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

Some properties of the kinetic equation for electron transport in semiconductors

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.
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). doi: 10.1063/1.1899210.

[2]

M. H. A. Davis, Markov Models and Optimization,, Monographs on Statistics and Applied Probability, (1993).

[3]

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

[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.

[5]

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

[6]

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

[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. doi: 10.1103/RevModPhys.55.645.

[8]

A. Jüngel, Transport Equations for Semiconductors,, vol. 773 of Lecture Notes in Physics, (2009). doi: 10.1007/978-3-540-89526-8.

[9]

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

[10]

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

[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. doi: 10.1007/s10955-011-0247-2.

[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. doi: 10.1108/03321641111101050.

[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. doi: 10.1016/j.camwa.2012.03.100.

[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. doi: 10.1051/m2an/2010051.

[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. doi: 10.3934/krm.2011.4.809.

[16]

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

[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), 1.

[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), 1.

[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. doi: 10.1023/B:JCEL.0000011426.11111.64.

[20]

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

[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). doi: 10.1063/1.3558887.

[22]

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

[23]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation,, Springer Series in Computational Mathematics, (2005).

[24]

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

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). doi: 10.1063/1.1899210.

[2]

M. H. A. Davis, Markov Models and Optimization,, Monographs on Statistics and Applied Probability, (1993).

[3]

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

[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.

[5]

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

[6]

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

[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. doi: 10.1103/RevModPhys.55.645.

[8]

A. Jüngel, Transport Equations for Semiconductors,, vol. 773 of Lecture Notes in Physics, (2009). doi: 10.1007/978-3-540-89526-8.

[9]

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

[10]

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

[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. doi: 10.1007/s10955-011-0247-2.

[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. doi: 10.1108/03321641111101050.

[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. doi: 10.1016/j.camwa.2012.03.100.

[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. doi: 10.1051/m2an/2010051.

[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. doi: 10.3934/krm.2011.4.809.

[16]

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

[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), 1.

[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), 1.

[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. doi: 10.1023/B:JCEL.0000011426.11111.64.

[20]

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

[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). doi: 10.1063/1.3558887.

[22]

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

[23]

S. Rjasanow and W. Wagner, Stochastic Numerics for the Boltzmann Equation,, Springer Series in Computational Mathematics, (2005).

[24]

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

[1]

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

[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]

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

[4]

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

[5]

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

[6]

Johannes Giannoulis. Transport and generation of macroscopically modulated waves in diatomic chains. Conference Publications, 2011, 2011 (Special) : 485-494. doi: 10.3934/proc.2011.2011.485

[7]

Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125

[8]

Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313

[9]

Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803

[10]

Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025

[11]

Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917

[12]

Piotr Zgliczyński. Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof. Journal of Computational Dynamics, 2015, 2 (1) : 95-142. doi: 10.3934/jcd.2015.2.95

[13]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305

[14]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57

[15]

Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations & Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39

[16]

Li Chen, Xiu-Qing Chen, Ansgar Jüngel. Semiclassical limit in a simplified quantum energy-transport model for semiconductors. Kinetic & Related Models, 2011, 4 (4) : 1049-1062. doi: 10.3934/krm.2011.4.1049

[17]

Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823

[18]

Maurizio Garrione, Manuel Zamora. Periodic solutions of the Brillouin electron beam focusing equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 961-975. doi: 10.3934/cpaa.2014.13.961

[19]

Xiaojun Zhou, Chunhua Yang, Weihua Gui. State transition algorithm. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1039-1056. doi: 10.3934/jimo.2012.8.1039

[20]

Tomoyuki Miyaji, Yoshio Tsutsumi. Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1633-1650. doi: 10.3934/cpaa.2018078

2017 Impact Factor: 1.219

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]