Journal of Computational Electronics

, Volume 17, Issue 1, pp 76–89 | Cite as

Three-dimensional optimal multi-level Monte–Carlo approximation of the stochastic drift–diffusion–Poisson system in nanoscale devices

  • Amirreza KhodadadianEmail author
  • Leila Taghizadeh
  • Clemens Heitzinger


The three-dimensional stochastic drift–diffusion–Poisson system is used to model charge transport through nanoscale devices in a random environment. Applications include nanoscale transistors and sensors such as nanowire field-effect bio- and gas sensors. Variations between the devices and uncertainty in the response of the devices arise from the random distributions of dopant atoms, from the diffusion of target molecules near the sensor surface, and from the stochastic association and dissociation processes at the sensor surface. Furthermore, we couple the system of stochastic partial differential equations to a random-walk-based model for the association and dissociation of target molecules. In order to make the computational effort tractable, an optimal multi-level Monte–Carlo method is applied to three-dimensional solutions of the deterministic system. The whole algorithm is optimal in the sense that the total computational cost is minimized for prescribed total errors. This comprehensive and efficient model makes it possible to study the effect of design parameters such as applied voltages and the geometry of the devices on the expected value of the current.


Stochastic drift–diffusion–Poisson system Multi-level Monte–Carlo Optimal method Uncertainty quantification Fluctuations Noise Transistor Nanowire 



The authors acknowledge support by FWF (Austrian Science Fund) START Project No. Y660 PDE Models for Nanotechnology. The authors also acknowledge discussions with Gerhard Tulzer.


  1. 1.
    Stern, E., Klemic, J.F., Routenberg, D.A., Wyrembak, P.N., Turner-Evans, D.B., Hamilton, A.D., LaVan, D.A., Fahmy, T.M., Reed, M.A.: Label-free immunodetection with CMOS-compatible semiconducting nanowires. Nature 445(7127), 519–522 (2007)CrossRefGoogle Scholar
  2. 2.
    Stern, E., Vacic, A., Rajan, N.K., Criscione, J.M., Park, J., Ilic, B.R., Mooney, D.J., Reed, M.A., Fahmy, T.M.: Label-free biomarker detection from whole blood. Nat. Nanotechnol. 5(2), 138–142 (2010)CrossRefGoogle Scholar
  3. 3.
    Tulzer, G., Heitzinger, C.: Fluctuations due to association and dissociation processes at nanowire-biosensor surfaces and their optimal design. Nanotechnology 26(2), 025502/1–9 (2015). CrossRefGoogle Scholar
  4. 4.
    Tulzer, G., Heitzinger, C.: Brownian-motion based simulation of stochastic reaction–diffusion systems for affinity based sensors. Nanotechnology 27(16), 165501/1–9 (2016). CrossRefGoogle Scholar
  5. 5.
    Heitzinger, C., Mauser, N.J., Ringhofer, C.: Multiscale modeling of planar and nanowire field-effect biosensors. SIAM J. Appl. Math. 70(5), 1634–1654 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Baumgartner, S., Heitzinger, C.: Existence and local uniqueness for 3D self-consistent multiscale models for field-effect sensors. Commun. Math. Sci 10(2), 693–716 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Heitzinger, C., Ringhofer, C.: Multiscale modeling of fluctuations in stochastic elliptic PDE models of nanosensors. Commun. Math. Sci. 12(3), 401–421 (2014). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Khodadadian, A., Hosseini, K., Manzour ol Ajdad, A., Hedayati, M., Kalantarinejad, R., Heitzinger, C.: Optimal design of nanowire field-effect troponin sensors. Comput. Biol. Med. 87, 46–56 (2017)CrossRefGoogle Scholar
  9. 9.
    Roy, G., Brown, A.R., Adamu-Lema, F., Roy, S., Asenov, A.: Simulation study of individual and combined sources of intrinsic parameter fluctuations in conventional nano-MOSFETs. IEEE Trans. Electron Devices 53(12), 3063–3070 (2006)Google Scholar
  10. 10.
    Seoane, N., Martinez, A., Brown, A.R., Barker, J.R., Asenov, A.: Current variability in Si nanowire MOSFETs due to random dopants in the source/drain regions: a fully 3-D NEGF simulation study. IEEE Trans. Electron Devices 56(7), 1388–1395 (2009)CrossRefGoogle Scholar
  11. 11.
    Taghizadeh, L., Khodadadian, A., Heitzinger, C.: The optimal multilevel Monte–Carlo approximation of the stochastic drift–diffusion–Poisson system. Comput. Methods Appl. Mech. Eng. 318, 739–761 (2017). MathSciNetCrossRefGoogle Scholar
  12. 12.
    Khodadadian, A., Taghizadeh, L., Heitzinger, C.: Optimal multilevel randomized quasi-Monte-Carlo method for the stochastic drift-diffusion-Poisson system. Comput. Methods Appl. Mech. Eng. 329, 480–497 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted korobov and sobolev spaces. J. Complex. 19(3), 301–320 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heitzinger, C., Taghizadeh, L.: Existence and local uniqueness for the stochastic drift–diffusion–Poisson system. (Submitted for publication) Google Scholar
  15. 15.
    Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. CRC Press, Boca Raton (1988)CrossRefzbMATHGoogle Scholar
  16. 16.
    Sano, N., Matsuzawa, K., Mukai, M., Nakayama, N.: On discrete random dopant modeling in drift–diffusion simulations: physical meaning of atomistic dopants. Microelectron. Reliab. 42(2), 189–199 (2002)CrossRefGoogle Scholar
  17. 17.
    Jiang, X.-W., Deng, H.-X., Luo, J.-W., Li, S.-S., Wang, L.-W.: A fully three-dimensional atomistic quantum mechanical study on random dopant-induced effects in 25-nm MOSFETs. IEEE Trans. Electron Devices 55(7), 1720–1726 (2008)CrossRefGoogle Scholar
  18. 18.
    Chen, D., Wei, G.-W.: Modeling and simulation of electronic structure, material interface and random doping in nano-electronic devices. J. Comput. Phys. 229(12), 4431–4460 (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Khodadadian, A., Heitzinger, C.: Basis adaptation for the stochastic nonlinear Poisson–Boltzmann equation. J. Comput. Electron. 15(4), 1393–1406 (2016)CrossRefGoogle Scholar
  20. 20.
    Patolsky, F., Lieber, C.M.: Nanowire nanosensors. Mater. Today 8(4), 20–28 (2005)CrossRefGoogle Scholar
  21. 21.
    Cliffe, K., Giles, M., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3–15 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51(1), 322–352 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Baumgartner, S., Heitzinger, C., Vacic, A., Reed, M.A.: Predictive simulations and optimization of nanowire field-effect PSA sensors including screening. Nanotechnology 24(22), 225503/1–9 (2013). CrossRefGoogle Scholar
  24. 24.
    Colinge, J.-P., et al.: FinFETs and Other Multi-Gate Transistors. Springer, Berlin (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Amirreza Khodadadian
    • 1
    Email author
  • Leila Taghizadeh
    • 1
  • Clemens Heitzinger
    • 1
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of Technology (TU Wien)ViennaAustria
  2. 2.School of Mathematical and Statistical SciencesArizona State UniversityTempeUSA

Personalised recommendations