A hierarchy of diffusive higher-order moment equations for semiconductors

SIAM Journal on Applied Mathematics

Volumn 68, Issue 1, 2007, Pages 171-198

  Jüngel, Ansgar ^a   Krause, Stefan ^b   Pietra, Paola ^c  

^a VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^b JOHANNES GUTENBERG UNIVERSITY   (Germany)

^c CNR   (Italy)

Author keywords

Chapman Enskog expansion; Energy transport model; Entropy maximization; Higher order moments; Moment method; Semiconductor Boltzmann equation

Indexed keywords

BOLTZMANN EQUATION; ENERGY TRANSFER; ENTROPY; PARTIAL DIFFERENTIAL EQUATIONS;

ENERGY-TRANSPORT MODEL; HIGHER-ORDER MOMENTS; MOMENT METHOD; SEMICONDUCTOR BOLTZMANN EQUATIONS;

SEMICONDUCTOR MATERIALS;

EID: 41849131777     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/070683313     Document Type: Article

References (55)

1. A. M. ANILE AND O. MUSCATO, Improved hydrodynamic model for carrier transport in semiconductors, Phys. Rev. B, 51 (1995), pp. 16728-16740.
2. A. M. ANILE, N. NIKIFORAKIS, V. ROMANO, AND G. RUSSO, Discretization of semiconductor device problems (II), in Handbook of Numerical Analysis XIII, W. H. A. Schilders and E. J. W. ter Maten, eds., Numerical Methods in Electromagnetics, North-Holland, Amsterdam, 2005.
3. A. M. ANILE AND S. PENNISI, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), pp. 13186-13193.
4. A. M. ANILE, G. RUSSO, AND V. ROMANO, Extended hydrodynamical model of carrier transport in semiconductors, SIAM J. Appl. Math., 61 (2000), pp. 74-101.
5. G. BACCARANI AND M. R. WORDEMAN, An investigation of steady-state velocity overshoot in silicon, Solid-State Electr., 29 (1982), pp. 970-977.
6. N. B. ABDALLAH AND P. DEGOND, On a hierarchy of macroscopic models for semiconductors, J. Math. Phys., 37 (1996), pp. 3306-3333.
7. N. B. ABDALLAH, P. DEGOND, AND S. GÉ NIEYS, An energy-transport model for semiconductors derived from the Boltzmann equation, J. Stat. Phys., 84 (1996), pp. 205-231.
8. K. BLØTEKJÆR, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), pp. 38-47.
9. D. CASSI AND B. RICCÒ, An analytical model of the energy distribution of hot electrons, IEEE Trans. Electron Devices, 37 (1990), pp. 1514-1521.
10. S. CHAPMAN AND T. G. COWLING, The Mathematical Theory of Non-Uniform Gases, Cambridge University Press, Cambridge, 1958.
11. D. CHEN, E. KAN, U. RAVAIOLI, C. SHU, AND R. DUTTON, An improved energy transport model including nonparabolicity and non-Maxwellian distribution effects, IEEE Electron Device Lett., 13 (1992), pp. 26-28.
12. J. F. COULOMBEL, F. GOLSE, AND T. GOUDON, Diffusion approximation and entropy-based moment closure for kinetic equations, Asymptot. Anal., 45 (2005), pp. 1-39.
13. 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), pp. 991-1015.
14. P. DEGOND, S. GÉNIEYS. AND A. JÜNGEL, Symmetrization and entropy inequality for general diffusion equations, C. R. Acad. Sci. Paris, 325 (1997), pp. 963-968.
15. 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), pp. 986-1007.
16. P. DEGOND, C. LEVERMORE, AND C. SCHMEISER, A note on the energy-transport limit of the semiconductor Boltzmann equation, in Proceedings of Transport in Transition Regimes (Minneapolis, 2000), IMA Vol. Math. Appl. 135, Springer, New York, 2004.
17. P. DEGOND AND C. RINGHOFER, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), pp. 587-628.
18. W. DREYER, Maximisation of the entropy in non-equilibrium, J. Phys. A, 20 (1987), pp. 6505-6517.
19. S. DE GROOT, Thermodynamik irreversibler Prozesse, Bibliographisches Institut, Mannheim, 1960.
20. P. DMITRUK, A. SAUL, AND L. REYNA, High electric field approximation to charge transport in semiconductor devices, Appl. Math. Lett., 5 (1992), pp. 99-102.
21. W. DREYER, M. JUNK, AND M. KUNIK, On the approximation of kinetic equations by moment systems, Nonlinearity, 14 (2001), pp. 881-906.
22. W. FANG AND K. ITO, Existence of stationary solutions to an energy drift-diffusion model for semiconductor devices, Math. Models Methods Appl. Sci., 11 (2001), pp. 827-840.
23. S. GADAU AND A. JÜNGEL, Finite-Element Approximation of the 3D Energy-Transport Models for Semiconductors, manuscript, 2007.
24. B. GEURTS, M. NEKOVEE, H. BOOTS, AND M. SCHUURMANS, Exact and moment equation modeling of electron transport in submicron structures, Appl. Phys. Lett., 59 (1991), pp. 1743-1745.
25. T. GRASSER, Non-parabolic macroscopic transport models for semiconductor device simulation, Phys. A, 349 (2005), pp. 221-258.
26. T. GRASSER, R. KOSIK. C. JUNGEMANN, H. KOSINA, AND S. SELBERHERR, Nonparabolic macroscopic transport models for device simulation based on bulk Monte Carlo data, J. Appl. Phys., 97 (2005), 093710.
27. T. GRASSER, H. KOSINA, M. GRITSCH, AND S. SELBERHERR, Using six moments of Boltzmann's equation for device simulation, J. Appl. Phys., 90 (2001), pp. 2389-2396.
28. T. GRASSER, H. KOSINA, C. HEITZINGER AND S. SELBERHERR, Characterization of the hot electron distribution function using six moments, J. Appl. Phys., 91 (2002), pp. 38693879.
29. T. GRASSER, H. KOSINA. AND S. SELBERHERR, Hot carrier effects within macroscopic transport models, Internat. J. High Speed Electr. Sys., 13 (2003), pp. 873-901.
30. T. GRASSER, T.-W. TANG. H. KOSINA. AND S. SELBERHERR, A review of hydrodynamic and energy-transport models for semiconductor device simulation, Proc. IEEE, 91 (2003), pp. 251-274.
31. J. GRIEPENTROG, An application o/ the implicit function theorem to an energy model of the semiconductor theory, Z. Angew. Math. Mech., 79 (1999), pp. 43-51.
32. S. HOLST, A. JUNGEL, AND P. PIETRA, A mixed finite-element discretization of the energytransport model for semiconductors, SIAM J. Sci. Comput., 24 (2003), pp. 2058-2075.
33. S. IHARA, Information Theory for Continuous Systems, World Scientific, Singapore, 1993.
34. J. JEROME, Analysis of Charge Transport. A Mathematical Study of Semiconductor Devices, Springer, Berlin, 1996.
35. M. JUNK, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), pp. 1143-1167.
36. A. JÜNGEL, Quasi-hydrodynamic Semiconductor Equations, Progr. Nonlinear Differential Equations, Birkhäuser, Basel, 2001.
37. A. JÜNGEL, D. MATTHES. AND J. MILIŠ IÓ, Derivation of new quantum hydrodynamic equations using entropy minimization, SIAM J. Appl. Math., 67 (2006), pp. 46-68.
38. M. JUNK AND V. ROMANO, Maximum entropy moment systems of the semiconductor Boltzmann equation using Kane's dispersion relation, Contin. Mech. Thermodyn., 17 (2004), pp. 247-267.
39. S. KAWASHIMA AND Y. SHIZUTA, On the normal form of the symmetric hyperbolic-parabolic systems associated with conservation laws, Tohoku Math. J., 40 (1988), pp. 449-464.
40. H. KREUZER, Nonequilibrium Thermodynamics and Its Statistical Foundation, Clarendon Press, Oxford, 1981.
41. C. LEVERMORE, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), pp. 1021-1065.
42. S. LIOTTA AND H. STRUCHTRUP, Moment equations for electrons in semiconductors: Comparison of spherical harmonics and full moments, Solid-State Electr., 44 (2000), pp. 95-103.
43. E. LYUMKIS, B. POLSKY. A. SHUR. AND P. VISOCKY, Transient semiconductor device simulation including energy balance equations, COMPEL, 11 (1992), pp. 311-325.
44. P. MARKOWICH, C. RINGHOFER, AND C. SCHMEISER, Semiconductor Equations, Springer, Vienna, 1990.
45. I. MÜLLER AND T. RUGGERI, Extended Thermodynamics, Springer, New York, 1993.
46. M. NEKOVEE, B. GEURTS, H. BOOTS, AND M. SCHUURMANS, Failure of extended-momentequation approaches to describe ballistic tmnsport in submicrometer structures, Phys. Rev. B, 45 (1992), pp. 6643-6651.
47. V. ROMANO, Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devices, Math. Methods Appl. Sci., 24 (2001), pp. 439-471.
48. M. RUDAN AND G. BACCARANI, On the structure and closure condition oj the hydrodynamical model, VLSI Design, 3 (1995), pp. 115-129.
49. C. SCHMEISER AND A. ZWIRCHMAYR, Convergence of moment methods for linear kinetic equations, SIAM J. Numer. Anal., 36 (1998), pp. 74-88.
50. J. SCHNEIDER, Entropic approximation in kinetic theory, ESAIM: Math. Mod. Numer. Anal., 38 (2004), pp. 541-561.
51. K.-I. SONODA, M. YAMAJI, K. TANIGUCHI, C. HAMAGUCHI, AND S. DUNHAM, Moment expansion approach to calculate impact ionization rate in submicron silicon devices, J. Appl. Phys., 80 (1996), pp. 5444-5448.
52. R. STRATTON, Diffusion of hot and cold electrons in semiconductor barriers, Phys. Rev., 126 (1962), pp. 2002-2014.
53. H. STRUCHTRUP, Extented moment method for electrons in semiconductors, Phys. A, 275 (2000), pp. 229-255.
54. H. STRUCHTRUP, Derivation of 13 moment equations for rarefied gas flow to second order accuracy for arbitrary interaction potentials, Multiscale Model. Simul., 3 (2005), pp. 221-243.
55. D. WOOLARD, H. TIAN, R. TREW, M. LITTLEJOHN, AND K. KIM, Hydrodynamic electrontransport model: Nonparabolic corrections to the streaming terms, Phys. Rev. B, 44 (1991), pp. 11119-11132.