Journal of Computational Electronics

, Volume 14, Issue 4, pp 859–863 | Cite as

Boundary conditions and the Wigner equation solution

  • Ivan DimovEmail author
  • Mihail Nedjalkov
  • Jean-Michel Sellier
  • Siegfried Selberherr


We consider the existence and uniqueness of the solution of the Wigner equation in the presence of boundary conditions. The equation, describing electron transport in nanostructures, is analyzed in terms of the Neumann series expansion of the corresponding integral form, obtained with the help of classical particle trajectories. It is shown that the mathematical aspects of the solution can not be separated from the physical attributes of the problem. In the presented analysis these two sides of the problem mutually interplay, which is of importance for understanding of the peculiarities of Wigner-quantum transport. The problem is first formulated as the long time limit of a general evolution process posed by initial and boundary conditions. Then the Wigner equation is reformulated as a second kind of a Fredholm integral equation which is of Volterra type with respect to the time variable. The analysis of the convergence of the corresponding Neumann series, sometimes called Liouville–Neumann series, relies on the assumption for reasonable local conditions obeyed by the kernel.


Wigner equation Monte Carlo method Boundary conditions Convergence 



This work is partially supported by the Bulgarian Science Fund under Grant DFNI I02/20, as well as by the EU AComIn Project FP7-REGPOT-2012-2013-1.


  1. 1.
    Dimov, I.: Monte Carlo Methods for Applied Scientists, p. 291. World Scientific, London (2008)zbMATHGoogle Scholar
  2. 2.
    Nedjalkov, M., Kosina, H., Selberherr, S., Ringhofer, C., Ferry, D.: Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices. Phys. Rev. B 70, 115319–115335 (2004)CrossRefGoogle Scholar
  3. 3.
    Li, R., Lu, T., Sun, Z.: Stationary Wigner equation with inflow boundary conditions: will a symmetric potential yield a symmetric solution? SIAM J. Appl. Math. 74, 885–897 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Rosati, R., Dolcini, F., Iotti, R.C., Rossi, F.: Wigner-function formalism applied to semiconductor quantum devices: failure of the conventional boundary condition scheme. Phys. Rev. B 88, 035401 (2013)CrossRefGoogle Scholar
  5. 5.
    Tatarskii, V.I.: The wigner representation of quantum mechanics. sov. phys. usp. 26, 311–327 (1983)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dias, N.C., Prata, J.N.: Admissible states in quantum phase space. Ann. Phys. 313, 110–146 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Frensley, W.: Wigner-function model of resonant-tunneling semiconductor device. Phys. Rev. B 36(3), 1570–1580 (1987)CrossRefGoogle Scholar
  8. 8.
    Nashed, M., Wahba, G.: Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind. Math. Comput. 28, 69–80 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    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. Springer, Berlin (2011)Google Scholar
  10. 10.
    Bertoni, A., Bordone, P., Ferrari, G., Giacobbi, N., Jacoboni, C.: Proximity effect of the contacts on electron transport in mesoscopic devices. J. Comput. Electr. 2, 137–140 (2003)CrossRefGoogle Scholar
  11. 11.
    Ferrari, G., Bordone, P., Jacoboni, C.: Electron dynamics inside short-coherence systems. Phys. Lett. A 356, 371–375 (2006)zbMATHCrossRefGoogle Scholar
  12. 12.
    Sellier, J.M., Nedjalkov, M., Dimov, I.: An introduction to applied quantum mechanics in the Wigner Monte Carlo formalism. Phys. Rep. 577, 1–34 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nedjalkov, M., Selberherr, S., Ferry, D.K., Vasileska, D., Dollfus, P., Querlioz, D., Dimov, I., Schwaha, P.: Physical scales in the Wigner-Boltzmann equation. Ann. Phys. 328, 220–237 (2013)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Ivan Dimov
    • 1
    Email author
  • Mihail Nedjalkov
    • 2
  • Jean-Michel Sellier
    • 1
  • Siegfried Selberherr
    • 2
  1. 1.IICTBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Institute for MicroelectronicsTU WienWienAustria

Personalised recommendations