American Institute of Mathematical Sciences

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September  2011, 4(3): 809-829. doi: 10.3934/krm.2011.4.809

Properties of the steady state distribution of electrons in semiconductors

Orazio Muscato 1, , Wolfgang Wagner 2, and  Vincenza Di Stefano 1,

1. 

Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale Andrea Doria 6 - 95125 Catania, Italy, Italy
2. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39 - 10117 Berlin, Germany

Received  October 2010 Revised  April 2011 Published  August 2011

This paper studies a Boltzmann transport equation with several electron-phonon scattering mechanisms, which describes the charge transport in semiconductors. The electric field is coupled to the electron distribution function via Poisson's equation. Both the parabolic and the quasi-parabolic band approximations are considered. The steady state behaviour of the electron distribution function is investigated by a Monte Carlo algorithm. More precisely, several nonlinear functionals of the solution are calculated that quantify the deviation of the steady state from a Maxwellian distribution with respect to the wave-vector. On the one hand, the numerical results illustrate known theoretical statements about the steady state and indicate directions for further studies. On the other hand, the nonlinear functionals provide tools that can be used in the framework of Monte Carlo algorithms for detecting regions in which the steady state distribution has a relatively simple structure, thus providing a basis for domain decomposition methods.
Keywords: electron transport, Monte Carlo algorithm., Boltzmann-Poisson equation, steady state distribution, semiconductors.
Mathematics Subject Classification: 82D37, 65C0.
Citation: Orazio Muscato, Wolfgang Wagner, Vincenza Di Stefano. Properties of the steady state distribution of electrons in semiconductors. Kinetic & Related Models, 2011, 4 (3) : 809-829. doi: 10.3934/krm.2011.4.809
References:
[1]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,", Springer Monographs in Mathematics, (2006).   Google Scholar

[2]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306.  doi: 10.1063/1.531567.  Google Scholar

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J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods,, J. Comput. Phys., 214 (2006), 55.  doi: 10.1016/j.jcp.2005.09.005.  Google Scholar

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Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3130.  doi: 10.1016/j.cma.2009.05.015.  Google Scholar

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M. V. Fischetti, S. E. Laux, P. M. Solomon and A. Kumar, Thirty years of Monte Carlo simulations of electronic transport in semiconductors: Their relevance to science and mainstream VLSI technology,, Journal of Computational Electronics, 3 (2004), 287.  doi: 10.1007/s10825-004-7063-8.  Google Scholar

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T. Grasser, H. Kosina, M. Gritsch and S. Selberherr, Using six moments of Boltzmann's transport equation for device simulation,, J. Appl. Phys., 90 (2001), 2389.  doi: 10.1063/1.1389757.  Google Scholar

[7]

T. Grasser, H. Kosina, C. Heitzinger and S. Selberherr, Characterization of the hot electron distribution function using six moments,, J. Appl. Phys., 91 (2002), 3869.  doi: 10.1063/1.1450257.  Google Scholar

[8]

C. Jacoboni and P. Lugli, "The Monte Carlo Method for Semiconductor Device Simulation,", Springer, (1989).   Google Scholar

[9]

C. Jacoboni and L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials,, Rev. Modern Phys., 55 (1983), 645.  doi: 10.1103/RevModPhys.55.645.  Google Scholar

[10]

C. Jungemann and B. Meinerzhagen, "Hierarchical Device Simulation. The Monte-Carlo Perspective,", Springer, (2003).   Google Scholar

[11]

A. Majorana, Equilibrium solutions of the non-linear Boltzmann equation for an electron gas in a semiconductor,, Nuovo Cimento B, 108 (1993), 871.  doi: 10.1007/BF02828734.  Google Scholar

[12]

A. Majorana, Trend to equilibrium of electron gas in a semiconductor according to the Boltzmann equation, Proceedings of the Fifteenth International Conference on Transport Theory, Part II (Göteborg, 1997),, Transport Theory Statist. Phys., 27 (1998), 547.  doi: 10.1080/00411459808205642.  Google Scholar

[13]

A. Majorana and C. Milazzo, Space homogeneous solutions of the linear semiconductor Boltzmann equation,, J. Math. Anal. Appl., 259 (2001), 609.  doi: 10.1006/jmaa.2001.7444.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).   Google Scholar

[15]

O. Muscato, W. Wagner and V. Di Stefano, Numerical study of the systematic error in Monte Carlo schemes for semiconductors,, M2AN Math. Model. Numer. Anal., 44 (2010), 1049.  doi: 10.1051/m2an/2010051.  Google Scholar

[16]

S. Rjasanow and W. Wagner, "Stochastic Numerics for the Boltzmann Equation,", Springer Series in Computational Mechanics, 37 (2005).   Google Scholar

[17]

V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices,, Materials Science and Engineering: R, 58 (2008), 228.   Google Scholar

show all references

References:
[1]

J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,", Springer Monographs in Mathematics, (2006).   Google Scholar

[2]

N. Ben Abdallah and P. Degond, On a hierarchy of macroscopic models for semiconductors,, J. Math. Phys., 37 (1996), 3306.  doi: 10.1063/1.531567.  Google Scholar

[3]

J. A. Carrillo, I. M. Gamba, A. Majorana and C.-W. Shu, 2D semiconductor device simulations by WENO-Boltzmann schemes: efficiency, boundary conditions and comparison to Monte Carlo methods,, J. Comput. Phys., 214 (2006), 55.  doi: 10.1016/j.jcp.2005.09.005.  Google Scholar

[4]

Y. Cheng, I. M. Gamba, A. Majorana and C.-W. Shu, A discontinuous Galerkin solver for Boltzmann-Poisson systems in nano devices,, Comput. Methods Appl. Mech. Engrg., 198 (2009), 3130.  doi: 10.1016/j.cma.2009.05.015.  Google Scholar

[5]

M. V. Fischetti, S. E. Laux, P. M. Solomon and A. Kumar, Thirty years of Monte Carlo simulations of electronic transport in semiconductors: Their relevance to science and mainstream VLSI technology,, Journal of Computational Electronics, 3 (2004), 287.  doi: 10.1007/s10825-004-7063-8.  Google Scholar

[6]

T. Grasser, H. Kosina, M. Gritsch and S. Selberherr, Using six moments of Boltzmann's transport equation for device simulation,, J. Appl. Phys., 90 (2001), 2389.  doi: 10.1063/1.1389757.  Google Scholar

[7]

T. Grasser, H. Kosina, C. Heitzinger and S. Selberherr, Characterization of the hot electron distribution function using six moments,, J. Appl. Phys., 91 (2002), 3869.  doi: 10.1063/1.1450257.  Google Scholar

[8]

C. Jacoboni and P. Lugli, "The Monte Carlo Method for Semiconductor Device Simulation,", Springer, (1989).   Google Scholar

[9]

C. Jacoboni and L. Reggiani, The Monte Carlo method for the solution of charge transport in semiconductors with applications to covalent materials,, Rev. Modern Phys., 55 (1983), 645.  doi: 10.1103/RevModPhys.55.645.  Google Scholar

[10]

C. Jungemann and B. Meinerzhagen, "Hierarchical Device Simulation. The Monte-Carlo Perspective,", Springer, (2003).   Google Scholar

[11]

A. Majorana, Equilibrium solutions of the non-linear Boltzmann equation for an electron gas in a semiconductor,, Nuovo Cimento B, 108 (1993), 871.  doi: 10.1007/BF02828734.  Google Scholar

[12]

A. Majorana, Trend to equilibrium of electron gas in a semiconductor according to the Boltzmann equation, Proceedings of the Fifteenth International Conference on Transport Theory, Part II (Göteborg, 1997),, Transport Theory Statist. Phys., 27 (1998), 547.  doi: 10.1080/00411459808205642.  Google Scholar

[13]

A. Majorana and C. Milazzo, Space homogeneous solutions of the linear semiconductor Boltzmann equation,, J. Math. Anal. Appl., 259 (2001), 609.  doi: 10.1006/jmaa.2001.7444.  Google Scholar

[14]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).   Google Scholar

[15]

O. Muscato, W. Wagner and V. Di Stefano, Numerical study of the systematic error in Monte Carlo schemes for semiconductors,, M2AN Math. Model. Numer. Anal., 44 (2010), 1049.  doi: 10.1051/m2an/2010051.  Google Scholar

[16]

S. Rjasanow and W. Wagner, "Stochastic Numerics for the Boltzmann Equation,", Springer Series in Computational Mechanics, 37 (2005).   Google Scholar

[17]

V. Sverdlov, E. Ungersboeck, H. Kosina and S. Selberherr, Current transport models for nanoscale semiconductor devices,, Materials Science and Engineering: R, 58 (2008), 228.   Google Scholar

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