P. Degond, A. J/ingel, and P. Pietra. Numerical discretization of energy-transport model for semiconduc- tors with non-parabolic band structure. SIAM J. $ci. Comp., 22:986-1007, 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... in non Newtonian ltration or the water ow through porous media (see [16] and the references therein) In this context, often single equations with n = 1 are considered (see [19] Systems of equations with n 1 arise, for instance, in non equilibrium thermodynamics [8] semiconductor modeling [9, 13] and alloy solidi cation processes [12] The porous medium equation (m 1) or the fast di usion equation (0 m 1) t (u 1=m ) u = 0; u 0; are included in (1) Furthermore, the p Laplace equation t u div(jruj p 2 ru) 0 is also included. Notice that the corresponding functions ....

P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energy-transport model for semiconductors with non-parabolic band structure. Submitted for publication, 1999.

....the di usion coecients are di erent. One example is: L = L ij ) 0 n 1 3 2 kB T 3 2 kB T 15 4 (k B T ) 2 : 18) In order to get this di usion matrix, we assumed the parabolic band approximation, Boltzmann statistics and a special ansatz for the momentum relaxation time (see [57] 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 ....

.... matrix, we assumed the parabolic band approximation, Boltzmann statistics and a special ansatz for the momentum relaxation time (see [57] 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, ....

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P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energy-transport model for semiconductors with non-parabolic band structure. Submitted for publication, 1999.

.... filtration or the water flow through porous media (see [Kal87] and the references therein) In this context, often single equations with n = 1 are considered (see [KnOt96] Systems of equations with n 1 arise, for instance, in non equilibrium thermodynamics [DGJ97] semiconductor modeling [DJP99, Jun96] and alloy solidification processes [HLR83] The porous medium equation (m 1) or the fast diffusion equation (0 m 1) t (u 1=m ) Gamma Deltau = 0; u 0; are included in (10) Furthermore, the p Laplace equation t u Gamma div(jruj p Gamma2 ru) 0 is also included. Notice that ....

....which form a system of strongly coupled, quasilinear parabolic equations for a charged fluid or gas, exposed to an electric field. They arise originally from non equilibrium thermodynamics and are used in many applications of charged particle transport, for instance in semiconductor theory [DJP99], in electro chemistry [Dey79] and alloy solidification processes [HLR83] In order to simplify the presentation, we consider a gas consisting only of negatively charged particles with particle density ae, internal energy density j, chemical potential , and temperature T . We assume that the ....

P. Degond, A. Jungel, P. Pietra. Numerical discretization of energytransport models for semiconductors with non-parabolic band structure. preprint, TU Berlin, Germany, (1999).

....applied potential. An important observation is that the current densities can be written in the drift diffusion form J i = g i (n; T ) x Gamma g i (n; T ) V x T ; i = 1; 2; 2.6) with g 1 = L 11 and g 2 = L 21 . This formulation is valid for any current densities coming from a SHE model (see [3]) In order to give analytical expressions for the diffusion coefficients and the energy relaxation term, in the variables n and T , we have to impose some physical assumptions: i) The energy band of the semiconductor crystal is spherical symmetric and a monotone function of the modulus k = j ....

....devices with doping concentrations below 10 19 cm Gamma3 . The fifth hypothesis is valid for silicon devices, in which ff = 0:5 (eV) Gamma1 and T is of the order 10 3 K. Under the assumptions (i) v) and up to second order terms in ffk B T , the functions g 1 and g 2 are given by [3] g i (n; T ) qn ff;fi i (k B T ) i ; i = 1; 2; 2.7) where the mobility ff;fi i is defined by ff;fi 1 = 0 1 Gamma 3(2 Gamma fi)ffk B T 1 15ffk B T=4 i T T 0 j Gamma1=2 Gammafi ; ff;fi 2 = 2 Gamma fi) 0 1 Gamma 3(3 Gamma fi)ffk B T 1 15ffk B T=4 i T T 0 ....

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P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energy-transport model for semiconductors with non-parabolic band structure. Submitted for publication,

....been used by Ringhofer [32] based on an entropy decaying property, and by Fourni6 [16] with compact schemes of fourth order. Bosisio et al. used mixed finite volume techniques [3] Mixed finite element methods applied to different formulations of the energy transport models have been employed in [13, 18, 19, 25, 29, 30] (see [5] for an overview) In the first part of the paper we make precise the definitions of Lij, n and W as functions of z and T for general and parabolic band structures (section 2.1) and give three different formulations of the energy transport model (i) in the primal entropy variables (see ....

.... Lij(l, T) n(l, T) and W(I, T) In [2] the following expressions have been derived: Lij( T) e T d(e)ei j 2e e Td, i,j = 1,2, n( T) e T e TN(e)de, d( l (k)lk 2 and = k) The energy relaxation term in the Fokker Planck approximation of the phonon collision operator is given by [13] W( T) e T oeN(e)2e T de (1 ) where To ) 0 is the (constant) lattice temperature d ) 0 is some constant. The advantages of the model e that the assumptions on the band diagr are rather general (the above hypotheses c be even weakened, see [2] d that analytic expressions c be ....

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P. Degond, A. J/ingel, and P. Pietra. Numerical discretization of energy-transport model for semiconduc- tors with non-parabolic band structure. SIAM J. $ci. Comp., 22:986-1007, 2000.

....P. R. China; and Fachbereich Mathematik und Statistik, Universit at Konstanz, 78457 Konstanz, Germany. E mail: tangs fmi.uni konstanz.de. 1 with a typical size of a few microns and moderately applied voltage [33] whereas energy transport models can also be used for certain submicron devices [14]. The hydrodynamic equations have been introduced by Bl tekj r [9] and subsequently thoroughly investigated by Baccarani and Wordeman [6] They can be derived from the Boltzmann equation by using a moment method. This yields usually a set of equations for the carrier density, momentum and energy ....

....velocity. We obtain the equations: n t divJ = 0; 14) J = T r (r(nT ) nrV ) 15) 3 2 nT t div 5 2 TJ rT = J rV 3 2 n(T 1) T ) 16) 2 V = N C; 17) where (T ) 0 T= T 1) Notice that this energy transport model is not of the general form derived in [14] expect for c = r = 0: For the derivation of the drift di usion model, we x the parameter 0 and let formally 0 in Eqs. 11) 13) to obtain T = 1 and n t divJ = 0; 18) J = rn nrV; 19) 2 V = n C: 20) 3 Numerical scheme In this section, we shall put the model into a more concise ....

P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energytransport model for semiconductors with non-parabolic band structure. SIAM J. Sci. Comp., 22:986-1007, 2000.

.... equation assuming dominating inelastic scattering mechanisms [2] or from the SHE model assuming dominant electron electron collisions [1] arrow 4) The advantage of the energy transport model is that explicit expressions for the di usion coecients can be given even for non parabolic band diagrams [7]. In Section 2 the energy transport model is presented, the physical assumptions on the band structure and the relaxation mechanism are given, and a high eld scaling is introduced. The limiting equation is a ( rst order) convection equation for the macroscopic electron density with eld dependent ....

....to d 0 =2 which is half of the low eld di usivity d 0 . We stress once more the fact that compared to the paper [3] we derive explicit expressions for the coecients in the drift di usion equation. Therefore, the proposed models can be solved numerically (using, for instance, the methods in [4, 7]) and the results can be compared to Monte Carlo simulations of the Boltzmann equation. The numerical simulation will be performed in a forthcoming publication. 3 2 Assumptions scaling Consider the dimensionless energy transport equations in the entropy variables =T and 1=T [1] t n div ....

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P. Degond, A. Jungel, and P. Pietra. Numerical discretization of energy-transport model for semiconductors with non-parabolic band structure. To appear in SIAM J. Sci. Comp., 2000.

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