^{1}, Y. Abdi

^{1,a)}and E. Arzi

^{1}

### Abstract

Electron transport and recombination in electrolyte-filled sensitized nanocrystalline solar cell was investigated using Monte-Carlo simulation. Multiple-trapping in an exponential tail of trap states was used as an electron transportmodel. For simulation of the recombination, a new approach based on Marcus theory of charge transfer was developed and utilized to simulate both linear and non-linear (trap-assisted) recombination of electrons with holes in the electrolyte. Monte-Carlo simulation results, based on this approach, reproduced the non-constant diffusion length, recently observed in several experimental works. All simulation results were compared with theoretical predictions of the Marcus theory of charge transfer. Based on this comparison, interestingly it was found that random walk electron lifetime is different from the one which is obtained experimentally by small-perturbation techniques. This result is similar to the well-known Darken equation that describes the difference between jump and chemical diffusion coefficient. An interpretation based on the transport-limited recombination picture was provided to describe this result. These simulations establish a clear picture that describes how the localized trap states contribute to the recombination, leading to the non-linear recombination kinetics in sensitized solar cells.

We would like to thank the Research Council of the University of Tehran for partial financial support. Partial financial support of the “Center of Excellence on the Structure and Physical Properties of Matter” of the University of Tehran is also acknowledged.

I. INTRODUCTION

II. THEORY

A. Diffusion coefficient and lifetime

B. Diffusion length

III. RANDOM-WALK SIMULATION

A. Network of nanoparticles

B. Transportmodel

C. Recombination model

IV. RESULTS AND DISCUSSION

V. SUMMARY AND CONCLUSION

### Key Topics

- Diffusion
- 49.0
- Solar cells
- 19.0
- Semiconductors
- 12.0
- Electronic transport
- 9.0
- Nanoparticles
- 9.0

##### B82B1/00

##### H01L27/142

##### H01L31/04

##### H02N6/00

## Figures

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).

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.

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 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 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.

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.

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.

## Tables

Constants used in simulations.

Constants used in simulations.

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