# The Child–Langmuir limit for semiconductors : a numerical validation

María-José Cáceres; José-Antonio Carrillo; Pierre Degond^{[1]}

- [1] MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France.

- Volume: 36, Issue: 6, page 1161-1176
- ISSN: 0764-583X

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topCáceres, María-José, Carrillo, José-Antonio, and Degond, Pierre. "The Child–Langmuir limit for semiconductors : a numerical validation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1161-1176. <http://eudml.org/doc/245878>.

@article{Cáceres2002,

abstract = {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.},

affiliation = {MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France.},

author = {Cáceres, María-José, Carrillo, José-Antonio, Degond, Pierre},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Boltzmann-Poisson system; Child-Langmuir limit; WENO schemes; semiconductor devices; detailed numerical comparisons; Child-Langmuir equations},

language = {eng},

number = {6},

pages = {1161-1176},

publisher = {EDP-Sciences},

title = {The Child–Langmuir limit for semiconductors : a numerical validation},

url = {http://eudml.org/doc/245878},

volume = {36},

year = {2002},

}

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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 6

SP - 1161

EP - 1176

AB - 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.

LA - eng

KW - Boltzmann-Poisson system; Child-Langmuir limit; WENO schemes; semiconductor devices; detailed numerical comparisons; Child-Langmuir equations

UR - http://eudml.org/doc/245878

ER -

## References

top- [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. Zbl0913.35130
- [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. Zbl0847.35009
- [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. Zbl0828.35131
- [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). Zbl0888.35114
- [8] N. Ben Abdallah, P. Degond and F. Méhats, The Child–Langmuir asymptotics for magnetized flows. Asymptot. Anal. 20 (1999) 97–13. Zbl0934.35183
- [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. Zbl0812.35137
- [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). Zbl1041.82524
- [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. Zbl0976.82053
- [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. Zbl0840.35082
- [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. Zbl0877.65065
- [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). Zbl0765.35001MR1063852
- [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. Zbl0927.65111
- [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.

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