Quantum Semiconductor Models

Chen, Li; Dreher, Michael

doi:10.1007/978-3-0348-0024-2_1

Cited by 8 publications

(12 citation statements)

References 108 publications

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“…However, the situation is less clear concerning the choice of boundary conditions for the particle density in quantum fluid models. In fact, many authors impose periodic boundary conditions [6,9,11,19,26,38], insulating boundary conditions [13], or they consider the whole-space problem [39]. Boundary conditions satisfying the Shapiro-Lopatinskii criterion have been examined in [12].…”

Section: Proof First We Multiply (32) By Hmentioning

confidence: 99%

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Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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Abstract. The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear thirdorder differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [10] from a Wigner equation using a moment method and a Chapman-Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier-Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo-Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier-Stokes system if the viscosity constant is smaller than the scaled Planck constant.Key words. Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent viscosity, global existence of solutions, viscous quantum hydrodynamic equations, third-order derivative, energy estimates. Gardner [20] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [21], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [10]. In this paper, we will reveal a connection between these two models by introducing an effective velocity variable, first used in capillary Korteweg-type models [5], and we will prove the global existence of weak solutions to the multidimensional initial-value problems for any finite-energy initial data. AMSIn the following, we describe the two dissipative quantum systems studied in this paper. The barotropic quantum Navier-Stokes equations for the particle density n and the particle velocity u read as

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Section: Proof First We Multiply (32) By Hmentioning

confidence: 99%

“…Concerning the multidimensional transient system, there exist only localin-time existence theorems [11,15]. We refer to the review [13] for more details. Up to now, there exist no global existence results for the multidimensional equations.…”

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confidence: 99%

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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“…The quantum mechanical effects enter the system mainly via the Bohm term B(n), which introduces third order spatial derivatives into the momentum balance equation, which complicates analytical studies of (1) considerably, compare [4][5][6]10] for results on the transient problem without barrier potential. Further analytical difficulties arise from the barrier potential V B having jumps, and in that situation the second equation of (1) must be understood in the distributional sense.…”

Section: Introductionmentioning

confidence: 99%

The combined viscous semi-classical limit for a quantum hydrodynamic system with barrier potential

Dreher

Schnur

2015

Journal of Mathematical Analysis and Applications

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We investigate the viscous model of quantum hydrodynamics, which describes the charge transport in a certain semiconductor. Quantum mechanical effects lead to third order derivatives, turning the stationary system into an elliptic system of mixed order in the sense of Douglis-Nirenberg. In the case most relevant to applications, the semiconductor device features a piecewise constant barrier potential. In the case of thermodynamic equilibrium, we obtain asymptotic expansions of interfacial layers of the particle density in neighbourhoods of the jump points of this barrier potential, and we present rigorous proofs of uniform estimates of the remainder terms in these asymptotic expansions.

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“…The first member of this family is the classical heat equation ∂ t n = n. The second one is the fourth-order Derrida-Lebowitz-Speer-Spohn (DLSS) equation, see (3) below, which is analyzed in [9,13]. This paper is concerned with the third family member, obtained from an expansion to orderh 4 (see [3,Appendix]), which reads as ∂ t n = div n∇ 1 2 ∂ 2 ij log n 2 + 1 n ∂ 2 ij n∂ 2 ij log n .…”

Section: Introductionmentioning

confidence: 99%

A multidimensional nonlinear sixth-order quantum diffusion equation

Bukal¹,

Jüngel²,

Matthes³

2013

Ann. Inst. H. Poincaré Anal. Non Linéaire

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This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.

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Quantum Semiconductor Models

Cited by 8 publications

References 108 publications

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Diffusive semiconductor moment equations using Fermi–Dirac statistics

The combined viscous semi-classical limit for a quantum hydrodynamic system with barrier potential

A multidimensional nonlinear sixth-order quantum diffusion equation

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