Renat Sibatov

Fractional kinetics in solids : anomalous charge transport in semiconductors, dielectrics, and nanosystems

Anomalous Diffusion Dispersive Transport in Semiconductors Anomalous Dielectric Relaxation Quantum Dot Systems.

FRACTIONAL PROCESSES: FROM POISSON TO BRANCHING ONE

Fractional generalizations of the Poisson process and branching Furry process are considered. The link between characteristics of the processes, fractional differential equations and Levy stable densities are discussed and used for the construction of the Monte Carlo algorithm for simulation of random waiting times in fractional processes. Numerical calculations are performed and limit distributions of the normalized variable Z = N/〈N〉 are found for both processes.

Memory regeneration phenomenon in dielectrics: the fractional derivative approach

Classical theory predicts that a capacitors charging current obeys the first-order differential equation and hence follows the exponential Debye law. However, there are many experimental results confirming the inverse-power Curie–von Schweidler law of the charging current. The principal difference between the Curie–von Schweidler law and the Debye law is the presence of memory: the process depends not only on initial conditions but also on the whole prehistory. We constructed and investigated the capacitor model that extends the fractional Westerlund model by accounting for the resistance of the capacitor. To follow the transition to classical Debye theory, we investigated the solution of the fractional equation for the order α close to 1. The calculations show that the solution obeys the exponential law up to some point of time independently of the prehistory and then changes its behavior to the inverse power law depending on the prehistory. Comparison with experimental data confirmed the existence of this effect. We named it the regenerated memory effect.

Truncated Lévy statistics for dispersive transport in disordered semiconductors

Abstract Probabilistic interpretation of transition from the dispersive transport regime to the quasi-Gaussian one in disordered semiconductors is given in terms of truncated Levy distributions. Corresponding transport equations with fractional order derivatives are derived. We discuss physical causes leading to truncated waiting time distributions in the process and describe influence of truncation on carrier packet form, transient current curves and frequency dependence of conductivity. Theoretical results are in a good agreement with experimental facts.

Fractional derivative formalism for non-destructive insulation diagnosis by polarization–depolarization current measurements

The fractional derivative method is considered as a possible formalism for the non-destructive insulation diagnosis by polarization and depolarization current measurements. The underlying motive is based on non-Debye relaxation processes clearly observed in long-time relaxation experiments with oil–paper, electrolytic and ultra-capacitors, transformers and other systems. Modes of relaxation at different time scales depend on charging prehistory and are sensitive to the state of a system. Fractional formalism provides useful parameters characterizing the state of the insulation system, which can be extracted from polarization–depolarization current measurements. Adequacy of the technique is examined by comparison of theoretical and experimental results for oil–paper, and ultra-capacitors.

Fractional Wave Equation for Dielectric Medium with Havriliak–Negami Response

The fractional generalizations of the relaxation equation and the wave equation in dielectrics with the response function of the Havriliak–Negami type are considered. The obtained fractional wave equation is concordant with the asymptotical equations derived by Tarasov VE (J Phys Condens Matter 20:145212, 2008) from Jonscher’s universal law. The explicit expression for the fractional operator in this equation is obtained and the Monte Carlo algorithm for calculation of actions of this operator and of the inverse one is constructed.

Interpreting data on solar cosmic ray fluxes via the fractional derivative method

Solar cosmic ray propagation through the interplanetary magnetic field is considered as a random process of particles traveling along magnetic lines at a finite velocity of free motion and with a free path distributed according to an inverse power law. The propagator is presented as a sum of direct (nonscattered) flux (singular part of solution) and multiple scattered flux (regular part). In the long-time asymptotic, the regular part is described by an equation with a fractional-order derivative. Using analytical expressions for the propagator, we numerically calculate fluxes of energetic particles accelerated by shock waves generated by solar flares. The presented model is in better agreement with Ulysses and Voyager 2 data than the Perri-Zimbardo model and may therefore be recommended for use in interpreting the results of further experiments.

ANOMALOUS KINETICS OF CHARGE CARRIERS IN DISORDERED SOLIDS: FRACTIONAL DERIVATIVE APPROACH

Anomalous (non-Gaussian) kinetics is often observed in various disordered materials, such as amorphous semiconductors, porous solids, polycrystalline films, liquid-crystalline materials, polymers, etc. Recently the anomalous relaxation-diffusion processes have been observed in nanoscale systems: nanoporous silicon, glasses doped by quantum dots, quasi-one-dimensional (1D) systems, arrays of colloidal quantum dots, and some others. The paper presents a review of new approach, based on fractional kinetic equations. We give a physical basis for some fractional equations deriving them from their classical counterparts by means of averaging over statistical ensemble of disordered media. We consider self-similarity as the main feature of these processes, and explain memory phenomena in frameworks of hidden variables conception.

A look at the cosmic ray anisotropy with the nonlocal relativistic transport approach

The Cosmic Ray anisotropy is a key element in the quest to find t he origin of the enigmatic particles. A well known problem is that, although most of the likely sources are in the Inner Galaxy, the direction from which the lowest energy particles (less t han about 1 PeV) come is largely from the Outer Galaxy. We show that this can be understood taking into account a possible reflection of charged particles by ’walls’ in the Interstellar Medium or/ and as a temporary phenomenon after the shock wave from the supernova explosion passed the Earth. This effect is too subtle to be explained by an ordinary diffusion theory and becomes apparent within the frames of the non-local relativistic transport theory, which involves conception s of free motion velocity and path lengths with probability distributions of non-exponential type ta ken for a turbulent interstellar medium.

On the problem of nondestructive diagnosis for quality assessment of electric insulation: A fractional calculus approach

Fractional calculus is used for theoretical description of a nondestructive diagnosis method for quality assessment of electric insulation in capacitors, transformers and other devices. The new analytical approach is based on deviation of the relaxation law from the exponential (Debye) one clearly observed in experiments on the long-time relaxation. Power modes of relaxation depends on charging prehistory and are sensitive to the state of an insulation system. As an example system, oil-paper insulation is considered. Fractional analysis of the measured polarization/depolarization currents is believed to provide indication of the general ageing status of the oil paper insulating system. Validity of the approach is confirmed by comparison of theoretical and experimental results for an oil paper capacitor.