Abstract
The quasineutral limit and the mixed layer problem of the bipolar drift-diffusion model for semiconductors with different mobilities are studied in one dimensional space in this paper. For the general smooth doping profile, the general initial data and the different mobilities of electrons and holes, the quasineutral limit is proven by the matched asymptotic expansion method of singular perturbation problem and the energy method.
Similar content being viewed by others
Data Availability Statement
This manuscript has no associated data.
References
Arnold, A., Markowich, P.A., Toscani, G.: On large time asymptotics for drift-diffusionpoisson systems. Transp. Theory and Stat. Phys. 29(3–5), 571–581 (2000)
Fang, W., Ito, K.: Global solution of the time-dependent drift-diffusion semiconductoe equations. J. Differential Equations 123, 523–566 (1995)
Gasser, I.: The initial time layer problem and the quasineutral limit in a nonlinear drift diffusion model for semiconductors. NoDEA Nonlinear Differential Equations Appl. 8(3), 237–249 (2001)
Gasser, I., Hsiao, L., Markowich, P.A., Wang, S.: Quasineutral limit of a nonlinear drift-diffusion model for semiconductor models. J. Mathematical Anal. And Appl. 168, 184–199 (2002)
Gasser, I., Levermore, D., Markowich, P.A., Schmeiser, C.: The initial time layer problem and the quasineutral limit in the semiconductor drift-diffusion model. European J. Appl. Math. 12, 497–512 (2001)
Hsiao, L., Wang, S.: Quasineutral limit of a nonlinear drift diffusion model for semiconductors: the fast diffusion case. Advances in Mathematics (China) 32(5), 615–622 (2003)
Hsiao, L., Wang, S.: Quasineutral limit of a time-dependent drift-diffusion-Poisson model for p-n junction semiconductor devices. J. Differential Equations 225(2), 411–439 (2006)
Hsiao, L., Ju, Q., Wang, S.: Quasi-neutral limit of the drift-diffusion model for semiconductors with general sign-changing doping profile. Science in China Series A-Mathematics 51(9), 1619–1630 (2008)
Jüngel, A.: Transport equations for semiconductors. Lecture Notes in Physics 773, Springer, Berlin Heidelberg 192, 32-43, (2009)
Jüngel, A., Peng, Y.J.: A hierarchy of hydrodynamic models for plasmas. Quasi-neutral limits in the drift-diffusion equations. Asymptotic anal. 28, 49–73 (2001)
Markowich, P.A., Ringhofer, C.A.: A singularly perturbed boundary value problem modelling a semiconductor device. SIAM J. Appl. Math. 44, 231–256 (1984)
Markowich, P.A., Ringhofer, C.A., Schmeiser, C.: Semiconductor Equations. Spinger Verlag, Wien New York (1990)
Ogawa, T., Yamamoto, M.: Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation. Math. Models and Methods in Appl. Sci. 19(6), 939–967 (2009)
Peng, Y.J.: Boundary layer analysis and quasineutral limits in the drift-diffusion equations. Math. Modelling and Numerical Anal. 35(2), 295–312 (2001)
Van Roosbroeck, W.: Theory of flow of electron and holes in germanium and other semiconductors. Bell System Technical Journal 29, 560–607 (1950)
Schmeiser, C., Wang, S.: Quasineutral limit of the drift-diffusion model for semiconductors with general initial data. Math. Models and Methods in Appl. Sci. 13, 463–470 (2003)
Wang, K., Wang, S.: Quasi-neutral limit to the drift-diffusion models for semiconductors with physical contact-insulating boundary conditions. J. Differential Equations 249, 3291–3311 (2010)
Wang, S., Wang, K.: Quasi-neutral limit to the drift-diffusion models for semiconductors with physical contact-insulating boundary conditions and the general sign-changing doping profile. Nonlinear Analysis 72(9–10), 3612–3626 (2010)
Wang, S., Wang, K.: The mixed layer problem and quasi-neutral limit of the drift-diffusion model for semiconductors. SIAM J. Math. Anal. 44(2), 699–717 (2012)
Wang, S., Xin, Z., Markowich, P.A.: Quasi-neutral limit of the drift-diffusion models for semiconductors: The case of general sign-changing doping profile. SIAM J. Math. Anal. 37, 1854–1889 (2006)
Yamamoto, M.: Asymptotic expansion of solutions to the drift-diffusion equation with large initial data. J. Math. Anal. and Appl. 369, 144–163 (2010)
Acknowledgements
We are grateful to Dr. Weizhu Bao from the Department of Mathematics of the National University of Singapore for meaningful discussions.
Funding
This work was supported by the National Natural Science Foundation of China (Nos. 11771031, 11831003, 11901021).
Author information
Authors and Affiliations
Contributions
The authors have no relevant financial or non-financial interests to disclose. All authors contributed to the study conception and design. Material preparation and analysis were performed by Chundi Liu and Shu Wang. The first draft of the manuscript was written by Chundi Liu and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing Interests
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, C., Wang, S. Mixed Layer Problem and Quasineutral Limit of the Bipolar Drift-Diffusion Model with Different Mobilities. Results Math 77, 193 (2022). https://doi.org/10.1007/s00025-022-01711-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-022-01711-7