Skip to main content
SpringerLink
Log in

Wigner Function Approach

  • Chapter
  • First Online:
Book cover Nano-Electronic Devices

Abstract

The Wigner function formalism has been introduced with an emphasis on basic theoretical aspects, and recently developed numerical approaches and applications for modeling and simulation of the transport of current carriers in electronic structures. Two alternative ways: the historical introduction of the function on top of the operator mechanics, and an independent formulation of the Wigner theory in phase space which then recovers the operator mechanics, demonstrate that the formalism provides an autonomous description of the quantum world.

The conditions of carrier transport in nano-electronic devices impose to extend this coherent physical picture by processes of interaction with the environment. Relevant becomes the Wigner–Boltzmann equation, derived for the case of interaction with phonons and impurities. The numerical aspects focus on two particle models developed to solve this equation. These models make the analogy between classical and Wigner transport pictures even closer: particles are merely classical, the only characteristics which carries the quantum information is a dimensionless quantity – affinity or sign.

The recent ground-breaking applications of the affinity method for simulation of typical nano-devices as the resonant tunneling diode and the ultra-short DG-MOSFET firmly establish the Wigner–Boltzmann equation as a bridge between coherent and semi-classical transport pictures. It became a basic route to understand the nano-device operation as an interplay between coherent and de-coherence phenomena. The latter, due to the environment: phonon field, contacts or defects, attempts to recover the classical transport picture.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. H. Weyl, “Quantenmechanik und Gruppentheorie,” Zeitschrift fr Physik, vol. 46, pp. 1–46, 1927.

    Article  Google Scholar 

  2. E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Physical Review, vol. 40, pp. 749–759, 1932.

    Article  MATH  Google Scholar 

  3. J. E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 99–124, 1949.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. I. Tatarskii, “The Wigner Representation of Quantum Mechanics,” Sov. Phys. Usp., vol. 26, pp. 311–327, 1983.

    Article  MathSciNet  Google Scholar 

  5. N. C. Dias and J. N. Prata, “Admissible states in quantum phase space,” Annals of Physics, vol. 313, pp. 110–146, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. K. Ferry, R. Akis, and J. P. Bird, “Einselection in action: decoherence and pointer states in open quantum dots,” Physical Review Letters, vol. 93, p. 026803, 2004.

    Article  Google Scholar 

  7. I. Knezevic, “Decoherence due to contacts in ballistic nanostructures,” Physical Review B, vol. 77, p. 125301, 2008.

    Article  Google Scholar 

  8. F. Buscemi, P. Bordone, and A. Bertoni, “Simulation of decoherence in 1D systems, a comparison between distinguishable- and indistinguishable-particle collisions,” Physica Status Solidi (c), vol. 5, pp. 139–142, 2008.

    Google Scholar 

  9. F. Buscemi, E. Cancellieri, P. Bordone, A. Bertoni, and C.Jacoboni, “Electron decoherence in a semiconductor due to electron-phonon scattering,” Physica Status Solidi (c), vol. 5, pp. 52–55, 2008.

    Google Scholar 

  10. D. Querlioz, J. Saint-Martin, A. Bournel, and P. Dollfus, “Wigner Monte Carlo simulation of phonon induced electron decoherence in semiconductor nanodevices,” Physical Review B, vol. 78, p. 165306, 2008.

    Article  Google Scholar 

  11. R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics. Wiley and Sons, 1975.

    Google Scholar 

  12. N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and C. Ringhofer, “Self-consistent study of resonant-tunneling diode,” Physical Review B, vol. 39, pp. 7720–7734, 1989.

    Article  Google Scholar 

  13. A. Gehring and H. Kosina, “Wigner-Function Based Simulation of Classic and Ballistic Transport in Scaled DG-MOSFETs Using the Monte Carlo Method,” Journal of Compuational Electronics, vol. 4, pp. 67–70, 2005.

    Article  Google Scholar 

  14. P. Carruthers and F. Zachariasen, “Quantum Collision Theory with Phase-Space Distributions,” Rev.Mod.Phys., vol. 55, no. 1, pp. 245–285, 1983.

    Google Scholar 

  15. B. Biegel and J. Plummer, “Comparison of self-consistency iteration options for the Wigner function method of quantum device simulation,” Physical Review B, vol. 54, pp. 8070–8082, 1996.

    Article  Google Scholar 

  16. W. Frensley, “Wigner-Function Model of Resonant-Tunneling Semiconductor Device,” Physical Review B, vol. 36, no. 3, pp. 1570–1580, 1987.

    Article  Google Scholar 

  17. W. Frensley, “Boundary conditions for open quantum systems driven far from equilibrium,” Reviews of Modern Physics, vol. 62, no. 3, pp. 745–789, 1990.

    Article  Google Scholar 

  18. K. Gullapalli, D. Miller, and D. Neikirk, “Simulation of quantum transport in memory-switching double-barrier quantum-well diodes,” Physical Review B, vol. 49, pp. 2622–2628, 1994.

    Article  Google Scholar 

  19. F. A. Buot and K. L. Jensen, “Lattice Weil-Wigner Formulation of Exact-Many Body Quantum-Transport Theory and Applications to Novel Solid-State Quantum-Based Devices,” Physical Review B, vol. 42, no. 15, pp. 9429–9457, 1990.

    Article  Google Scholar 

  20. R. K. Mains and G. I. Haddad, “Wigner function modeling of resonant tunneling diodes with high peak-to-valley ratios,” Journal of Applied Physics, vol. 64, pp. 5041–5044, 1988.

    Article  Google Scholar 

  21. D. Querlioz, H. N. Nguyen, J. Saint-Martin, A. Bournel, S. Galdin-Retailleau, and P. Dollfus, “Wigner-Boltzmann Monte Carlo approach to nanodevice simulation: from quantum to semiclassical transport,” Journal of Computational Electronics, vol. 8, pp. 324–335, 2009.

    Article  Google Scholar 

  22. M. Nedjalkov, “Wigner transport in presence of phonons: Particle models of the electron kinetics,” in From Nanostructures to Nanosensing Applications, Proceedings of the International School of Physics ‘Enrico Fermi’ (A. P. A. D’Amico, G. Balestrino, ed.), vol. 160, (Amsterdam), pp. 55–103, IOS Press, 2005.

    Google Scholar 

  23. F. Rossi, C.Jacoboni, and M.Nedjalkov, “A Monte Carlo Solution of the Wigner Transport Equation,” Semiconductor Sci. Technology, vol. 9, pp. 934–936, 1994.

    Article  Google Scholar 

  24. P. Bordone, M. Pascoli, R. Brunetti, A. Bertoni, and C. Jacoboni, “Quantum transport of electrons in open nanostructures with the Wigner-function formalism,” Physical Review B, vol. 59, no. 4, pp. 3060–3069, 1999.

    Article  Google Scholar 

  25. I. Levinson, “Translational invariance in uniform fields and the equation for the density matrix in the Wigner representation,” Soviet Phys. JETP, vol. 30, no. 2, pp. 362–367, 1970.

    MathSciNet  Google Scholar 

  26. J. R. Barker and D. K. Ferry, “Self-Scattering Path-Variable Formulation of High Field, Time-Dependent Quantum Kinetic Equations for Semiconductor Transport in the Finite-Collision-Duration Regime,” Physical Review Letters, vol. 42, no. 26, pp. 1779–1781, 1979.

    Article  Google Scholar 

  27. M. Nedjalkov, D. Vasileska, D. Ferry, C. Jacoboni, C. Ringhofer, I. Dimov, and V. Palankovski, “Wigner transport models of the electron-phonon kinetics in quantum wires,” Physical Review B, vol. 74, pp. 035311–1–035311–18, July 2006.

    Google Scholar 

  28. J. Schilp, T. Kuhn, and G. Mahler, “Electron-phonon quantum kinetics in pulse-excited semiconductors: Memory and renormalization effects,” Physical Review B, vol. 50, no. 8, pp. 5435–5447, 1994.

    Article  Google Scholar 

  29. C. Fuerst, A. Leitenstorfer, A. Laubereau, and R. Zimmermann, “Quantum Kinetic Electron-Phonon Interaction in GaAs: Energy Nonconserving Scattering Events and Memory Effects,” Physical Review Letters, vol. 78, pp. 3733–3736, 1997.

    Article  Google Scholar 

  30. P. Bordone, D. Vasileska, and D. Ferry, “Collision-Duration Time for Optical-Phonon Emission in Semiconductors,” Physical Review B, vol. 53, no. 7, pp. 3846–3855, 1996.

    Article  Google Scholar 

  31. T. Kuhn and F. Rossi, “Monte Carlo Simulation of Ultrafast Processes in Photoexcited Semiconductors: Coherent and Incoherent Dynamics,” Physical Review B, vol. 46, pp. 7496–7514, 1992.

    Article  Google Scholar 

  32. F. Rossi and T. Kuhn, “Theory of Ultrafast Phenomena in Photoexcited Semiconductors,” Reviews of Modern Physics, vol. 74, pp. 895–950, July 2002.

    Article  Google Scholar 

  33. K. Thornber, “High-field electronic conduction in insulators,” Solid-State Electron., vol. 21, pp. 259–266, 1978.

    Article  Google Scholar 

  34. J. Barker and D. Ferry, “On the Physics and Modeling of Small Semiconductor Devices–I,” Solid-State Electron., vol. 23, pp. 519–530, 1980.

    Article  Google Scholar 

  35. M. V. Fischetti, “Monte Carlo Solution to the Problem of High-Field Electron Heating in SiO 2,” Physical Review Letters, vol. 53, no. 3, p. 1755, 1984.

    Google Scholar 

  36. C. Jacoboni, A. Bertoni, P. Bordone, and R. Brunetti, “Wigner-function Formulation for Quantum Transport in Semiconductors: Theory and Monte Carlo Approach,” Mathematics and Computers in Simulations, vol. 55, no. 1-3, pp. 67–78, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  37. P. Bordone, A. Bertoni, R. Brunetti, and C. Jacoboni, “Monte Carlo simulation of quantum electron transport based on Wigner paths,” Mathematics and Computers in Simulation, vol. 62, p. 307, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Lipavski, F. Khan, F. Abdolsalami, and J. Wilkins, “High-Field Transport in Semiconductors. I. Absence of the Intra-Collisional Field Effect,” Physical Review B, vol. 43, no. 6, pp. 4885–4896, 1991.

    Google Scholar 

  39. T. Gurov, M. Nedjalkov, P. Whitlock, H. Kosina, and S. Selberherr, “Femtosecond relaxation of hot electrons by phonon emission in presence of electric field,” Physica B, vol. 314, pp. 301–304, 2002.

    Article  Google Scholar 

  40. M. Nedjalkov, D. Vasileska, E. Atanassov, and V. Palankovski, “Ultrafast Wigner Transport in Quantum Wires,” Journal of Computational Electronics, vol. 6, pp. 235–238, 2007.

    Article  Google Scholar 

  41. C. Ringhofer, M. Nedjalkov, H. Kosina, and S. Selberherr, “Semi-Classical Approximation of Electron-Phonon Scattering beyond Fermi’s Golden Rule,” SIAM Journal of Applied Mathematics, vol. 64, pp. 1933–1953, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Herbst, M. Glanemann, V. Axt, and T. Kuhn, “Electron-phonon quantum kinetics for spatially inhomogenenous excitations,” Phisical Review B, vol. 67, pp. 195305–1–195305–18, 2003.

    Google Scholar 

  43. P. Bordone, A. Bertoni, R. Brunetti, and C. Jacoboni, “Monte Carlo Simulation of Quantum Electron Transport Based on Wigner Paths,” Mathematics and Computers in Simulation, vol. 62, pp. 307–314, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  44. R. Brunetti, C. Jacoboni, and F. Rossi, “Quantum theory of transient transport in semiconductors: A Monte Carlo approach,” Physical Review B, vol. 39, pp. 10781–10790, May 1989.

    Article  Google Scholar 

  45. B. K. Ridley, Quantum processes in semiconductors. Oxford University Press, fourth ed., 1999.

    Google Scholar 

  46. K.-Y. Kim and B. Lee, “On the high order numerical calculation schemes for the Wigner transport equation,” Solid-State Electronics, vol. 43, pp. 2243–2245, 1999.

    Article  Google Scholar 

  47. Y. Yamada, H. Tsuchiya, and M. Ogawa, “Quantum Transport Simulation of Silicon-Nanowire Transistors Based on Direct Solution Approach of the Wigner Transport Equation,” IEEE Trans. Electron Dev., vol. 56, pp. 1396–1401, 2009.

    Article  Google Scholar 

  48. S. Barraud, “Phase-coherent quantum transport in silicon nanowires based on Wigner transport equation: Comparison with the nonequilibrium-Green-function formalism,” Journal of Applied Physics, vol. 106, p. 063714, 2009.

    Article  Google Scholar 

  49. H. Tsuchiya and U. Ravaioli, “Particle Monte Carlo Simulation of Quantum Phenomena in Semiconductor Devices,” J.Appl.Phys., vol. 89, pp. 4023–4029, April 2001.

    Google Scholar 

  50. R. Sala, S. Brouard, and G. Muga, “Wigner Trajectories and Liouville’s theorem,” J. Chem. Phys., vol. 99, pp. 2708–2714, 1993.

    Article  Google Scholar 

  51. P. Vitanov, M. Nedjalkov, C. Jacoboni, F. Rossi, and A. Abramo, “Unified Monte Carlo Approach to the Boltzmann and Wigner Equations,” in Advances in Parallel Algorithms (Bl. Sendov and I. Dimov, eds.), pp. 117–128, IOS Press, 1994.

    Google Scholar 

  52. D. Ferry, R. Akis, and D. Vasileska, “Quantum Effect in MOSFETs: Use of an Effective Potential in 3D Monte Carlo Simulation of Ultra-Schort Channel Devices,” Int.Electron Devices Meeting, pp. 287–290, 2000.

    Google Scholar 

  53. L. Shifren, R. Akis, and D. Ferry, “Correspondence Between Quantum and Classical Motion: Comparing Bohmian Mechanics with Smoothed Effective Potential Approach,” Phys.Lett.A, vol. 274, pp. 75–83, 2000.

    Google Scholar 

  54. S. Ahmed, C. Ringhofer, and D. Vasileska, “An Effective Potential Aprroach to Modeling 25nm MOSFET Devices,” Journal of Computational Electronics, vol. 2, pp. 113–117, 2003.

    Article  Google Scholar 

  55. C. Ringhofer, C. Gardner, and D. Vasileska, “An Effective Potentials and Quantum Fluid Models: A Thermodynamic Approach,” Journal of High Speed Electronics and Systems, vol. 13, pp. 771–801, 2003.

    Article  Google Scholar 

  56. S. Haas, F. Rossi, and T. Kuhn, “Generalized Monte Carlo approach for the study of the coherent ultrafast carrier dynamics in photoexcited semiconductors,” Physical Review B, vol. 53, no. 12, pp. 12855–12868, 1996.

    Article  Google Scholar 

  57. M. Nedjalkov, I. Dimov, F. Rossi, and C. Jacoboni, “Convergency of the Monte Carlo Algorithm for the Wigner Quantum Transport Equation,” Journal of Mathematical and Computer Modelling, vol. 23, no. 8/9, pp. 159–166, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  58. K. L. Jensen and F. A. Buot, “The Methodology of Simulating Particle Trajectories Trough Tunneling Structures Using a Wigner Distribution Approach,” IEEE Trans.Electron Devices, vol. 38, no. 10, pp. 2337–2347, 1991.

    Article  Google Scholar 

  59. H. Tsuchiya and T. Miyoshi, “Simulation of Dynamic Particle Trajectories through Resonant-Tunneling Structures based upon Wigner Distribution Function,” Proc. 6th Int. Workshop on Computational Electronics IWCE6, Osaka, pp. 156–159, 1998.

    Google Scholar 

  60. M. Pascoli, P. Bordone, R. Brunetti, and C. Jacoboni, “Wigner Paths for Electrons Interacting with Phonons,” Physical Review B, vol. B 58, pp. 3503–3506, 1998.

    Article  Google Scholar 

  61. V. Sverdlov, A. Gehring, H. Kosina, and S. Selberherr, “Quantum transport in ultra-scaled double-gate MOSFETs: A Wigner function-based Monte Carlo approach,” Solid-State Electronics, vol. 49, pp. 1510–1515, 2005.

    Article  Google Scholar 

  62. D. Querlioz, J. Saint-Martin, V. N. Do, A. Bournel, and P. Dollfus, “A Study of Quantum Transport in End-of-Roadmap DG-MOSFETs Using a Fully Self-Consistent Wigner Monte Carlo Approach,” IEEE Trans. Nanotechnology, vol. 5, pp. 737–744, 2006.

    Article  Google Scholar 

  63. D. Querlioz, J. Saint-Martin, V. N. Do, A. Bournel, and P. Dollfus, “Fully quantum self-consistent study of ultimate DG-MOSFETs including realistic scattering using a Wigner Monte-Carlo approach,” Int. Electron Device Meeting Tech. Dig. (IEDM), pp. 941–944, 2006.

    Google Scholar 

  64. L. Shifren and D. K. Ferry, “A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation,” Journal of Computational Electronics, vol. 1, pp. 55–58, 2002.

    Article  Google Scholar 

  65. D. Querlioz, P. Dollfus, V. N. Do, A. Bournel, and V. L. Nguyen, “An improved Wigner Monte-Carlo technique for the self-consistent simulation of RTDs,” Journal of Computational Electronics, vol. 5, pp. 443–446, 2006.

    Article  Google Scholar 

  66. D. Querlioz and P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices - A particle description of quantum transport and decoherence. ISTE-Wiley, 2010.

    Google Scholar 

  67. M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer, and D. K. Ferry, “Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices,” Physical Review B, vol. 70, p. 115319, 2004.

    Article  Google Scholar 

  68. A. Bertoni, P. Bordone, G. Ferrari, N. Giacobbi, and C. Jacoboni, “Proximity effect of the contacts on electron transport in mesoscopic devices,” Journal of Computational Electronics, vol. 2, pp. 137–140, 2003.

    Article  Google Scholar 

  69. C. Jacoboni and L. Reggiani, “The Monte Carlo Method for the Solution of Charge Transport in Semiconductors with Applications to Covalent Materials,” Rev.Mod.Phys., vol. 55, no. 3, pp. 645–705, 1983.

    Google Scholar 

  70. H. Kosina, “Wigner function approach to nano device simulation,” International Journal of Computational Science and Engineering, vol. 2, no. 3/4, pp. 100 – 118, 2006.

    Article  Google Scholar 

  71. S. Ermakow, Die Monte-Carlo-Methode und verwandte Fragen. München, Wien: R. Oldenburg Verlag, 1975.

    MATH  Google Scholar 

  72. J. Hammersley and D. Handscomb, Monte Carlo Methods. New York: John Wiley, 1964.

    MATH  Google Scholar 

  73. H. Kosina and M. Nedjalkov, Handbook of Theoretical and Computational Nanotechnology, vol. 10, ch. Wigner Function Based Device Modeling, pp. 731–763. Los Angeles: American Scientific Publishers, 2006.

    Google Scholar 

  74. M. Nedjalkov, R. Kosik, H. Kosina, and S. Selberherr, “Wigner Transport Through Tunneling Structures - Scattering Interpretation of the Potential Operator,” in Proc. Simulation of Semiconductor Processes and Devices, (Kobe, Japan), pp. 187–190, Publication Office Business Center for Academic Societies Japan, 2002.

    Google Scholar 

  75. H. Kosina, M. Nedjalkov, and S. Selberherr, “A Monte Carlo Method Seamlessly Linking Classical and Quantum Transport Calculations,” Journal of Compuational Electronics, vol. 2, no. 2-4, pp. 147–151, 2003.

    Article  Google Scholar 

  76. H. Kosina, V. Sverdlov, and T. Grasser, “Wigner Monte Carlo Simulation: Particle Annihilation and Device Applications,” in Proc. Simulation of Semiconductor Processes and Devices, (Monterey, CA, USA), pp. 357–360, Institute of Electrical and Electronics Engineers, Inc., Sept. 2006.

    Google Scholar 

  77. R. Tsu and L. Esaki, “Tunneling in a finite superlattice,” Appl. Phys. Lett., vol. 22, pp. 562–564, 1973.

    Article  Google Scholar 

  78. L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double barriers,” Appl. Phys. Lett., vol. 24, pp. 593–595, 1974.

    Article  Google Scholar 

  79. T. J. Shewchuk, P. C. Chapin, P. D. Coleman, W. Kopp, R. Fischer, and H. Morkoç, “Resonant Tunneling Oscillations in a GaAs-AlxGa1-xAs Heterostructure at Room-Temperature,” Appl. Phys. Lett., vol. 46, pp. 508–510, 1985.

    Article  Google Scholar 

  80. H. Mizuta and T. Tanoue, The physics and applications of resonant tunnelling diodes. Cambridge University Press, 1995.

    Google Scholar 

  81. G. Iannaccone, G. Lombardi, M. Macucci, and B. Pellegrini, “Enhanced Shot Noise in Resonant Tunneling: Theory and Experiment,” Phys. Rev. Lett., vol. 80, pp. 1054–1057, 1998.

    Article  Google Scholar 

  82. Y. M. Blanter and M. Büttiker, “Transition from sub-Poissonian to super-Poissonian shot noise in resonant quantum wells,” Phys. Rev. B, vol. 59, pp. 10217–10226, 1999.

    Article  Google Scholar 

  83. W. Song, E. E. Mendez, V. Kuznetsov, and B. Nielsen, “Shot noise in negative-differential-conductance devices,” Appl. Phys. Lett., vol. 82, pp. 1568–1570, 2003.

    Article  Google Scholar 

  84. S. S. Safonov, A. K. Savchenko, D. A. Bagrets, O. N. Jouravlev, Y. V. Nazarov, E. H. Linfield, and D. A. Ritchie., “Transition from sub-Poissonian to super-Poissonian shot noise in resonant quantum wells,” Phys. Rev. Lett., vol. 91, p. 136801, 2003.

    Google Scholar 

  85. X. Oriols, A. Trois, and G. Blouin, “Self-consistent simulation of quantum shot noise in nanoscale electron devices,” Appl. Phys. Lett., vol. 85, pp. 3596–3598, 2004.

    Article  Google Scholar 

  86. V. Y. Aleshkin, L. Reggiani, N. V. Alkeev, V. E. Lyubchenko, C. N. Ironside, J. M. L. Figueiredo, and C. R. Stanley, “Coherent approach to transport and noise in double-barrier resonant diodes,” Phys. Rev. B, vol. 70, p. 115321, 2004.

    Article  Google Scholar 

  87. V. N. Do, P. Dollfus, and V. L. Nguyen, “Transport and noise in resonant tunneling diode using self-consistent Green’s function calculation,” J. Appl. Phys., vol. 100, p. 093705, 2006.

    Article  Google Scholar 

  88. T. J. Park, Y. K. Lee, S. K. Kwon, J. H. Kwon, and J. Jang, “Resonant tunneling diode made of organic semiconductor superlattice,” Appl. Phys. Lett., vol. 89, p. 151114, 2006.

    Article  Google Scholar 

  89. T. Kanazawa, R. Fujii, T. Wada, Y. Suzuki, M. Watanabe, and M. Asada, “Room temperature negative differential resistance of CdF2/CaF2 double-barrier resonant tunneling diode structures grown on Si(100) substrates,” Appl. Phys. Lett., vol. 90, p. 092101, 2007.

    Article  Google Scholar 

  90. M. V. Petrychuk, A. E. Belyaev, A. M.Kurakin, S. V. Danylyuk, N. Klein, and S. A. Vitusevich, “Mechanisms of current formation in resonant tunneling AlN/GaN heterostructures,” Appl. Phys. Lett., vol. 91, p. 222112, 2007.

    Article  Google Scholar 

  91. J.-P. Colinge, “Multiple-gate SOI MOSFETs,” Solid-State Electronics, vol. 48, pp. 897–905, 2004.

    Article  Google Scholar 

  92. J. Saint-Martin, A. Bournel, and P. Dollfus, “Comparison of multiple-gate MOSFET architectures using Monte Carlo simulation,” Solid-State Electronics, vol. 50, pp. 94–101, 2006.

    Article  Google Scholar 

  93. http://www.itrs.net/reports.html.

    Google Scholar 

  94. D. J. Frank, R. H. Dennard, E. Nowak, P. M. Solomon, Y. Taur, and H. S. P. Wong, “Device scaling limits of Si MOSFETs and their application dependencies,” Proc. IEEE, vol. 89, pp. 259–288, 2001.

    Article  Google Scholar 

  95. P. Dollfus, A. Bournel, S. Galdin, S. Barraud, and P. Hesto, “Effect of discrete impurities on electron transport in ultrashort MOSFET using 3-D MC simulation,” IEEE Trans. Electron Devices, vol. 51, pp. 749–756, 2004.

    Article  Google Scholar 

  96. T. Skotnicki, “Materials and device structures for sub-32 nm CMOS nodes,” Microelectronic Engineering, vol. 84, pp. 1845–1852, 2007.

    Article  Google Scholar 

  97. D. Reid, C. Millar, G. Roy, S. Roy, and A. Asenov, “Analysis of threshold voltage distribution due to random dopants: a 100 000-sample 3-D simulation study,” IEEE Trans. Electron Devices, vol. 56, pp. 2255–2263, 2009.

    Article  Google Scholar 

  98. J. Widiez, J. Lolivier, M. Vinet, T. Poiroux, B. Previtali, F. Daugé, M. Mouis, and S. Deleonibus, “Experimental evaluation of gate architecture influence on DG SOI MOSFETs performance,” IEEE Trans. Electron Devices, vol. 52, pp. 1772–1779, 2005.

    Article  Google Scholar 

  99. M. Vinet, T. Poiroux, J. Widiez, J. Lolivier, B. Previtali, C. Vizioz, B. Guillaumot, Y. L. Tiec, P. Besson, B. Biasse, F. Allain, M. Casse, D. Lafond, J.-M. Hartmann, Y. Morand, J. Chiaroni, and S. Deleonibus, “Bonded planar double-metal-gate NMOS transistors down to 10 nm,” IEEE Electron Device Lett., vol. 26, pp. 317–319, 2005.

    Article  Google Scholar 

  100. J. Widiez, T. Poiroux, M. Vinet, M. Mouis, and S. Deleonibus, “Experimental comparison between Sub-0.1-μm ultrathin SOI single- and double-gate MOSFETs: Performance and Mobility,” IEEE Trans. Nanotechnol., vol. 5, pp. 643–648, 2006.

    Article  Google Scholar 

  101. V. Barral, T. Poiroux, M. Vinet, J. Widiez, B. Previtali, P. Grosgeorges, G. L. Carval, S. Barraud, J.-L. Autran, D. Munteanu, and S. Deleonibus, “Experimental determination of the channel backscattering coefficient on 10-70 nm-metal-gate Double-Gate transistors,” Solid-State Electronics, vol. 51, pp. 537–542, 2007.

    Article  Google Scholar 

  102. J. Saint-Martin, A. Bournel, V. Aubry-Fortuna, F. Monsef, C. Chassat, and P. Dollfus, “Monte Carlo simulation of double gate MOSFET including multi sub-band description,” J. Computational Electronics, vol. 5, pp. 439–442, 2006.

    Article  Google Scholar 

  103. A. Bournel, V. Aubry-Fortuna, J. Saint-Martin, and P. Dollfus, “Device performance and optimization of decananometer long double gate MOSFET by Monte Carlo simulation,” Solid-State Electronics, vol. 51, pp. 543–550, 2007.

    Article  Google Scholar 

  104. M. Vinet, T. Poiroux, C. Licitra, J. Widiez, J. Bhandari, B. Previtali, C. Vizioz, D. Lafond, C. Arvet, P. Besson, L. Baud, Y. Morand, M. Rivoire, F. Nemouchi, V. Carron, and S. Deleonibus, “Self-aligned planar double-gate MOSFETs by bonding for 22-nm node, with metal gates, high-κ dielectrics, and metallic source/drain,” IEEE Electron Device Lett., vol. 30, pp. 748–750, 2009.

    Article  Google Scholar 

  105. E. Joos, Decoherence and the Appearance of a Classical World in Quantum Theory. Springer-Verlag, 2003.

    Google Scholar 

  106. D. Querlioz, “Phnomnes quantiques et dcohrence dans les nano-dispositifs semiconducteurs : tude par une approche Wigner Monte Carlo,” PhD Dissertation, Univ. Paris-Sud, Orsay, 2008.

    Google Scholar 

  107. M. V. Fischetti and S. E. Laux, “Monte Carlo study of electron transport in silicon inversion layers,” Phys. Rev. B, vol. 48, pp. 2244–2274, 1993.

    Article  Google Scholar 

  108. J. Saint-Martin, A. Bournel, F. Monsef, C. Chassat, and P. Dollfus, “Multi sub-band Monte Carlo simulation of an ultra-thin double gate MOSFET with 2D electron gas,” Semicond. Sci. Techn., vol. 21, pp. L29–L31, 2006.

    Article  Google Scholar 

  109. L. Lucci, P. Palestri, D. Esseni, L. Bergagnini, and L. Selmi, “Multisubband Monte Carlo Study of Transport, Quantization, and Electron-Gas Degeneration in Ultrathin SOI n-MOSFETs,” IEEE Trans. Electron Devices, vol. 54, pp. 1156–1164, 2007.

    Article  Google Scholar 

  110. D. Querlioz, J. Saint-Martin, K. Huet, A. Bournel, V. Aubry-Fortuna, C. Chassat, S. Galdin-Retailleau, and P. Dollfus, “On the Ability of the Particle Monte Carlo Technique to Include Quantum Effects in Nano-MOSFET Simulation,” IEEE Trans. Electron Devices, vol. 54, pp. 2232–2242, 2007.

    Article  Google Scholar 

  111. F. Monsef, P. Dollfus, S. Galdin-Retailleau, H. J. Herzog, and T. Hackbarth, “Electron transport in Si/SiGe modulation-doped heterostructures using Monte Carlo simulation,” J. Appl. Phys., vol. 95, pp. 3587–3593, 2004.

    Article  Google Scholar 

  112. S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy, and O. L. Krivanek, “Surface roughness at the Si(100)-SiO2 interface,” Phys. Rev. B, vol. 32, pp. 8171–8186, 1985.

    Article  Google Scholar 

  113. H. Sakaki, T. Noda, K. Hirakawa, M. Tanaka, and T. Matsusue, “Interface roughness scattering in GaAs/AlAs quantum wells,” Appl. Phys. Lett., vol. 51, pp. 1934–1936, 1987.

    Article  Google Scholar 

  114. D. Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically based modeling of low field electron mobility in ultrathin single- and double-gate SOI n-MOSFETs,” IEEE Trans. Electron Devices, vol. 50, pp. 2445–2455, 2003.

    Article  Google Scholar 

  115. V. N. Do, “Modelling and simulation of quantum electronic transport in semiconductor nanometer devices,” PhD Dissertation, Univ. Paris-Sud, Orsay, 2007.

    Google Scholar 

  116. J. Saint-Martin, A. Bournel, and P. Dollfus, “On the ballistic transport in nanometer-scaled DG MOSFETs,” IEEE Trans. Electron Devices, vol. 51, pp. 1148–1155, 2004.

    Article  Google Scholar 

Download references

Acknowledgments

This work has been partially supported by the: European Community through Projects PULLNANO (FP6-IST-026828), SINANO (FP6-IST-506844) and NANOSIL (FP7-ICT-216171); French Agence Nationale de la Recherche through Project MODERN (ANR-05-NANO-02); Austrian Science Fund through Projects FWF-P21685, P17285-N02 and F2509 (SFB 025 IR-ON).

Author information

Authors and Affiliations

  1. Institute of Microelectronics, TU Vienna, Vienna, Austria

    M. Nedjalkov

Authors
  1. M. Nedjalkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Querlioz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Dollfus
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. H. Kosina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Nedjalkov .

Editor information

Editors and Affiliations

  1. Dept. Electrical Engineering &, Center for Solid State, Arizona State University, Tempe, 85287-5706, Arizona, USA

    Dragica Vasileska

  2. , Dept. of Electrical Engineering, Arizona State University, Tempe, 85287-5706, Arizona, USA

    Stephen M. Goodnick

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H. (2011). Wigner Function Approach. In: Vasileska, D., Goodnick, S. (eds) Nano-Electronic Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8840-9_5

Download citation

  • DOI: https://doi.org/10.1007/978-1-4419-8840-9_5

  • Published:

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-8839-3

  • Online ISBN: 978-1-4419-8840-9

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation