P. Degond and C. Schmeiser. Macroscopic models for semiconductor heterostructures. Article in preparation, 1997. 24  Home/Search   Document Details and Download   Summary   Related Articles   Check

....models which take into account quantum e ects are, for instance, the quantum 1 2 hydrodynamic, the quantum energy transport and the quantum drift di usion equations. Other semiconductor models, for instance kinetic models, so called SHE models or high eld models, can be found in the literature [34, 61, 163, 185, 186], but in this paper we only discuss the above mentioned models. 11 transport model Quantum energy12 DYNAMIC Energy transport Quantum Quantum hydroCLASSICAL MODELS KINETIC MODELS Hydrodynamic model model Standard driftdiffusion model Quantum driftdiffusion model dynamic model ....

....for details) The corresponding model has been considered by Chen et al. 40] Di erent momentum relaxation time approximations give di erent di usion matrices [57] Note that both di usion matrices (17) and (18) are symmetric and positive de nite. Related energy transport models are derived in [61]. The derivation of the energy transport model from the Boltzmann equation is explained in more detail in [19, 136] In the physical and mathematical literature, the energy transport equations are investigated numerically since several years. They are discretized by using extensions of the ....

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P. Degond and C. Schmeiser. Macroscopic models for semiconductor heterostructures. J. Math. Phys., 39:1-30, 1998.

....is supplemented with the initial condition 8x 2 Omega ; k 2 B; f ff (0; x; k) f ff in (x; k) 2.5) where the rescaled mean free path ff belongs to ]0; 1] and tends to 0. 3 The boundary conditions are described by a scattering operator relating the incoming and outgoing part of f , as in [16]: 8t 2 IR ; x 2 Omega ; k 2 B Gamma (x) f ff (t; x; k) Z B (x) R(k 0 k) ffi( k) Gamma (k 0 ) f ff (t; x; k 0 )dk 0 ; 2.6) where B Sigma (x) ae k 2 B; Sigmar k (k) Delta (x) 0 oe , x) is the outward unit normal at x 2 Omega Gamma and R(k 0 k) ....

....leads to Z B Gamma (x) G( k) jr k (k) Delta (x)j f(t; x; k) dk = Z B (x) G( k) jr k (k) Delta (x)j f(t; x; k) dk (2.24) for all f satisfying (2.6) and all functions G. Equation (2. 23) is a reciprocity relation resulting from the time reversibility of the microscopic dynamics, see [16] or [9] and references therein. We refer to [6] for a detailed physical interpretation of this framework, as well as for a discussion of the relevant bibliography. 6 2.2 The result Let us first introduce the following definition. Definition 1: We say that f ff is a weak solution of (2.2) ....

P. Degond, C. Schmeiser, Macroscopic models for semiconductor heterostructures, article in preparation, (1997). 26

....in nonequilibrium thermodynamics [18, x53] and is also used in the drift di usion model for semiconductors (i.e. T = const. where w i is called quasi Fermi potential. Recently, this transformation appears naturally in the derivation of an energy transport model for semiconductor heterostructures [12]. The symmetrization property of the transformation (1.14) is also observed by Albinus [4] in the case of the energy transport model. The system (1.7) 1.13) is mathematically investigated here for the rst time. ii) Entropy function. To derive a priori estimates we use the entropy function ....

P. Degond and C. Schmeiser. Macroscopic models for semiconductor heterostructures. Article in preparation, 1997. 24

....of the extrapolation length [5] from the solution of a half space problem for a simplified, stationary kinetic equation [12] In this work, we extend the results of [17] to other macroscopic models. The derivation of higher order transmission conditions for heterojunctions is the subject of [6]. For situations where kinetic effects are important in parts of the bulk material, domain decomposition strategies are natural, where the Boltzmann equation is solved in regions with strong kinetic effects and a macroscopic model is used whereever it is accurate enough. Then the two models need ....

P. Degond, C. Schmeiser, Macroscopic models for semiconductor heterostructures, preprint, Univ. Paul Sabatier, Toulouse, 1997.

....[15, x53] and is also used in the drift diffusion model for semiconductors (i.e. T = const. where w i is called quasi Fermi potential (see examples 1 and 2) Recently, this transformation appears naturally in the derivation of an energy transport model for semiconductor heterostructures [14]. The symmetrization property of the transformation (1.12) is also observed by Albinus [2] in the case of the energy transport model (example 1) The system (1.6) 1.9) is mathematically investigated here for the first time. The plan of the paper is as follows. In Section 2, we define the notion ....

P. Degond, C. Schmeiser, Macroscopic models for semiconductor heterostructures, article in preparation (1997).

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