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Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution
Properties of the steady state distribution of electrons in semiconductors
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 |
References:
[1] |
J. Banasiak and L. Arlotti, "Perturbations of Positive Semigroups with Applications,", Springer Monographs in Mathematics, (2006).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).
|
[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. |
[16] |
S. Rjasanow and W. Wagner, "Stochastic Numerics for the Boltzmann Equation,", Springer Series in Computational Mechanics, 37 (2005).
|
[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).
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations,", Springer-Verlag, (1990).
|
[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. |
[16] |
S. Rjasanow and W. Wagner, "Stochastic Numerics for the Boltzmann Equation,", Springer Series in Computational Mechanics, 37 (2005).
|
[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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