Quantum Drift-diffusion Model Research Articles

Quantum hydrodynamic models for the two-band Kane system

— We consider Kane’s quantum model for two-band charged particles, composed of two Schrodinger-like equations coupled by a k · p term which describes interband tunneling. Using the WKB method we derive the hydrodynamic equations both for a zero-temperature and for a nonzero-temperature two-band quantum fluid. In the latter case, we introduce a family of closure relations and write in full details the simplest two-band, isentropic, fluid-dynamical model. Finally, introducing appropriate relaxation terms and performing drift-diffusive limits, we obtain the corresponding quantum drift-diffusion models for zero and nonzero-temperature quantum fluids.

Numerical simulation of tunneling effects in nanoscale semiconductor devices using quantum corrected drift-diffusion models

In this article, we deal with the numerical approximation of a quantum drift-diffusion model capable of describing tunneling effects through the thin oxide barrier in nanoscale semiconductor devices. We propose a novel reformulation of the mathematical model that allows a natural generalization of the Gummel decoupled algorithm, widely adopted in the case of the drift-diffusion system. Then, we address the finite element discretization of the linearized problems obtained after decoupling, and we prove well-posedness and a discrete maximum principle for the solution of the continuity equations. Finally, we validate the physical accuracy and the numerical stability of the proposed algorithms on the simulation of a real-life nanoscale device.

Quantum Energy-Transport and Drift-Diffusion Models

We show that Quantum Energy-Transport and Quantum Drift-Diffusion models can be derived through diffusion limits of a collisional Wigner equation. The collision operator relaxes to an equilibrium defined through the entropy minimization principle. Both models are shown to be entropic and exhibit fluxes which are related with the state variables through spatially non-local relations. Thanks to an h expansion of these models, h2 perturbations of the Classical Energy-Transport and Drift-Diffusion models are found. In the Drift-Diffusion case, the quantum correction is the Bohm potential and the model is still entropic. In the Energy-Transport case however, the quantum correction is a rather complex expression and the model cannot be proven entropic.

Entropic Discretization of a Quantum Drift-Diffusion Model

This paper is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. In a first step, we introduce an implicit semi-discretization in time which possesses some interesting properties: this system is well-posed, it preserves the positivity of the density, the total charge is conserved, and it is entropic (a free energy is dissipated). Then, after a discretization of the space variable, we define a numerical scheme which has the same properties and is equivalent to a convex minimization problem. These results are illustrated by some numerical simulations.

Numerical approximation of a quantum drift-diffusion model

This Note is devoted to the discretization and numerical simulation of a new quantum drift-diffusion model that was recently derived. We define an implicit numerical scheme which is equivalent to a convex minimization problem and which preserves the physical properties of the continuous model: charge conservation, positivity of the density and dissipation of an entropy. We illustrate these results by some numerical simulations. To cite this article: S. Gallego, F. Méhats, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Multidimensional Discretization of the Stationary Quantum Drift-Diffusion Model for Ultrasmall MOSFET Structures

This paper describes a new approach to construct a multidimensional discretization scheme of quantum drift-diffusion (QDD) model (or density gradient model) arising in MOSFET structures. The discretization is performed for the stationary QDD equations replaced by an equivalent form, employing an exponential transformation of variables. A multidimensional discretization scheme is constructed by making use of an exponential-fitting method in a class of conservative difference schemes, applying the finite-volume method, which leads to a consistent generalization of the Scharfetter-Gummel expression to the nonlinear Sturm-Liouville type equation. The discretization method is evaluated in a variety of MOSFET structures, including a double-gate MOSFET with thin body layer. The discretization method provides numerical stability and accuracy for carrier transport simulations with quantum confinement effects in ultrasmall MOSFET structures.

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.

On the Minimal Energy State of a Mixture of Charged Classical and Quantum Fluids

The paper concerns the equilibrium state of ultra small semiconductor devices. Due to the quantum drift diffusion model, electrons and holes behave as a mixture of charged quantum fluids. Typically the involved scaled Planck's constants of holes, ξ, is significantly smaller than the scaled Planck's constant of electrons. By setting formally ξ = 0 a well-posed differential-algebraic system arises. Existence and uniqueness of an equilibrium solution is proved. A rigorous asymptotic analysis shows that this equilibrium solution is the limit (in a rather strong sense) of quantum systems as ξ → 0. In particular the ground state energies of the quantum systems converge to the ground state energy of the differential-algebraic system as ξ → 0.

A REVIEW ON THE QUANTUM DRIFT DIFFUSION MODEL

ABSTRACT We consider the quantum drift diffusion model for semiconductor devices and collect recent results on the stationary and transient equations. The stationary model including generation–recombination terms is studied for bipolar devices and the transient equations are considered in the unipolar case. We cover several topics, such as existence and uniqueness of solutions, asymptotic limits and convergence of a nonlinear iteration scheme in the stationary case as well convergence of a positivity preserving semidiscretization of the transient equations and the linear stability of stationary states.

The Linearized Transient Quantum Drift Diffusion Model — Stability of Stationary States

The transient quantum drift diffusion model is derived from the isothermal quantum hydrodynamic model via a zero relaxation time limit. This yields a fourth-order nonlinear parabolic equation for the electron density, which is self-consistently coupled to the Poisson equation for the electrostatic potential. A stability analysis of the linearized system is performed by means of Hilbert space methods, which rely on a generalized Poincaré-type inequality. In the quantum case more states are linearly stable than in the classical one. The linear stability of an implicit Euler discretization for the nonlinear equations is proven and numerical results for a diode are presented, indicating that no maximum principle holds for the linearized system.

The Stationary Current-Voltage Characteristics of the Quantum Drift-Diffusion Model

This paper is concerned with numerical algorithms for the bipolar quantum drift-diffusion model. For the thermal equilibrium case, a quasi-gradient method minimizing the energy functional is introduced and strong convergence is proved. The computation of current-voltage characteristics is performed by means of an extended Gummel-iteration. It is shown that the involved fixed point mapping is a contraction for small applied voltages. In this case the model equations are uniquely solvable and convergence of the proposed iteration scheme follows. Numerical simulations of a one-dimensional resonant tunneling diode are presented. The computed current-voltage characteristics are in good qualitative agreement with experimental measurements. The appearance of negative differential resistances is verified for the first time in a quantum drift-diffusion model.

On the stationary quantum drift-diffusion model

A bipolar Quantum Drift Diffusion Model including generation-recombination terms is considered. Existence of solutions is proven for a general setting including the case of vanishing particle densities at some parts of the boundary. The proof is based on a Schauder fixed point iteration combined with a minimization procedure. It is proven that, contrary to the classical drift-diffusion model, vacuum can only appear at the boundary. In the case of nonvanishing boundary data, the semiclassical limit is carried out rigorously. The variational structure of the model allows to prove strong \(H^1\) convergence of particle densities, Fermi levels and electrostatic potential.

Optimal Control of the Stationary Quantum Drift-Diffusion Model

In this work an optimal control problem for a stationary quantum drift diffusion (QDD) model is analyzed. This QDD model contains four space-dependent observables: The non-negative particle density of electrons, the electrostatic potential, the quantum quasi-Fermi potential and the current density. The goal is to optimize the shape of quantum barriers in a quantum diode. Existence of optimal solutions is proved. Moreover, first-order necessary optimality conditions are derived.

A class of functional inequalities and their applications to fourth-order nonlinear parabolic equations

We study a class of fourth order nonlinear parabolic equations which include the thin-film equation and the quantum drift-diffusion model as special cases. We investigate these equations by first developing functional inequalities of the type $ \int_\Omega u^{2\gamma-\alpha-\beta}\Delta u^\alpha\Delta u^\beta dx \geq c\int_\Omega|\Delta u^\gamma |^2dx $, which seem to be of interest on their own right.