# Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Tempered Fractional Stable Distributions

## 3. Weak Scattering in Mesoscopic Fractal Wires

## 4. Incoherent Sequential Tunneling Through Wire with Lévy-Type Disorder

## 5. Concluding Remarks

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- Pook, W.; Janßen, M. Multifractality and scaling in disordered mesoscopic systems. Zeitschrift für Physik B Condens. Matter
**1991**, 82, 295–298. [Google Scholar] [CrossRef] - Hegger, H.; Huckestein, B.; Hecker, K.; Janssen, M.; Freimuth, A.; Reckziegel, G.; Tuzinski, R. Fractal conductance fluctuations in gold nanowires. Phys. Rev. Lett.
**1996**, 77, 3885. [Google Scholar] [CrossRef] [PubMed] - Barthelemy, P.; Bertolotti, J.; Wiersma, D.S. A Lévy flight for light. Nature
**2008**, 453, 495. [Google Scholar] [CrossRef] [PubMed] - Kohno, H.; Yoshida, H. Multiscaling in semiconductor nanowire growth. Phys. Rev. E
**2004**, 70, 062601. [Google Scholar] [CrossRef] [PubMed] - Kohno, H. Self-organized nanowire formation of Si-based materials. In One-Dimensional Nanostructures; Springer: Berlin, Germany, 2008; pp. 61–78. [Google Scholar]
- Raboutou, A.; Peyral, P.; Lebeau, C.; Rosenblatt, J.; Burin, J.P.; Fouad, Y. Fractal vortices in disordered superconductors. Phys. A Stat. Mech. Appl.
**1994**, 207, 271–279. [Google Scholar] [CrossRef] - Beenakker, C.; Groth, C.; Akhmerov, A. Nonalgebraic length dependence of transmission through a chain of barriers with a Lévy spacing distribution. Phys. Rev. B
**2009**, 79, 024204. [Google Scholar] [CrossRef] - Falceto, F.; Gopar, V.A. Conductance through quantum wires with Lévy-type disorder: Universal statistics in anomalous quantum transport. Europhys. Lett.
**2011**, 92, 57014. [Google Scholar] [CrossRef] - Sibatov, R.T. Distribution of the conductance of a linear chain of tunnel barriers with fractal disorder. JETP Lett.
**2011**, 93, 503–507. [Google Scholar] [CrossRef] - Fernández-Marín, A.; Méndez-Bermúdez, J.; Gopar, V.A. Photonic heterostructures with Lévy-type disorder: Statistics of coherent transmission. Phys. Rev. A
**2012**, 85, 035803. [Google Scholar] [CrossRef] - Amanatidis, I.; Kleftogiannis, I.; Falceto, F.; Gopar, V.A. Conductance of one-dimensional quantum wires with anomalous electron wave-function localization. Phys. Rev. B
**2012**, 85, 235450. [Google Scholar] [CrossRef] - Burioni, R.; di Santo, S.; Lepri, S.; Vezzani, A. Scattering lengths and universality in superdiffusive Lévy materials. Phys. Rev. E
**2012**, 86, 031125. [Google Scholar] [CrossRef] [PubMed] - Fernández-Marín, A.A.; Méndez-Bermúdez, J.; Carbonell, J.; Cervera, F.; Sánchez-Dehesa, J.; Gopar, V. Beyond Anderson localization in 1D: Anomalous localization of microwaves in random waveguides. Phys. Rev. Lett.
**2014**, 113, 233901. [Google Scholar] [CrossRef] [PubMed] - Kotulski, M. Asymptotic distributions of continuous-time random walks: A probabilistic approach. J. Stat. Phys.
**1995**, 81, 777–792. [Google Scholar] [CrossRef] - Montroll, E.W.; Weiss, G.H. Random walks on lattices. II. J. Math. Phys.
**1965**, 6, 167–181. [Google Scholar] [CrossRef] - Kolokoltsov, V.; Korolev, V.; Uchaikin, V. Fractional stable distributions. J. Math. Sci.
**2001**, 105, 2569–2576. [Google Scholar] [CrossRef] - Koponen, I. Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. Phys. Rev. E
**1995**, 52, 1197. [Google Scholar] [CrossRef] [PubMed] - Rosiński, J. Tempering stable processes. Stoch. Process. Their Appl.
**2007**, 117, 677–707. [Google Scholar] [CrossRef] [Green Version] - Mantegna, R.N.; Stanley, H.E. Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. Phys. Rev. Lett.
**1994**, 73, 2946. [Google Scholar] [CrossRef] - Sibatov, R.T.; Uchaikin, V.V. Truncated Lévy statistics for dispersive transport in disordered semiconductors. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 4564. [Google Scholar] [CrossRef] - Sibatov, R.T.; Morozova, E.V. Tempered fractional model of transient current in organic semiconductor layers. In Theory and Applications of Non-Integer Order Systems; Springer: Berlin, Germany, 2017. [Google Scholar]
- Saichev, A.I.; Zaslavsky, G.M. Fractional kinetic equations: solutions and applications. Chaos Interdiscip. J. Nonlinear Sci.
**1997**, 7, 753–764. [Google Scholar] [CrossRef] [Green Version] - Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep.
**2000**, 339, 1–77. [Google Scholar] [CrossRef] - Uchaikin, V.V.; Zolotarev, V.M. Chance and Stability: Stable Distributions and Their Applications; Walter de Gruyter: Berlin, Germany, 1999. [Google Scholar]
- Uchaikin, V.V.; Sibatov, R.T. Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics, and Nanosystems; World Scientific: Singapore, 2013. [Google Scholar]
- Sabzikar, F.; Meerschaert, M.M.; Chen, J. Tempered fractional calculus. J. Comput. Phys.
**2015**, 293, 14–28. [Google Scholar] [CrossRef] [PubMed] - Gajda, J.; Magdziarz, M. Fractional Fokker–Planck equation with tempered α-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E
**2010**, 82, 011117. [Google Scholar] [CrossRef] [PubMed] - Kumar, A.; Vellaisamy, P. Inverse tempered stable subordinators. Stat. Probab. Lett.
**2015**, 103, 134–141. [Google Scholar] [CrossRef] [Green Version] - Alrawashdeh, M.S.; Kelly, J.F.; Meerschaert, M.M.; Scheffler, H.P. Applications of inverse tempered stable subordinators. Comput. Math. Appl.
**2017**, 73, 892–905. [Google Scholar] [CrossRef] [Green Version] - Cartea, Á.; del Castillo-Negrete, D. Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E
**2007**, 76, 041105. [Google Scholar] [CrossRef] - Beenakker, C.W. Random-matrix theory of quantum transport. Rev. Mod. Phys.
**1997**, 69, 731. [Google Scholar] [CrossRef] - Dorokhov, O. Transmission coefficient and the localization length of an electron in N bound disordered chains. JETP Lett.
**1982**, 36, 318–321. [Google Scholar] - Mello, P.; Pereyra, P.; Kumar, N. Macroscopic approach to multichannel disordered conductors. Ann. Phys.
**1988**, 181, 290–317. [Google Scholar] [CrossRef] - Muttalib, K.; Klauder, J. Generalized Fokker–Planck equation for multichannel disordered quantum conductors. Phys. Rev. Lett.
**1999**, 82, 4272. [Google Scholar] [CrossRef] - Beenakker, C.; Rejaei, B. Nonlogarithmic repulsion of transmission eigenvalues in a disordered wire. Phys. Rev. Lett.
**1993**, 71, 3689. [Google Scholar] [CrossRef] [PubMed] - Haubold, H.J.; Mathai, A.M.; Saxena, R.K. Mittag–Leffler functions and their applications. J. Appl. Math.
**2011**, 2011, 298628. [Google Scholar] [CrossRef] - Valkó, P.P.; Abate, J. Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Comput. Math. Appl.
**2004**, 48, 629–636. [Google Scholar] [CrossRef] - Uchaikin, V.V. Self-similar anomalous diffusion and Lévy-stable laws. Phys. Uspekhi
**2003**, 46, 821. [Google Scholar] [CrossRef]

**Figure 1.**Evolution of conductance distribution for the case of weak scattering in a quasi-fractal wire characterized by a tempered-stable distribution of distances between scatterers (

**left panel**). Distributions for different values of $\alpha $ (

**right panel**).

**Figure 2.**Scaling of $\langle -lnG\rangle $ vs. $L/l$ for different values of $\alpha $ and $\gamma $.

**Figure 3.**Average resistance (

**a**,

**c**) and relative fluctuations (

**b**,

**d**) for a quantum wire of length L. Distances between barriers in the wire are distributed according to the tempered Lévy stable law with parameters $\alpha $ and $\mu =0.01$. Resistances of individual barriers are distributed according to the tempered $\beta $-stable law with $\beta =0.5$ or 1.0, and $\nu =0.01$.

**Figure 4.**(

**a**) Average conductance of a quantum wire of length L in the case of sequential incoherent tunneling. Barrier resistances are distributed according to the $\beta $-stable law. The spatial distribution of barriers in the wire is regular $\alpha =1$, the mean distance is $l=0.1$ nm, ${\rho}_{\beta}=1$. (

**b**) Relative fluctuations for different values of $\beta $.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sibatov, R.T.; Sun, H.
Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder. *Fractal Fract.* **2019**, *3*, 47.
https://doi.org/10.3390/fractalfract3040047

**AMA Style**

Sibatov RT, Sun H.
Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder. *Fractal and Fractional*. 2019; 3(4):47.
https://doi.org/10.3390/fractalfract3040047

**Chicago/Turabian Style**

Sibatov, Renat T., and HongGuang Sun.
2019. "Tempered Fractional Equations for Quantum Transport in Mesoscopic One-Dimensional Systems with Fractal Disorder" *Fractal and Fractional* 3, no. 4: 47.
https://doi.org/10.3390/fractalfract3040047