Top: schematic illustration of the energy diagram in a trap-contained semiconductor in contact with the electrolyte, under open-circuit condition. Depending on the electron energy, its wave function is extended (in CB) or localized (in a trap). Gaussian shape density of states of holes has also been depicted in the figure (see Eq. (6)). Bottom: A nanoparticle, containing traps and electrons. An electron in CB can recombine with the probability of . An electron in a trap is recombined with the probability of , provided that the trap is being near the surface of the nanoparticle (in the dashed region).
Log of Electron lifetime versus Fermi-level under LR situation, at . Dots are the lifetime obtained by RW simulation (). Dashed line is the small perturbation lifetime () calculated via Eq. (7) with . Coincidence between two results is obtained after the rescaling of by . Inset graph shows the same figure but in linear scale.
Electron lifetime versus Fermi-level, at three different temperatures under NLR situation (LR lifetime of Figure 2 has been shown again for comparison). Points are the lifetime obtained by the RW simulations, and lines are the MT theoretical predictions (Eq. (7)) rescaled by . At high Fermi-level and at a fixed temperature, NLR lifetime approaches to the LR lifetime, which can be seen for . Due to the logarithmic scale, error bars of the RW results are smaller than the size of the markers.
Electron diffusion coefficient versus Fermi-level, at three different temperatures under LR and NLR situations. Mechanism of the recombination does not affect the electron transport, and therefore for , LR and NLR diffusion coefficient are the same. Strong dependence of the diffusion coefficient on the Fermi-level shows the well-known anomalous transport in sensitized solar cells. In agreement with the Darken equation (4), MT rescaled coefficients () coincide with the RW simulation results (). Due to the logarithmic scale, error bars of the RW results are smaller than the size of the markers.
Electron diffusion length versus Fermi-level at three different temperatures under both LR and NLR situations. Points are the direct RW results that completely coincide with the empty squares, obtained indirectly from RW simulations. MT prediction of the Eq. (12) is also depicted for each temperature. Under LR, diffusion length is constant at all . Under NLR situation, diffusion length is reduced, because in addition to the CB, electron can recombine via the trap states too. At high , electron spends less time in the traps, and therefore NLR results approach to the LR diffusion length.
Distribution of the electron travelling distance (distance travelled before recombination) under LR, obtained from RW simulation at and . Distribution has been fitted to the exponential part of the Eq. (9). The fitting has led to the value of , which is in agreement with the LR diffusion length obtained, shown in Figure 5.
Constants used in simulations.
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