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A Wigner Potential Decomposition in the Signed-Particle Monte Carlo Approach

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11189)


The description of the electron evolution, provided by the Wigner equation, involves a force-less Liouville operator, which is associated with particles moving over Newtonian trajectories, and a Wigner potential operator associated with generation of positive and negative particles. These concepts can be combined to develop stochastic algorithms for solving the Wigner equation, consolidated by the so-called signed particle approach. We investigate the option to split the Wigner potential into two parts and to approximate one of them by a classical force term. The purpose is two-fold: First, we search for ways to simplify the numerical complexity involved in the simulation of the Wigner equation. Second, such a term offers a way to a self-consistent coupling of the Wigner and the Poisson equations. The particles in the signed-particle approach experience a force through the classical component of the potential. A cellular automaton algorithm is used to update the discrete momentum of the accelerated particles, which is then utilized along with the Wigner-based generation/annihilation processes. The effect of the approximation on generic physical quantities such as current and density are investigated for different cut-off wavenumbers (wavelengths), and the results are promising for a self-consistent solution of the Wigner and Poisson equations.


  • Wigner function
  • Potential splitting
  • Signed-particle approach

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  • DOI: 10.1007/978-3-030-10692-8_29
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  1. Ellinghaus, P.: Two-Dimensional Wigner Monte Carlo Simulation for Time-Resolved Quantum Transport with Scattering. Dissertation, Institute for Microelectronics, TU Wien (2016)

    Google Scholar 

  2. Gehring, A., Kosina, H.: Wigner function-based simulation of quantum transport in scaled DG-MOSFETs using a Monte Carlo method. J. Comput. Electron. 4(1–2), 67–70 (2005)

    CrossRef  Google Scholar 

  3. Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H.: Wigner function approach. In: Vasileska, D., Goodnick, S. (eds.) Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling, pp. 289–358. Springer, New York (2011).

    CrossRef  Google Scholar 

  4. Querlioz, D., Dollfus, P.: The Wigner Monte Carlo Method for Nanoelectronic Devices. A Particle Description of Quantum Transport and Decoherence. Wiley, Hoboken (2010)

    MATH  Google Scholar 

  5. Gonzalez, R.C., Woods, R.E.: Digital Image Processing, 3rd edn, pp. 154–155. Prentice Hall, New York (2008)

    Google Scholar 

  6. Ellinghaus, P., Nedjalkov, M., Selberherr, S.: Optimized particle regeneration scheme for the Wigner Monte Carlo method. In: Dimov, I., Fidanova, S., Lirkov, I. (eds.) NMA 2014. LNCS, vol. 8962, pp. 27–33. Springer, Cham (2015).

    CrossRef  Google Scholar 

  7. ViennaWD - Wigner Ensemble Monte Carlo Simulator.

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This research has been supported by the Austrian Science Fund through the project FWF-P29406-N30.

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Correspondence to Majid Benam .

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Benam, M., Nedjalkov, M., Selberherr, S. (2019). A Wigner Potential Decomposition in the Signed-Particle Monte Carlo Approach. In: Nikolov, G., Kolkovska, N., Georgiev, K. (eds) Numerical Methods and Applications. NMA 2018. Lecture Notes in Computer Science(), vol 11189. Springer, Cham.

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