Wigner Function Approach

Chapter

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.

Keywords

Wigner function Wigner-Boltzmann equation Monte Carlo Quantum particles De-coherence 

References

  1. 1.
    H. Weyl, “Quantenmechanik und Gruppentheorie,” Zeitschrift fr Physik, vol. 46, pp. 1–46, 1927.CrossRefGoogle Scholar
  2. 2.
    E. Wigner, “On the quantum corrections for thermodynamic equilibrium,” Physical Review, vol. 40, pp. 749–759, 1932.MATHCrossRefGoogle Scholar
  3. 3.
    J. E. Moyal, “Quantum mechanics as a statistical theory,” Proceedings of the Cambridge Philosophical Society, vol. 45, pp. 99–124, 1949.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    V. I. Tatarskii, “The Wigner Representation of Quantum Mechanics,” Sov. Phys. Usp., vol. 26, pp. 311–327, 1983.MathSciNetCrossRefGoogle Scholar
  5. 5.
    N. C. Dias and J. N. Prata, “Admissible states in quantum phase space,” Annals of Physics, vol. 313, pp. 110–146, 2004.MathSciNetMATHCrossRefGoogle Scholar
  6. 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.CrossRefGoogle Scholar
  7. 7.
    I. Knezevic, “Decoherence due to contacts in ballistic nanostructures,” Physical Review B, vol. 77, p. 125301, 2008.CrossRefGoogle Scholar
  8. 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. 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. 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.CrossRefGoogle Scholar
  11. 11.
    R. Balescu, Equilibrium and Nonequilibrium Statistical Mechanics. Wiley and Sons, 1975.Google Scholar
  12. 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.CrossRefGoogle Scholar
  13. 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.CrossRefGoogle Scholar
  14. 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. 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.CrossRefGoogle Scholar
  16. 16.
    W. Frensley, “Wigner-Function Model of Resonant-Tunneling Semiconductor Device,” Physical Review B, vol. 36, no. 3, pp. 1570–1580, 1987.CrossRefGoogle Scholar
  17. 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.CrossRefGoogle Scholar
  18. 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.CrossRefGoogle Scholar
  19. 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.CrossRefGoogle Scholar
  20. 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.CrossRefGoogle Scholar
  21. 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.CrossRefGoogle Scholar
  22. 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. 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.CrossRefGoogle Scholar
  24. 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.CrossRefGoogle Scholar
  25. 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.MathSciNetGoogle Scholar
  26. 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.CrossRefGoogle Scholar
  27. 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. 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.CrossRefGoogle Scholar
  29. 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.CrossRefGoogle Scholar
  30. 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.CrossRefGoogle Scholar
  31. 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.CrossRefGoogle Scholar
  32. 32.
    F. Rossi and T. Kuhn, “Theory of Ultrafast Phenomena in Photoexcited Semiconductors,” Reviews of Modern Physics, vol. 74, pp. 895–950, July 2002.CrossRefGoogle Scholar
  33. 33.
    K. Thornber, “High-field electronic conduction in insulators,” Solid-State Electron., vol. 21, pp. 259–266, 1978.CrossRefGoogle Scholar
  34. 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.CrossRefGoogle Scholar
  35. 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. 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.MathSciNetMATHCrossRefGoogle Scholar
  37. 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.MathSciNetMATHCrossRefGoogle Scholar
  38. 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. 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.CrossRefGoogle Scholar
  40. 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.CrossRefGoogle Scholar
  41. 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.MathSciNetMATHCrossRefGoogle Scholar
  42. 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. 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.MathSciNetMATHCrossRefGoogle Scholar
  44. 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.CrossRefGoogle Scholar
  45. 45.
    B. K. Ridley, Quantum processes in semiconductors. Oxford University Press, fourth ed., 1999.Google Scholar
  46. 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.CrossRefGoogle Scholar
  47. 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.CrossRefGoogle Scholar
  48. 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.CrossRefGoogle Scholar
  49. 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. 50.
    R. Sala, S. Brouard, and G. Muga, “Wigner Trajectories and Liouville’s theorem,” J. Chem. Phys., vol. 99, pp. 2708–2714, 1993.CrossRefGoogle Scholar
  51. 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. 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. 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. 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.CrossRefGoogle Scholar
  55. 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.CrossRefGoogle Scholar
  56. 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.CrossRefGoogle Scholar
  57. 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.MathSciNetMATHCrossRefGoogle Scholar
  58. 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.CrossRefGoogle Scholar
  59. 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. 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.CrossRefGoogle Scholar
  61. 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.CrossRefGoogle Scholar
  62. 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.CrossRefGoogle Scholar
  63. 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. 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.CrossRefGoogle Scholar
  65. 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.CrossRefGoogle Scholar
  66. 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. 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.CrossRefGoogle Scholar
  68. 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.CrossRefGoogle Scholar
  69. 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. 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.CrossRefGoogle Scholar
  71. 71.
    S. Ermakow, Die Monte-Carlo-Methode und verwandte Fragen. München, Wien: R. Oldenburg Verlag, 1975.MATHGoogle Scholar
  72. 72.
    J. Hammersley and D. Handscomb, Monte Carlo Methods. New York: John Wiley, 1964.MATHGoogle Scholar
  73. 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. 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. 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.CrossRefGoogle Scholar
  76. 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. 77.
    R. Tsu and L. Esaki, “Tunneling in a finite superlattice,” Appl. Phys. Lett., vol. 22, pp. 562–564, 1973.CrossRefGoogle Scholar
  78. 78.
    L. L. Chang, L. Esaki, and R. Tsu, “Resonant tunneling in semiconductor double barriers,” Appl. Phys. Lett., vol. 24, pp. 593–595, 1974.CrossRefGoogle Scholar
  79. 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.CrossRefGoogle Scholar
  80. 80.
    H. Mizuta and T. Tanoue, The physics and applications of resonant tunnelling diodes. Cambridge University Press, 1995.Google Scholar
  81. 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.CrossRefGoogle Scholar
  82. 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.CrossRefGoogle Scholar
  83. 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.CrossRefGoogle Scholar
  84. 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. 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.CrossRefGoogle Scholar
  86. 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.CrossRefGoogle Scholar
  87. 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.CrossRefGoogle Scholar
  88. 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.CrossRefGoogle Scholar
  89. 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.CrossRefGoogle Scholar
  90. 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.CrossRefGoogle Scholar
  91. 91.
    J.-P. Colinge, “Multiple-gate SOI MOSFETs,” Solid-State Electronics, vol. 48, pp. 897–905, 2004.CrossRefGoogle Scholar
  92. 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.CrossRefGoogle Scholar
  93. 93.
    http://www.itrs.net/reports.html.Google Scholar
  94. 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.CrossRefGoogle Scholar
  95. 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.CrossRefGoogle Scholar
  96. 96.
    T. Skotnicki, “Materials and device structures for sub-32 nm CMOS nodes,” Microelectronic Engineering, vol. 84, pp. 1845–1852, 2007.CrossRefGoogle Scholar
  97. 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.CrossRefGoogle Scholar
  98. 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.CrossRefGoogle Scholar
  99. 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.CrossRefGoogle Scholar
  100. 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.CrossRefGoogle Scholar
  101. 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.CrossRefGoogle Scholar
  102. 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.CrossRefGoogle Scholar
  103. 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.CrossRefGoogle Scholar
  104. 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.CrossRefGoogle Scholar
  105. 105.
    E. Joos, Decoherence and the Appearance of a Classical World in Quantum Theory. Springer-Verlag, 2003.Google Scholar
  106. 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. 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.CrossRefGoogle Scholar
  108. 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.CrossRefGoogle Scholar
  109. 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.CrossRefGoogle Scholar
  110. 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.CrossRefGoogle Scholar
  111. 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.CrossRefGoogle Scholar
  112. 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.CrossRefGoogle Scholar
  113. 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.CrossRefGoogle Scholar
  114. 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.CrossRefGoogle Scholar
  115. 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. 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.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institute of MicroelectronicsTU ViennaViennaAustria

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