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Multiband quantum transport models for semiconductor devices

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Transport Phenomena and Kinetic Theory

Abstract

The modeling of semiconductor devices, which is a very active and intense field of research, has to keep up with the speed at which the fabrication technology proceeds; the devices of the last generations have become increasingly smaller, reaching a size so small that quantum effects dominate their behaviour. Quantum effects such as resonant tunneling and other size-quantized effects cannot be described by classical or semiclassical theories; they need a full quantum description [Fre90, JAC92, KKFR89, MRS90, RBJ91, RBJ92]. A very important feature, which has appeared in the devices of the last generation and which requires a full quantum treatment, is the presence of the interband current: a contribution to the total current which arises from transitions between the conduction and the valence band states. Resonant interband tunneling diodes (RITDs) are examples of semiconductor devices which exploit this phenomenon; they are of big importance in nanotechnology for their applications to high-speed and miniaturized systems [YSDX91, SX89]. In the band diagram structure of these diodes there is a small region where the valence band edge lies above the conduction band edge (valence quantum well), making interband resonance possible.

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References

  1. Alì, G., Frosali, G.: Quantum hydrodynamic models for the two-band Kane system, Nuovo Cimento B, 120(12), 1279–1298 (2005).

    Google Scholar 

  2. Alì, G., Frosali, G., and Manzini, C.: On the drift-diffusion model for a two-band quantum fluid at zero-temperature. Ukrainian Math. J., 57(6), 723–730 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  3. Allaire, G., Piatnitski, A.: Homogenization of the Schrödinger equation and effective mass theorems. Comm. Math. Phys., 258(1), 1–22 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  4. Barletti, L.: A mathematical introduction to the Wigner formulation of quantum mechanics. Boll. Unione Mat. Ital., B, 6-B, 693–716 (2003).

    MathSciNet  Google Scholar 

  5. Barletti, L.: Wigner envelope functions for electron transport in semiconductor devices. Transport Theory Statist. Phys., 32(3&4), 253–277 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  6. Barletti, L.: A “spinorial” Wigner function describing the two-band kp dynamics of electrons in crystals. In: Primicerio, M., Spigler, R., and Valente, V. (eds.) Applied and Industrial Mathematics in Italy. World Scientific, Singapore (2005).

    Google Scholar 

  7. Barletti, L.: On the thermal equilibrium of a quantum system described by a two-band Kane Hamiltonian. Nuovo Cimento B, 119(12), 1125–1140 (2004).

    Google Scholar 

  8. Barletti, L.: Quantum moment equations for a two-band k.p Hamiltonian. Boll. Unione Mat. Ital., B, 8-B, 103–121 (2005).

    MathSciNet  Google Scholar 

  9. Barletti, L., Demeio, L.: Wigner function approach to multiband transport in semiconductor devices, Proc. VI Congresso SIMAI, Chia Laguna (CA-Italy) May 27–31, 2002. Versione su CD-rom.

    Google Scholar 

  10. Bensoussan, A., Lions, J.-L., and Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978).

    MATH  Google Scholar 

  11. Bordone, P., Pascoli, M., Brunetti, R., Bertoni, A., and Jacoboni, C.: Quantum transport of electrons in open nanostructures with the Wigner function formalism, Phys. Rev. B, 59, 3060–3069 (1999).

    Article  Google Scholar 

  12. Borgioli, G., Frosali, G., and Zweifel, P. F.: Wigner approach to the two-band Kane model for a tunneling diode. Transport Theory Statist. Phys. 32(3&4), 347–366 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  13. Buot, F. A.: Method for calculating TrH n in solid-state theory. Phys. Rev. B, 10, 3700–3705 (1974).

    Article  Google Scholar 

  14. Buot, F. A.: Magnetic susceptibility of interacting free and Bloch electrons. Phys. Rev. B, 14, 3310–3328 (1976).

    Article  Google Scholar 

  15. Buot, F. A.: Direct construction of path integrals in the lattice-space multiband dynamics of electrons in a solid. Phys. Rev. A, 33, 2544–2562 (1986).

    Article  Google Scholar 

  16. Buot, F. A., Jensen, K. L.: Lattice Weyl-Wigner formulation of exact many-body quantum-transport theory and applications to novel solid-state quantum-based devices. Phys. Rev. B, 42, 9429–9457 (1990).

    Article  Google Scholar 

  17. Burt, M. G.: The justification for applying the effective-mass approximation to microstructure. J. Phys: Condens. Matter, 4, 6651–6690 (1992).

    Article  Google Scholar 

  18. Degond, P., Ringhofer, C.: Quantum moment hdrodynamics and the entropy principle. Journal Stat. Phys., 112(3), 587–628 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  19. Demeio, L., Barletti, L., Bertoni, A., Bordone, P., and Jacoboni, C.: Wigner function approach to multiband transport in semiconductors. Physica B, 314, 104–107 (2002).

    Article  Google Scholar 

  20. Demeio, L., Barletti, L., Bordone P., and Jacoboni, C.: Wigner function for multiband transport in semiconductors. Transport Theory Statist. Phys., 32(3&4), 307–325 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  21. Demeio, L., Bordone, P., and Jacoboni, C.: Numerical and analytical applications of multiband transport in semiconductors. Proc. XXIII Symposium on Rarefied Gas Dynamics, Whistler, BC, Canada, July 20–25, 2002, pp. 92–98 (AIP Conference Proceedings vol. 663, New York, 2003).

    Google Scholar 

  22. Demeio, L., Bordone, P., and Jacoboni, C.: Multi-band, non-parabolic Wigner function approach to electron transport in semiconductors. Internal Report N. 3/2003, Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, April 2003, Transport Theory Statist. Phys., (to appear).

    Google Scholar 

  23. Feynman, R. P.: Statistical Mechanics: A Set of Lectures, Addison-Wesley, Reading (1972).

    Google Scholar 

  24. Frensley, W. R.: Boundary conditions for open quantum systems far from equilibrium. Rev. Mod. Phys., 62, 745–791(1990).

    Article  Google Scholar 

  25. Frosali, G., Morandi, O.: A quantum kinetic approach for modeling a two-band resonant tunneling diode. Transport Theory Statist. Phys. (submitted).

    Google Scholar 

  26. Gardner, C. L.: The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54, 409–427 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  27. Gasser, I., Markowich, P. A., and Unterreiter, A.: Quantum hydrodynamics. In Proceedings of the SPARCH GdR Conference, held in St. Malo (1995).

    Google Scholar 

  28. Iafrate, G. J., Krieger, J. B.: Quantum transport for Bloch electrons for inhomogeneous electric fields. Phys. Rev. B, 40, 6144–6148 (1989).

    Article  Google Scholar 

  29. Jacoboni, C.: Comparison between quantum and classical results in hot-electron transport, Semicomd. Sci. Technol., 7, B6–B11 (1992).

    Article  Google Scholar 

  30. Jacoboni, C., Brunetti, R., Bordone, P., and Bertoni, A.: Quantum transport and its simulation with the Wigner function approach. In Brennan, K., Paul Ruden, P. (eds.) Topics in High Field Transport in Semiconductors. World Scientific, Singapore (2001), pp. 25–61.

    Google Scholar 

  31. Jüngel, A.: Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, Basel (2001).

    MATH  Google Scholar 

  32. Kane, E. O.: Energy band structure in p-type Germanium and Silicon. J. Phys. Chem. Solids, 1, 82–89 (1956).

    Article  Google Scholar 

  33. Kefi, J.: Analyse mathématique et numérique de modèles quantiques pour les semiconducteurs. PhD Thesis, Université Toulouse III-Paul Sabatier (2003).

    Google Scholar 

  34. Kluksdahl, N. C., Kriman, A. M., Ferry, D. K., and Ringhofer, C.: Self-consistent study of the resonant-tunneling diode. Phys. Rev. B, 39(11), 7720–7735 (1989).

    Article  Google Scholar 

  35. Krieger, J. B., Iafrate, G. J.: Quantum transport for bloch electrons in a spatially homogeneous electric field. Phys. Rev. B, 35, 9644–9658 (1987).

    Article  Google Scholar 

  36. Landau, L. D., Lifshitz, E. M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford (1977).

    Google Scholar 

  37. Luttinger, J. M., Kohn, W.: Motion of electrons and holes in perturbed periodic fields. Phys. Rev. II, 97, 869–882 (1955).

    Article  MATH  Google Scholar 

  38. Markowich, P. A., Mauser, N. J., and Poupaud, F.: A Wigner function approach to (semi)classical limits: Electrons in a periodic potential. J. Math. Phys., 35(3), 1066–1094 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  39. Markowich, P. A., Ringhofer, Ch. A., and Schmeiser, Ch.: Semiconductor Equations. Springer-Verlag, Wien (1990).

    MATH  Google Scholar 

  40. Modugno, M., Morandi, O.: A multiband envelope function model for quantum transport in a tunneling diode. Phys. Rev. B, 71, 235331 (2005).

    Article  Google Scholar 

  41. Poupaud, F., Ringhofer, C.: Semi-classical limits in a crystal with exterior potentials and effective mass theorems. Comm. Partial Differential Equations 21(11–12), 1897–1918 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  42. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. Academic Press, New York (1972).

    Google Scholar 

  43. Rossi, F., Brunetti, R., and Jacoboni, C.: An introduction to charge quantum transport in semiconductors and numerical approaches. In Ferry, D. K., Barker, J. R. (eds.) Granular Nanoelectronics. Plenum Press, New York (1991), pp. 43–61.

    Google Scholar 

  44. Rossi, F., Brunetti, R., and Jacoboni, C.: Quantum Transport, in Shah, J. (ed.) Hot Carriers in Semiconductors Nanostructures: Physics and Applications. Academic Press, San Diego (1992), pp. 153–188.

    Google Scholar 

  45. Sweeney M., Xu, J. M.: Resonant interband tunnel diodes. Appl. Phys. Lett., 54(6), 546–548 (1989).

    Article  Google Scholar 

  46. Wannier, G. H.: Dynamics of band electrons in electric and magnetic fields. Rev. Mod. Phys., 34, 645–655 (1962).

    Article  MathSciNet  Google Scholar 

  47. Wenckebach, W. T.: Essential of Semiconductor Physics. John Wiley & Sons, Chichester (1999).

    Google Scholar 

  48. Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev., 40, 749–759 (1932).

    Article  MATH  Google Scholar 

  49. Yang, R. Q., Sweeny, M., Day, D., and Xu, J. M.: Interband tunneling in heterostructure tunnel diodes. IEEE Transactions on Electron Devices, 38(3), 442–446 (1991).

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, I-50134, Firenze, Italy

    Luigi Barletti

  2. Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche 1, I-60131, Ancona, Italy

    Lucio Demeio

  3. Dipartimento di Matematica “G.Sansone”, Università di Firenze, Via S. Marta 3, I-50139, Firenze, Italy

    Giovanni Frosali

Authors
  1. Luigi Barletti
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  2. Lucio Demeio
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  3. Giovanni Frosali
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Editors and Affiliations

  1. Dipartimento di Matematica, Politecnico di Torino, 32 Piazza Leonardo da Vinci, 20133, Milano, Italy

    Carlo Cercignani

  2. Dipartimento di Matematica “F. Castorati”, Università degli Studi di Pavia, 1 via Ferrata, 27100, Pavia, Italy

    Ester Gabetta

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Barletti, L., Demeio, L., Frosali, G. (2007). Multiband quantum transport models for semiconductor devices. In: Cercignani, C., Gabetta, E. (eds) Transport Phenomena and Kinetic Theory. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4554-0_4

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  • DOI: https://doi.org/10.1007/978-0-8176-4554-0_4

  • Publisher Name: Birkhäuser Boston

  • Print ISBN: 978-0-8176-4489-5

  • Online ISBN: 978-0-8176-4554-0

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