# American Institute of Mathematical Sciences

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

## 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

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.
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
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##### References:
 [1] Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955 [2] Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 [3] N. Ben Abdallah, M. Lazhar Tayeb. Diffusion approximation for the one dimensional Boltzmann-Poisson system. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1129-1142. doi: 10.3934/dcdsb.2004.4.1129 [4] Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2021  doi: 10.3934/fods.2021034 [5] Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159 [6] Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81 [7] Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 [8] Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014 [9] Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic & Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048 [10] Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055 [11] Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789 [12] La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 [13] Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335 [14] Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683 [15] Marco A. Fontelos, Lucía B. Gamboa. On the structure of double layers in Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1939-1967. doi: 10.3934/dcdsb.2012.17.1939 [16] Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic & Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032 [17] Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159 [18] Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395 [19] Taposh Kumar Das, Óscar López Pouso. New insights into the numerical solution of the Boltzmann transport equation for photons. Kinetic & Related Models, 2014, 7 (3) : 433-461. doi: 10.3934/krm.2014.7.433 [20] Laurent Bernis, Laurent Desvillettes. Propagation of singularities for classical solutions of the Vlasov-Poisson-Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 13-33. doi: 10.3934/dcds.2009.24.13

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