The Child-Langmuir limit for semiconductors : a numerical validation
Cáceres, María-José ; Carrillo, José-Antonio ; Degond, Pierre1
1 MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France.
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 6, pp. 1161-1176.

The Boltzmann-Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child-Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child-Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child-Langmuir regime by performing detailed numerical comparisons between the simulation of the Boltzmann-Poisson system and the Child-Langmuir equations in test problems.

DOI : 10.1051/m2an:2003011
Classification : 35L65, 65M99, 82D37
Mots-clés : Boltzmann-Poisson system, Child-Langmuir limit, WENO schemes, semiconductor devices
Cáceres, María-José  ; Carrillo, José-Antonio  ; Degond, Pierre 1

1 MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France. 
@article{M2AN_2002__36_6_1161_0,
     author = {C\'aceres, Mar{\'\i}a-Jos\'e and Carrillo, Jos\'e-Antonio and Degond, Pierre},
     title = {The {Child-Langmuir} limit for semiconductors : a numerical validation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1161--1176},
     publisher = {EDP-Sciences},
     volume = {36},
     number = {6},
     year = {2002},
     doi = {10.1051/m2an:2003011},
     zbl = {1028.35102},
     mrnumber = {1958663},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2003011/}
}
TY  - JOUR
AU  - Cáceres, María-José
AU  - Carrillo, José-Antonio
AU  - Degond, Pierre
TI  - The Child-Langmuir limit for semiconductors : a numerical validation
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2002
SP  - 1161
EP  - 1176
VL  - 36
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2003011/
DO  - 10.1051/m2an:2003011
LA  - en
ID  - M2AN_2002__36_6_1161_0
ER  - 
%0 Journal Article
%A Cáceres, María-José
%A Carrillo, José-Antonio
%A Degond, Pierre
%T The Child-Langmuir limit for semiconductors : a numerical validation
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2002
%P 1161-1176
%V 36
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2003011/
%R 10.1051/m2an:2003011
%G en
%F M2AN_2002__36_6_1161_0
Cáceres, María-José; Carrillo, José-Antonio; Degond, Pierre. The Child-Langmuir limit for semiconductors : a numerical validation. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 6, pp. 1161-1176. doi : 10.1051/m2an:2003011. http://archive.numdam.org/articles/10.1051/m2an:2003011/

[1] F. Alabau, K. Hamdache and Y.J. Peng, Asymptotic analysis of the transient Vlasov-Poisson system for a plane diode. Asymptot. Anal. 16 (1998) 25-48. | Zbl

[2] H.U. Baranger and J.W. Wilkins, Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends. Phys. Rev. B 36 (1987) 1487-1502.

[3] N. Ben Abdallah, The Child-Langmuir regime for electron transport in a plasma including a background of positive ions. Math. Models Methods Appl. Sci. 4 (1994) 409-438.

[4] N. Ben Abdallah, Convergence of the Child-Langmuir asymptotics of the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 27 (1996) 92-109. | Zbl

[5] N. Ben Abdallah, Étude de modèles asymptotiques de transport de particules chargées: Asymptotique de Child-Langmuir. Ph.D. thesis.

[6] N. Ben Abdallah and P. Degond, The Child-Langmuir law for the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 26 (1995) 364-398. | Zbl

[7] N. Ben Abdallah and P. Degond, The Child-Langmuir law in the kinetic theory of charged particles: semiconductors models. Mathematical problems in semiconductor physics, Rome (1993) 76-102. Longman, Harlow, Pitman Res. Notes Math. Ser. 340 (1995). | Zbl

[8] N. Ben Abdallah, P. Degond and F. Méhats, The Child-Langmuir asymptotics for magnetized flows. Asymptot. Anal. 20 (1999) 97-13. | Zbl

[9] N. Ben Abdallah, P. Degond and C. Schmeiser, On a mathemaical model of hot-carrier injection in semiconductors. Math. Methods Appl. Sci. 17 (1994) 1193-1212. | Zbl

[10] J.A. Carrillo, I.M. Gamba, O. Muscato and C.-W. Shu, Comparison of Monte Carlo and deterministic simulations of a silicon diode. IMA series (to be published). | Zbl

[11] J.A. Carrillo, I.M. Gamba and C.-W. Shu, Computational macroscopic approximations to the 1-D relaxation-time kinetic system for semiconductors. Phys. D 146 (2000) 289-306. | Zbl

[12] P. Degond and P.A. Raviart, An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law. Asymptot. Anal. 4 (1991) 187-214. | Zbl

[13] P. Degond and P.A. Raviart, On a penalization of the Child-Langmuir emission condition for the one-dimensional Vlasov-Poisson equation. Asymptot. Anal. 6 (1992) 1-27.

[14] G. Jiang and C.-W. Shu, Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1996) 202-228. | Zbl

[15] I. Langmuir and K.T. Compton, Electrical discharges in gases: Part II, fundamental phenomena in electrical discharges. Rev. Modern Phys. 3 (1931) 191-257.

[16] P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer, New York (1990). | MR | Zbl

[17] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (A. Quarteroni Ed.). Springer, Lecture Notes in Math. 1697 (1998) 325-432. | Zbl

[18] M.S. Shur and L.F. Eastman, Ballistic transport in semiconductors at low temperature for low-power high-speed logic. IEEE Trans. Electron Dev. ED-26 (1979) 1677-1683.

[19] M.S. Shur and L.F. Eastman, Near ballistic transport in GaAs devices at 77 K. Solid-State Electron 24 (1991) 11-18.

Cité par Sources :