Journal of Mathematical Chemistry

, Volume 56, Issue 3, pp 728–746 | Cite as

The charge-carrier mobility in disordered organic materials: the long-range one-dimensional diffusion with the memory effect

Original Paper
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Abstract

The transport of charge carriers in disordered organic materials is considered based on the techniques of generalized Langevin equation. We simulate the one-dimensional diffusion of a charge in the ensemble of molecular chains interacting with the acoustic phonon subsystem of bulk environment. The random local charge transitions between chain links are mutually correlated. The full computation of the zero-field charge mobility for the N, N-di(1-naphthyl)-N, N-diphenyl-(1,1-biphenyl)-4,4-diamine (\(\alpha \)-NPD) is performed as an illustration. Several models for the probabilities of local transitions are tested. The individual local diffusion constants are randomly varied along a molecular chain within several orders of magnitude. The stationary diffusion regime establishes for every chain the temperature-dependent partial charge mobility as a frequency-dependent complex-valued response function. It is averaged over the chain ensemble. The computational scheme is simple and efficient. The importance of the memory effect depends on specific properties of a given material. This dependence in terms of the system parameters is discussed.

Keywords

Charge transport in semiconductors Spatially correlated disorder Generalized Langevin equation Retarded memory effect 
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Notes

Acknowledgements

The authors are grateful to Dr. A.V. Odinokov for providing the input files used in Ref. [13]. This work was supported by the Russian Science Foundation, Contract No. 14-43-00052.

References

  1. 1.
    I.I. Fishchuk, A. Kadashchuk, H. Bässler, S. Nešpůrek, Nondispersive polaron transport in disordered organic solids. Phys. Rev. B 67, 224303 (2003)CrossRefGoogle Scholar
  2. 2.
    H. Bässler, E.V. Emelianova, Steady-State Photoconduction in Amorphous Organic Solids (Springer, Berlin, 2010), pp. 259–300Google Scholar
  3. 3.
    J.L. Brédas, D. Beljonne, V. Coropceanu, J. Cornil, Charge-transfer and energy-transfer processes in \(\pi \)-conjugated oligomers and polymers: A molecular picture. Chem. Rev 104(11), 4971–5004 (2004)CrossRefGoogle Scholar
  4. 4.
    V. Coropceanu, J. Cornil, D.A. da Silva Filho, Y. Olivier, R. Silbey, J.L. Brédas, Charge transport in organic semiconductors. Chem. Rev. 107(4), 926–952 (2007)CrossRefGoogle Scholar
  5. 5.
    J. Nelson, J.J. Kwiatkowski, J. Kirkpatrick, J.M. Frost, Modeling charge transport in organic photovoltaic materials. Acc. Chem. Res. 42(11), 1768–1778 (2009)CrossRefGoogle Scholar
  6. 6.
    S.D. Baranovskii, O. Rubel, F. Jansson, R. Österbacka, Description of Charge Transport in Disordered Organic Materials (Springer, Berlin, 2010), pp. 1–28Google Scholar
  7. 7.
    V. Rühle, A. Lukyanov, F. May, M. Schrader, T. Vehoff, J. Kirkpatrick, B. Baumeier, D. Andrienko, Microscopic simulations of charge transport in disordered organic semiconductors. J. Chem. Theory Comput 7(10), 3335–3345 (2011)CrossRefGoogle Scholar
  8. 8.
    R. Marcus, N. Sutin, Electron transfers in chemistry and biology. Biochim. Biophys. Acta (BBA) Rev. Bioenerg 811(3), 265–322 (1985)CrossRefGoogle Scholar
  9. 9.
    A. Miller, E. Abrahams, Impurity conduction at low concentrations. Phys. Rev. 120, 745–755 (1960)CrossRefGoogle Scholar
  10. 10.
    P. Kordt, J.J.M. van der Holst, M. Al Helwi, W. Kowalsky, F. May, A. Badinski, C. Lennartz, D. Andrienko, Modeling of organic light emitting diodes: From molecular to device properties. Adv. Funct. Mater 25(13), 1955–1971 (2015).  https://doi.org/10.1002/adfm.201403004 CrossRefGoogle Scholar
  11. 11.
    H. Bässler, Charge transport in disordered organic photoconductors a monte carlo simulation study. Phys. Status Solidi (B) 175(1), 15–56 (1993)CrossRefGoogle Scholar
  12. 12.
    S.V. Novikov, Hopping charge transport in organic materials. Russ. J. Electrochem 48(4), 388–400 (2012)CrossRefGoogle Scholar
  13. 13.
    M.V. Alfimov, A.A. Bagaturyants, M.V. Bogdanova, I.A. Valuev, V.E. Velikhov, A.V. Gavrikov, A.A. Knizhnik, D.N. Krasikov, A.V. Odinokov, B.V. Potapkin, Multiscale modeling volt ampere specifications for single-layer organic devices. Russ. Nanotechnol 10, 102–107 (2015)Google Scholar
  14. 14.
    V. Rodin, F. Symalla, V. Meded, P. Friederich, D. Danilov, A. Poschlad, G. Nelles, F. von Wrochem, W. Wenzel, Generalized effective-medium model for the carrier mobility in amorphous organic semiconductors. Phys. Rev. B 91, 155203 (2015)CrossRefGoogle Scholar
  15. 15.
    O. Stenzel, C. Hirsch, T. Brereton, B. Baumeier, D. Andrienko, D. Kroese, V. Schmidt, A general framework for consistent estimation of charge transport properties via random walks in random environments. Multiscale Model. Simul. 12(3), 1108–1134 (2014).  https://doi.org/10.1137/130942504 CrossRefGoogle Scholar
  16. 16.
    B. Baumeier, O. Stenzel, C. Poelking, D. Andrienko, V. Schmidt, Stochastic modeling of molecular charge transport networks. Phys. Rev. B 86, 184202 (2012).  https://doi.org/10.1103/PhysRevB.86.184202 CrossRefGoogle Scholar
  17. 17.
    M.V. Basilevsky, A.V. Odinokov, S.V. Titov, E.A. Mitina, Golden rule kinetics of transfer reactions in condensed phase: The microscopic model of electron transfer reactions in disordered solid matrices. J. Chem. Phys 139(23), 234102 (2013)CrossRefGoogle Scholar
  18. 18.
    M.V. Basilevsky, E.A. Bormotova, Charge transfer kinetics in non-polar media including a local molecular mode: the temperature dependence in wide ranges. J. Math. Chem. 55, 941–960 (2017)CrossRefGoogle Scholar
  19. 19.
    R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics. Springer Series in Solid-State Sciences (Springer, Berlin, 2012)Google Scholar
  20. 20.
    C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer Series in Synergetics (Springer-Verlag, Berlin, 1985)Google Scholar
  21. 21.
    B. Fain, Theory of Rate Processes in Condensed Media, Lecture Notes in Chemistry. (Springer, Berlin, 2012)Google Scholar
  22. 22.
    U. Weiss, Quantum Dissipative Systems, Series in Modern Condensed Matter Physics (World Scientific, Singapore, 2012)Google Scholar
  23. 23.
    S.L.M. Van Mensfoort, V. Shabro, R.J. De Vries, R.A.J. Janssen, R. Coehoorn, Hole transport in the organic small molecule material -NPD: evidence for the presence of correlated disorder. J. Appl. Phys. 107(11), 113710 (2010)CrossRefGoogle Scholar
  24. 24.
    M.V. Basilevsky, A.V. Odinokov, K.G. Komarova, Charge-transfer mobility parameters in photoelectronic devices: The advanced millerabrahams computation. J. Phys. Chem. B 119(24), 7430–7438 (2015)CrossRefGoogle Scholar
  25. 25.
    E.W. Dijkstra, A note on two problems in connexion with graphs. Num. Math. 1(1), 269–271 (1959)CrossRefGoogle Scholar
  26. 26.
    J. Jortner, M. Bixon, Advances in Chemical Physics, Volume 106, Part 1: Electron Transfer – From Isolated Molecules to Biomolecules (Wiley, Hoboken, 1999)Google Scholar
  27. 27.
    R.R. Dogonadze, A.M Kuznetsov, VINITI, Moscow (1973). (in Russian) Google Scholar
  28. 28.
    J. Ulstrup, Charge Transfer Processes in Condensed Media, Lecture notes in chemistry (Springer-Verlag, Berlin, 1979)Google Scholar
  29. 29.
    J.E. Norton, J.L. Brédas, Polarization energies in oligoacene semiconductor crystals. J. Am. Chem. Soc 130(37), 12377–12384 (2008)CrossRefGoogle Scholar
  30. 30.
    D.P. McMahon, A. Troisi, Evaluation of the external reorganization energy of polyacenes. J. Phys. Chem. Lett 1(6), 941–946 (2010)CrossRefGoogle Scholar
  31. 31.
    F. Brown-Altvater, T. Rangel, J.B. Neaton, Ab initio phonon dispersion in crystalline naphthalene using van der waals density functionals. Phys. Rev. B 93, 195206 (2016)CrossRefGoogle Scholar
  32. 32.
    I. Natkaniec, E.L. Bokhenkov, B. Dorner, J. Kalus, G.A. Mackenzie, G.S. Pawley, U. Schmelzer, E.F. Sheka, Phonon dispersion in d 8 -naphthalene crystal at 6k. J. Phys. C Solid State Phys 13(23), 4265 (1980)CrossRefGoogle Scholar
  33. 33.
    V.K. Jindal, J. Kalus, E.L. Bokhenkov, S.L. Chaplot, B. Dorner, I. Natkaniec, G.S. Pawley, E.F. Sheka, Temperature dependence of the phonon frequencies in deuterated anthracene. J. Phy. C Solid State Phys. 15(36), 7283 (1982)CrossRefGoogle Scholar
  34. 34.
    B. Dorner, E.L. Bokhenkov, S.L. Chaplot, J. Kalus, I. Natkaniec, G.S. Pawley, U. Schmelzer, E.F. Sheka, The 12 external and the 4 lowest internal phonon dispersion branches in d 10 -anthracene at 12k. J. Phys. C Solid State Phys. 15(11), 2353 (1982)CrossRefGoogle Scholar
  35. 35.
    S. Naka, H. Okada, H. Onnagawa, Y. Yamaguchi, T. Tsutsui, Carrier transport properties of organic materials for EL device operation. Synth. Met. 111112, 331–333 (2000)CrossRefGoogle Scholar
  36. 36.
    T. Matsushima, C. Adachi, Enhancing hole transports and generating hole traps by doping organic hole-transport layers with p-type molecules of 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane. Thin Solid Films 517(2), 874–877 (2008)CrossRefGoogle Scholar
  37. 37.
    P. Friederich, F. Symalla, V. Meded, T. Neumann, W. Wenzel, Ab initio treatment of disorder effects in amorphous organic materials: Toward parameter free materials simulation. J. Chem. Theory Comput. 10(9), 3720–3725 (2014)CrossRefGoogle Scholar
  38. 38.
    P.E. Parris, M. Kuś, D.H. Dunlap, V.M. Kenkre, Nonlinear response theory: Transport coefficients for driving fields of arbitrary magnitude. Phys. Rev. E 56, 5295–5305 (1997)CrossRefGoogle Scholar
  39. 39.
    P.E. Parris, V.M. Kenkre, D.H. Dunlap, Nature of charge carriers in disordered molecular solids: Are polarons compatible with observations? Phys. Rev. Lett. 87, 126601 (2001)CrossRefGoogle Scholar
  40. 40.
    S.V. Novikov, D.H. Dunlap, V.M. Kenkre, P.E. Parris, A.V. Vannikov, Essential role of correlations in governing charge transport in disordered organic materials. Phys. Rev. Lett. 81, 4472–4475 (1998)CrossRefGoogle Scholar
  41. 41.
    P. Kordt, S. Stodtmann, A. Badinski, M. Al Helwi, C. Lennartz, D. Andrienko, Parameter-free continuous drift-diffusion models of amorphous organic semiconductors. Phys. Chem. Chem. Phys. 17, 22778–22783 (2015).  https://doi.org/10.1039/C5CP03605D CrossRefGoogle Scholar
  42. 42.
    J.J. Kwiatkowski, J. Nelson, H. Li, J.L. Bredas, W. Wenzel, C. Lennartz, Simulating charge transport in tris(8-hydroxyquinoline) aluminium (Alq3). Phys. Chem. Chem. Phys. 10, 1852–1858 (2008)CrossRefGoogle Scholar
  43. 43.
    A. Fuchs, T. Steinbrecher, M.S. Mommer, Y. Nagata, M. Elstner, C. Lennartz, Molecular origin of differences in hole and electron mobility in amorphous alq3-a multiscale simulation study. Phys. Chem. Chem. Phys. 14, 4259–4270 (2012)CrossRefGoogle Scholar
  44. 44.
    M. Bouhassoune, S. van Mensfoort, P. Bobbert, R. Coehoorn, Carrier-density and field-dependent charge-carrier mobility in organic semiconductors with correlated gaussian disorder. Org. Electron. 10(3), 437–445 (2009)CrossRefGoogle Scholar
  45. 45.
    P. Kordt, D. Andrienko, Modeling of spatially correlated energetic disorder in organic semiconductors. J. Chem. Theory Comput. 12(1), 36–40 (2016).  https://doi.org/10.1021/acs.jctc.5b00764. PMID: 26588022CrossRefGoogle Scholar
  46. 46.
    A. Lukyanov, D. Andrienko, Extracting nondispersive charge carrier mobilities of organic semiconductors from simulations of small systems. Phys. Rev. B 82, 193202 (2010).  https://doi.org/10.1103/PhysRevB.82.193202 CrossRefGoogle Scholar
  47. 47.
    P. Kordt, T. Speck, D. Andrienko, Finite-size scaling of charge carrier mobility in disordered organic semiconductors. Phys. Rev. B 94, 014208 (2016).  https://doi.org/10.1103/PhysRevB.94.014208 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.The Photochemistry Center of the Russian Academy of SciencesMoscowRussia
  2. 2.Karpov Institute of Physical ChemistryMoscowRussia

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