Skip to main content
SpringerLink
Log in

On iterative schemes for a stationary problem to a quantum drift diffusion model

  • Published:
Journal of Computational Electronics Aims and scope Submit manuscript

Abstract

In this paper, we propose a new iterative scheme to obtain a stationary solution to a quantum drift diffusion (QDD) model for semiconductors in a multi-dimensional space. The model consists of two fourth order nonlinear elliptic equations for the carrier densities and the nonlinear Poisson equation for the electrostatic potential. The present scheme adopts the Gummel method to decouple the system. For the computational efficiency, this scheme also employs an exponential transformation of variables to preserve positivity of the carrier densities and an inner iteration to couple the quantum potentials with the electrostatic potential. The present scheme is constructed to improve the other schemes developed by Ancona, de Falco and Odanaka. We show the advantage of the present scheme on computational costs in comparison to the other schemes for simulations of current-voltage characteristics for a double-gate MOSFET.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. Datta, S.: Nanoscale device modeling: the Green’s function method. Superlattices Microstruct. 28, 253–278 (2000)

    Article  Google Scholar 

  2. Frensley, W.: Wigner-function model of a resonant-tunneling semiconductor device. Phys. Rev. B 26, 1570–1580 (1987)

    Article  Google Scholar 

  3. Ancona, M.G., Tiersten, H.F.: Macroscopic physics of the silicon inversion layer. Phys. Rev. B 35(15), 7959–7965 (1987)

    Article  Google Scholar 

  4. Jüngel, A.: Transport Equations for Semiconductors. Lect. Notes Phys., vol. 773. Springer, Berlin (2009)

    Book  Google Scholar 

  5. Degond, P., Mehats, F., Ringhofer, C.: Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 118, 625–665 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gardner, C.L.: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54(2), 407–427 (1994)

    Article  Google Scholar 

  7. Ancona, M.G.: Finite-difference schemes for the density gradient equations. J. Comput. Electron. 1(3), 435–443 (2002)

    Article  Google Scholar 

  8. de Falco, C., Gatti, E., Lacaita, A.L., Sacco, R.: Quantum-corrected drift-diffusion models for transport in semiconductor devices. J. Comput. Phys. 204, 533–561 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Odanaka, S.: Multidimensional discretization of the stationary quantum drift-diffusion model for ultrasmall MOSFET structures. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 23(6), 837–842 (2004)

    Article  Google Scholar 

  10. Odanaka, S.: A high-resolution method for quantum confinement transport simulations in MOSFET. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 26(1), 80–85 (2007)

    Article  Google Scholar 

  11. Shimada, T., Odanaka, S.: A numerical method for a transient quantum drift-diffusion model arising in semiconductor devices. J. Comput. Electron. 7(4), 485–493 (2008)

    Article  Google Scholar 

  12. Ancona, M.G., Yu, Z., Dutton, R.W., Vande Voorde, P.J., Cao, M., Vook, V.: Density-gradient analysis of MOS tunneling. IEEE Trans. Electron Devices 47, 2310–2319 (2000)

    Article  Google Scholar 

  13. Pinnau, R., Unterreiter, A.: The stationary current-voltage characteristics of the quantum drift-diffusion model. SIAM J. Numer. Anal. 37, 211–245 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Biegel, B.A., Ancona, M.G., Rafferty, C.S., Yu, Z.: Efficient multi-dimensional simulation of quantum confinement effects in advanced MOS devices. NAS Technical Report NAS-04-008 (2004)

  15. Shao, X., Yu, Z.: Negative gate-overlap in nanoscaled DG-MOSFETs with asymmetric gate bias. J. Comput. Electron. 5(4), 389–392 (2006)

    Article  Google Scholar 

  16. Ben Abdallah, N., Unterreiter, A.: On the stationary quantum drift-diffusion model. Z. Angew. Math. Phys. 49, 251–275 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jüngel, A., Li, H.-L., Matsumura, A.: The relaxation-time limit in the quantum hydrodynamic equations for semiconductors. J. Differ. Equ. 225(2), 440–464 (2006)

    Article  MATH  Google Scholar 

  18. Nishibata, S., Shigeta, N., Suzuki, M.: Asymptotic behaviors and classical limits of solutions to a quantum drift-diffusion model for semiconductors. Math. Models Methods Appl. Sci. 20, 909–936 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gummel, H.K.: A self-consistent iterative scheme for one-dimensional steady state transistor calculations. IEEE Trans. Electron Devices ED-11(10), 455–465 (1964)

    Article  Google Scholar 

  20. Scharfetter, D.L., Gummel, H.K.: Large-signal analysis of a silicon read diode oscillator. IEEE Trans. Electron Devices ED-16(1), 64–77 (1969)

    Article  Google Scholar 

  21. Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, Wien (1984)

    Book  Google Scholar 

  22. Bank, R.E., Rose, D.J., Fichtner, W.: Numerical methods for semiconductor device simulation. IEEE Trans. Electron Devices ED-30(9), 1031–1041 (1983)

    Article  Google Scholar 

  23. Tang, T.-W., Wang, X., Li, Y.: Discretization scheme for the density gradient equation and effect of boundary conditions. J. Comput. Electron. 1(3), 389–393 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to express gratitude to Professor Shinji Odanaka and Professor Shinya Nishibata for fruitful discussions.

Author information

Authors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Yoshikatsu Noda

Authors
  1. Yoshikatsu Noda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yoshikatsu Noda.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Noda, Y. On iterative schemes for a stationary problem to a quantum drift diffusion model. J Comput Electron 11, 385–396 (2012). https://doi.org/10.1007/s10825-012-0418-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10825-012-0418-7

Keywords

Search

Navigation