Quantum Drift-diffusion Model Research Articles

Conservation laws and solutions of a quantum drift-diffusion model for semiconductors

A non-linear system of partial differential equations describing a quantum drift-diffusion model for semiconductor devices is investigated by methods of group analysis. An infinite number of conservation laws associated with symmetries of the model are found. These conservation laws are used for representing the system of equations under consideration in the conservation form. Exact solutions provided by the method of conservation laws are discussed. These solutions are different from invariant solutions.

Quantum drift-diffusion model for IMPATT devices

Quantum correction is necessary on the classical drift-diffusion (CLDD) model to predict the accurate behavior of high frequency performance of ATT devices at frequencies greater than 200 GHz when the active layer of the device shrinks in the range of 150---350 nm. In the present work, a quantum drift-diffusion model for impact avalanche transit time (IMPATT) devices has been developed by incorporating appropriate quantum mechanical corrections based on density-gradient theory which macroscopically takes into account important quantum mechanical effects such as quantum confinement, quantum tunneling, etc. into the CLDD model. Quantum potentials (synonymous as Bohm potentials) have been incorporated in the current density equations as necessary quantum mechanical corrections for the analysis of millimeter-wave (mm-wave) and Terahertz (THz) IMPATT devices. It is observed that the large-signal (L-S) performance of the device is degraded due to the incorporation of quantum corrections into the model when the frequency of operation increases above 200 GHz; while the effect of quantum corrections are negligible for the devices operating at lower mm-wave frequencies.

Group analysis of the drift–diffusion model for quantum semiconductors

In the present paper a quantum drift–diffusion model describing semi-conductor devices is considered. New conservation laws for the model are computed and used to construct exact solutions.

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.

Existence and semiclassical limit for the quantum drift-diffusion model

In this paper, we construct a weak solution to the unipolar quantum drift-diffusion model in a bounded domain with Neumann boundary conditions, thereby extending an existence result in Gianazza et al. (Arch Rational Mech Anal 194:133–220, 2009) to the case where the boundary of the domain is only Lipschitz. We also obtain the limit of the solution as the scaled Planck constant $$\varepsilon $$ in the problem goes to $$0$$ .

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.

On the Multidimensional Bipolar Isothermal Quantum Drift-Diffusion Model

The bipolar isothermal quantum drift-diffusion model in two or three space dimensions with initial value and periodic boundary conditions is investigated. The global existence of weak solution to the problem is obtained by using semi-discretizing in time and entropy estimate. Furthermore, it is shown that the solution to the problem exponentially approaches its mean value as time increases to infinity by using a series of inequality technique.

An Inverse Problem in Quantum Statistical Physics

We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential.

ASYMPTOTIC BEHAVIORS AND CLASSICAL LIMITS OF SOLUTIONS TO A QUANTUM DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS

This paper discusses a time global existence, asymptotic behavior and a singular limit of a solution to the initial boundary value problem for a quantum drift-diffusion model of semiconductors over a one-dimensional bounded domain. Firstly, we show a unique existence and an asymptotic stability of a stationary solution for the model. Secondly, it is shown that the time global solution for the quantum drift-diffusion model converges to that for a drift-diffusion model as the scaled Planck constant tends to zero. This singular limit is called a classical limit. Here these theorems allow the initial data to be arbitrarily large in the suitable Sobolev space. We prove them by applying an energy method.

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.

Weak solutions to the stationary quantum drift-diffusion model

The existence of weak solutions to the stationary quantum drift-diffusion equations for semiconductor devices is investigated. The proof is based on minimization procedure of non-linear functional and Schauder fixed-point theorem. Furthermore, the semiclassical limit ε → 0 from the quantum drift-diffusion model to the classical drift-diffusion model is discussed.

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.

The bipolar quantum drift-diffusion model

A fourth order parabolic system, the bipolar quantum drift-diffusion model in semiconductor simulation, with physically motivated Dirichlet-Neumann boundary condition is studied in this paper. By semidiscretization in time and compactness argument, the global existence and semiclassical limit are obtained, in which semiclassical limit describes the relation between quantum and classical drift-diffusion models. Furthermore, in the case of constant doping, we prove the weak solution exponentially approaches its constant steady state as time increases to infinity.

Semiclassical limit for bipolar quantum drift-diffusion model

Semiclassical limit to the solution of transient bipolar quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical bipolar drift-diffusion model. In addition, the authors also prove the existence of weak solution.

The semiclassical limit in the quantum drift-diffusion model

Semiclassical limit to the solution of isentropic quantum drift-diffusion model in semiconductor simulation is discussed. It is proved that the semiclassical limit of this solution satisfies the classical drift-diffusion model. In addition, we also proved the global existence of weak solutions.

A Sixth-Order Nonlinear Parabolic Equation for Quantum Systems

The global-in-time existence of weak nonnegative solutions to a sixth-order nonlinear parabolic equation in one space dimension with periodic boundary conditions is proved. The equation arises from an approximation of the quantum drift-diffusion model for semiconductors and describes the evolution of the electron density in the semiconductor crystal. The existence result is based on two techniques. First, the equation is reformulated in terms of exponential and power variables, which allows for the proof of nonnegativity of solutions. The existence of solutions to an approximate equation is shown by fixed point arguments. Second, a priori bounds uniformly in the approximation parameters are derived from the algorithmic entropy construction method which translates systematic integration by parts into polynomial decision problems. The a priori estimates are employed to show the exponential time decay of the solution to the constant steady state in the $L^1$ norm with an explicit decay rate. Furthermore, som...

Quantum-corrected drift-diffusion models: Solution fixed point map and finite element approximation

This article deals with the analysis of the functional iteration, denoted Generalized Gummel Map (GGM), proposed in [C. de Falco, A.L. Lacaita, E. Gatti, R. Sacco, Quantum-Corrected Drift-Diffusion Models for Transport in Semiconductor Devices, J. Comp. Phys. 204 (2) (2005) 533–561] for the decoupled solution of the Quantum Drift-Diffusion (QDD) model. The solution of the problem is characterized as being a fixed point of the GGM, which permits the establishment of a close link between the theoretical existence analysis and the implementation of a numerical tool, which was lacking in previous non-constructive proofs [N.B. Abdallah, A. Unterreiter, On the stationary quantum drift-diffusion model, Z. Angew. Math. Phys. 49 (1998) 251–275, R. Pinnau, A. Unterreiter, The stationary current–voltage characteristics of the quantum drift-diffusion model, SIAM J. Numer. Anal. 37 (1) (1999) 211–245]. The finite element approximation of the GGM is illustrated, and the main properties of the numerical fixed point map (discrete maximum principle and order of convergence) are discussed. Numerical results on realistic nanoscale devices are included to support the theoretical conclusions.