The Vlasov (or Liouville) equation of the previous chapter does not take into account short-range particle interactions, like collisions of the particles with other particles or with the crystal lattice. In this chapter, we extend the Vlasov equation to include scattering mechanisms which leads to the Boltzmann equation. We present only a phenomenological derivation. For rigorous results, we refer to [1, Sect. 1.5.3] and [2, Chap. 4].
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References
H. Babovsky. Die Boltzmann-Gleichung. Teubner, Stuttgart, 1998.
C. Cercignani, R. Illner, and M. Pulvirenti. The Mathematical Theory of Dilute Gases. Springer, New York, 1994.
L. Boltzmann. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Sitzungsberichte Akad. Wiss. Wien 66 (1872), 275–370. Translation: Further studies on the thermal equilibrium of gas molecules. In: S. Brush (ed.), Kinetic Theory, Vol. 2, 88–174. Pergamon Press, Oxford, 1966.
T. Carleman. Sur la théorie de l’équation intégro-différentielle de Boltzmann. Acta Mathematica 60 (1933), 91–146.
L. Arkeryd. On the Boltzmann equation. Arch. Rat. Mech. Anal. 45 (1971), 1–34.
R. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann. Math. 130 (1989), 321–366.
F. Golse, B. Perthame, P.-L. Lions, and R. Sentis. Regularity of the moments of the solution of a transport equation. J. Funct. Anal. 76 (1988), 110–125.
P. Gérard. Solutions globales du problème de Cauchy pour l’équation de Boltzmann (d’après R. Di Perna et P.-L. Lions). Séminaire Bourbaki, Vol. 1988–89, Astérisque 161–162, Exp. No. 699 (1989), 257–281.
P.-L. Lions. Global solutions of kinetic models and related problems. In: C. Cercignani and M. Pulvirenti (eds.), Nonequilibrium Problems in Many-Particle Systems, Lecture Notes in Math. 1551, 58–86. Springer, Berlin, 1992.
C. Villani. A review of mathematical topics in collisional kinetic theory. In: S. Friedlander and D. Serre (eds.), Handbook of Mathematical Fluid Dynamics, Vol. 1, 71–305. Elsevier, Amsterdam, 2002.
F. Poupaud. On a system of nonlinear Boltzmann equations of semiconductors physics. SIAM J. Appl. Math. 50 (1990), 1593–1606.
F. Mustieles. Global existence of solutions for the nonlinear Boltzmann equation of semiconductor physics. Rev. Mat. Iberoamer. 6 (1990), 43–59.
F. Mustieles. Global existence of weak solutions for a system of nonlinear Boltzmann equations in semiconductor physics. Math. Meth. Appl. Sci. 14 (1991), 139–153.
H. Andréasson. Global existence of smooth solutions in three dimensions for the semiconductor Vlasov-Poisson-Boltzmann equation. Nonlin. Anal.: Theory Meth. Appl. 28 (1990), 1193–1211.
A. Majorana and S. Marano. Space homogeneous solutions to the Cauchy problem for semiconductor Boltzmann equations. SIAM J. Math. Anal. 28 (1997), 1294–1308.
A. Majorana and S. Marano. On the Cauchy problem for spatially homogeneous semiconductor Boltzmann equations: existence and uniqueness. Annali Math. 184 (2005), 275–296.
A. Majorana and C. Milazzo. Space homogeneous solutions of the linear semiconductor Boltzmann equation. J. Math. Anal. Appl. 259 (2001), 609–629.
G. Bird. Molecular Gas Dynamics and Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994.
M. Fischetti and S. Laux. Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects. Phys. Rev. B 38 (1988), 9721–9745.
K. Nanbu. Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases. J. Phys. Soc. Japan 52 (1983), 2042–2049.
A. Gnudi, D. Ventura, and G. Baccarani. Modeling impact ionization in a BJT by means of a spherical harmonics expansion of the Boltzmann equation. IEEE Trans. Computer-Aided Design 12 (1993), 1706–1713.
N. Goldsman, L. Henrickson, and J. Frey. A physics-based analytical/numerical solution to the Boltzmann transport equation for use in device simulation. Solid State Electr. 34 (1991), 389–396.
C. Gray and H. Ralph. Solution of Boltzmann’s equation for semiconductors using a spherical harmonic expansion. J. Phys. C: Solid State Phys. 5 (1972), 55–62.
C. Buet. A discrete velocity scheme for the Boltzmann operator of rarefied gas dynamics. Transp. Theory Stat. Phys. 25 (1996), 33–60.
L. Pareschi and B. Perthame. A Fourier spectral method for homogeneous Boltzmann equations. Transp. Theory Stat. Phys. 25 (1996), 369–383.
E. Gabetta, L. Pareschi, and G. Toscani. Relaxation schemes for nonlinear kinetic equations. SIAM J. Numer. Anal. 34 (1997), 2168–2194.
C. Auer, A. Majorana, and F. Schürrer. Numerical schemes for solving the non-stationary Boltzmann–Poisson system for two-dimensional semiconductor devices. In: T. Goudon, E. Sonnendrücker, and D. Talay (eds.), ESAIM: Proceedings 15 (2005), 75–86.
M. Cáceres, J. A. Carrillo, and A. Majorana. Deterministic simulation of the Boltzmann–Poisson system in GaAs-based semiconductors. SIAM J. Sci. Comput. 27 (2006), 1981–2009.
M. Galler and F. Schürrer. A direct multigroup-WENO solver for the 2D non-stationary Boltzmann–Poisson system for GaAs devices: GaAs-MESFET. J. Comput. Phys. 212 (2006), 778–797.
G. Ossig and F. Schürrer. Simulation of non-equilibrium electron transport in silicon quantum wires. I. Comput. Electr. 7 (2008), 367–370.
V. Aristov. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer, Dordrecht, 2001.
L. Pareschi. Computational methods and fast algorithms for Boltzmann equations. In: N. Bellomo (ed.), Lecture Notes on the Discretization of the Boltzmann Equation, Series Adv. Math. Appl. Sci. 63, Chapter 7. World Scientific, Singapore, 2003.
K. Brennan. The Physics of Semiconductors. Cambridge University Press, Cambridge, 1999.
H. Grahn. Introduction to Semiconductor Physics. World Scientific, Singapore, 1999.
M. Lundstrom. Fundamentals of Carrier Transport. 2nd edition, Cambridge University Press, Cambridge, 2000.
K. Seeger. Semiconductor Physics. An Introduction. Springer, Berlin, 2004.
V. Gantmakher and Y. Levinson. Carrier Scattering in Metals and Semiconductors. North Holland, New York, 1987.
B. Ridley. Quantum Processes in Semiconductors. Clarendon Press, Oxford, 1982.
W. Zawadzki. Mechanics of electron scattering in semiconductors. In: T. Moss (ed.), Handbook of Semiconductors, Vol. 1, Chapter 12. North-Holland, New York, 1982.
W. Wenckebach. Essentials of Semiconductor Physics. John Wiley & Sons, Chichester, 1999.
N. Ben Abdallah and P. Degond. On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37 (1996), 3308–3333.
L. Reggiani (ed.). Hot Electron Transport in Semiconductors. Springer, Berlin, 1985.
C. Kittel. Introduction to Solid State Physcis. 7th edition, John Wiley & Sons, New York, 1996.
N. Ben Abdallah, P. Degond, and S. Génieys. An energy-transport model for semiconductors derived from the Boltzmann equation. J. Stat. Phys. 84 (1996), 205–231.
N. Ashcroft and N. Mermin. Solid State Physics. Sanners College, Philadelphia, 1976.
A. Majorana. Equilibrium solutions of the non-linear Boltzmann equations for an electron gas in a semiconductor. Il Nuovo Cimento B 108 (1993), 871–877.
F. Poupaud. Mathematical theory of kinetic equations for transport modelling in semiconductors. In: B. Perthame (ed.), Advances in Kinetic Theory and Computing: Selected Papers, Ser. Adv. Math. Appl. Sci. 22, 141–168. World Scientific, Singapore, 1994.
Y. Sone. Kinetic Theory and Fluid Dynamics. Birkhäuser, Boston, 2002.
P. Markowich, C. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, Vienna, 1990.
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Jüngel, A. (2009). Scattering Models. In: Transport Equations for Semiconductors. Lecture Notes in Physics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89526-8_4
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