Quantum Drift Research Articles

On local well-posedness of the thin-film equation via the Wasserstein gradient flow

A local existence and uniquness of the gradient flow of one dimensional Dirichlet energy on the Wasserstein space is proved. The proofs are based on a relaxation of displacement convexity in the Wasserstein space and can be applied to a family of higher order energy functionals which are not displacement convex in the standard sense. As the result a local well-posedness of the corresponding nonlinear evolution equations including the thin-film equation and the quantum drift diffusion equation are proved.

Infinite speed of support propagation for the Derrida–Lebowitz–Speer–Spohn equation and quantum drift–diffusion models

We show that weak solutions of the Derrida–Lebowitz–Speer–Spohn (DLSS) equation display infinite speed of support propagation. We apply our method to the case of the quantum drift–diffusion equation which augments the DLSS equation with a drift term and possibly a second-order diffusion term. The proof is accomplished using weighted entropy estimates, Hardy’s inequality and a family of singular weight functions to derive a differential inequality; the differential inequality shows exponential growth of the weighted entropy, with the growth constant blowing up very fast as the singularity of the weight becomes sharper. To the best of our knowledge, this is the first example of a nonnegativity-preserving higher-order parabolic equation displaying infinite speed of support propagation.

Classical solutions to the one-dimensional stationary quantum drift–diffusion model

The existence of classical solutions to the one-dimensional stationary quantum drift–diffusion model for semiconductor devices is investigated. The proof is based on an exponential variable transformation and the Leray–Schauder fixed-point theorem. Furthermore, the uniqueness of solutions to the isothermal model is proved, if the current density and the Planck constant are sufficiently small.

On iterative schemes for a stationary problem to a quantum drift diffusion model

In this paper, we propose a new iterative scheme to obtain a stationary solution to a quantum drift diffusion (QDD) model for semiconductors in a multi-dimensional space. The model consists of two fourth order nonlinear elliptic equations for the carrier densities and the nonlinear Poisson equation for the electrostatic potential. The present scheme adopts the Gummel method to decouple the system. For the computational efficiency, this scheme also employs an exponential transformation of variables to preserve positivity of the carrier densities and an inner iteration to couple the quantum potentials with the electrostatic potential. The present scheme is constructed to improve the other schemes developed by Ancona, de Falco and Odanaka. We show the advantage of the present scheme on computational costs in comparison to the other schemes for simulations of current-voltage characteristics for a double-gate MOSFET.

Boundary layer analysis in the semiclassical limit of a quantum drift–diffusion model

We study a singularly perturbed elliptic second order system in one space variable as it appears in a stationary quantum drift–diffusion model of a semiconductor. We prove the existence of solutions and their uniqueness as minimizers of a certain functional and determine rigorously the principal part of an asymptotic expansion of a boundary layer of those solutions. We prove analytical estimates of the remainder terms of this asymptotic expansion, and confirm by means of numerical simulations that these remainder estimates are sharp.

New solutions for the quantum drift–diffusion model of semiconductors

In this paper new symmetry reductions and exact solutions are found for the one-dimensional quantum drift–diffusion model for semiconductors based on the Bohm potential. The symmetry reductions are derived by using the nonclassical method developed by Bluman and Cole. Further reductions are obtained by means of other types of symmetry reductions or by ansatz-based reductions. In particular, several types of exact solutions are derived: kinks, k-hump compactons and elliptic traveling waves. The new solutions can display several types of coherent structures.

A Family of Nonlinear Fourth Order Equations of Gradient Flow Type

Global existence and long-time behavior of solutions to a family of nonlinear fourth order evolution equations on R d are studied. These equations constitute gradient flows for the perturbed information functionals with respect to the L 2-Wasserstein metric. The value of α ranges from α = 1/2, corresponding to a simplified quantum drift diffusion model, to α = 1, corresponding to a thin film type equation.

Vortex structures in dense electron–positron–ion plasmas

A linear dispersion relation for electrostatic quantum drift and acoustic waves has been found for dense electron–positron–ion magnetoplasmas. Both the fermion and thermal temperature effects have been considered for electrons and positrons. In the nonlinear regime, a stationary solution in the form of dipolar vortices has been obtained. For illustration, the results were applied to the astrophysical plasma of the atmosphere of neutron stars/pulsars.

A GLOBALLY CONVERGENT GUMMEL MAP FOR OPTIMAL DOPANT PROFILING

We study a generalized Gummel iteration for the solution of an abstract optimal semiconductor design problem, which covers a wide range of semiconductor models. The algorithm is to exploit the special structure of the KKT system and it can be interpreted as a descent algorithm for an appropriately defined cost functional. This allows for a convergence proof which does not need the assumption of small biasing voltages. The algorithm is explicitly stated for the (quantum) drift diffusion model, the energy transport model and the microscopic Schrödinger–Poisson model.

Drift ion acoustic solitons in an inhomogeneous 2-D quantum magnetoplasma

Linear and nonlinear propagation characteristics of quantum drift ion acoustic waves are investigated in an inhomogeneous two-dimensional plasma employing the quantum hydrodynamic (QHD) model. In this regard, the dispersion relation of the drift ion acoustic waves is derived and limiting cases are discussed. In order to study the drift ion acoustic solitons, nonlinear quantum Kadomstev–Petviashvilli (KP) equation in an inhomogeneous quantum plasma is derived using the drift approximation. The solution of quantum KP equation using the tangent hyperbolic (tanh) method is also presented. The variation of the soliton with the quantum Bohm potential, the ratio of drift to soliton velocity in the co-moving frame, v * u , and the increasing magnetic field are also investigated. It is found that the increasing number density decreases the amplitude of the soliton. It is also shown that the fast drift soliton (i.e., v * > u ) decreases whereas the slow drift soliton (i.e., v * < u ) increases the amplitude of the soliton. Finally, it is shown that the increasing magnetic field increases the amplitude of the quantum drift ion acoustic soliton. The stability of the quantum KP equation is also investigated. The relevance of the present investigation in dense astrophysical environments is also pointed out.

Electrostatic drift modes in quantum pair plasmas

Electrostatic drift waves in a nonuniform quantum magnetized electron-positron (pair) plasma are investigated. An explicit and straightforward analytical expression of the fluctuation frequency is presented. The effects induced by quantum fluctuations, density gradients, and magnetic field inhomogeneity on the wave frequencies are discussed and a purely quantum drift mode appears. The present analytical investigations are relevant to dense astrophysical objects as well as laboratory ultracold plasmas.

Electrostatic drift vortices in quantum magnetoplasmas

A nonlinear equation similar to the Hasegawa–Mima equation for drift waves in dense quantum plasmas has been obtained. Therefore, electrostatic drift vortices can appear in nonuniform dense magnetoplasmas. These nonlinear structures can be of dipolar form as well as street-type vortices. It is noticed that these structures are formed on very short spatial scales. Further, it is discussed that quantum drift waves can couple with quantum ion acoustic waves, like classical plasmas, if the ion parallel dynamics is also considered.

Convergent finite element discretizations of the density gradient equation for quantum semiconductors

We study nonlinear finite element discretizations for the density gradient equation in the quantum drift diffusion model. In particular, we give a finite element description of the so-called nonlinear scheme introduced by Ancona. We prove the existence of discrete solutions and provide a consistency and convergence analysis, which yields the optimal order of convergence for both discretizations. The performance of both schemes is compared numerically, in particular, with respect to the influence of approximate vacuum boundary conditions.

A functional iteration for the quantum drift–diffusion model: existence analysis and numerical approximation

AbstractThis communication describes a functional iteration technique, denoted Generalized Gummel Map (GGM), for the solution of the Quantum Drift–Diffusion (QDD) model. The assumptions needed to establish existence of a fixed point for the GGM are described and their plausibility is assessed via physical arguments supported by one–dimensional numerical experiments. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Simulation of a resonant tunneling diode using an entropic quantum drift-diffusion model

We present an entropic Quantum Drift Diffusion model (eQDD). Some links between this model and other existing models are exhibited, especially with the Classical Drift-Diffusion (CDD) model, the Density Gradient (DG) model and the Schr¨odinger-Poisson model (SP). Numerical results show that the model captures important features concerning the modeling of a resonant tunneling diode. To finish, some comparisons between the models stated above are realized.

An algorithmic construction of entropies in higher-order nonlinear PDEs

A new approach to the construction of entropies and entropy productions for a large class of nonlinear evolutionary PDEs of even order in one space dimension is presented. The task of proving entropy dissipation is reformulated as a decision problem for polynomial systems. The method is successfully applied to the porous medium equation, the thin film equation and the quantum drift–diffusion model. In all cases, an infinite number of entropy functionals together with the associated entropy productions is derived. Our technique can be extended to higher-order entropies, containing derivatives of the solution, and to several space dimensions. Furthermore, logarithmic Sobolev inequalities can be obtained.

The relaxation-time limit in the quantum hydrodynamic equations for semiconductors

The relaxation-time limit from the quantum hydrodynamic model to the quantum drift–diffusion equations in R 3 is shown for solutions which are small perturbations of the steady state. The quantum hydrodynamic equations consist of the isentropic Euler equations for the particle density and current density including the quantum Bohm potential and a momentum relaxation term. The momentum equation is highly nonlinear and contains a dispersive term with third-order derivatives. The equations are self-consistently coupled to the Poisson equation for the electrostatic potential. The relaxation-time limit is performed both in the stationary and the transient model. The main assumptions are that the steady-state velocity is irrotational, that the variations of the doping profile and the velocity at infinity are sufficiently small and, in the transient case, that the initial data are sufficiently close to the steady state. As a by-product, the existence of global-in-time solutions to the quantum drift–diffusion model in R 3 close to the steady-state is obtained.

Uniform Convergence of an Exponentially Fitted Scheme for the Quantum Drift Diffusion Model

We analyze an exponentially fitted finite element scheme for the unipolar quantum drift diffusion model in one-dimensional space. The existence of discrete solutions is shown under very mild assumptions, and convergence of a subsequence is proved by compactness arguments. The scheme is constructed in such a way that it reduces in the semiclassical limit to the well-known Scharfetter--Gummel discretization for the classical drift diffusion model. We derive uniform error bounds which allow for the semiclassical limit on the discrete level. Numerical tests underlining the analytical results are presented.

A Scharfetter–Gummel Type Discretization of the Quantum Drift Diffusion Model

AbstractWe derive a generalized Scharfetter–Gummel discretization of the Quantum Drift Diffusion Model in one space dimension. The scheme relies on the introduction of a generalized potential which identifies the quantum drift term. Further, the new scheme is stable in the semiclassical limit recovering the SG scheme for the classical drift diffusion equations. Numerical results for a ballistic diode are presented.

Effect of fullerene on the photogeneration and transport of charge carriers in triphenylamine-containing polyimides

Based on studies of the quantum yield and drift mobility of charge carriers in thin films of fullerene-containing dye-sensitized polyimides performed in wide ranges of electric-field strengths ((1–10)×105 V cm−1) and temperatures (77–350 K), a mechanism of the interaction of fullerene molecules with aromatic fragments of polyimides and dye anions is suggested. It was found that fullerene, being an efficient sensitizer of the photoeffect in aromatic triphenylamine-containing polyimides, reduces by more than an order of magnitude the photosensitivity and mobility of charge carriers in dye-sensitized polyimides, restoring these parameters to values typical of unsensitized polyimides.