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

Abstract

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.

Optoelectronic Studies of Methylammonium Lead Iodide Perovskite Solar Cells with Mesoporous TiO2: Separation of Electronic and Chemical Charge Storage, Understanding Two Recombination Lifetimes, and the Evolution of Band Offsets during J–V Hysteresis

“…The OF is given by (-S+1) where S is the slope of log(τ) vs log(Q) in figure 5d. 44,45 We can make this calculation for the short and long lifetimes and total or above baseline charge (table 1). If the implied recombination current is not physically reasonable, then the specific TPV decay is unlikely to be a measure of recombination in the cell.…”

Physical electrochemistry of nanostructured devices

“…Additionally changes may occur in the local electrostatic level associated with self-charging of the phase or interaction of the carriers. 55 In contrast, for metals and highly doped semiconductors, the DOS at the Fermi level is very high. Significant changes in the density of carriers, as an effect of an applied potential, are not possible, because the Fermi level is pinned at a fixed energy level.…”

“…13,68,72,[129][130][131][132][133][134][135] Using quasi-equilibrium arguments, the variations of both diffusion coefficient and lifetime were attributed to the statistics of electrons in the material, which deviates from dilution, as described by thermodynamic factors. 136 The varying D n was recognized as a chemical diffusion coefficient, 55,127 and the correlation between variations of D n and t n , 135 was explained by a common origin of their variations in an exponential distribution in the bandgap. 95,136 The application of the multiple trapping model in DSC will be described in detail in section 3.4.…”

“…142 These ideas have been vastly applied in surface diffusion 143 and in the simulation of model systems consisting of interacting particles diffusing on the lattice. 144,145 Recently, the chemical diffusion coefficient has been found quite useful for rationalizing the transport of electrons in metal-oxide nanostructures, 55,127 and it allows also the formulation of the generalized Einstein relation in a very clear way, as discussed later. All these concepts have been well known for a long time in the field of electrochemistry of inorganic solids.…”

“…We first consider a simple model that introduces some important concepts related to the finite occupation of electron sites. 55 It is a 3-dimensional solid composed of a cubic lattice of weakly interacting quantum dots of radius R and separation d, with a single electronic state at the energy level E 0 , that can be occupied at most by one carrier, as indicated in Fig. 16(a).…”