Kinetic theory of hopping transport

Schmidlin, F. W.

doi:10.1080/13642818008245405

Cited by 66 publications

(26 citation statements)

References 33 publications

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“…In part, this may be due to the influence of a particular equivalence between MT and CTRW. Schmidlin, Noolandi, and others [3,[15][16][17] showed that the models, under certain conditions, yield similar results [15]. But the general structure of diffusion equations in the MT model has not been derived yet, and this is the main purpose of this Letter.…”

mentioning

confidence: 85%

Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous-Time Random Walk

Bisquert

2003

Phys. Rev. Lett.

116

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We investigate the macroscopic diffusion of carriers in the multiple-trapping (MT) regime, in relation with electron transport in nanoscaled heterogeneous systems, and we describe the differences, as well as the similarities, between MT and the continuous-time random walk (CTRW). Diffusion of free carriers in MT can be expressed as a generalized continuity equation based on fractional time derivatives, while the CTRW model for diffusive transport generalizes the constitutive equation for the carrier flux.

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mentioning

confidence: 85%

Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous-Time Random Walk

Bisquert

2003

Phys. Rev. Lett.

116

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“…Kinetic-Transport Formalism. We state the kinetictransport equations in the MT model for a single kind of trap, consisting on the equations of conservation for free and trapped electrons and Fick's law for the free electrons: 17 Here N L is the total density of localized sites (per unit volume),…”

Section: Chemical Diffusion Coefficientmentioning

confidence: 99%

“…2,5,6,9,12 The key feature of the MT framework is the restriction that only free electrons contribute to the diffusion current. 17 Diffusion by direct hopping between localized states is also possible in materials with a wide distribution of traps, but this mechanism will not be considered in this paper. For trapping and recombination, the ideas formulated by Rose 19 can be adapted to DSSC as indicated in ref 11.…”

Section: Introductionmentioning

confidence: 99%

Interpretation of the Time Constants Measured by Kinetic Techniques in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells

2004

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The processes of charge separation, transport, and recombination in dye-sensitized nanocrystalline TiO 2 solar cells are characterized by certain time constants. These are measured by small perturbation kinetic techniques, such as intensity modulated photocurrent spectroscopy (IMPS), intensity modulated photovoltage spectroscopy (IMVS), and electrochemical impedance spectroscopy (EIS). The electron diffusion coefficient, D n , and electron lifetime, τ n , obtained by these techniques are usually found to depend on steady-state Fermi level or, alternatively, on the carrier concentration. We investigate the physical origin of such dependence, using a general approach that consists on reducing the general multiple trapping kinetic-transport formalism, to a simpler diffusion formalism, which is valid in quasi-static conditions. We describe in detail a simple kinetic model for diffusion, trapping, and interfacial charge transfer of electrons, and we demonstrate the compensation of trap-dependent factors when forming steady-state quantities such as the diffusion length, L n , or the electron conductivity, σ n .

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“…18 Here, we assume that the conduction band states of a crystalline semiconductor constitute the extended states. Figure 1b illustrates the microscopic events in diffusion of conduction band electrons, with number density n c , in a nanoporous semiconductor with a localized level in the band gap, of energy E L , with number density of trapped electrons n L .…”

Section: Diffusion Of Free Electronsmentioning

confidence: 99%

“…5,10,13,16,17 MT adopts the restriction that only free electrons are able to displace, whereas a distribution of localized sites traps the electrons and releases them thermally to the transport states. 18 In another paper, we have discussed this approach. 1 We define a quasistatic measurement as that in which a common equilibrium of free and localized electrons persists even as the Fermi level, µ j n , varies with time, and we show quite generally 1 that the quasistatic condition allows the reduction of MT framework to the conventional diffusion equations, with the "effective" diffusion coefficient, D n , given by where n c is the concentration of free (conduction band) electrons, D 0 is the diffusion coefficient in the trap-free system, and n L is the total density of localized electrons in traps.…”

Section: Introductionmentioning

confidence: 99%

Chemical Diffusion Coefficient of Electrons in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells

2004

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The properties and physical interpretation of the electron diffusion coefficient, D n , observed in nanostructured semiconductors and dye-sensitized solar cells by small perturbation electrical and electrooptical techniques are investigated. The chemical diffusion coefficient is defined as the product of a thermodynamic factor (that accounts for the effect of nonideal statistics in a gradient of chemical potential) and a jump diffusion coefficient (that depends on average hopping distances and rates). The thermodynamic and kinetic similarities, as well as the differences, between interacting ions in the lattice and electrons in nanostructured semiconductors are discussed. From these considerations, the chemical diffusion coefficient of electrons for multiple trapping transport in nanoporous dye solar cells can be calculated indirectly, and the results are in agreement with the reduction of the kinetic-transport equations in the quasistatic approximation. The chemical diffusion coefficient in several models for electrons is discussed: for a wide but stationary distribution of sites energies, giving the Fermi-level dependent D n [Fisher et al. J. Phys. Chem. B 2000, 104, 949] and for the enhancement of the diffusivity by effect of electrostatic shift of energy levels [Vanmaekelbergh et al. J. Phys. Chem. B 1999, 103, 747]. This last model is shown to correspond to the mean-field approximation to interaction in a lattice gas. A general expression containing all of these cases is provided, and several important consequences of the distinction between macroscopic diffusion and single particle random walk are pointed out.

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Kinetic theory of hopping transport

Cited by 66 publications

References 33 publications

Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous-Time Random Walk

Fractional Diffusion in the Multiple-Trapping Regime and Revision of the Equivalence with the Continuous-Time Random Walk

Interpretation of the Time Constants Measured by Kinetic Techniques in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells

Chemical Diffusion Coefficient of Electrons in Nanostructured Semiconductor Electrodes and Dye-Sensitized Solar Cells

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