Direct and inverse problems in dispersive time-of-flight photocurrent revisited

Regular Article
Part of the following topical collections:
  1. Topical issue: Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook


Using the fact that the continuous time random walk (CTRW) scheme is a random process subordinated to a simple random walk under the operational time given by the number of steps taken by the walker up to a given time, we revisit the problem of strongly dispersive transport in disordered media, which first lead Scher and Montroll to introducing the power law waiting time distributions. Using a subordination approach permits to disentangle the complexity of the problem, separating the solution of the boundary value problem (which is solved on the level of normal diffusive transport) from the influence of the waiting times, which allows for the solution of the direct problem in the whole time domain (including short times, out of reach of the initial approach), and simplifying strongly the analysis of the inverse problem. This analysis shows that the current traces do not contain information sufficient for unique restoration of the waiting time probability densities, but define a single-parametric family of functions that can be restored, all leading to the same photocurrent forms. The members of the family have the power-law tails which differ only by a prefactor, but may look astonishingly different at their body. The same applies to the multiple trapping model, mathematically equivalent to a special limiting case of CTRW.


  1. 1.
    H. Scher, E. Montroll, Phys. Rev. B 12, 2455 (1975)ADSCrossRefGoogle Scholar
  2. 2.
    R.C.I. MacKenzie, C.S. Shuttle, G.F. Dibb, N. Treat, E.V. Hauff, M.J. Robb, C.J. Hawker, M.L. Chabinyc, J. Nelson, J. Phys. Chem. C 117, 12407 (2013)CrossRefGoogle Scholar
  3. 3.
    E. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)ADSCrossRefGoogle Scholar
  4. 4.
    H. Scher, M.F. Shlesinger, J.T. Bendler, Phys. Today 44, 26 (1991)ADSCrossRefGoogle Scholar
  5. 5.
    V.M. Kenkre, R.S. Knox, Phys. Rev. B 9, 5297 (1974)ADSCrossRefGoogle Scholar
  6. 6.
    J. Klafter, R. Silbey, Phys. Rev. Lett. 44, 55 (1980)ADSCrossRefGoogle Scholar
  7. 7.
    T. Tiedje, A. Rose, Solid State Commun. 37, 49 (1980)ADSCrossRefGoogle Scholar
  8. 8.
    D. Monroe, Phys. Rev. Lett. 54, 146 (1985)ADSCrossRefGoogle Scholar
  9. 9.
    J. Lorrmann, M. Ruf, D. Vocker, V. Dyakonov, C. Deibel, J. Appl. Phys. 115, 183702 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    M. Dentz, A. Cortis, H. Scherm, B. Berkowitz, Adv. Water Res. 27, 155 (2004)CrossRefGoogle Scholar
  11. 11.
    Y. Edery, A. Guadagnini, H. Scher, B. Berkowitz, Water Resour. Res. 50, 1490 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    M. Brinza, G.J. Adriaenssens, J. Optolectron. Adv. Mater. 8, 2028 (2006)Google Scholar
  13. 13.
    F.W. Schmidlin, Phil. Mag. B 41, 535 (1980)ADSCrossRefGoogle Scholar
  14. 14.
    T. Nagase, K.-H. Kishimoto, H. Naito, J. Appl. Phys. 86, 5026 (1999)ADSCrossRefGoogle Scholar
  15. 15.
    J. Klafter, I.M. Sokolov, First Steps in Random Walks: From Tools to Applications (Oxford Univ. Press, Oxford, 2011)Google Scholar
  16. 16.
    A.P. Tyutnev, P.E. Parris, V.S. Saenko, Chem. Phys. 457, 122 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    B.W. Philippa, R.D. White, R.E. Robson, Phys. Rev. E 84, 041138 (2011)ADSCrossRefGoogle Scholar
  18. 18.
    A. Hirao, H. Nishizawa, M. Sugiuchi, Phys. Rev. Lett. 75, 1787 (1995)ADSCrossRefGoogle Scholar
  19. 19.
    G.F. Seynhaeve, R.P. Barclay, G.J. Adriaenssens, J.M. Marshall, Phys. Rev. B 39, 10196 (1989)ADSCrossRefGoogle Scholar
  20. 20.
    M.E. Scharfe, Phys. Rev. B 2, 5025 (1970)ADSCrossRefGoogle Scholar
  21. 21.
    A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series: Special Functions (Russ.) (Nauka, Moscow, 1983)Google Scholar
  22. 22.
    I.M. Sokolov, A.V. Chechkin, J. Klafter, Acta Phys. Pol. B 35, 1323 (2004)ADSGoogle Scholar
  23. 23.
    I.M. Sokolov, Phys. Rev. E 66, 041101 (2002)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    I.M. Sokolov, J. Klafter, Chaos 15, 026103 (2005)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    K.L. Kuhlman, Numer. Algor. 63, 339 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    R.T. Sibatov, V.V. Uchaikin, Commun. Nonlinear Sci. Numer. Simulat. 16, 4564 (2011)ADSCrossRefGoogle Scholar
  27. 27.
    I.M. Sokolov, Soft Matter 8, 9043 (2012)ADSCrossRefGoogle Scholar
  28. 28.
    R. Richert, L. Pautmeier, H. Bassler, Phys. Rev. Lett. 63, 547 (1989)ADSCrossRefGoogle Scholar
  29. 29.
    I.M. Sokolov, R. Metzler, Phys. Rev. E 67, 010101 (2003)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Departament de Química Física, Universitat de BarcelonaCataloniaSpain
  2. 2.Institut für Physik, Humboldt Universität zu BerlinBerlinGermany

Personalised recommendations