Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 6, pp 2957–2977 | Cite as

p-Laplace thermistor modeling of electrothermal feedback in organic semiconductor devices

  • Matthias LieroEmail author
  • Thomas Koprucki
  • Axel Fischer
  • Reinhard Scholz
  • Annegret Glitzky


In large-area organic light-emitting diodes (OLEDs), spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors, we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered. We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions, we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder’s fixed-point theorem.

Mathematics Subject Classification

35J92 65M08 35D30 35G60 35J57 35Q79 80M12 80A20 


p-Laplace Stationary thermistor model Nonlinear coupled system Finite-volume approximation Existence and boundedness Self-heating Arrhenius-like conductivity law Organic light-emitting diode 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bergemann K.J., Krasny R., Forrest S.R.: Thermal properties of organic light-emitting diodes. Organ. Electron. 13(9), 1565–1568 (2012)CrossRefGoogle Scholar
  2. 2.
    Bourbaki, N.: Éléments de mathématique. Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d’une mesure, Espaces L p, Deuxième édition revue et augmentée. Actualités Scientifiques et Industrielles, No. 1175, Hermann, Paris (1965)Google Scholar
  3. 3.
    Bradji A., Herbin R.: Discretization of coupled heat and electrical diffusion problems by finite-element and finite-volume methods. IMA J. Numer. Anal. 28(3), 469–495 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Cimatti G.: Remark on existence and uniqueness for the thermistor problem under mixed boundary conditions. Quart. Appl. Math 47(1), 117–121 (1989)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Cimatti G.: A remark on the thermistor problem with rapidly growing conductivity. Appl. Anal. 80(1-2), 133–140 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Diening L., Harjulehto P., Hästö P., Růžička M.: Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)Google Scholar
  7. 7.
    Diening L., Nägele P., Růžička M.: Monotone operator theory for unsteady problems in variable exponent spaces. Complex Var. Elliptic Equ. 57(11), 1209–1231 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Eleuteri M.: Hölder continuity results for a class of functionals with non standard growth. Artic. Ric. Mat. 7(8), 129–157 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handbook of Numerical Analysis, Vol. VII, North-Holland, Amsterdam, pp. 713–1020 (2000)Google Scholar
  10. 10.
    Fiaschi A., Knees D., Reichelt S.: Global higher integrability of minimizers of variational problems with mixed boundary conditions. J. Math. Anal. Appl. 401(1), 269–288 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Fischer A., Koprucki T., Gärtner K., Tietze M.L., Brückner J., Lüssem B., Leo K., Glitzky A., Scholz R.: Feel the heat: nonlinear electrothermal feedback in organic LEDs. Adv. Funct. Mater. 24(22), 3366–3366 (2014)CrossRefGoogle Scholar
  12. 12.
    Fischer A., Pahner P., Lüssem B., Leo K., Scholz R., Koprucki T., Fuhrmann J., Gärtner K., Glitzky A.: Self-heating effects in organic semiconductor crossbar structures with small active area. Org. Electron. 13(11), 2461–2468 (2012)CrossRefGoogle Scholar
  13. 13.
    Fischer A., Pahner P., Lüssem B., Leo K., Scholz R., Koprucki T., Gärtner K., Glitzky A.: Self-heating, bistability, and thermal switching in organic semiconductors. Phys. Rev. Lett. 110, 126601 (2013)CrossRefGoogle Scholar
  14. 14.
    Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974, Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Band 38 (1974)Google Scholar
  15. 15.
    Gröger K.: A W 1,p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283(4), 679–687 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Harjulehto P., Hästö P., Lê Út. V., Nuortio M.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72(12), 4551–4574 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Keskin A.Ü.: A simple analog behavioural model for NTC thermistors including selfheating effect. Sens. Actuat. A Phys. 118(2), 244–247 (2005)CrossRefGoogle Scholar
  18. 18.
    Kohári, Z., Pohl, L.: How thermal environment affects OLEDs’ operational characteristics. 28 th IEEE SEMI-THERM Symposium, p. 331 (2012)Google Scholar
  19. 19.
    Lindqvist, P.: Notes on the p-Laplace equation, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 102, University of Jyväskylä, Jyväskylä (2006)Google Scholar
  20. 20.
    Mitrinović, D.S.: Analytic inequalities. In cooperation with P. M. Vasić. Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer, New York (1970)Google Scholar
  21. 21.
    Park J., Lee J., Noh Y.: Optical and thermal properties of large-area OLED lightings with metallic grids. Org. Electron. 13, 184–194 (2012)CrossRefGoogle Scholar
  22. 22.
    Pohl, L., Kollar, E.: Extension of the sunred algorithm for electrothermal simulation and its application in failure analysis of large area (organic) semiconductor devices. THERMINIC 17th Workshop on Thermal Investigation of ICs and Systems, Sept. 2011, Paris, 2011, pp. 1–6 (2011)Google Scholar
  23. 23.
    Schöll E.: Nonequilibrium Phase Transitions in Semiconductors: Self-Organization Induced by Generation and Recombination Processes. Springer Series in Synergetics. Springer, Berlin (1987)CrossRefGoogle Scholar
  24. 24.
    Shaw M.P., Mitin V.V., Schöll E., Grubin H.L.: The Physics of Instabilities in Solid State Electron Devices. Plenum Press, New York (1992)CrossRefGoogle Scholar
  25. 25.
    Slawinski M., Bertram D., Heuken M., Kalisch H., Vescan A.: Electrothermal characterization of large-area organic light-emitting diodes employing finite-element simulation. Org. Electron. 12(8), 1399–1405 (2011)CrossRefGoogle Scholar
  26. 26.
    Slawinski M., Weingarten M., Heuken M., Vescan A., Kalisch H.: Investigation of large-area OLED devices with various grid geometries. Org. Electron. 14(10), 2387–2391 (2013)CrossRefGoogle Scholar
  27. 27.
    van Mensfoort S.L.M., Coehoorn R.: Effect of Gaussian disorder on the voltage dependence of the current density in sandwich-type devices based on organic semiconductors. Phys. Rev. B 78, 085207 (2008)CrossRefGoogle Scholar
  28. 28.
    Xie Hong., Allegretto W.: \({C^{\alpha}({\overline{\Omega}})}\) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM J. Math. Anal. 22(6), 1491–1499 (1991)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Yang L., Wei B., Zhang J.: Transient thermal characterization of organic light-emitting diodes. Semicond. Sci. Technol. 27(10), 105011 (2012)CrossRefGoogle Scholar
  30. 30.
    Zeidler, E.: Nonlinear functional analysis and its applications. Nonlinear Monotone Operators (Translated from the German by the author and Leo F. Boron), II/B, Springer, New York (1990)Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Matthias Liero
    • 1
    Email author
  • Thomas Koprucki
    • 1
  • Axel Fischer
    • 2
  • Reinhard Scholz
    • 2
  • Annegret Glitzky
    • 1
  1. 1.Weierstrass InstituteBerlinGermany
  2. 2.Institut für Angewandte PhotophysikTU DresdenDresdenGermany

Personalised recommendations