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Energy-Transport Equations

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Book cover Transport Equations for Semiconductors

Part of the book series: Lecture Notes in Physics ((LNP,volume 773))

The drift-diffusion equations are derived by the moment method by employing only the zeroth-order moment \(\langle M\rangle = \int_B M\mathrm{d}k/4\pi^3\), where the Maxwellian M describes the equilibrium state. As explained in Sect. 2.4, we obtain more general diffusion equations by taking into account higher-order moments. The energy-transport equations are derived by choosing the moments \(n=\langle M\rangle\)(particle density) and \(ne=\langle E(k)M\rangle\)(energy density), where\(E(k)\)is the energy band. The results of Sect. 2.4 are valid only for a simple BGK collision operator. In this chapter, we will assume more realistic scattering terms including elastic, carrier–carrier, and inelastic collision processes. In the following we proceed as in [1] and [2].

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References

  1. N. Ben Abdallah and P. Degond. On a hierarchy of macroscopic models for semiconductors. J. Math. Phys. 37 (1996), 3308–3333.

    Google Scholar 

  2. P. Degond, C. Levermore, and C. Schmeiser. A note on the energy-transport limit of the semiconductor Boltzmann quation. In: N. Ben Abdallah et al. (eds.), Proceedings of Transport in Transition Regimes (Minneapolis, 2000), IMA Math. Appl. 135, 137–153. Springer, New York, 2004.

    Google Scholar 

  3. 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.

    Google Scholar 

  4. R. Stratton. Diffusion of hot and cold electrons in semiconductor barriers. Phys. Rev. 126 (1962), 2002–2014.

    Article  ADS  Google Scholar 

  5. Y. Apanovich, E. Lyumkis, B. Polski, A. Shur, and P. Blakey. A comparison of energy balance and simplified hydrodynamic models for GaAs simulation. COMPEL 12 (1993), 221–230.

    MATH  Google Scholar 

  6. M. Rudan, A. Gnudi, and W. Quade. A generalized approach to the hydrodynamic model of semiconductor equations. In: G. Baccarani (ed.), Process and Device Modeling for Microelectronics, 109–154. Elsevier, Amsterdam, 1993.

    Google Scholar 

  7. D. Chen, E. Kan, U. Ravaioli, C. Shu, and R. Dutton. An improved energy transport model including nonparabolicity and non-Maxwellian distribution effects. IEEE Electr. Device Lett. 13 (1992), 26–28.

    Article  ADS  Google Scholar 

  8. A. Forghieri, R. Guerrieri, P. Ciampolini, A. Gnudi, M. Rudan, and G. Baccarani. A new discretization strategy of the semiconductor equations comprising momentum and energy balance. IEEE Trans. Computer-Aided Design Integr. Circuits Sys. 7 (1988), 231–242.

    Article  Google Scholar 

  9. K. Souissi, F. Odeh, H. Tang, and A. Gnudi. Comparative studies of hydrodynamic and energy transport models. COMPEL 13 (1994), 439–453.

    MATH  MathSciNet  Google Scholar 

  10. D. Woolard, H. Tian, R. Trew, M. Littlejohn, and K. Kim. Hydrodynamic electron-transport: Nonparabolic corrections to the streaming terms. Phys. Rev. B 44 (1991), 11119–11132.

    Article  ADS  Google Scholar 

  11. W. Allegretto and H. Xie. Nonisothermal semiconductor systems. In: X. Liu and D. Siegel (eds.), Comparison Methods and Stability Theory. Lect. Notes Pure Appl. Math. 162, 17–24. Marcel Dekker, New York, 1994.

    Google Scholar 

  12. P. Degond, S. Génieys, and A. Jüngel. A system of parabolic equations in nonequilibrium thermodynamics including thermal and electrical effects. J. Math. Pures Appl. 76 (1997), 991–1015.

    MATH  MathSciNet  Google Scholar 

  13. P. Degond, S. Génieys, and A. Jüngel. A steady-state system in nonequilibrium thermodynamics including thermal and electrical effects. Math. Meth. Appl. Sci. 21 (1998), 1399–1413.

    Article  MATH  Google Scholar 

  14. A. Jüngel. Regularity and uniqueness of solutions to a parabolic system in nonequilibrium thermodynamics. Nonlin. Anal. 41 (2000), 669–688.

    Article  MATH  Google Scholar 

  15. W. Fang and K. Ito. Existence of stationary solutions to an energy drift-diffusion model for semiconductor devices. Math. Models Meth. Appl. Sci. 11 (2001), 827–840.

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Griepentrog. An application of the implicit function theorem to an energy model of the semiconductor theory. Z. Angew. Math. Mech. 79 (1999), 43–51.

    Article  MATH  MathSciNet  Google Scholar 

  17. L. Chen and L. Hsiao. The solution of Lyumkis energy transport model in semiconductor science. Math. Meth. Appl. Sci. 26 (2003), 1421–1433.

    Article  MATH  MathSciNet  Google Scholar 

  18. L. Chen, L. Hsiao, and Y. Li. Large time behavior and energy relaxation time limit of the solutions to an energy transport model in semiconductors. J. Math. Anal. Appl. 312 (2005), 596–619.

    Article  MATH  MathSciNet  Google Scholar 

  19. Y. Apanovich, P. Blakey, R. Cottle, E. Lyumkis, B. Polsky, A. Shur, and A. Tcherniaev. Numerical simulations of submicrometer devices including coupled nonlocal transport and nonisothermal effects. IEEE Trans. Electr. Devices 42 (1995), 890–897.

    Article  ADS  Google Scholar 

  20. M. Fournié. Numerical discretization of energy-transport model for semiconductors using high-order compact schemes. Appl. Math. Letters 15 (2002), 727–734.

    Article  Google Scholar 

  21. C. Ringhofer. An entropy-based finite difference method for the energy transport system. Math. Models Meth. Appl. Sci. 11 (2001), 769–796.

    Article  MATH  MathSciNet  Google Scholar 

  22. F. Bosisio, R. Sacco, F. Saleri, and E. Gatti. Exponentially fitted mixed finite volumes for energy balance models in semiconductor device simulation. In: H. Bock et al. (eds.), Proceedings of ENUMATH 97, 188–197. World Scientific, Singapore, 1998.

    Google Scholar 

  23. P. Degond, A. Jüngel, and P. Pietra. Numerical discretization of energy-transport models for semiconductors with nonparabolic band structure. SIAM J. Sci. Comput. 22 (2000), 986–1007.

    Article  MATH  MathSciNet  Google Scholar 

  24. S. Gadau and A. Jüngel. A 3D mixed finite-element approximation of the semiconductor energy-transport equations. SIAM J. Sci. Comput. 31, (2008), 1120–1140.

    Google Scholar 

  25. S. Holst, A. Jüngel, and P. Pietra. An adaptive mixed scheme for energy-transport simulations of field-effect transistors. SIAM J. Sci. Comput. 25 (2004), 1698–1716.

    Article  MATH  MathSciNet  Google Scholar 

  26. C. Lab and P. Caussignac. An energy-transport model for semiconductor heterostructure devices: Application to AlGaAs/GaAs MODFETs. COMPEL 18 (1999), 61–76.

    MATH  Google Scholar 

  27. A. Marrocco and P. Montarnal. Simulation de modèles ‘‘energy transport’’ á l’aide des éléments finis mixtes. C. R. Acad. Sci. Paris, Sér. I 323 (1996), 535–541.

    MATH  MathSciNet  Google Scholar 

  28. J. Jerome and C.-W. Shu. Energy models for one-carrier transport in semiconductor devices. In: W. Coughran et al. (eds.), Semiconductors, Part II, IMA Math. Appl. 59, 185–207. Springer, New York, 1994.

    Google Scholar 

  29. S. de Groot and P. Mazur. Nonequilibrium Thermodynamics. Dover Publications, New York, 1984.

    Google Scholar 

  30. H. Kreuzer. Nonequilibrium Thermodynamics and Its Statistical Foundation. Clarondon Press, Oxford, 1981.

    Google Scholar 

  31. H. W. Alt and S. Luckhaus. Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Lundstrom. Fundamentals of Carrier Transport. 2nd edition, Cambridge University Press, Cambridge, 2000.

    Book  Google Scholar 

  33. K. Brennan. The Physics of Semiconductors. Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  34. C. Schmeiser and A. Zwirchmayr. Elastic and drift-diffusion limits of electron–phonon interaction in semiconductors. Math. Models Meth. Appl. Sci. 8 (1998), 37–53.

    Article  MATH  MathSciNet  Google Scholar 

  35. E. Lyumkis, B. Polsky, A. Shur, and P. Visocky. Transient semiconductor device simulation including energy balance equation. COMPEL 11 (1992), 311–325.

    MATH  Google Scholar 

  36. A. Jüngel. Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, Basel, 2001.

    MATH  Google Scholar 

  37. Y. Choi and R. Lui. Multi-dimensional electrochemistry model. Arch. Rat. Mech. Anal. 130 (1995), 315–342.

    Article  MATH  MathSciNet  Google Scholar 

  38. Z. Deyl (ed.). Electrophoresis: A Survey of Techniques and Applications. Elsevier, Amsterdam, 1979.

    Google Scholar 

  39. A. Bermudez and C. Saguez. Mathematical formulation and numerical solution of an alloy solidification problem. In: A. Fasano (ed.), Free Boundary Problems: Theory and Applications, Vol.\ 1, 237–247. Pitman, Boston, 1983.

    Google Scholar 

  40. R. Hills, D. Loper, and P. Roberts. A thermodynamically consistent model of a mushy zone. Quart. J. Mech. Appl. Math. 36 (1983), 505–539.

    Article  MATH  Google Scholar 

  41. S. de Groot. Thermodynamik irreversibler Prozesse. Bibliographisches Institut, Mann\-heim, 1960.

    Google Scholar 

  42. G. Albinus. A thermodynamically motivated formulation of the energy model of semiconductor devices. Preprint No.\ 210, WIAS Berlin, Germany, 1995.

    Google Scholar 

  43. S. Kawashima and Y. Shizuta. On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J., II.\ Ser. 40 (1988), 449–464.

    Article  MATH  MathSciNet  Google Scholar 

  44. P. Degond, S. Génieys, and A. Jüngel. Symmetrization and entropy inequality for general diffusion equations. C. R. Acad. Sci. Paris, Sér. I 325 (1997), 963–968.

    MATH  ADS  Google Scholar 

  45. S. Holst, A. Jüngel, and P. Pietra. A mixed finite-element discretization of the energy-transport equations for semiconductors. SIAM J. Sci. Comput. 24 (2003), 2058–2075.

    Article  MATH  MathSciNet  Google Scholar 

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  1. TU Wien Inst. Analysis und Scientific Computing, Wiedner Hauptstr. 8-10, 1040 Wien, Austria

    Ansgar Jüngel (Prof. Dr)

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  1. Ansgar Jüngel
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Jüngel, A. (2009). Energy-Transport Equations. In: Transport Equations for Semiconductors. Lecture Notes in Physics, vol 773. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89526-8_6

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  • DOI: https://doi.org/10.1007/978-3-540-89526-8_6

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