Optimization of the Deterministic Solution of the Discrete Wigner Equation

  • Johann CervenkaEmail author
  • Paul Ellinghaus
  • Mihail Nedjalkov
  • Erasmus Langer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9374)


The development of novel nanoelectronic devices requires methods capable to simulate quantum-mechanical effects in the carrier transport processes. We present a deterministic method based on an integral formulation of the Wigner equation, which considers the evolution of an initial condition as the superposition of the propagation of particular fundamental contributions.

Major considerations are necessary, to overcome the memory and time demands typical for any quantum transport method. An advantage of our method is that it is perfectly suited for parallelization due to the independence of each fundamental contribution. Furthermore, a dramatic speed-up of the simulations has been achieved due to a preconditioning of the resulting equation system.

To evaluate this deterministic approach, the simulation of a Resonant Tunneling Diode, will be shown.


  1. 1.
    Kim, K.Y., Lee, B.: On the high order numerical calculation schemes for the wigner transport equation. Solid-State Electron. 43, 2243–2245 (1999)CrossRefGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Griffiths, D.: Introduction to Quantum Mechanics. Pearson Prentice Hall, Upper Saddle River (2005)Google Scholar
  4. 4.
    Kosik, R.: Numerical challenges on the road to nanoTCAD. Ph.D. thesis, Institut für Mikroelektronik (2004)Google Scholar
  5. 5.
    Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H.: Wigner function approach. Nano-electronic Devices; Semiclassical and Quantum Transport Modeling, pp. 289–358. Springer, New York (2011)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Sellier, J.M.D., Nedjalkov, M., Dimov, I., Selberherr, S.: A Benchmark study of the Wigner Monte Carlo method. Monte Carlo Meth. Appl. 20(1), 43–51 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dimov, I.T.: Monte Carlo Methods for Applied Scientists. World Scientific, Singapore (2008)zbMATHGoogle Scholar
  9. 9.
    Sudiarta, I.W., Geldart, D.J.W.: Solving the Schrödinger equation using the finite difference time domain method. J. Phys. A: Math. Theor. 40(8), 1885 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Fu, Y., Willander, M.: Electron wave-packet transport through nanoscale semiconductor device in time domain. J. Appl. Phys. 97(9), 094311 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Johann Cervenka
    • 1
    Email author
  • Paul Ellinghaus
    • 1
  • Mihail Nedjalkov
    • 1
  • Erasmus Langer
    • 1
  1. 1.Institute for MicroelectronicsTU WienViennaAustria

Personalised recommendations