Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T07:05:51.957Z Has data issue: false hasContentIssue false
  • English
  • Français

The Child–Langmuir limit for semiconductors:a numerical validation

Published online by Cambridge University Press:  15 January 2003

María-José Cáceres ,
José-Antonio Carrillo  and
Pierre Degond
María-José Cáceres
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain. caceresg@ugr.es., carrillo@ugr.es.
José-Antonio Carrillo
Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain. caceresg@ugr.es., carrillo@ugr.es.
Pierre Degond
Affiliation:
MIP, UMR CNRS 5640, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France. degond@mip.ups-tlse.fr.
Get access

Abstract

The Boltzmann–Poisson system modeling the electron flow in semiconductorsis 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-voltagecharacteristic curves by means of a scalingprocedure 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 theBoltzmann–Poisson system and the Child–Langmuir equations in testproblems.

Keywords

Boltzmann-Poisson systemChild-Langmuir limitWENO schemessemiconductor devices.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1161 - 1176
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alabau, F., Hamdache, K. and Peng, Y.J., Asymptotic analysis of the transient Vlasov-Poisson system for a plane diode. Asymptot. Anal. 16 (1998) 25-48.
Baranger, H.U. and Wilkins, J.W., Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends. Phys. Rev. B 36 (1987) 1487-1502. CrossRef
Ben Abdallah, N., 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.
Ben Abdallah, N., Convergence of the Child-Langmuir asymptotics of the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 27 (1996) 92-109. CrossRef
N. Ben Abdallah, Étude de modèles asymptotiques de transport de particules chargées: Asymptotique de Child-Langmuir. Ph.D. thesis.
Ben Abdallah, N. and Degond, P., The Child-Langmuir law for the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 26 (1995) 364-398. CrossRef
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).
Ben Abdallah, N., Degond, P. and Méhats, F., The Child-Langmuir asymptotics for magnetized flows. Asymptot. Anal. 20 (1999) 97-13.
Ben Abdallah, N., Degond, P. and Schmeiser, C., On a mathemaical model of hot-carrier injection in semiconductors. Math. Methods Appl. Sci. 17 (1994) 1193-1212.
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).
Carrillo, J.A., Gamba, I.M. and Shu, C.-W., Computational macroscopic approximations to the 1-D relaxation-time kinetic system for semiconductors. Phys. D 146 (2000) 289-306. CrossRef
Degond, P. and Raviart, P.A., An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law. Asymptot. Anal. 4 (1991) 187-214.
Degond, P. and Raviart, P.A., On a penalization of the Child-Langmuir emission condition for the one-dimensional Vlasov-Poisson equation. Asymptot. Anal. 6 (1992) 1-27.
Jiang, G. and Shu, C.-W., Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126 (1996) 202-228. CrossRef
Langmuir, I. and Compton, K.T., Electrical discharges in gases: Part II, fundamental phenomena in electrical discharges. Rev. Modern Phys. 3 (1931) 191-257. CrossRef
P.A. Markowich, C.A. Ringhofer and C. Schmeiser, Semiconductor Equations. Springer, New York (1990).
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.
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.
Shur, M.S. and Eastman, L.F., Near ballistic transport in GaAs devices at 77 K. Solid-State Electron 24 (1991) 11-18. CrossRef