Influence of the Network Geometry on Electron Transport in Nanoparticle Networks
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Influence of the Network Geometry on Electron Transport in Nanoparticle Networks K. D. Benkstein, N. Kopidakis, J. van de Lagemaat, and A. J. Frank National Renewable Energy Laboratory Golden, CO 80401, U.S.A. ABSTRACT Computer simulations are applied to understand the influence of network geometry on the electron transport dynamics in random nanoparticle networks, and the predicted results are compared with those measured in one class of random nanoparticle networks: dye-sensitized nanocrystalline TiO2 solar cells. The model is applicable to all classes of random nanoparticle networks, such as highly disordered quantum dot arrays. The random nanoparticle networks are simulated by the step-wise condensation of a diffusion-limited aggregate. The fractal dimension of the nanoparticle films was estimated from the simulations to be 2.28, which is in quantitative agreement with gas-sorption measurements of TiO2 nanoparticle films. Electron transport on the computer-generated networks is simulated by random walk. The experimental measurements and random-walk simulations are found to be in quantitative agreement. For both a power-law dependence of the electron diffusion coefficient D on the film porosity P is found as described by the relation: D ∝ | P-Pc |µ.. This power-law relation can also be derived from percolation theory, although only qualitatively. The critical porosity Pc (percolation threshold) and the conductivity exponent µ are found to be 0.76 ± 0.01 and 0.82 ± 0.05, respectively. It is estimated that during their respective transit through 50 and 75% porous 10-µm thick films as employed in the dye-cell, the average number of particles visited by electrons increases by 10-fold, from 106 to 107. INTRODUCTION Electron transport in mesoporous semiconductors has generally been modeled from the perspective that network topology has no influence on electron transport [1-5]. It is typically assumed that transport is only trap limited and that electrons diffuse in three dimensions, restricted only by the macroscopic dimensions of the film and electrostatic interaction with the electrolyte (ambipolar diffusion [6]). A recent study has shown, however, that there is a strong relation between the film porosity and the average coordination number of particles in a network [7]. An important implication of this study is that film morphology can affect the path length of electrons in the film. It has also been reported that the number of interconnects between particles [8] and the extent of overlap between neighboring particles [9] influence the electron transport rate. A comparison between real and simulated films suggest that real nanoparticle film can be accurately depicted as a random network with a random number of interconnections at each particle, which may be likened to a “hub” [7,10]. In this paper, we highlight a recent study [10] that describes the first clear evidence that the network geometry strongly influences the electron transport dynamics in mesoporous TiO2 nanoparticle films and show that percolati
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