Journal of Computational Electronics

, Volume 14, Issue 4, pp 888–893 | Cite as

The Wigner equation in the presence of electromagnetic potentials

  • Mihail Nedjalkov
  • Josef Weinbub
  • Paul Ellinghaus
  • Siegfried Selberherr
S.I. : Wigner functions

Abstract

An analysis of the possible formulations of the Wigner equation under a general gauge for the electric field is presented with an emphasis on the computational aspects of the problem. The numerical peculiarities of those formulations enable alternative computational strategies based on existing numerical methods applied in the Wigner formalism, such as finite difference or stochastic particle methods. The phase space formulation of the problem along with certain relations to classical mechanics offers an insight about the role of the gauge transforms in quantum mechanics.

Keywords

Wigner function Electromagnetic potentials Gauge transform 

References

  1. 1.
    Lorenz, L.: On the identity of the vibrations of light with electrical currents. Philos. Mag. 34, 287–301 (1867)Google Scholar
  2. 2.
    Bloch, F.: Über die quantenmechanik der elektronen in kristallgittern. Zeitschrift für Physik 52, 555–600 (1929). doi:10.1007/BF01339455 CrossRefGoogle Scholar
  3. 3.
    Wannier, G.H.: Wave functions and effective Hamiltonian for Bloch electrons in an electric field. Phys. Rev. 117, 432–439 (1960). doi:10.1103/PhysRev.117.432 MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Houston, W.V.: Acceleration of electrons in a crystal lattice. Phys. Rev. 57, 184–186 (1940). doi:10.1103/PhysRev.57.184 MathSciNetCrossRefGoogle Scholar
  5. 5.
    Rossi, F.: Bloch oscillations and Wannier–Stark localization in semiconductor superlattices. In: Schöll, E. (ed.) Theory of Transport Properties of Semiconductor Nanostructures. Electronic Materials Series, vol. 4, pp. 283–320. Springer, Berlin (1998). doi:10.1007/978-1-4615-5807-1_9
  6. 6.
    Wigner, E.: On the quantum correction for thermodynamic equilibrium. Phys. Rev. Lett. 40, 749–759 (1932). doi:10.1103/PhysRev.40.749 Google Scholar
  7. 7.
    Kubo, R.: Wigner representation of quantum operators and its applications to electrons in a magnetic field. J. Phys. Soc. Jpn. 11, 2127–2139 (1964). doi:10.1143/JPSJ.19.2127 CrossRefGoogle Scholar
  8. 8.
    Stratonovich, R.L.: Kalibrovochno-invariantnyj analog raspredeleniya Wignera (in Russian). (Gauge-invariant analog of the Wigner distribution). Doklady Akademii Nauk SSSR 109, 72–75 (1956)MathSciNetGoogle Scholar
  9. 9.
    Haas, F., Zamanian, J., Marklund, M., Brodin, G.: Fluid moment hierarchy equations derived from gauge invariant quantum kinetic theory. N. J. Phys. 12, 073027 (2010). doi:10.1088/1367-2630/12/7/073027 CrossRefGoogle Scholar
  10. 10.
    Serimaa, O.T., Javanainen, J., Varro, S.: Gauge independent Wigner functions: general formulation. Phys. Rev. A 33, 2913–2927 (1986). doi:10.1103/PhysRevA.33.2913 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H.: Wigner function approach. In: Vasileska, D., Goodnick, S.M. (eds.) Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling, pp. 289–358. Springer, New York (2011). doi:10.1007/978-1-4419-8840-9_5
  12. 12.
    Frensley, W.R.: Wigner-function model of a resonant-tunneling semiconductor device. Phys. Rev. B 36, 1570–1580 (1987). doi:10.1103/PhysRevB.36.1570 CrossRefGoogle Scholar
  13. 13.
    Shifren, L., Ferry, D.K.: A Wigner function based ensemble Monte Carlo approach for accurate incorporation of quantum effects in device simulation. J. Comput. Electron. 1, 55–58 (2002). doi:10.1023/A:1020711726836
  14. 14.
    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). doi:10.1103/PhysRevB.70.115319 CrossRefGoogle Scholar
  15. 15.
    Querlioz, D., Dollfus, P., Do, V.N., Bournel, A., Nguyen, V.L.: An improved Wigner Monte-Carlo technique for the self-consistent simulation of RTDs. J. Comput. Electron. 5, 443–446 (2006). doi:10.1007/s10825-006-0044-3 CrossRefGoogle Scholar
  16. 16.
    Dorda, A., Schürrer, F.: A WENO-solver combined with adaptive momentum discretization for the Wigner transport equation and its application to resonant tunneling diodes. J. Comput. Phys. 284, 95–116 (2015). doi:10.1016/j.jcp.2014.12.026 MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Mihail Nedjalkov
    • 1
  • Josef Weinbub
    • 1
  • Paul Ellinghaus
    • 1
  • Siegfried Selberherr
    • 1
  1. 1.Institute for MicroelectronicsTU WienViennaAustria

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