On Some Extension of Energy-Drift-Diffusion Models: Gradient Structure for Optoelectronic Models of Semiconductors

  • Alexander Mielke
  • Dirk Peschka
  • Nella Rotundo
  • Marita Thomas
Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

Abstract

We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon-densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.

Notes

Acknowledgements

This research was partially supported by DFG via project B4 in SFB 787 and by the Einstein Foundation Berlin via the Matheon project OT1 in ECMath. The authors are grateful to M. Liero, A. Glitzky and T. Koprucki for fruitful discussions.

References

  1. 1.
    Albinus, G., Gajewski, H., Hünlich, R.: Thermodynamic design of energy models of semiconductor devices. Nonlinearity 15(2), 367 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bandelow, U., Gajewski, H., Hünlich, R.: Fabry-Perot Lasers: Thermodynamics-Based Modeling, pp. 63–85. Springer, New York (2005)Google Scholar
  3. 3.
    Chuang, S.: Physics of Optoelectronic Devices. Wiley Series in Pure and Applied Optics, vol. 22. Wiley, New York (1995)Google Scholar
  4. 4.
    Glitzky, A., Mielke, A.: A gradient structure for systems coupling reaction–diffusion effects in bulk and interfaces. Z. Angew. Math. Phys. 64(1), 29–52 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Jordan, R., Kinderlehrer, D., Otto, F.: The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29(1), 1–17 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Landau, L., Lifshitz, E.: Statistical Physics, vol. 5: Course of Theoretical Physics. Pergamon Press, Oxford (1980)Google Scholar
  7. 7.
    Mielke, A.: A gradient structure for reaction–diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24(4), 1329–1346 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Mielke, A.: On thermodynamical couplings of quantum mechanics and macroscopic systems. In: Mathematical Results in Quantum Mechanics: Proceedings of the QMath12 Conference, pp. 331–348. World Scientific (2015)Google Scholar
  9. 9.
    Mittnenzweig, M., Mielke, A.: An entropic gradient structure for Lindblad equations and GENERIC for quantum systems coupled to macroscopic models. arXiv preprint (2016). ArXiv:1609.05765Google Scholar
  10. 10.
    Öttinger, H.C.: Beyond Equilibrium Thermodynamics. Wiley, Hoboken, NJ (2005)CrossRefGoogle Scholar
  11. 11.
    Otto, F.: Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory. Arch. Ration. Mech. Anal. 141(1), 63–103 (1998)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Otto, F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26(1–2), 101–174 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Peschka, D., Thomas, M., Glitzky, A., Nürnberg, R., Gärtner, K., Virgilio, M., Guha, S., Schroeder, T., Capellini, G., Koprucki, T.: Modeling of edge-emitting lasers based on tensile strained germanium microstrips. IEEE Photonics J. 7(3), 1–15 (2015)CrossRefGoogle Scholar
  14. 14.
    Wachutka, G.K.: Rigorous thermodynamic treatment of heat generation and conduction in semiconductor device modeling. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 9(11), 1141–1149 (1990)CrossRefGoogle Scholar
  15. 15.
    Würfel, P.: The chemical potential of radiation. J. Phys. C Solid State Phys. 15(18), 3967 (1982)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  • Alexander Mielke
    • 1
  • Dirk Peschka
    • 1
  • Nella Rotundo
    • 1
  • Marita Thomas
    • 1
  1. 1.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

Personalised recommendations