Skip to main content
SpringerLink
Log in

Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

This paper is concerned with the global well-posedness and relaxation-time limits for the solutions in the full quantum hydrodynamic model, which can be used to analyze the thermal and quantum influences on the transport of carriers in semiconductor devices. For the Cauchy problem in ℝ3, we prove the global existence, uniqueness and exponential decay estimate of smooth solutions, when the initial data are small perturbations of an equilibrium state. Moreover, we show that the solutions converge into that of the simplified quantum energy-transport model and the quantum drift-diffusion model for the moment relaxation limit, and the moment and energy relaxation limit, respectively.

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

References

  1. X. Chen: The isentropic quantum drift-diffusion model in two or three dimensions. Z. Angew. Math. Phys. 60 (2009), 416–437.

    Article  MathSciNet  CAS  Google Scholar 

  2. X. Chen, L. Chen: Initial time layer problem for quantum drift-diffusion model. J. Math. Anal. Appl. 343 (2008), 64–80.

    Article  MathSciNet  Google Scholar 

  3. X. Chen, L. Chen, H. Jian: Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model. Nonlinear Anal., Real World Appl. 10 (2009), 1321–1342.

    Article  MathSciNet  Google Scholar 

  4. L. Chen, Q. Ju: Existence of weak solution and semiclassical limit for quantum drift-diffusion model. Z. Angew. Math. Phys. 58 (2007), 1–15.

    Article  MathSciNet  Google Scholar 

  5. J. Dong: Mixed boundary-value problems for quantum hydrodynamic models with semiconductors in thermal equilibrium. Electron. J. Differ. Equ. 2005 (2005), Article ID 123, 8 pages.

  6. C. L. Gardner: The quantum hydrodynamic model for semiconductor devices. SIAM J. Appl. Math. 54 (1994), 409–427.

    Article  ADS  MathSciNet  Google Scholar 

  7. U. Gianazza, G. Savaré, G. Toscani: The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation. Arch. Ration. Mech. Anal. 194 (2009), 133–220.

    Article  MathSciNet  Google Scholar 

  8. M. P. Gualdani, A. Jüngel, G. Toscani: A nonlinear fourth-order parabolic equation with nonhomogeneous boundary conditions. SIAM. J. Math. Anal. 37 (2006), 1761–1779.

    Article  MathSciNet  Google Scholar 

  9. F. Huang, H.-L. Li, A. Matsumura: Existence and stability of steady-state of one-dimensional quantum hydrodynamic system for semiconductors. J. Differ. Equations 225 (2006), 1–25.

    Article  ADS  MathSciNet  Google Scholar 

  10. Y. Jia, H. Li: Large-time behavior of solutions of quantum hydrodynamic model for semiconductors. Acta Math. Sci., Ser. B, Engl. Ed. 26 (2006), 163–178.

    Article  MathSciNet  Google Scholar 

  11. A. Jüngel: A steady-state quantum Euler-Poisson system for potential flows. Commun. Math. Phys. 194 (1998), 463–479.

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Jüngel, H. Li: Quantum Euler-Poisson systems: Existence of stationary states. Arch. Math., Brno 40 (2004), 435–456.

    MathSciNet  Google Scholar 

  13. A. Jüngel, H. Li: Quantum Euler-Poisson systems: Global existence and exponential decay. Q. Appl. Math. 62 (2004), 569–600.

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  15. A. Jüngel, D. Matthes, J. P. Milišić: Derivation of new quantum hydrodynamic equations using entropy minimization. SIAM J. Appl. Math. 67 (2006), 46–68.

    Article  MathSciNet  Google Scholar 

  16. A. Jüngel, J. P. Milišić: A simplified quantum energy-transport model for semiconductors. Nonlinear Anal., Real World Appl. 12 (2011), 1033–1046.

    Article  MathSciNet  Google Scholar 

  17. A. Jüngel, R. Pinnau: A positivity-preserving numerical scheme for a nonlinear fourth order parabolic system. SIAM J. Numer. Anal. 39 (2001), 385–406.

    Article  MathSciNet  Google Scholar 

  18. A. Jüngel, I. Violet: The quasineutral limit in the quantum drift-diffusion equations. Asymptotic Anal. 53 (2007), 139–157.

    MathSciNet  Google Scholar 

  19. Y.-H. Kim, S. Ra, S.-C. Kim: Asymptotic behavior of strong solutions of a simplified energy-transport model with general conductivity. Nonlinear Anal., Real World Appl. 59 (2021), Article ID 103261, 18 pages.

  20. S. Klainerman, A. Majda: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981), 481–524.

    Article  ADS  MathSciNet  Google Scholar 

  21. H. Li, P. Marcati: Existence and asymptotic behavior of multi-dimensional quantum hydrodynamic model for semiconductors. Commun. Math. Phys. 245 (2004), 215–247.

    Article  ADS  MathSciNet  Google Scholar 

  22. H. Li, G. Zhang, M. Zhang, C. Hao: Long-time self-similar asymptotic of the macroscopic quantum models. J. Math. Phys. 49 (2008), Article ID 073503, 14 pages.

  23. J. Mao, F. Zhou, Y. Li: Some limit analysis in a one-dimensional stationary quantum hydrodynamic model for semiconductors. J. Math. Anal. Appl. 364 (2010), 186–194.

    Article  MathSciNet  Google Scholar 

  24. P. A. Markowich, C. A. Ringhofer, C. Schmeiser: Semiconductor Equations. Springer, Vienna, 1990.

    Book  Google Scholar 

  25. L. Nirenberg: On elliptic partial differential equations. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 13 (1959), 115–162.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. S. Nishibata, M. Suzuki: Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits. J. Differ. Equations 244 (2008), 836–874.

    Article  ADS  MathSciNet  Google Scholar 

  28. S. Ra, H. Hong: The existence, uniqueness and exponential decay of global solutions in the full quantum hydrodynamic equations for semiconductors. Z. Angew. Math. Phys. 72 (2021), Article ID 107, 32 pages.

  29. J. Ri, S. Ra: Solution to a multi-dimensional isentropic quantum drift-diffusion model for bipolar semiconductors. Electron. J. Differ. Equ. 2018 (2018), Article ID 200, 19 pages.

  30. J. Simon: Compact sets in the space Lp(0, T; B). Ann. Mat. Pura Appl. (4) 146 (1986), Article ID 146, 32 pages.

  31. G. Zhang, H.-L. Li, K. Zhang: Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors. J. Differ. Equations 245 (2008), 1433–1453.

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgment

The authors would like to thank the anonymous referees for helpful comments and suggestions, which greatly improved the quality of the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Sciences, Unjong District, 355, Pyongyang, Democratic People’s Republic of Korea

    Sungjin Ra

  2. Institute of Mathematics, State Academy of Sciences, Unjong District, 355, Pyongyang, Democratic People’s Republic of Korea

    Hakho Hong

Authors
  1. Sungjin Ra
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hakho Hong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sungjin Ra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ra, S., Hong, H. Relaxation-time limits of global solutions in full quantum hydrodynamic model for semiconductors. Appl Math 69, 113–137 (2024). https://doi.org/10.21136/AM.2023.0039-23

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.21136/AM.2023.0039-23

Keywords

MSC 2020

Search

Navigation