

Finite element solution of the fundamental equations of semiconductor devices. II

Miloš Zlámal (2001)

Applications of Mathematics

In part I of the paper (see Zlámal [13]) finite element solutions of the nonstationary semiconductor equations were constructed. Two fully discrete schemes were proposed. One was nonlinear, the other partly linear. In this part of the paper we justify the nonlinear scheme. We consider the case of basic boundary conditions and of constant mobilities and prove that the scheme is unconditionally stable. Further, we show that the approximate solution, extended to the whole time interval as a piecewise...

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

Claire Chainais-Hillairet, Jian-Guo Liu, Yue-Jun Peng (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal...

Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis

Claire Chainais-Hillairet, Jian-Guo Liu, Yue-Jun Peng (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal...

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