Skip to main content

Distributed-memory parallelization of the Wigner Monte Carlo method using spatial domain decomposition


The Wigner Monte Carlo method, based on the generation and annihilation of particles, has emerged as a promising approach to treat transient problems of quantum electron transport in nanostructures. Tackling these simulations in multiple spatial dimensions demands a parallelized approach to facilitate a practical application of the method in order to investigate realistic problems, due to the otherwise exorbitant execution-times and memory requirements. Because of the annihilation step, a straight-forward parallelization of the Wigner Monte Carlo code is not possible, since sub-ensembles of particles can not be treated independently. Moreover, the large memory requirements of the annihilation procedure presents challenges when working in a distributed-memory setting. A solution to this problem is presented here with a parallelization approach using a spatial domain decomposition, implemented using the message passing interface. The presented benchmark results, based on standard one-dimensional examples, exhibit a good efficiency in the scalability of not only speed, but also memory consumption, which is paramount for the simulation of realistic devices.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  1. If a sub-ensemble is big enough to yield a statistically representative solution to the simulation task, the ’parallelization’ simply amounts to a simultaneous repetition of the same experiment on different computational units, the results of which are averaged.

  2. In this context a node refers to a computer, which is part of a larger cluster.

  3. For the remainder of this work, the term process refers to an MPI process.


  1. Amoroso, S., Gerrer, L., Asenov, A., Sellier, J., Dimov, I., Nedjalkov, M.,Selberherr, S.: Quantum insights in gate oxide charge-trapping dynamics in nanoscale MOSFETs. In: Simulation of Semiconductor Processes and Devices (SISPAD), 2013 International Conference on, pp. 25–28 (2013). doi:10.1109/SISPAD.2013.6650565

  2. Hager, G., Wellein, G.: Introduction to High Performance Computing for Scientists and Engineers. CRC Press, Boca Raton (2010)

    Book  Google Scholar 

  3. Vienna, W.D.:

  4. Nedjalkov, M., Book chapter: Wigner transport in presence of phonons: particle models of the electron kinetics, in from nanostructures to nanosensing applications. In: A. Paoletti, A. D’Amico, G. Ballestrino (eds.) Proceedings of the International School of Physics ”Enrico Fermi”, vol. 160, pp. 55–103. IOS Press, Amsterdam (2005). doi:10.3254/1-58603-527-4-55

  5. Querlioz, D., Dollfus, P.: The Wigner Monte Carlo Method for Nanoelectronic Devices—A Particle Description of Quantum Transport and Decoherence. ISTE-Wiley, London (2010)

    MATH  Google Scholar 

  6. Nedjalkov, M., Vasileska, D., Ferry, D.K., Jacoboni, C., Ringhofer, C., Dimov, I., Palankovski, V.: Wigner transport models of the electron-phonon kinetics in quantum wires. Phys. Rev. B 74, 035311 (2006). doi:10.1103/PhysRevB.74.035311

    Article  Google Scholar 

  7. Nedjalkov, M., Querlioz, D., Dollfus, P., Kosina, H. Review chapter: Wigner function approach. In: D. Vasileska, S. Goodnick (eds.) Nano-Electronic Devices: Semiclassical and Quantum Transport Modeling, pp. 289 - 358. Springer-Verlag, New York (2011), ISBN: 978-1-4419-8839-3. doi:10.1007/978-1-4419-8840-9_5. Invited

  8. Tatarskii, V.I.: The Wigner representation of quantum mechanics. Sov. Phys. Usp. 26, 311 (1983). doi:10.1070/PU1983v026n04ABEH004345

    Article  MathSciNet  Google Scholar 

  9. Ravaioli, U., Osman, M.A., Ptz, W., Kluksdahl, N., Ferry, D.K.: Investigation of ballistic transport through resonant-tunnelling quantum wells using Wigner function approach, Physica B+C 134(13), 36 (1985). doi:10.1016/0378-4363(85)90317-1

  10. Frensley, W.: Wigner-function model of resonant-tunneling semiconductor device. Phys. Rev. B 36(3), 1570 (1987). doi:10.1103/PhysRevB.36.1570

    Article  Google Scholar 

  11. Kluksdahl, N.C., Poetz, W., Ravaiolli, U., Ferry, D.K.: Wigner function study of a double quantum well resonant-tunneling diode. Superlattices Microstruct. 3, 41 (1987)

    Article  Google Scholar 

  12. Kluksdahl, N.C., Kriman, A.M., Ferry, D.K., Ringhofer, C.: Self-consistent study of resonant-tunneling diode. Phys. Rev. B 39, 7720 (1989). doi:10.1103/PhysRevB.39.7720

    Article  Google Scholar 

  13. Frensley, W.: Boundary conditions for open quantum systems driven far from equilibrium. Rev. Mod. Phys. 62(3), 745 (1990). doi:10.1103/RevModPhys.62.745

    Article  Google Scholar 

  14. Rupp, K., Grasser, K.T., Jüngel, A.: On the feasibility of spherical harmonics expansions of the Boltzmann transport equation for three-dimensional device geometries. In: Proceedings of the IEEE International Electron Devices Meeting (IEDM) (2011). doi:10.1109/IEDM.2011.6131667

  15. Peikert, V., Schenk, A.: A wavelet method to solve high-dimensional transport equations in semiconductor devices. In: 2011 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), pp. 299–302 (2011). doi:10.1109/SISPAD.2011.6035029

  16. Vitanov, P., Nedjalkov, M., Jacoboni, C., Rossi, F., Abramo, A.: Unified Monte Carlo approach to the Boltzmann and Wigner equations. In: Sendov, Bl, Dimov, I. (eds.) Advances in Parallel Algorithms, pp. 117–128. IOS Press, Amsterdam (1994)

    Google Scholar 

  17. Rossi, F., Jacoboni, C., Nedjalkov, M.: A Monte Carlo solution of the Wigner transport equation. Semicond. Sci. Technol. 9, 934 (1994). doi:10.1088/0268-1242/9/5S/143

    Article  Google Scholar 

  18. Sala, R., Brouard, S., Muga, J.G.: Wigner trajectories and Liouvilles theorem. J. Chem. Phys. 99(4), 2708 (1993). doi:10.1063/1.465232

    Article  Google Scholar 

  19. Bordone, P., Bertoni, A., Brunetti, R., Jacoboni, C.: Monte Carlo simulation of quantum electron transport based on Wigner paths. 3rd IMACS Seminar on Monte Carlo Methods. Math. Comput. Simulat. 62(36), 307 (2003). doi:10.1016/S0378-4754(02)00241-0

  20. Shifren, L., Ferry, D.: A wigner function based ensemble monte carlo approach for accurate incorporation of quantum effects in device simulation. J. Comput. Electron. 1(1–2), 55 (2002). doi:10.1023/A:1020711726836

    Article  Google Scholar 

  21. Shifren, L., Ringhofer, C., Ferry, D.: A Wigner function-based quantum ensemble Monte Carlo study of a resonant tunneling diode. IEEE Trans. Electron. Devices 50(3), 769 (2003). doi:10.1109/TED.2003.809434

    Article  Google Scholar 

  22. 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

    Article  Google Scholar 

  23. Nedjalkov, M., Schwaha, P., Selberherr, S., Sellier, J.M., Vasileska, D.: Wigner quasi-particle attributes—an asymptotic perspective. Appl. Phys. Lett. 102(16), 163113 (2013). doi:10.1063/1.4802931

    Article  Google Scholar 

  24. Nedjalkov, M., Vasileska, D.: Semi-discrete 2D Wigner-particle approach. J. Comput. Electron. 7(3), 222 (2008). doi:10.1007/s10825-008-0197-3

    Article  Google Scholar 

  25. Ellinghaus, P., Nedjalkov, M., Selberherr, S.: Optimized particle regeneration scheme for the Wigner Monte Carlo method. In: Abstracts of the Eighth International Conference on Numerical Methods and Applications (NMA) (2014)

  26. Ellinghaus, P., Nedjalkov, M., Selberherr, S.: The Wigner Monte Carlo method for accurate semiconductor device simulation. In: 2014 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), pp. 113–116 (2014)

  27. Los, V., Los, N.: Exact solution of the one-dimensional time-dependent Schrödinger equation with a rectangular well/barrier potential and its applications. Theo. Math. Phys. 177(3), 1706 (2013). doi:10.1007/s11232-013-0128-8

    Article  MATH  MathSciNet  Google Scholar 

Download references


The research leading to these results has received funding from: the Austrian Science Fund (FWF) through the grant P23296, the European Commission under FP7 project AComIn (FP7 REGPOT-2012-2013-1), as well as by the Bulgarian National Science Fund (NSF) under Grant DCVP 02/1. The computational results presented have been achieved in part using the Vienna Scientific Cluster (VSC).

Author information

Authors and Affiliations


Corresponding author

Correspondence to Paul Ellinghaus.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ellinghaus, P., Weinbub, J., Nedjalkov, M. et al. Distributed-memory parallelization of the Wigner Monte Carlo method using spatial domain decomposition. J Comput Electron 14, 151–162 (2015).

Download citation

  • Published:

  • Issue Date:

  • DOI:


  • Wigner
  • Monte Carlo
  • Message passing interface
  • Domain decomposition
  • Parallel
  • Memory-distributed