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Open AccessArticle

Fractal Generalization of the Scher–Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids

Moscow Institute of Physics and Technology, 9 Institutskiy Per., 141701 Dolgoprudny, Moscow Region, Russia
Mathematics 2020, 8(11), 1991; https://doi.org/10.3390/math8111991
Received: 5 October 2020 / Revised: 4 November 2020 / Accepted: 5 November 2020 / Published: 8 November 2020
(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)
The Scher–Montroll model successfully describes subdiffusive photocurrents in homogeneously disordered semiconductors. The present paper generalizes this model to the case of fractal spatial disorder (self-similar random distribution of localized states) under the conditions of the time-of-flight experiment. Within the fractal model, we calculate charge carrier densities and transient current for different cases, solving the corresponding fractional-order equations of dispersive transport. Photocurrent response after injection of non-equilibrium carriers by the short laser pulse is expressed via fractional stable distributions. For the simplest case of one-sided instantaneous jumps (tunneling) between neighboring localized states, the dispersive transport equation contains fractional Riemann–Liouville derivatives on time and longitudinal coordinate. We discuss the role of back-scattering, spatial correlations induced by quenching of disorder, and spatiotemporal non-locality produced by the fractal trap distribution and the finite velocity of motion between localized states. We derive expressions for the photocurrent and transit time that allow us to determine the fractal dimension of the distribution of traps and the dispersion parameter from the time-of-flight measurements. View Full-Text
Keywords: continuous time random walk; fractal; photocurrent; nanotube; anomalous diffusion; fractional equation; time-of-flight continuous time random walk; fractal; photocurrent; nanotube; anomalous diffusion; fractional equation; time-of-flight
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MDPI and ACS Style

Sibatov, R.T. Fractal Generalization of the Scher–Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids. Mathematics 2020, 8, 1991. https://doi.org/10.3390/math8111991

AMA Style

Sibatov RT. Fractal Generalization of the Scher–Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids. Mathematics. 2020; 8(11):1991. https://doi.org/10.3390/math8111991

Chicago/Turabian Style

Sibatov, Renat T. 2020. "Fractal Generalization of the Scher–Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids" Mathematics 8, no. 11: 1991. https://doi.org/10.3390/math8111991

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