Modeling drive currents and leakage currents: a dynamic approach

  • Wim Magnus
  • Fons Brosens
  • Bart Sorée


The dynamics of electrons and holes propagating through the nano-scaled channels of modern semiconductor devices can be seen as a widespread manifestation of non-equilibrium statistical physics and its ruling principles. In this respect both the devices that are pushing conventional CMOS technology towards the final frontiers of Moore’s law and the upcoming set of alternative, novel nanostructures grounded on entirely new concepts and working principles, provide an almost unlimited playground for assessing physical models and numerical techniques emerging from classical and quantum mechanical non-equilibrium theory.

In this paper we revisit the Boltzmann as well as the Wigner–Boltzmann equation which offers a valuable platform to study transport of charge carriers taking part in drive currents. We focus on a numerical procedure that regained attention recently as an alternative tool to solve the time-dependent Boltzmann equation for inhomogeneous systems, such as the channel regions of field-effect transistors, and we discuss its extension to the Wigner–Boltzmann equation.

Furthermore, we pay attention to the calculation of tunneling leakage currents. The latter typically occurs in nano-scaled transistors when part of the carrier distribution sustaining the drive current is found to tunnel into the gate due the presence of an ultra-thin insulating barrier separating the gate from the channel region. In particular, we discuss the paradox related to the very existence of leakage currents established by electrons occupying quasi-bound states, while the (real) wave functions of the latter cannot carry net currents.

Finally, we describe a simple model to resolve the paradox as well as to estimate gate currents provided the local carrier generation rates largely exceed the tunneling rates.


Wigner–Boltzmann equation Characteristic curves Carrier transport Gate leakage currents 


  1. 1.
    Devreese, J.T., Evrard, R.: On the momentum distribution of electrons in polar semiconductors for high electric field. Phys. Stat. Sol. (b) 78, 85 (1976) CrossRefGoogle Scholar
  2. 2.
    Devreese, J.T., Evrard, R., Kartheuser, E.: Note on the solution of the Boltzmann equation for electron-LO phonon scattering. Phys. Stat. Sol. (b) 90, K73–K76 (1978) CrossRefGoogle Scholar
  3. 3.
    Brosens, F., Devreese, J.T.: Time-dependent momentum distribution of polarons at arbitrary temperature and electric field. Phys. Stat. Sol. (b) 111, 433–696 (1982) CrossRefGoogle Scholar
  4. 4.
    Brittin, W.E., Chappell, W.L.: The Wigner distribution function and second quantization in phase space. Rev. Mod. Phys. 34, 620–627 (1962) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bordone, P., Pascoli, M., Brunetti, R., Bertoni, A., Jacoboni, C., Abramo, A.: Quantum transport of electrons in open nanostructures with the Wigner-function formalism. Phys. Rev. B 59, 3060–3069 (1999) CrossRefGoogle Scholar
  6. 6.
    Jacoboni, C., Bertoni, A., Bordone, P., Giacobbi, N.: Simulation of Wigner function transport in tunneling and quantum structures. In: Technical Proceedings of the Fifth International Conference on Modeling and Simulation of Microsystems, Nanotech 2002—MSM 2002, p. 474 Google Scholar
  7. 7.
    Balaban, S.N., Pokatilov, E.P., Fomin, V.M., Gladilin, V.N., Devreese, J.T., Magnus, W., Schoenmaker, W., Van Rossum, M., Sorée, B.: Quantum transport in a cylindrical sub-0.1 μm silicon-based MOSFET. Solid-State Electron. 46, 435 (2002) CrossRefGoogle Scholar
  8. 8.
    Croitoru, M.D., Gladilin, V.N., Fomin, V.M., Devreese, J.T., Magnus, W., Schoenmaker, W., Sorée, B.: Quantum transport in a nanosize silicon-on-insulator metal-oxide-semiconductor field-effect transistor. J. Appl. Phys. 93, 1230–1240 (2003) CrossRefGoogle Scholar
  9. 9.
    Croitoru, M.D., Gladilin, V.N., Fomin, V.M., Devreese, J.T., Magnus, W., Schoenmaker, W., Sorée, B.: Quantum transport in a nanosize double-gate metal-oxide-semiconductor field-effect transistor. J. Appl. Phys. 96, 2305–2310 (2004) CrossRefGoogle Scholar
  10. 10.
    Nedjalkov, M., Kosina, H., Selberherr, S., Ringhofer, C., Ferry, D.K.: Unified particle approach to Wigner–Boltzmann transport in small semiconductor devices. Phys. Rev. B 70, 115319 (2004), and references therein CrossRefGoogle Scholar
  11. 11.
    Querlioz, D., Saint-Martin, J., Do, V.-N., Bournel, A., Dollfus, P.: A study of quantum transport in end-of-roadmap DG-MOSFETs using a fully self-consistent Wigner Monte Carlo approach. IEEE Trans. Nanotechnol. 5, 737–744 (2006) CrossRefGoogle Scholar
  12. 12.
    Croitoru, M.D., Gladilin, V.N., Fomin, V.M., Devreese, J.T., Magnus, W., Schoenmaker, W., Sorée, B.: Quantum transport in an ultra-thin SOI MOSFET: Influence of the channel thickness on the IV characteristics. Solid State Commun. (2008). doi: 10.1016/j.ssc.2008.04.025 Google Scholar
  13. 13.
    Fischetti, M.V.: Master-equation approach to the study of electronic transport in small semiconductor devices. Phys. Rev. B 59, 4901–4917 (1999) CrossRefGoogle Scholar
  14. 14.
    Kubo, R.: Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. J. Phys. Soc. Jpn. 12, 570–586 (1957) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Greenwood, D.A.: The Boltzmann equation in the theory of electrical conduction in metals. Proc. Phys. Soc. Lond. 71, 585–596 (1957) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Zubarev, D.N., Sheperd, P.J., Gray, P.: Nonequilibrium Statistical Thermodynamics. Consultants Bureau, New York (1974) Google Scholar
  17. 17.
    Bløtekjaer, K.: Transport equations for electrons in two-valley semiconductors. IEEE Trans. Electron. Dev. ED-17, 38–47 (1970) CrossRefGoogle Scholar
  18. 18.
    Rudan, M., Odeh, F.: Multi-dimensional discretization scheme for the hydrodynamic of velocity overshoot effects in Si and GaAs devices. COMPEL 1, 65–87 (1982) Google Scholar
  19. 19.
    Rudan, M., Reggiani, S., Gnani, E., Baccarani, G.: A coherent extension of the transport equations in semiconductors incorporating the quantum correction—Part I: Single-particle dynamics. IEEE Trans. Nanotechnol. 4, 495–502 (2005) CrossRefGoogle Scholar
  20. 20.
    Rudan, M., Reggiani, S., Gnani, E., Baccarani, G.: A coherent extension of the transport equations in semiconductors incorporating the quantum correction—Part II: Collective transport. IEEE Trans. Nanotechnol. 4, 503–509 (2005) CrossRefGoogle Scholar
  21. 21.
    Selberherr, S.: Device modeling and physics. Phys. Scr. T. 35, 293–298 (1991) CrossRefGoogle Scholar
  22. 22.
    Kosina, H., Langer, E., Selberherr, S.: Device modelling for the 1990s. Microelectron. J. 26, 217–233 (1995) CrossRefGoogle Scholar
  23. 23.
    Schenk, A.: Advanced Physical Models for Silicon Device Simulation. Springer, Berlin (1998), Chap. 1 MATHGoogle Scholar
  24. 24.
    Cook, R.K., Frey, J.: An efficient technique for two-dimensional simulation of model of semiconductor devices. COMPEL 4, 149–183 (1986) Google Scholar
  25. 25.
    Chen, D., Sangiorgi, E., Pinto, M.R., Kan, E.C., Ravaioli, U., Dutton, R.W.: An improved energy transport model including non-parabolicity non-Maxwellian distribution effects. IEEE Trans. Electron. Dev. 13, 26–28 (1992) CrossRefGoogle Scholar
  26. 26.
    Peeters, F.M., Devreese, J.T.: Nonlinear conductivity in polar semiconductors: Alternative derivation of the Thornber-Feynman theory. Phys. Rev. B 23, 1936 (1981) CrossRefGoogle Scholar
  27. 27.
    Lei, X.L., Ting, C.S.: Theory on nonlinear electron transport for solids in a strong electric field. Phys. Rev. B 30, 4809 (1984) CrossRefGoogle Scholar
  28. 28.
    Lei, X.L., Ting, C.S.: Two-dimensional balance equations in nonlinear electronic transport and application to GaAs-GaAlAs heterojunctions. J. Appl. Phys. 58, 2270 (1985) CrossRefGoogle Scholar
  29. 29.
    Lei, X.L., Ting, C.S.: Green’s-function approach to nonlinear electronic transport for an electron–impurity–phonon system in a strong electric field, transport and application to GaAs-GaAlAs heterojunctions. Phys. Rev. B 32, 1112 (1985) CrossRefGoogle Scholar
  30. 30.
    Lei, X.L.: Balance equations for hot electron transport in an arbitrary energy band. Phys. Stat. Sol. (b) 170, 519 (1992) CrossRefGoogle Scholar
  31. 31.
    Lei, X.L., Horing, N.J.M.: Balance equation approach to hot-carrier transport in semiconductors. Int. J. Mod. Phys. B 6, 805–936 (1992) CrossRefGoogle Scholar
  32. 32.
    Lei, X.L.: Hydrodynamic balance-equations for electron-transport and thermoelectric-power in an arbitrary energy-band. Phys. Stat. Sol. (b) 192, K1 (1995) CrossRefGoogle Scholar
  33. 33.
    Jacoboni, C., Reggiani, L.: The Monte-Carlo method for the solution of charge transport in semiconductors with applications to covalent materials. Rev. Mod. Phys. 55, 645–705 (1983) CrossRefGoogle Scholar
  34. 34.
    Fischetti, M.V., Laux, S.E.: Monte-Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects. Phys. Rev. B 38, 9721–9745 (1988) CrossRefGoogle Scholar
  35. 35.
    Jacoboni, C., Lugli, P.: The Monte-Carlo method for semiconductor simulation. In: Selberherr, S. (ed.) Computational Microelectronics. Springer, Wien (1989) Google Scholar
  36. 36.
    Sverdlov, V., Kosina, H., Grasser, T., Selberherr, S.: Self-consistent Wigner Monte Carlo simulations of current in emerging nanodevices: role of tunneling and scattering. In: 28th International Conference on the Physics of Semiconductors (ICPS 2006). doi: 10.1063/1.2730425
  37. 37.
    Keldysh, L.V.: Diagram techniques for nonequilibrium processes. JETP Sov. Phys. 20, 1-18–1026 (1965) MathSciNetGoogle Scholar
  38. 38.
    Datta, S.: Electronic Transport in Mesoscopic Systems, p. 293. Cambridge University Press, Cambridge (1995) Google Scholar
  39. 39.
    Mathews, J., Walker, R.L.: Mathematical Methods of Physics. Benjamin, Elmsford (1964), Chap. 8 MATHGoogle Scholar
  40. 40.
    Geurts, B.J.: Modelling transport in submicron structures using the relaxation time Boltzmann equation. J. Phys., Condens. Matter 3, 9447–9458 (1991) CrossRefGoogle Scholar
  41. 41.
    Cáceres, M.J., Carrillo, J.A., Goudon, T.: Equilibrium rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Commun. Part. Differ. Equ. 28, 969–989 (2003) MATHCrossRefGoogle Scholar
  42. 42.
    Cáceres, M.J., Carrillo, J.A., Gamba, I., Majorana, A., Shu, C.-W.: DSMC versus WENO-BTE: A double gate MOSFET example. J. Comput. Electron. 5, 471–474 (2006) CrossRefGoogle Scholar
  43. 43.
    Brosens, F., Magnus, W.: Carrier transport in nanodevices: revisiting the Boltzmann and Wigner distribution functions. Phys. Stat. Sol. (b) (2008). doi: 10.1002/pssb.200844424 Google Scholar
  44. 44.
    Vlasov, A.A.: The vibrational properties of an electron gas. Sov. Phys. Usp. 10, 721–733 (1968). doi: 10.1070/PU1968v010n06ABEH003709 CrossRefGoogle Scholar
  45. 45.
    Natori, K.: Ballistic metal-oxide-semiconductor field effect transistor. J. Appl. Phys. 76, 4879–4890 (1994) CrossRefGoogle Scholar
  46. 46.
    Rhew, J.-H., Lundstrom, M.S.: A numerical study of ballistic transport in a nanoscale MOSFET. Solid-State Electron. 92, 1899–1906 (2002) Google Scholar
  47. 47.
    Rhew, J.-H., Lundstrom, M.S.: Drift-diffusion equation for ballistic transport in nanoscale metal-oxide-semiconductor field effect transistors. J. Appl. Phys. 92, 5196–5202 (2002) CrossRefGoogle Scholar
  48. 48.
    Fenton, E.: Electrical and chemical potentials in a quantum-mechanical conductor. Superlattices Microstruct. 16, 87 (1994) CrossRefGoogle Scholar
  49. 49.
    Kamenev, A., Kohn, W.: Landauer conductance without two chemical potentials. Phys. Rev. B 63, 155304 (2001) CrossRefGoogle Scholar
  50. 50.
    Sorée, B., Magnus, W., Schoenmaker, W.: Conductance quantization and dissipation. Phys. Lett. A 310, 322–328 (2003) MATHCrossRefGoogle Scholar
  51. 51.
    Fowler, R.H., Nordheim, L.W.: Electron emission in intense electric fields. Proc. R. Soc. A 119, 173–181 (1928) CrossRefGoogle Scholar
  52. 52.
    Hendriks, M., Magnus, W., van de Roer, T.G.: Accurate modelling of the accumulation region of a double barrier resonant tunneling diode. Solid-State Electron. 39, 703 (1996) CrossRefGoogle Scholar
  53. 53.
    Bardeen, J.: Tunnelling from a many-particle point of view. Phys. Rev. Lett. 6, 57–59 (1961) CrossRefGoogle Scholar
  54. 54.
    Breit, G., Wigner, E.P.: Capture of slow neutrons. Phys. Rev. 49, 519 (1936) MATHCrossRefGoogle Scholar
  55. 55.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics (Non-relativistic Theory), p. 441. Pergamon, London (1958) Google Scholar
  56. 56.
    Sune, J., Olivio, P., Ricco, B.: Self-consistent solution of the Poisson and Schrödinger equations in accumulated semiconductor–insulator interfaces. J. Appl. Phys. 70, 337 (1991) CrossRefGoogle Scholar
  57. 57.
    Ghatak, A.K., Thyagarajan, K., Shenoy, M.R.: A novel numerical technique for solving the one-dimensional Schroedinger equation using matrix approach—application to quantum well structures. Quantum Electron. 24, 1524–1531 (1988) CrossRefGoogle Scholar
  58. 58.
    Magnus, W., Schoenmaker, W.: Full quantum mechanical model for the charge distribution and the leakage currents in ultra-thin metal-insulator-semiconductor capacitors. J. Appl. Phys. 88, 5833–5842 (2000) CrossRefGoogle Scholar
  59. 59.
    Magnus, W., Schoenmaker, W.: On the calculation of gate tunneling currents in ultra-thin metal-insulator-semiconductor capacitors. Microelectron. Reliability 41, 31–35 (2001) CrossRefGoogle Scholar
  60. 60.
    Pourghaderi, M.A., Magnus, W., Sorée, B., Meuris, M., De Meyer, K., Heyns, M.: Tunneling-lifetime model for metal-oxide-semiconductor structures. Phys. Rev. B 80, 085315 (2009) CrossRefGoogle Scholar
  61. 61.
    Shockley, W., Read, W.T. Jr.: Statistics of the recombinations of holes and electrons. Phys. Rev. 87, 835–842 (1952) MATHCrossRefGoogle Scholar
  62. 62.
    Fossum, J.G., Lee, D.S.: A physical model for the dependence of carrier lifetime on doping density in nondegenerate silicon. Solid-State Electron. 25, 741–747 (1982) CrossRefGoogle Scholar
  63. 63.
    Fossum, J.G., Mertens, R.P., Lee, D.S., Nijs, J.F.: Carrier recombination and lifetime in highly doped silicon. Solid-State Electron. 26, 569–576 (1983) CrossRefGoogle Scholar
  64. 64.
    Clerc, R., Ghibaudo, G., Pananakakis, G.: Bardeen’s approach for tunneling evaluation in MOS structures. Solid-State Electron. 46, 1039–1044 (2002) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media LLC 2009

Authors and Affiliations

  1. 1.IMECLeuvenBelgium
  2. 2.Physics DepartmentUniversiteit AntwerpenAntwerpenBelgium

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