Semiconductor Equations

Markowich, Peter A.; Ringhofer, Christian; Schmeiser, Christian

doi:10.1007/978-3-7091-6961-2

Cited by 1,484 publications

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“…For fixed values of the scaled wave-number Ã, scaled Planck's constant À and parameters « and ¬, the equations (8)(9) are the parametric equations for the corresponding Nyquist diagram. This curve is obtained by varying ª ¾´ ½ ½µ.…”

Section: Three-stream Equilibriummentioning

confidence: 99%

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Stability analysis of a three-stream quantum-plasma equilibrium

Haas¹,

Manfredi²,

Goedert³

2003

Braz. J. Phys.

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We apply the Nyquist method to analyze the stability of small amplitude electrostatic waves in quantum plasmas described by the Wigner-Poisson system. The technique is presented in detail to assess the stability of a threestream equilibrium distribution function. The topology of the Nyquist diagrams is substantially changed with increasing quantum diffraction effects. I IntroductionThe topic of quantum plasmas has received considerable attention in the recent literature [1]- [8]. The main reason for this is the increasing importance of quantum effects in the manufacturing process and performance of today's micro-electronic devices. For instance, no classical transport model can adequately incorporate quantum effects such as resonant tunnelling, which is the basis for the operation of devices like the resonant tunnelling diode. Accordingly, there has been much work dedicated to the development of quantum transport models. One of such models is provided by the Wigner-Poisson system [9,10], which is the main concern of the present paper. We address the question of the stability of small amplitude electrostatic waves described by the Wigner-Poisson system. In the formal classical limit, the Wigner-Poisson system goes into the Vlasov-Poisson system, with linear stability properties much better understood than those of the corresponding quantum model. One natural question in this regard is: how do quantum effects modify the linear stability properties of typical classical plasma equilibria? Here we address this question in detail for the case of a three-stream plasma equilibrium.The subject of quantum plasma stability is a subtle one. In [11] it was shown that two counter-streaming beams generate a linearly unstable mode that is not present in the corresponding classical situation. In the opposite direction, for large wave-numbers quantum effects are responsible for the suppression of all unstable modes. The extra details appearing in the quantum context can be traced back to the subtle nature of quantum wave-particle interactions in a plasma. In [12], the quantum plasma instability question is addressed by means of the Nyquist method. Let ´ µ Ö´ µ · ´ µ ¼ be the dispersion relation for the quantum plasma, where and are the frequency and wave-number of linear waves, and where the dispersion function is decomposed into its real and imaginary parts. In the Nyquist method, the stability of a mode of fixed wave-number is checked by varying (taken as a real variable) from minus to plus infinity. Drawing the corresponding curve in the Ö ¢ plane, we obtain the socalled Nyquist diagrams. The number of unstable modes is equal to the number of times the Nyquist diagram encircles the origin. For Vlasov-Poisson plasmas, the Nyquist method shows that monotonically decreasing equilibrium functions of the energy are stable against small amplitude perturbations. Also, in this classical case the sign of the associated Penrose functional [13,14] determines the linear stability of symmetric equilibria with at most two maxima. For Wigner...

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Section: Three-stream Equilibriummentioning

confidence: 99%

“…The topic of quantum plasmas has received considerable attention in the recent literature [1]- [8]. The main reason for this is the increasing importance of quantum effects in the manufacturing process and performance of today's micro-electronic devices.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a three-stream quantum-plasma equilibrium

Haas¹,

Manfredi²,

Goedert³

2003

Braz. J. Phys.

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“…The overall problem is then solved by a fixed-point procedure for the unknown N s , as for the standard drift-diffusion-Poisson problem [18,25,26]. The global-in-time existence relies heavily on an entropy estimate.…”

Section: Notations and Strategy Of The Proofmentioning

confidence: 99%

“…Step 2. The surface potential V s being known, solve the following parabolic equation for the unknownN s : 25) with the boundary conditionN 26) and the initial valueN…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Diffusive Transport of Partially Quantized Particles: Existence, Uniqueness and Long-Time Behaviour

Abdallah¹,

Méhats²,

Vauchelet³

2006

Proceedings of the Edinburgh Mathematical Society

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A self-consistent model for charged particles, accounting for quantum confinement, diffusive transport and electrostatic interaction is considered. The electrostatic potential is a solution of a threedimensional Poisson equation with the particle density as the source term. This density is the product of a two-dimensional surface density and that of a one-dimensional mixed quantum state. The surface density is the solution of a drift-diffusion equation with an effective surface potential deduced from the fully three-dimensional one and which involves the diagonalization of a one-dimensional Schrödinger operator. The overall problem is viewed as a two-dimensional drift-diffusion equation coupled to a Schrödinger-Poisson system. The latter is proven to be well posed by a convex minimization technique. A relative entropy and an a priori L 2 estimate provide sufficient bounds to prove existence and uniqueness of a global-in-time solution. In the case of thermodynamic equilibrium boundary data, a unique stationary solution is proven to exist. The relative entropy allows us to prove the convergence of the transient solution towards it as time grows to infinity. Finally, the low-order approximation of the relative entropy is used to prove that this convergence is exponential in time.

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“…The standard or low injection model has been mathematically and numerically investigated in many papers, see [33], [34] and references therein. Existence and uniqueness of weak solutions was shown.…”

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confidence: 99%

On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

Díaz

Galiano

Jüngel

2001

Nonlinear Analysis: Real World Applications

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Abstract. A drift-diffusion model for semiconductors with nonlinear diffusion is considered. The model consists of two quasilinear degenerate parabolic equations for carrier densities and the Poisson equation for electric potential. We assume Lipschitz continuous nonlinearities in the drift and generation-recombination terms.Existence of weak solutions is proven by using a regularization technique. Uniqueness of solutions is proven when either the diffusion term ϕ is strictly increasing and solutions have spatial derivatives in L 1 (Q T ) or when ϕ is non-decreasing and a suitable entropy condition is fullfilled by the electric potential.

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Semiconductor Equations

Abstract: This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustration, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.

Cited by 1,484 publications

References 0 publications

Stability analysis of a three-stream quantum-plasma equilibrium

Stability analysis of a three-stream quantum-plasma equilibrium

Diffusive Transport of Partially Quantized Particles: Existence, Uniqueness and Long-Time Behaviour

On a quasilinear degenerate system arising in semiconductors theory. Part I: Existence and uniqueness of solutions

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