Monte Carlo Random Walk Simulation of Electron Transport in Dye

Article pubs.acs.org/JPCC

Monte Carlo Random Walk Simulation of Electron Transport in Dye-Sensitized Nanocrystalline Solar Cells: Influence of Morphology and Trap Distribution M. Ansari-Rad, Y. Abdi,* and E. Arzi Nano-Physics Research Laboratory, Department of Physics, University of Tehran, Tehran, Iran ABSTRACT: Electron transport in the porous nanostructured titanium oxide, as a main concern in the dye-sensitized solar cells, was investigated by random walk simulation. Geometrically disordered nanoparticle networks with random distribution of energy was generated and utilized for simulations. Dependency of the diffusion coefficient (Def) on the nanoparticles size, grains connectivity, and the network porosity was completely studied in two cases: traps are placed mainly on the surface or in the volume of the nanoparticles. It was shown that the Def is independent of nanoparticle size for both surface and volume diffusion. We have shown in this study that increasing the Def with the particle size is a consequence of the trap-filling (electronic) effect and not a geometrical effect. The role of electronic effect in electron transport in nanoporous materials has been highlighted in this work.

1. INTRODUCTION Electron transport in nanostructured semiconductor materials is a very important and interesting problem from both theoretical and technological point of view. Recently, some new nanostructured photovoltaic devices were fabricated in which the efficient electronic transport is responsible for their overall performance.1,2 Among these, dye-sensitized solar cells3,4 (DSCs, also called Gratzel cells) are the most promising candidates, in which transport occurs in an n-type, wide band gap mesoporous nanocrystalline TiO2 film. Traditionally, DSCs include a dye-sensitized highly porous (50−60%) TiO2 film on a transparent conductive oxide (TCO) electrode, an electrolyte containing I−/I3− redox couple filling the pores of the TiO2, and a platinum-coated counter electrode placed on the top of the TiO2 electrode. The mesoporous electrodes are commonly prepared by sintering the TiO2 nanoparticles on a TCO substrate, which results in an interconnected network of nanoparticles. The electric power conversion in all types of the solar cells is occurred due to three successive steps: electron−hole (e−h) generation, e−h separation, and charge transport.5 In DSCs, charge transport includes the diffusion of ions through the electrolyte to counter electrode and the electron transport through the TiO2 layer toward the external circuit. Because the nanoparticles in electrolyte-filled mesoporous TiO 2 are surrounded by electrolyte ions, there is not any remarkable band bending in TiO2. Consequently, no macroscopic electric field exists in the film, and electronic transport is believed to be carried out only by diffusion.6−8 Measured electron diffusion coefficient, in mesoporous nanocrystalline TiO2, is several orders of magnitude smaller than that measured in single-crystal bulk sample (0.4 cm2 s−1).9 Because of such extremely slow transport, the recombination of e−h becomes an important problem. Therefore, the electron © 2011 American Chemical Society

diffusion coefficient is a key parameter that needs to be assessed to optimize the performance of the DSCs.10 Electron transport in the porous materials also shows other anomalous features such as quasi-Fermi-level-dependent diffusion coefficient11−13 and dispersive diffusion of nonthermalized electrons.14,15 This anomalous transport has been explained by multiple-trapping (MT) model, in which the diffusion of conduction band (CB) electrons is affected by the trapping-detrapping events.11,16−18 The time spent by an electron in a trap is given by a waiting time distribution (WTD). On the basis of the MT transport model, the continuous-time random walk (RW) with a power law WTD, ψ ∝ t−1−β (0 < β < 1), has been successfully utilized to explain anomalous diffusion and calculate the contribution of traps in the diffusion coefficient.14,19,20 There are different opinions about the precise nature of the trap and their dominant spatial location. Exponentially decreasing tail of defect states below the CB are usually believed to be accounted for traps with power-law WTD.14,19,21 Such static traps can reproduce anomalous transport features. They have also been used to obtain the temperature-dependent diffusion coefficient, but the achieved results are not consistent with the experimental data.20 As mentioned above, the bulk trap-free diffusion coefficient, D0, reduces to a trap-induced diffusion coefficient, Dn, as a consequence of static traps. Structural (geometrical) disorder of TiO2 electrode can also vary the diffusion coefficient to an effective one, Def. Here we have shown how the geometrical disorder of the TiO2 film affects the electron diffusion coefficient. Received: August 17, 2011 Revised: December 27, 2011 Published: December 30, 2011 3212

dx.doi.org/10.1021/jp207907b | J. Phys. Chem. C 2012, 116, 3212−3218

The Journal of Physical Chemistry C

Article

Cass et al.22,23 have used Monte Carlo random walk model in an ordered trap-contained network of TiO2 nanoparticles. They have studied the effect of the electrical connectivity between the nanoparticles (θr) on the transport properties of the electrons. Anta et al.24 have utilized multielectron RW simulation to investigate the effect of nanoparticle size (r) on the electron transport in a random packing network of nanoparticles. They have used an exponential distribution for trap energies. They have introduced two models, named r2 model in which traps are placed on the surface of the nanoparticles, and the r3 model, in which traps are distributed throughout the whole material. Their results show that only the r2 model depicts an increase in the diffusion coefficient by increasing the particle size (as obtained by experiments).25,26 For TiO2 electrode, the dependency of the diffusion coefficient on porosity (P) is believed27−29 to be well-described by the percolation theory at the vicinity of the critical porosity, Pc, at which Def diminishes to zero (δ is the conductivity exponent that controls the speed of transition Def → 0)30

Def ∝ |P − Pc|δ

2. MONTE CARLO RANDOM WALK SIMULATIONS 2.1. Theory. The gradient of the electrochemical potential, η, in the semiconductors has the main contribution in the calculation of the charge current. Electrochemical potential for CB electrons can be written as η = μ − eφ, where μ and φ are the chemical and electrical potentials, respectively. μ is related to the local density of CB electrons, nc as ⎛n ⎞ μ = μ0 + kT ln⎜ c ⎟ ⎝ Nc ⎠

(2) 5,8

So the total charge flux becomes

j = − D0∇nc + ncuc∇φ (3) Here μ0 is the electron affinity, nc is called the effective density of state of CB, and D0 and uc are the diffusion coefficient and mobility of the CB electrons, respectively. The first and second terms in eq 3 correspond to the diffusion and drift current (negligible in DSCs) of the CB electrons, respectively. In the MT model, the continuity equation, considering the trapping rate (Rt) and detrapping rate (Rdt), can be stated as follows17

(1)

∂nc = − ∇· j − R t + R dt (4) ∂t Under the quasi-static conditions, eq 4 reduces to a simple form with a corrected diffusion coefficient, Dn, of17

Some experimental works27−29 suggest that Pc is around 0.75, which is consistent with the prediction of the percolation theory. Also, for the exponent, the values between 0.5 and 2.2 were reported. Anta et al.24 have obtained Pc ≈ 0.68 and 0.14 < δ < 0.79 in their simulation results. As previously noted, because of trap-filling (electronic) effect, diffusion coefficient of the electrons is strongly Fermilevel-dependent. Hence, to end up with a thorough description about the geometry, it is very important to isolate the geometrical effect from the electronic effect, which can be achieved by studying the transport in dif ferent geometry at the same Fermi level. In this Article, by RW simulation of DSCs under open circuit condition, we will show that the electron diffusion coefficient depends on particle size if the measurement is made at constant electron density. This observation made it possible to establish that traps are most probably located on the surface of the nanoparticles.24−26,31 We now carry out simulations at a constant Fermi level, resembling an experiment under potentiostatic control, and we show that the diffusion coefficient is constant. Therefore, we must be accurate in using the general expression of “the diffusion coefficient increases with particle size”. We will discuss that the results for surface diffusion are consistent with the MT prediction in exponential CB tail, only when the volume densities in MT equations are replaced by surface ones, which is, in contrast with previous consideration.25,31 We have also investigated the porosity and sintering effect by RW simulation. Our results show that if the Fermi level is kept constant at different porosities, then the value of Pc ≈ 0.78 provides a good approximation for both surface and volume diffusion. Conductivity exponent, however, changes from ∼0.9 for surface diffusion to ∼1.3 for volume one. These results shed some light on the precise way that the geometrical factors affect the diffusion coefficient. For all of the simulations carried out in this study, we have used the single-electron approximation and fixed the Fermi level manually. Despite the highly time-consuming multielectron simulations, this approximation permits us to make a good statistical averaging. This approximation is very appropriate for investigating the geometrical dependency of the transport. Also, it can readily be utilized for modeling the cell at potentiostatic control.

−1 ⎛ ∂n t ⎞ Dn = ⎜1 + ⎟ D0 ∂nc ⎠ ⎝

(5)

where nt is the local density of electrons in traps and the total electron density is given by n = nc + nt. For trap energy with the exponential distribution of

g (E ) =

⎛ E − Ec ⎞ Nt exp⎜ ⎟ kT0 ⎝ kT0 ⎠

(6)

diffusion coefficient, Dn, can be written as14,17,18

Dn =

Nc 1/ α

α(Nt)

n(1 −α)/ αD0

(7)

where Nt, kT0, and Ec are the density of traps, characteristic energy of traps, and the CB edge energy level (E < Ec), respectively, and α = T/T0. The above expression is only a consequence of power-law WTD with α = β.14,19,20,25 To take into account the effect of geometry, we introduce the effective diffusion coefficient, Def, by a geometry-dependent correction factor Cg as28,29 −1 ⎛ ∂n t ⎞ Def = Cg(r , θr , P) × ⎜1 + ⎟ D0 ∂nc ⎠ ⎝

(8)

Here r, θr, and P are the nanoparticle radius, nanoparticle overlap, and the film porosity, respectively. (See the next section.) This equation explains the contribution of geometry and electronic effects in calculating the diffusion coefficient (included in terms Cg(r,θr,P) and (1 + ∂nt/∂nc)−1, respectively). The main purpose of this Article is to explore the explicit form of Cg. 2.2. Simulation Implementation. Recently Anta et al.24 have used a random−cluster model32,33 to simulate the structural disorder of mesoporous TiO2. In our work, we have 3213



Article

Figure 1. (a) Schematic of the algorithm used to generate the porous network of the nanoparticles and (b) a realistic network produced by this algorithm with the parameters of r = 10 nm and θr = θR = 0.1. The picture has been visualized by the VMD software.

simulation and therefore can be used to model an experiment under potentiostatic control in an arbitrary illumination intensity. Finally, it provides a fine way to explore the geometrical dependency of the diffusion coefficient (or any other dynamical parameter) in DSCs by running the simulation for different geometry at a same Fermi level. In the beginning of the simulation, the electron is placed randomly in a trap with the coordinate x0⃗ . The waiting (detrapping) time of the electron in ith trap, with energy of Ei, is given by (Ec = 0)

modified the model to simulate the mesoporous network of TiO2 nanoparticles. The algorithm is represented in Figure 1. Part a of this Figure is a schematic of the procedure, and part b shows a visualization of a network produced by the algorithm. As shown in this Figure, the procedure is started by putting a sphere (nanoparticle) with radius r at the origin. Then, spheres with the same radius are being brought randomly around the initial sphere to cover the initial one’s surface. We have placed the spheres one by one on the condition that they have to be at a minimum distance d from at least one other sphere. The maximum overlap among two spheres can be defined as θr = 1 − d/2r. This process is repeated so that several shells of spheres are added to the initial sphere to obtain a bigger sphere with a radius of R. We set R = 4r(1 − θr), that is, approximately equivalent to the radius of sphere surrounded by two shells of nanoparticles. In the next step, spheres with radius R are being brought together in the same manner as primary spheres with an overlap parameter of θR. Finally, a box with size L = 8R(1 − θR)/√3, containing these secondary spheres, is repeated periodically in 3D space to avoid finite size effects. Typical value used for θr and θR is 0.1.24 In the next step, static traps were placed into the random cluster based on two models:24 r2 (surface diffusion) and r3 (volume diffusion) models, as already described. In the r2 model, traps, with a surface density of Nts, are placed only on the surface of the nanoparticles, and in the r3 model, traps are embedded into the nanoparticles with a volume density of Ntv. The energies of the traps are taken from the exponential distribution of eq 6. Single-electron RW simulation was used to simulate electron transport via these traps. Trap occupancy was simulated by a truncated distribution18,33 in which all traps below the manually fixed Fermi level Ef are inaccessible for electrons. It has been shown that the single-electron simulation can reproduce multielectron simulation. The single-electron approximation has three important advantages. First, it has the great advantage of the huge time saving. (A single run of multielectron simulation needs 40 h of CPU time.34) Also, it provides an easy way to introduce a desired Fermi level in

ti =

⎛ E ⎞ 1 ln(p) exp⎜ − i ⎟ ⎝ kT ⎠ v0

(9)

where p is a random number uniformly distributed between 0 and 1. v0 is the attempt to jump frequency, and its value depends on the mechanism that causes detrapping. Although electron−phonon interaction is considered to be responsible for electron detrapping,35 v0 is often considered as a fitting parameter. In each simulation step, the electron is moved randomly to an adjacent trap in the neighborhood region with a cutoff distance of rcut, and the corresponding detrapping time of ti is added to the total time of simulation. For simplicity, only the adjacent traps located in this cutoff area are considered as the destination sites.24,34 From a physics point of view, and based on the MT picture, the electron in a localized state (trap), is detrapped by interaction with the phonons into the extended state and after some time, is retrapped, by another trap. The time spent by electron in extended state is considered to be negligible in comparison with waiting time of electron in the localized states.15 For a normal diffusion, after well enough long time t, the diffusion coefficient can be calculated from the expression

⟨[xt⃗ − x0⃗ ]2 ⟩ = 6Djt

(10)

Dj is the jump diffusion coefficient (Dj = α Def17,18,36), and the symbol ⟨⟩ denotes a statistical average. In this work, the 3214



Article

averaging has been carried out over a 2000 times sampling to reduce the statistical error. The values of constants used for the simulations are given as below

v0 = 2 THz, T = 300 K, T0 = 600 − 800 K, Nts = 1 nm−2, Ntv = 0.3 nm−3, rcut = 2.5 nm The value of v024 determines the time scale of the simulation. As previously mentioned, it can be treated as a free parameter. The values of trap characteristic temperature, T0, and the trap density, Ntv, are given in accordance with the data reported in literatures, which are obtained by fitting the theoretical prediction of the MT model to the experimental data.37

3. RESULTS AND DISCUSSION 3.1. Film Characterization. The porosity P (= Vvoids/V) and the surface roughness ξ (= S/V) are two important parameters in the generated networks. Practically P and ξ can be obtained by computing the total number of traps per film volume, Nttot. In r3 model, Nttot is proportional to (1 − P). Also, in r2 model, Nttot is proportional to the surface roughness of ξ. Therefore, for each generated network of nanoparticles, one can distribute traps based on r2 and r3 models and then compute Nttot in each case to obtain ξ and P, respectively. For comparing our generated networks with those reported in refs 25 and 24, we computed ξ and P for radii 10 < r < 18 nm while fixing θr = θR = 0.1. An averaging on 500 networks was carried out for each given radius. (A few minutes of CPU time was needed to generate and characterize a network.) In quantitative agreement with previous works, an averaged porosity of 58% was obtained for all radii. Also, for surface roughness, from the best fit, we found ξ ≈ r −1.04±0.01, that is, in excellent agreement with simple geometrical consideration. It was reported that by varying the temperature of the autoclaving step in the nanoparticles synthesis, the nanoparticles with desired radius can be achievable.31,38 Traditionally, films with porosities between 50 and 60% are used in cell fabrication. This range of porosities is believed to be appropriate for both maximum dye loading and efficient charge transport. For exploring this issue, we have scanned a wide range of porosities via changing the overlap fractions θr, θR, or both, and for each case, the corresponding ξ was computed. The results are plotted in Figure 2 for r = 10 nm. In the low porosities, film is predominantly filled by the matter, and the surface area decreases considerably. In the high porosities, there is not enough matter to produce high surface area. Therefore, as can be seen in Figure 2, there is a maximum in the curves. Interestingly, the maximum surface roughness takes place in the porosity range of 44−52%. Highest ξ (119 μm−1) was obtained by reducing the porosity to 44% by increasing the θr up to 0.22 and keeping θr = 0.1. These parameters can be adjusted experimentally by applying different pressures prior to the sintering step28,29 and also by controlling the sintering condition. Because the lower porosities also facilitate the electron transport, it seems to be worthwhile to investigate the possibility of our result experimentally. 3.2. Effect of Surface Roughness on Electron Transport. The effect of surface roughness, ξ, (∼r−1) on the electron diffusion coefficient Dj was investigated by RW simulation for both surface and bulk diffusion for radii 10 < r < 18 nm. To set apart electronic effect and focus on the geometry effect only, we carried out simulations at a same Fermi-level Ef. Results are

Figure 2. Surface roughness, ξ, versus porosity, P. Different porosities were created by changing the overlap fraction. Upper curve: θr = 0.1 and 0.04 < θR < 0.27. Lower curve: θR = 0.1 and 0.04 < θr < 0.22. Middle curve: 0.05 < θr = θR < 0.19. Each point has been obtained by averaging on 500 samples. The error bars fall within the marker size.

shown in Figure 3. As demonstrated in this Figure, the Dj is approximately constant while varying r, for both r2 and r3 models.

Figure 3. Diffusion coefficient Df versus surface roughness ξ at constant Ef = −0.55 eV and T0 = 600 K. As shown, the Dj is approximately constant. Solid line (bold line) shows the prediction of MT model for surface diffusion when surface (volume) density is used in eq 11.

Our result for r2 model seems to be in contradiction with recent reports,25,26,31 in which it was observed that increasing the particle size (i.e., a decrease in ξ) leads to an increase in the electron diffusion coefficient, but it must be noted that these observations have been resulted from measurements at constant photocharge density. Photocharge density is defined as nph = Q/eV, where Q is the saturated collected charge in a usual photocurrent transient measurement.20 Because in the r2 model the total trap density decreases with increasing the particle size, at a constant photocharge density it leads to an increase in the Fermi level. Consequently, it leads to a greater Dj. Hence we can conclude that experimental increase in Dj with the particle size is a consequence of electronic filling effect and not a 3215



Article

geometrical effect. In other words, it can be concluded that Cg(r,θr,P) = Cg(θr,P). We also can conclude another important result from the data obtained by the r2 model. For this, we rewrite eq 7 as below, which is a more appropriate form in the case of constant Ef17

Dn =

Nc ⎛1 − α ⎞ exp⎜ E ⎟D ⎝ kT f ⎠ 0 αNt

(11)

where Nt is the trap density. There are two choices for substituting the Nt in the r2 model; one can interpret Nt as surface density of traps, Nts, and the other choice is the total volume density of traps, Nttot (= ξNts). Substituting the Nttot leads to obtain Dn ≈ ξ−1 (bold line in Figure 3), and substituting the Nts leads to obtain Dn ≈ ξ0 (solid line in Figure 3). As can be seen in Figure.3, our RW result is consistent with the results obtained by substitution of Nts in eq 11. Also in eq 7, which is the more appropriate form for the case of constant photocharge density, there are two choices for (n, Nt). They can be interpreted as the volume densities (nph, Nttot) or surface densities (nphξ−1, Nts). Volume and surface densities lead to achieve Dn ≈ ξ−1/α and Dn ≈ ξ1−1/α, respectively. Our simulation result, in the case of constant photocharge density for r2 model, is depicted in Figure 4. As shown in this Figure,

Figure 5. Surface diffusion coefficient Dj as a function of the film porosity P at constant Ef = −0.55 eV and T0 = 600 K. Porosity was changed by varying θr and (or) θR.

remarkable that different methods, in which the same porosities with different surface roughnesses are produced, lead to the same diffusion coefficients. It confirms again our former conclusion that the diffusion coefficient is independent of film roughness ξ. The effect of nanoparticles connectivity has been completely investigated previously by Cass et al.22,23 As demonstrated in Figure 5, the effect of connectivity (here represented by θr) can be interpreted in the context of porosity. In other words, by separating the electronic effect from the geometrical effect, the diffusion coefficient can be described only by porosity as

Cg(r , θr , P) = Cg(P) ∝ |P − Pc|δ Critical porosity of Pc and conductivity exponent, δ, have been computed for both surface and volume diffusion via fitting eq 1 to diffusion data at constant Ef. Results are shown in Figure 6.

Figure 4. Variation of the diffusion coefficient, Dj, with surface roughness ξ at constant nph = 2.5 × 1015 cm−3 and T0 = 800 K. Solid line (bold line) shows the prediction of MT model for surface diffusion when surface (volume) densities are used in eq 7.

RW simulation result again is consistent with the theoretical formula corresponding to the surface densities (solid line in Figure 4) and not volume densities (bold line). It is worth mentioning that using the volume density for r2 model, as done in refs 25 and 31, leads to the conclusion that even at a same Fermi level, diffusion coefficient changes with particle size. Then, it seems to be a good idea to repeat the experimental work of ref 25 at a same Ef. The result will be a good test for capability of the current RW simulation to simulate correctly electron transport in DSCs. 3.3. Effect of Porosity and Connectivity on Electron Transport. As explained in Section 3.1, different porosities were established by changing θr, θR, or both. Simulations at constant Ef of −0.55 eV were done at various porosities for both r2 and r3 models. Results are shown in Figure 5 for r2 model (results for r3 model (not shown) are similar). As expected, diffusion coefficient decreases by increasing the porosity. It is

Figure 6. Diffusion coefficient Dj versus porosity ξ for both surface and volume diffusion at constant Ef = −0.55 eV and T0 = 600 K. Lines show the fitted curves corresponding to eq 1. The critical porosity for both surface and volume diffusion is ∼78%.

The fitting parameters of Pc and δ are variable in this fitting. The fitting of Figure 6 led to Pc = 78.29 ± 5.08 and 78.19 ± 9.95% for surface and volume diffusion, respectively. The conductivity exponent of δ obtained is 0.927 ± 0.097 and 1.303 ± 0.178 for surface and volume diffusion, respectively. The value of surface conductivity exponent is close to that of 3216



Article

Benkstein et al.27 derived from RW simulation on a trap-free porous network, but the percolation theory30 for RW on the infinite cluster predicts an exponent around 1.5 that seems to be closest to our volume exponent estimate. Exponents around 0.5 to 0.827 and 1.8 to 2.228 were reported in the experimental studies. Hence it seems that further works are needed to elucidate this problem, especially with further regards to electronic effect. Because of the implementation of constant Fermi-level in our simulations, our estimate for Pc and δ is independent of T0. Anta and Morales-Flórez24 have carried out a number of simulations at constant total electron density and observed that interestingly in this case δ is dependent on T0. This result confirms our main conclusion in this Article that for having a thorough description of the geometry effects on transport (Cg(r,θr,P)) the electronic effects ((1 + ∂nt/∂nc)−1), must be kept constant. In a percolating cluster, when porosity goes to its critical value Pc, a well-known feature called dispersive dif f usion30 appears. In this range of porosities, the steady-state diffusion coefficient is smaller than its initial values. Also, when the porosity goes to Pc the time spent by random walker to reach the steady-state diffusion coefficient increases.24,27 The results of such studies are represented in Figure 7. It can be seen from this Figure that

This result was independent of the spatial location of traps (surface or volume diffusion), but in a constant photocharge density the diffusion coefficient for surface diffusion was proportional to roughness as ξ1−1/α. It was shown that RW results for surface diffusion are consistent with the MT prediction only when the surface densities are used in theoretical formulas. By introducing a geometrical factor as Cg(r,θr,P), it is shown that the geometry dependency of the diffusion coefficient can be described only as a function of the porosity (Cg(r,θr,P) = (Cg(P)).

■

AUTHOR INFORMATION

Corresponding Author

*Tel/Fax: +98 21 61118610. E-mail: [email protected].

■

ACKNOWLEDGMENTS We would like to thank the Research Council of the University of Tehran for partial financial support.

■

REFERENCES

(1) Soga, T. Nanostructured Materials for Solar Energy Conversion, 1st ed.; Elsevier Science: Amsterdam, The Netherlands, 2006. (2) Gregg, B. A. J. Phys. Chem. B 2003, 107, 4688−4698. (3) O’Regan, B.; Grätzel, M. Nature 1991, 353, 737−740. (4) Hagfeldt, A.; Boschloo, G.; Sun, L.; Kloo, L.; Pettersson, H. Chem. Rev. 2010, 110, 6595−6663. (5) Würfel, P. Physics of Solar Cells: From Principles to New Concepts; Wiley-VCH: Germany, 2005. (6) Sölbrand, A.; Lindstrom, H.; Hagfeldt, A.; Lindquist, S. E.; Södergren, S. J. Phys. Chem. B 1997, 101, 2514−2518. (7) van de Lagemaat, J.; Park, N. G.; Frank, A. J. J. Phys. Chem. B 2000, 104, 2044−2052. (8) Peter, L. M. Phys. Chem. Chem. Phys. 2007, 9, 2630−2642. (9) Forro, L.; Chauvet, O.; Emin, D.; Zuppiroli, L.; Berger, H.; Lévy, F. J. J. Appl. Phys. 1994, 75, 633−635. (10) Wang, M.; Chen, P.; Humphry-Baker, R.; Zakeeruddin, S. M.; Gratzel, M. Chem. Phys. Chem. 2009, 10, 290−299. (11) Frank, A. J.; Kopidakis, N.; van de Lagemaat, J. Coord. Chem. Rev. 2004, 248, 1165−1179. (12) Sölbrand, A.; Henningsson, A.; Södergren, S.; Lindström, H.; Hagfeldt, A.; Lindquist, S. E. J. Phys. Chem. B 1999, 103, 1078−1083. (13) Wang, M.; Chamberland, N.; Breau, L.; Moser, J. E.; HumphryBaker, R.; Marsan, B.; Zakeeruddin, S. M.; Gratzel, M. Nature Chem. 2010, 2, 385−389. (14) van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2001, 105, 11194−11205. (15) van de Lagemaat, J.; Zhu, K.; Benkstein, K. D.; Frank, A. J. Inorg. Chim. Acta 2008, 361, 620−626. (16) Nelson, J.; Chandler, R. E. Coord. Chem. Rev. 2004, 248, 1181− 1194. (17) Bisquert, J.; Vikhrenko, V. S. J. Phys. Chem. B 2004, 108, 2313− 2322. (18) Anta, J. A.; Mora-Seró, I.; Dittrich, Th.; Bisquert, J. Phys. Chem. Chem. Phys. 2008, 10, 4478−4485. (19) Nelson, J. Phys. Rev. B 1999, 59, 15374−15380. (20) Kopidakis, N.; Benkstein, K. D.; van de Lagemaat, J.; Frank, A. J. Phys. Rev. B 2006, 73, 045326−045433. (21) Barzykin, A. V.; Tachiya, M. J. Phys. Chem. B 2002, 106, 4356− 4363. (22) Cass, M. J.; Qiu, F. L.; Walker, A. B.; Fisher, A. C.; Peter, L. M. J. Phys. Chem. B 2003, 107, 113−119. (23) Cass, M. J.; Walker, A. B.; Martinez, D.; Peter, L. M. J. Phys. Chem. B 2005, 109, 5100−5107. (24) Anta, J. A.; Morales-Flórez, V. J. Phys. Chem. C 2008, 112, 10287−10293. (25) Kopidakis, N.; Neale, N. R.; Zhu, K.; van de Lagemaat, J.; Frank, A. J. Appl. Phys. Lett. 2005, 87, 202106.

Figure 7. Evolution of the normalized surface diffusion coefficient, Dj(t)/Dj(t0), with time for three different porosities. Simulations were carried out at constant Ef = −0.55 eV and T0 = 600 K.

for Ef = −0.55 eV and T0 = 600 K the time to reach steady state is ∼10 ms for P = 74%, and Dj drops down to one third of its initial value in the steady state. It must be mentioned that a much more intense dispersive transport occurs for nonthermalized electrons,14,15,18 but this kind of dispersivity cannot be realized by our single-electron simulation in which the Fermi level has been fixed at the beginning of the simulation.

4. SUMMARY AND CONCLUSIONS A mesoporous material containing the nanoparticles with radius of r was produced. Different porosity and surface roughness were obtained by changing the nanoparticles overlap. It is shown that the porosities between 44 and 52% are more suitable for both faster transport and higher dye loading than commonly used porosities of 50−60%. As-produced porous networks were then utilized for electron-transport simulations. Monte Carlo random walk simulations in trap-contained nanoparticles showed that in a same Fermi level the diffusion coefficient remains constant by varying the surface roughness. 3217



Article

(26) Nakade, S.; Saito, Y.; Kubo, W.; Kitamura, T.; Wada, Y.; Yanagida, S. J. Phys. Chem. B 2003, 107, 8607−8611. (27) Benkstein, K. D.; Kopidakis, N.; van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2003, 107, 7759−7767. (28) Dittrich, Th.; Ofir, A.; Tirosh, S.; Grinis, L.; Zaban, A. Appl. Phys. Lett. 2006, 88, 182110. (29) Ofir, A.; Dor, S.; Grinis, L.; Zaban, A.; Dittrich, Th.; Bisquert, J. J. Chem. Phys. 2008, 128, 064703. (30) Ben-Avraham., D.; Havlin, S. Diffusion and Reactions in Fractals and Disordered Systems; Cambridge University Press: Cambridge, U.K., 2005. (31) Zhu, K.; Kopidakis, N.; Neale, N. R.; van de Lagemaat, J.; Frank, A. J. J. Phys. Chem. B 2006, 110, 25174−25180. (32) Morales-Flórez, V.; De La Rosa-Fox, N.; Piñero, M.; Esquivias, L. J. Sol-Gel Sci. Technol. 2005, 35, 203−210. (33) Morales-Flórez, V.; Piñero, M.; De La Rosa-Fox, N.; Esquivias, L.; Anta, A. J.; Primera, J. J. Non-Cryst. Solids 2008, 354, 193−198. (34) Gonzalez-Vazquez, J. P.; Anta, J. A.; Bisquert. J. Phys. Chem. C 2010, 114, 8552−8558. (35) Baranovski, S. Charge Transport in Disordered Solids with Applications in Electronics; John Wiley & Sons: Chichester, England, 2006. (36) Bisquert, J. J. Phys. Chem. B 2004, 108, 2323−2332. (37) Anta, J. A.; Mora-Seró, I.; Dittrich, Th.; Bisquert, J. J. Phys. Chem. C 2007, 111, 13997−14000. (38) Zaban, A.; Aruna, S. T.; Tirosh, S.; Gregg, B. A.; Mastai, Y. J. Phys. Chem. B 2000, 104, 4130−4133.

3218


Monte Carlo Random Walk Simulation of Electron Transport in Dye

Recommend Documents