Fractional Kinetics in Solids : Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems.

The standard (Markovian) transport model based on the Boltzmann equation cannot describe some non-equilibrium processes called anomalous that take place in many disordered solids. Causes of anomality lie in non-uniformly scaled (fractal) spatial heterogeneities, in which particle trajectories take c...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Uchaikin, Vladimir
Otros Autores: Sibatov, Renat
Formato: Electrónico eBook
Idioma:Inglés
Publicado: Singapore : World Scientific, 2012.
Temas:
Chemical kinetics > Mathematics.
Electric discharges > Mathematical models.
Electron transport > Mathematical models.
Fractional calculus.
Semiconductors > Electric properties.
Solid state physics > Mathematics.
Cinétique chimique > Mathématiques.
Décharges électriques > Modèles mathématiques.
Électrons > Transport > Modèles mathématiques.
Dérivées fractionnaires.
Physique de l'état solide > Mathématiques.
Electric discharges > Mathematical models
Electron transport > Mathematical models
Fractional calculus
Semiconductors > Electric properties
Solid state physics > Mathematics
Acceso en línea:Texto completo
Tabla de Contenidos:
  • Preface
  • 1. Statistical grounds
  • 1.1 Levy stable statistics
  • 1.1.1 Generalized limit theorems
  • 1.1.2 Two subclasses of stable distributions
  • 1.1.3 Fractional stable distributions
  • 1.1.4 Self-similar processes: Brownian motion and Levy motion
  • 1.1.5 Space-fractional equations
  • 1.2 Random flight models
  • 1.2.1 Continuous time random flights
  • 1.2.2 Counting process for number of jumps
  • 1.2.3 The Poisson process
  • 1.2.4 The Fractional Poisson process
  • 1.2.5 Simulation of waiting times
  • 1.3 Some properties of the fractional Poisson process
  • 1.3.1 The nth arrival time distribution
  • 1.3.2 The fractional Poisson distribution
  • 1.3.3 Limit fractional Poisson distributions
  • 1.3.4 Fractional Furry process
  • 1.3.5 Time-fractional equation
  • 1.4 Random flights on a one-dimensional Levy-Lorentz gas
  • 1.4.1 One-dimensional Levy-Lorentz gas
  • 1.4.2 The flight process on the fractal gas
  • 1.4.3 Propagators
  • 1.4.4 Fractional equation for flights on fractal
  • 1.5 Subdiffusion
  • 1.5.1 Integral equations of diffusion in a medium with traps
  • Necessary and sufficient condition for subdiffusion
  • 1.5.2 Differential equations of subdiffusion
  • 1.5.3 Subdiffusion distribution density
  • 1.5.4 Analysis of subdiffusion distributions
  • 1.5.5 Discussion
  • 2. Fractional kinetics of dispersive transport
  • 2.1 Macroscopic phenomenology
  • 2.1.1 A role of phenomenology in studying complex systems
  • 2.1.2 Universality of transient current curves
  • 2.1.3 From self-similarity to fractional derivatives
  • 2.1.4 From transient current to waiting time distribution
  • 2.2 Microscopic backgrounds of dispersive transport
  • 2.2.1 From the Scher-Montroll model to fractional derivatives
  • 2.2.2 Physical basis of the power-law waiting time distribution
  • 2.2.3 Multiple trapping regime
  • 2.2.4 Hopping conductivity.
  • 2.2.5 Bassler's model of Gaussian disorder
  • 2.3 Fractional formalism of multiple trapping
  • 2.3.1 Prime statements
  • 2.3.2 Multiple trapping regime and Arkhipov-Rudenko approach
  • 2.3.3 Fractional equations for delocalized carriers
  • 2.3.4 Fractional equation for the total concentration
  • 2.3.5 Two-state dynamics
  • 2.3.6 Delocalized carrier concentration
  • 2.3.7 Percolation and fractional kinetics
  • 2.3.8 The case of Gaussian disorder
  • 2.4 Some applications
  • 2.4.1 Dispersive diffusion
  • 2.4.2 Photoluminescence decay
  • 2.4.3 Including recombination
  • 2.4.4 Including generation
  • 2.4.5 Bipolar dispersive transport
  • 2.4.6 The family of fractional dispersive transport equations
  • 3. Transient processes in disordered semiconductor structures
  • 3.1 Time-of-flight method
  • 3.1.1 Transient current in disordered semiconductors
  • 3.1.2 Transient current for truncated waiting time distributions
  • 3.1.3 Distributed dispersion parameter
  • 3.1.4 Transient current curves in case of Gaussian disorder
  • 3.1.5 Percolation in porous semiconductors
  • 3.1.6 Non-stationary radiation-induced conductivity
  • 3.2 Non-homogeneous distribution of traps
  • 3.2.1 Non-uniform spatial distribution of localized states
  • 3.2.2 Multilayer structures
  • 3.2.3 The "disordered
  • crystalline" semiconductor structure
  • 3.3 Transient processes in a diode under dispersive transport conditions
  • 3.3.1 Turning on by the current step
  • 3.3.2 Turning off by interruption of circuit
  • 3.4 Frequency properties of disordered semiconductor structures
  • 3.4.1 Frequency dependence of conductivity
  • 3.4.2 A diode at dispersive transport conditions
  • 4. Fractional kinetics in quantum dots and wires
  • 4.1 Fractional optics of quantum dots
  • 4.1.1 Off- and on-intervals statistics
  • 4.1.2 Physical mechanisms of power law blinking
  • 4.1.3 Two-state renewal model.
  • 4.1.4 Fractional blinking process
  • 4.1.4.1 Total fluorescence time distribution
  • 4.1.5 Photon counts distribution
  • 4.2 Charge kinetics in colloidal quantum dot arrays
  • 4.2.1 Fractional currents in colloidal quantum dot array
  • 4.2.2 Modification of the Scher-Montroll model
  • 4.2.3 Current decay in the modified model
  • 4.2.4 Interdot disorder
  • 4.2.5 Monte Carlo simulation
  • 4.3 Conductance through fractal quantum conductors
  • 4.3.1 Weak localization (scattering)
  • 4.3.2 Sequential incoherent tunneling
  • 5. Fractional relaxation in dielectrics
  • 5.1 The relaxation problem
  • 5.1.1 The relaxation functions
  • 5.1.2 Non-Debye empirical laws
  • 5.1.3 Superposition model
  • 5.1.4 Stochastic interpretations of the universal relaxation law
  • 5.1.5 Random activation energy model
  • 5.2 Fractional approach
  • 5.2.1 Fractional derivatives for relaxation problem
  • 5.2.2 Polar dielectrics: model of rotational subdiffusion
  • 5.2.3 A prehistory contribution
  • 5.2.4 Green's function
  • 5.2.4.1 The first representation
  • 5.2.4.2 The second representation
  • 5.3 The Cole-Cole kinetics
  • 5.3.1 Fractional generalization of the Ohm's law
  • 5.3.2 Numerical demonstration of the memory effect
  • 5.3.2.1 Mittag-Leffer representation
  • 5.3.2.2 Monte Carlo calculations
  • 5.3.3 Polarization-depolarization currents
  • 5.3.4 Radiation-induced dielectric effect in polymers
  • 5.3.5 Hysteresis in ferroelectric ceramics
  • 5.4 The Havriliak-Negami kinetics
  • 5.4.1 The Cole-Davidson response
  • 5.4.2 Fractional kinetics and Havriliak-Negami response
  • 5.4.3 Stochastic inversion of the Havriliak-Negami operator
  • 5.4.4 Three-power term approximation of the HN-relaxation
  • 5.4.5 Pass-through conductivity and Raicu's response
  • 5.4.6 Fractional waves in the HN dielectrics
  • 5.5 The Kohlrausch-Williams-Watts kinetics
  • 5.5.1 The KWW relaxation function.
  • 5.5.2 Levy-stable statistics and KWW relaxation
  • 5.5.2.1 Relaxation in glassy materials
  • 5.5.2.2 Quantum decay theory
  • 5.5.3 Fractional equation for KWW relaxation
  • 6. The scale correspondence principle
  • 6.1 Finity and infinity
  • 6.2 Intermediate space-asymptotics
  • 6.3 Intermediate time-asymptotics
  • 6.4 Concluding remarks
  • Appendix A One-sided stable laws
  • Appendix B Fractional stable distributions
  • Appendix C Fractional operators: main properties
  • C.1 Axiomatics (Ross, 1975)
  • C.2 Interrelations between fractional operators
  • C.3 The law of exponents
  • C.4 Differentiation of a product
  • C.5 Integration by parts
  • C.6 Generalized Taylor series
  • C.7 Expression of fractional derivatives through the integers
  • C.8 Indirect differentiation: chain rule
  • C.9 Fractional powers of operators and Levy stable variables
  • Bibliography
  • Index.

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