American Institute of Mathematical Sciences

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December  2013, 6(4): 955-967. doi: 10.3934/krm.2013.6.955

Some properties of the kinetic equation for electron transport in semiconductors

Wolfgang Wagner 1,

1. 

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

Received  June 2013 Revised  September 2013 Published  November 2013

The paper studies the kinetic equation for electron transport in semiconductors. New formulas for the heat generation rate are derived by analyzing the basic scattering mechanisms. In addition, properties of the steady state distribution are discussed and possible extensions of the deviational particle Monte Carlo method to the area of electron transport are proposed.
Keywords: Electron transport equation, Monte Carlo algorithm., semiconductors, steady state distribution, heat generation.
Mathematics Subject Classification: 82D37, 65C0.
Citation: 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
References:
[1]

L. L. Baker and N. G. Hadjiconstantinou, Variance reduction for Monte Carlo solutions of the Boltzmann equation, Phys. Fluids, 17 (2005), 051703. doi: 10.1063/1.1899210.  Google Scholar

[2]

M. H. A. Davis, Markov Models and Optimization, Monographs on Statistics and Applied Probability, 49. Chapman & Hall, London, 1993.  Google Scholar

[3]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.  Google Scholar

[4]

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), 287-293. Google Scholar

[5]

T. M. M. Homolle and N. G. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701(1-4). Google Scholar

[6]

C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer, New York, 1989. doi: 10.1007/978-3-7091-6963-6.  Google Scholar

[7]

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), 645-705. doi: 10.1103/RevModPhys.55.645.  Google Scholar

[8]

A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[9]

A. Majorana, Trend to equilibrium of electron gas in a semiconductor according to the Boltzmann equation, Transport Theory Statist. Phys., 27 (1998), 547-571. doi: 10.1080/00411459808205642.  Google Scholar

[10]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[11]

O. Muscato and V. Di Stefano, An energy transport model describing heat generation and conduction in silicon semiconductors, J. Stat. Phys., 144 (2011), 171-197. doi: 10.1007/s10955-011-0247-2.  Google Scholar

[12]

O. Muscato and V. Di Stefano, Heat generation and transport in nanoscale semiconductor devices via Monte Carlo and hydrodynamic simulations, COMPEL, 30 (2011), 519-537. doi: 10.1108/03321641111101050.  Google Scholar

[13]

O. Muscato, V. Di Stefano and W. Wagner, A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation, Comput. Math. Appl., 65 (2013), 520-527. doi: 10.1016/j.camwa.2012.03.100.  Google Scholar

[14]

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), 1049-1068. doi: 10.1051/m2an/2010051.  Google Scholar

[15]

O. Muscato, W. Wagner and V. Di Stefano, Properties of the steady state distribution of electrons in semiconductors, Kinetic and Related Models, 4 (2011), 808-829. doi: 10.3934/krm.2011.4.809.  Google Scholar

[16]

C. Ni, Z. Aksamija, J. Y. Murthy and U. Ravaioli, Coupled electro-thermal simulation of MOSFETs, Journal of Computational Electronics, 11 (2012), 93-105. Google Scholar

[17]

J.-P. M. Péraud and N. G. Hadjiconstantinou, Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations, Phys. Rev. B, 84 (2011), 205331(1-15). Google Scholar

[18]

J.-P. M. Péraud and N. G. Hadjiconstantinou, An alternative approach to efficient simulation of micro/nanoscale phonon transport, Applied Physics Letters, 101 (2012), 153114(1-4). Google Scholar

[19]

N. J. Pilgrim, W. Batty and R. W. Kelsall, Electrothermal Monte Carlo simulations of InGaAs/AlGaAs HEMTs, Journal of Computational Electronics, 2 (2003), 207-211. doi: 10.1023/B:JCEL.0000011426.11111.64.  Google Scholar

[20]

E. Pop, S. Sinha and K. E. Goodson, Heat generation and transport in nanometer-scale transistors, Proceedings of the IEEE, 94 (2006), 1587-1601. doi: 10.1109/JPROC.2006.879794.  Google Scholar

[21]

G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606. doi: 10.1063/1.3558887.  Google Scholar

[22]

K. Raleva, D. Vasileska, S. M. Goodnick and M. Nedjalkov, Modeling thermal effects in nanodevices, IEEE Transactions on Electron Devices, 55 (2008), 1306-1316. doi: 10.1109/TED.2008.921263.  Google Scholar

[23]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer Series in Computational Mathematics, 37. Springer-Verlag, Berlin, 2005.  Google Scholar

[24]

W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods Appl., 14 (2008), 191-268. doi: 10.1515/MCMA.2008.010.  Google Scholar

show all references

References:
[1]

L. L. Baker and N. G. Hadjiconstantinou, Variance reduction for Monte Carlo solutions of the Boltzmann equation, Phys. Fluids, 17 (2005), 051703. doi: 10.1063/1.1899210.  Google Scholar

[2]

M. H. A. Davis, Markov Models and Optimization, Monographs on Statistics and Applied Probability, 49. Chapman & Hall, London, 1993.  Google Scholar

[3]

A. Eibeck and W. Wagner, Stochastic interacting particle systems and nonlinear kinetic equations, Ann. Appl. Probab., 13 (2003), 845-889. doi: 10.1214/aoap/1060202829.  Google Scholar

[4]

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), 287-293. Google Scholar

[5]

T. M. M. Homolle and N. G. Hadjiconstantinou, Low-variance deviational simulation Monte Carlo, Phys. Fluids, 19 (2007), 041701(1-4). Google Scholar

[6]

C. Jacoboni and P. Lugli, The Monte Carlo Method for Semiconductor Device Simulation, Springer, New York, 1989. doi: 10.1007/978-3-7091-6963-6.  Google Scholar

[7]

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), 645-705. doi: 10.1103/RevModPhys.55.645.  Google Scholar

[8]

A. Jüngel, Transport Equations for Semiconductors, vol. 773 of Lecture Notes in Physics, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.  Google Scholar

[9]

A. Majorana, Trend to equilibrium of electron gas in a semiconductor according to the Boltzmann equation, Transport Theory Statist. Phys., 27 (1998), 547-571. doi: 10.1080/00411459808205642.  Google Scholar

[10]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[11]

O. Muscato and V. Di Stefano, An energy transport model describing heat generation and conduction in silicon semiconductors, J. Stat. Phys., 144 (2011), 171-197. doi: 10.1007/s10955-011-0247-2.  Google Scholar

[12]

O. Muscato and V. Di Stefano, Heat generation and transport in nanoscale semiconductor devices via Monte Carlo and hydrodynamic simulations, COMPEL, 30 (2011), 519-537. doi: 10.1108/03321641111101050.  Google Scholar

[13]

O. Muscato, V. Di Stefano and W. Wagner, A variance-reduced electrothermal Monte Carlo method for semiconductor device simulation, Comput. Math. Appl., 65 (2013), 520-527. doi: 10.1016/j.camwa.2012.03.100.  Google Scholar

[14]

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), 1049-1068. doi: 10.1051/m2an/2010051.  Google Scholar

[15]

O. Muscato, W. Wagner and V. Di Stefano, Properties of the steady state distribution of electrons in semiconductors, Kinetic and Related Models, 4 (2011), 808-829. doi: 10.3934/krm.2011.4.809.  Google Scholar

[16]

C. Ni, Z. Aksamija, J. Y. Murthy and U. Ravaioli, Coupled electro-thermal simulation of MOSFETs, Journal of Computational Electronics, 11 (2012), 93-105. Google Scholar

[17]

J.-P. M. Péraud and N. G. Hadjiconstantinou, Efficient simulation of multidimensional phonon transport using energy-based variance-reduced Monte Carlo formulations, Phys. Rev. B, 84 (2011), 205331(1-15). Google Scholar

[18]

J.-P. M. Péraud and N. G. Hadjiconstantinou, An alternative approach to efficient simulation of micro/nanoscale phonon transport, Applied Physics Letters, 101 (2012), 153114(1-4). Google Scholar

[19]

N. J. Pilgrim, W. Batty and R. W. Kelsall, Electrothermal Monte Carlo simulations of InGaAs/AlGaAs HEMTs, Journal of Computational Electronics, 2 (2003), 207-211. doi: 10.1023/B:JCEL.0000011426.11111.64.  Google Scholar

[20]

E. Pop, S. Sinha and K. E. Goodson, Heat generation and transport in nanometer-scale transistors, Proceedings of the IEEE, 94 (2006), 1587-1601. doi: 10.1109/JPROC.2006.879794.  Google Scholar

[21]

G. A. Radtke, N. G. Hadjiconstantinou and W. Wagner, Low-noise Monte Carlo simulation of the variable hard sphere gas, Phys. Fluids, 23 (2011), 030606. doi: 10.1063/1.3558887.  Google Scholar

[22]

K. Raleva, D. Vasileska, S. M. Goodnick and M. Nedjalkov, Modeling thermal effects in nanodevices, IEEE Transactions on Electron Devices, 55 (2008), 1306-1316. doi: 10.1109/TED.2008.921263.  Google Scholar

[23]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation, Springer Series in Computational Mathematics, 37. Springer-Verlag, Berlin, 2005.  Google Scholar

[24]

W. Wagner, Deviational particle Monte Carlo for the Boltzmann equation, Monte Carlo Methods Appl., 14 (2008), 191-268. doi: 10.1515/MCMA.2008.010.  Google Scholar

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