Momentum-Based Transport Models

  • Christoph Jungemann
  • Bernd Meinerzhagen
Part of the Computational Microelectronics book series (COMPUTATIONAL)


The MC model is very CPU intensive and simpler but more CPU efficient models based on balance equations have been developed. The most widely used momentum-based models are the drift-diffusion (DD) and the hydrodynamic (HD) models [7.1–7.13]. Both models are derived by applying different degrees of approximation to balance equations of the type (2.39). The DD model is the simplest momentum-based model and consists of balance equations for the particle and current densities. Thus, only the first two moments of the distribution function are calculated instead of the full distribution function and a large fraction of the information contained in the distribution funtion is lost. On the other hand, the dimentionality of the problem is reduced by the integration of the k-space and the CPU efficiency is improved by orders of magnitude. In the case of the HD model the first four moments are considered including the particle density, current density, particle gas temperature, and the energy current density. This already enables the simulation of nonlocal effects,like the velocity overshoot, which have a strong impact on the device behavior of modern deep sub-micron devices.


Spectral Intensity Einstein Relation Donor Doping Velocity Overshoot Inverse Mass 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [7.1]
    W. V. Roosbroeck, “Theory of the flow of electrons and holes in germanium and other semiconductors”Bell System Technical Journalpp. 561–607, 1950.Google Scholar
  2. [7.2]
    W. Shockley, Electrons and Holes in Semiconductors, van Nostrand, Princeton, New Jersey, 1950.Google Scholar
  3. [7.3]
    R. Stratton, “Diffusion of hot and cold electrons in semiconductor barriers”Phys. Rev vol. 126, pp. 2002–2013, 1962.CrossRefGoogle Scholar
  4. [7.4]
    A. H. Marshak and K. M. van Vliet, “Electrical current in solids with position-dependent band structure”Solid-State Electron. vol. 21, pp. 417–427, 1978.CrossRefGoogle Scholar
  5. [7.5]
    W. L. Engl, H. K. Dirks, and B. Meinerzhagen, “Device modeling”Proc. IEEEvol. 71, pp. 10–33, 1983.CrossRefGoogle Scholar
  6. [7.6]
    S. Selberherr, Analysis and Simulation of Semiconductor Devices Springer, Wien, 1984.CrossRefGoogle Scholar
  7. [7.7]
    K. Blotekj er, “Transport equations for electrons in two-valley semiconductors”IEEE Trans. Electron Devices vol. 17, no. 1, pp. 38–47, 1970.CrossRefGoogle Scholar
  8. [7.8]
    R. K. Cook and J. Frey, “An efficient technique for two-dimensional simulation of velocity overshoot effects in Si and GaAs devices”COMPEL Int. J. Comput. Math. Electr. Electron. Eng. vol. 1, pp. 65–87, 1982.CrossRefGoogle Scholar
  9. [7.9]
    M. Fukuma and R. H. Uebbing, “Two-dimensional MOSFET simulation with energy transport phenomena”, in IEDM 1984, pp. 621–624.Google Scholar
  10. [7.10]
    W. Hänsch and M. Miura-Mattausch, “A new current relation for hot electron transport”, in Proc. NASECODE IV 1985, pp. 311–314.Google Scholar
  11. [7.11]
    B. Meinerzhagen and W. L. Engl, “The influence of the thermal equilibrium approximation on the accuracy of classical two-dimensional numerical modeling of silicon submicrometer MOS transistors”IEEE Trans. Electron Devices vol. 35, no. 5, pp. 689–697, 1988.CrossRefGoogle Scholar
  12. [7.12]
    R. Thoma, A. Emunds, B.Meinerzhagen, H. J. Peifer, and W. L. Engl, “Hydrodynamic equations for semiconductors with nonparabolic bandstructures”IEEE Trans. Electron Devices vol. 38, no. 6, pp. 1343–1352, 1991.CrossRefGoogle Scholar
  13. [7.13]
    R. Thoma, “Entwicklung eines hydrodynamischen Gleichungssystems zur Beschreibung von Halbleiterbauelementen”, Doktorarbeit, RWTH Aachen, 1991, Aachen.Google Scholar
  14. [7.14]
    C. Jungemann, B. Neinhüs, and B. Meinerzhagen, “Investigation of the local force approximation in numerical device simulation by full-band Monte Carlo simulation”VLSI Design vol. 13, no. 1–4, pp. 1–4, 2001.CrossRefGoogle Scholar
  15. [7.15]
    B. Neinhüs, S. Decker, P. Graf, F. M. Bufler, and B. Meinerzhagen, “Consistent hydrodynamic and Monte-Carlo simulation of SiGe HBTs based on table models for the relaxation times”VLSI Design vol. 8, pp. 387–391, 1998.CrossRefGoogle Scholar
  16. [7.16]
    H. BenekingHigh Speed Semiconductor DevicesChapman & Hall, London, 1994.Google Scholar
  17. [7.17]
    W. Shockley, John A. Copeland, and R. P. James, “The impedance field method of noise calculation in active semiconductor devices”, in Quantum theory of atoms, molecules and solid state P. O. Lowdin, Ed., pp. 537–563. Academic Press, 1966.Google Scholar
  18. [7.18]
    F. Bonani, G. Ghione, M. R. Pinto, and R. K. Smith, “An efficient approach to noise analysis through multidimentional physics-based models”IEEE Trans. Electron Devices vol. 45, no. 1, pp. 261–269, 1998.CrossRefGoogle Scholar
  19. [7.19]
    F. Bonani and G. Ghione, “Generation-recombination noise modelling in semiconductor devices through population or approximate equivalent current density fluctuations”Solid-State Electron. vol. 43, pp. 285–295, 1999.CrossRefGoogle Scholar
  20. [7.20]
    F. Bonani and G. Ghione, “Noise modeling for PDE based device simulations”, in Proc. ICNF, Gainesville, FL, 2001, vol. 16, pp. 631–636.Google Scholar
  21. [7.21]
    P. Shiktorov, E. Starikov, V. Gruzinskis, L. Varani, J.-C. Vaissiere, J.-P. Nougier, and L. Reggiani, “Admittance field method for the calculation of the spectral density of current fluctuations”Fluctuation and Noise Letters vol. 1, no. 1, pp. R1–R11, 2001.CrossRefGoogle Scholar
  22. [7.22]
    K. M. van Vliet, “Markov approach to density fluctuations due to transport and scattering. I. Mathematical formalism”J. Math. Phys. vol. 12, pp. 1981–1998, 1971.MATHCrossRefGoogle Scholar
  23. [7.23]
    Sh. KoganElectronic Noise and Fluctuations in Solids, Cambridge University Press, Cambridge, New York, Melbourne, 1996.CrossRefGoogle Scholar
  24. [7.24]
    S. M. Kogan, “Equations for the correlation functions using a generalized Keldysh technique” Phys. Rev. A vol. 44, pp. 8072–8082, 1991.CrossRefGoogle Scholar
  25. [7.25]
    H. S. Min, “A unified theory of noise in nondegenerate semiconductors”J. Appl. Phys. vol. 61, pp. 4549–4565, 1987.CrossRefGoogle Scholar
  26. [7.26]
    C. Jungemann and B. Meinerzhagen, “Analysis of the stochastic error of station-ary Monte Carlo device simulations”,IEEE Trans. Electron Devices vol. 48, no. 5, pp. 985–992, 2001.CrossRefGoogle Scholar
  27. [7.27]
    F. Bonani and G. Ghione Noise in Semiconductor Devices, Modeling and Simulation Advanced Microelectronics. Springer, Berlin, Heidelberg, New York, 2001.Google Scholar
  28. [7.28]
    P. Shiktorov, E. Starikov, V. Gruzinskis, T. Gonzalez, J. Mateos, D. Pardo, L. Reggiani, L. Varani, and J. C. Vaissere, “Langevin forces and generalized transfer fields for noise modeling in deep submicron devices”IEEE Trans. Electron Devices vol. 47, no. 10, pp. 1992–1998, 2000.CrossRefGoogle Scholar
  29. [7.29]
    C. Jungemann, B. Neinhüs, S. Decker, and B. Meinerzhagen, “Hierarchical 2D RF noise simulation of Si and SiGe devices by Langevin-type DD and HD models based on MC generated noise parameters”, in IEDM Tech. Dig., Washington (USA), 2001, pp. 481–484.Google Scholar
  30. [7.30]
    L. Reggiani, E. Starikov, P. Shiktorov, V. Gruzinskis, and L. Varani, “Modelling of small-signal response and electronic noise in semiconductor high-field transport”Semicond. Sci. Technol. vol. 12, pp. 141–156, 1997.CrossRefGoogle Scholar
  31. [7.31]
    A. Papoulis, Probability,Random Variables and Stochastic Processes, Mc GrawHill, 3rd edition, 1991.Google Scholar
  32. [7.32]
    M. V. Fischetti, N. Sano, S. E. Laux, and K. Natori, “Full-band-structure theory of high-field transport and impact ionization of electrons and holes in Ge, Si, and GaAs”IEEE J. Tech. Comp.Aided Designno.3, 1997.Google Scholar
  33. [7.33]
    J.-P. Nougier, “Fluctuations and noise of hot carriers in semiconducor materials and devices” IEEE Trans. Electron Devices vol. ED-41, no. 11, pp. 2034–2049, 1994.CrossRefGoogle Scholar
  34. [7.34]
    P. Shiktorov, V. Gruzinskis, E. Starikov, T. González, J. Mateos, D. Pardo, L. Reggiani, and L.Varani, “On the spectral strength of the noise source entering the transfer impedance method”Appl. Phys. Lett. vol. 71, pp. 3093–3095, 1997.CrossRefGoogle Scholar
  35. [7.35]
    G. RöpkeStatistische Mechanik für das Nichtgleichgewicht Physik-Verlag, Weinheim, 1987.MATHGoogle Scholar
  36. [7.36]
    A. BlumElektronisches Rauschen B. G. Teubner, Stuttgart, 1996.Google Scholar
  37. [7.37]
    E. V. Sukhorukov and D. Loss, “Noise in multiterminal diffuse conductors: Universality, nonlocality, and exchange effects”Phys. Rev. B vol. 59, pp. 13054–13066,1999CrossRefGoogle Scholar
  38. [7.38]
    J. C. Hensel, H. Hasegawa, and M. Nakayama, “Cyclotron resonance in uniaxially stressed silicon. II. Nature of the covalent bond”Phys. Rev. A vol. 138, pp. 225–238, 1965.Google Scholar
  39. [7.39]
    D. Nuernbergk, “Simulation des Transportverhaltens in Si/Si1_xGex /SiHeterobipolartransistoren”, Dissertation, Technische Universität Ilmenau, Ilmenau, 1999, (H. Utz Verlag Wissenschaft, München: 1999).Google Scholar
  40. [7.40]
    A. van der ZielNoise in Solid State Devices and Circuits John Wiley and Sons, Cansada, 1986.Google Scholar
  41. [7.41]
    F. M. Klaassen and J. Prins, “Thermal noise of MOS transistors”Philips Res. Repts pp. 505–514, 1967.Google Scholar
  42. [7.42]
    P. Klein, “An analytical thermal noise model of deep submicron MOSFET’s”IEEE Electron Device Lett. vol. 20, no. 8, pp. 399–401, 1999.CrossRefGoogle Scholar
  43. [7.43]
    A. J. Scholten, H. J. Tromp, L. F. Tiemeijer, R. van Langevelde, R. J. Havens, P. W. H. de Vreede, R. F. M. Roes, P. H. Woerlee, A. H. Montree,, and D. B. M. Klaassen, “Accurate thermal noise model for deep-submicron CMOS”IEDM Tech. Dig. pp. 155–158, 1999.Google Scholar
  44. [7.44]
    J.-S. Goo, C. Choi, F. Danneville, E. Morifuji, H. Sasaki Momose, Z. Yu, H. Iwai, T. H. Lee, and R. W. Dutton,“An accurate and efficient high frequency noise simulation technique for deep submicron MOSFETs”IEEE Trans. Electron Devicesvol. 47 no. 12, pp. 2410–2419, 2000.Google Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Christoph Jungemann
    • 1
  • Bernd Meinerzhagen
    • 1
  1. 1.Institut für Theoretische Elektrotechnik und MikroelektronikUniversität BremenBremenGermany

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