Geometrical effects on the electron residence time in semiconductor nano-particles.

Geometrical effects on the electron residence time in semiconductor nano-particles Hakimeh Koochi and Fatemeh Ebrahimi Citation: The Journal of Chemical Physics 141, 094702 (2014); doi: 10.1063/1.4894136 View online: http://dx.doi.org/10.1063/1.4894136 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Comparison of crystal growth and thermoelectric properties of n-type Bi-Se-Te and p-type Bi-Sb-Te nanocrystalline thin films: Effects of homogeneous irradiation with an electron beam J. Appl. Phys. 115, 214311 (2014); 10.1063/1.4881676 Enhanced thermoelectric properties of Mg2Si by addition of TiO2 nanoparticles J. Appl. Phys. 111, 023701 (2012); 10.1063/1.3675512 High near-infrared transparency and carrier mobility of Mo doped In 2 O 3 thin films for optoelectronics applications J. Appl. Phys. 106, 063716 (2009); 10.1063/1.3224946 Effect of plasma and thermal annealing on optical and electronic properties of SnO 2 substrates used for a- Si solar cells J. Appl. Phys. 92, 620 (2002); 10.1063/1.1481192 Study of Hall and effective mobilities in pseudomorphic Si 1−x Ge x p -channel metal–oxide–semiconductor fieldeffect transistors at room temperature and 4.2 K J. Appl. Phys. 82, 5210 (1997); 10.1063/1.366385

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THE JOURNAL OF CHEMICAL PHYSICS 141, 094702 (2014)

Geometrical effects on the electron residence time in semiconductor nano-particles Hakimeh Koochi1 and Fatemeh Ebrahimi1,2,a) 1 2

Physics Department, University of Birjand, Birjand 97175-615, Iran Solar Energy Research Group, University of Birjand, Birjand, Iran

(Received 11 May 2014; accepted 18 August 2014; published online 3 September 2014) We have used random walk (RW) numerical simulations to investigate the influence of the geometry on the statistics of the electron residence time τ r in a trap-limited diffusion process through semiconductor nano-particles. This is an important parameter in coarse-grained modeling of charge carrier transport in nano-structured semiconductor films. The traps have been distributed randomly on the surface (r2 model) or through the whole particle (r3 model) with a specified density. The trap energies have been taken from an exponential distribution and the traps release time is assumed to be a stochastic variable. We have carried out (RW) simulations to study the effect of coordination number, the spatial arrangement of the neighbors and the size of nano-particles on the statistics of τ r . It has been observed that by increasing the coordination number n, the average value of electron residence time, τ r rapidly decreases to an asymptotic value. For a fixed coordination number n, the electron’s mean residence time does not depend on the neighbors’ spatial arrangement. In other words, τ r is a porosity-dependence, local parameter which generally varies remarkably from site to site, unless we are dealing with highly ordered structures. We have also examined the effect of nano-particle size d on the statistical behavior of τ r . Our simulations indicate that for volume distribution of traps, τ r scales as d2 . For a surface distribution of traps τ r increases almost linearly with d. This leads to the prediction of a linear dependence of the diffusion coefficient D on the particle size d in ordered structures or random structures above the critical concentration which is in accordance with experimental observations. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4894136] I. INTRODUCTION

Nanostructured semiconductor films like porous nanocrytaline TiO2 are low cost materials of great importance because of their potential for applications in photovoltaic and optoelectronic devices, specially in dye-sensitized solar cells (DSSC).1, 2 In a DSSC, a thin film of nano-porous TiO2 covered by dye molecules3 is sandwiched between two electrodes. The film consists of nano-size particles and its morphology depends on the fabrication and preparation processes. The roughness of the nanoporous film, and the interconnected nano-particle network provides a large internal surface area. This feature allows for a large number of the light harvesting dye molecules attached to the body,4 and therefore increases the number of injected photo electrons. The exact nature of charge transport in this complex structure has not been yet understood thoroughly and still remains as a challenge. Generally speaking, in a semiconductor, the electrical current density Jc is the sum of two different terms: A drift motion induced by the local electric field E and a diffusive motion arises from the spatial variation in charge carrier’s concentration ρ, Jc = μc ρE − Dc ∇ρ,

(1)

where μc and Dc are the carrier’s mobility and its diffusion coefficient, respectively. In the case of liquid junca) Author to whom correspondence should be addressed. Electronic mail:

[email protected]

0021-9606/2014/141(9)/094702/8/$30.00

tions, the injected electrons within the nano-particles are subjected to a local screening effect by the surrounding electrolyte which facilitates their percolation through the film. Therefore, no drift term appears in the transport equation and transport occurs by diffusion only.5 It is believed that this diffusion process is limited mainly by the “traps,” the electron localized states with their existence is the main consequence of disorder in the electronic structure of the material.6 Specific numerical techniques have been developed to deal with the energy and morphological disorder of these materials.3, 7–16 Experimental and numerical works indicate that the performance of DSSC and similar devices is strongly dependent on the charge collection efficiency of the film.17–20 The nanostructured network is the host of photo-generated electrons which are excited in the dye molecules. The photo-generated electrons must travel across the film to reach the transparent conductive oxide layer. In other words, the film provides the conductive pathways from the interior to the surface where the collecting electrode injects them to the external circuit. In this regard, the best film’s morphology is the one with faster transport and less re-combinations, which minimizes the charge and energy losses. Optimization of the morphology of these structures for reaching higher energy conversion efficiency is a subject of interest.21 Several methods have been developed to prepare new structures with an arbitrary morphology with the aim of improving the performance of the devices.17, 19, 22–27

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Traditionally, the TiO2 films are fabricated from randomly oriented anatase nano-crystals. A coarse-grained transport model which focuses exclusively on the effect of the morphology on electron diffusion in random percolating clusters of nano-particles has been devised by Benkenstein et al.8 It was the first model which indicated the significance of the network geometry on electron transport in meso-porous TiO2 films. The authors of Ref. 8 adapted the random walk approach and estimated the diffusion coefficient for the disordered random packing of TiO2 nano-particles. In the case of an ordinary diffusion process in three dimensions, the rootmean-squared-average displacement R is related to the electron collection time τ t via R 2 = 6Dc τt .

(2)

Assuming that τ r , the electron’s residence time to be porosityindependent, the collection time τ t can be easily computed according to the expression: τt = Mj τ r where Mj is the number of particles, through which each electron must travel from its point of origin in one side of the simulation box to its final destination (collecting electrode) at the opposite side. As the main result, a power law dependence on porosity for the diffusion coefficient has been predicted which is almost in quantitative agreement with transient photo-current measurements and the predictions of percolating theory.28 The applicability of this model beyond the percolation limit has been discussed by Ofir et al.27 As mentioned earlier, random percolating clusters of nano-particles are not the only possible morphology for common applications. Preparation of organized structures,17 partially ordered structures,19 or even totally ordered structures23 for the sake of improving the energy conversion efficiency is a subject of interest. In this context, generalization or if necessary, modification of the current transport models, especially the coarse-grained models which are fast and reliable can provide the researchers with a tool to compare the efficiency of different structures. Clearly, the concept of electron resident time is central to any coarse-grained model which deal with electron transport in nano-structured semiconductor. A qualitative estimation of this parameter is therefore, of great importance. It is believed that within each grain there exists a number of traps randomly distributed in the surface or through the volume.13, 29 Besides, every nano-particle is surrounded by a number of neighboring particles which connect it to the entire structure. Each of these neighbors can act as the entrance or the exit for the moving carrier. In this regard, the electron moves between nano-particles can be considered as another percolation process, but this time on the sub-particle scale. In the transport model of Ref. 8, all the complexities regarding the charge transport within each nano-particle (finescale transport) have been represented by the parameter τ r . The value of τ r itself has been assumed a phenomenological quantity which can be estimated by fitting predicted values of Dc with experimental results. So far, most of the numerical studies have been mainly focused on the estimation of charge carrier diffusion coefficient, a property of the whole material. In this work, our aim is to estimate numerically the impact of local geometry on the

charge carrier residence time which is an average property of a nano-particle. With this intention, we adapted the method introduced by Anta and Morales-Florez13 for a single nanoparticle considering its local geometry. We have carried out the RW simulations (within the nano particles) and investigated the effects of particle size, the coordination number and the spatial arrangement of the neighbors on the statistical behavior of electron residence time. In addition to increase the speed of simulations, the other advantage of the coarse-grained model is to separate the role of each geometrical factor even without calculation of diffusion coefficient. As an important example, we will see that with a knowledge of electron residence time it is possible to compare the diffusion coefficient of porous semiconductors consists of nano-particles of different sizes, just with the help of simple scaling arguments and percolation theory. The paper organizes as follows. After Introduction we explain the numerical method and the details of simulation works in Sec. II. The results of our simulation for r3 and r2 models are presented and discussed in Sec. III. The paper is concluded at Sec. IV.

II. SIMULATION METHODOLOGY

Random Walk Numerical Simulation (RWNS) has become a very popular tool to study charge transport in nanostructured materials and devices.7, 9, 13, 30 The RWNS method has been extended to achieve a fine and realistic modeling of electron transport in nano-structured devices which generally contains both morphological and energetic disorder in nanostructures.13 We have adapted this fine-scale RW method with multiple-trapping transport to estimate the mean residence time for the diffusion of electron from one particle to one of its nearest neighbor all with the same diameter(size) d. In a nano-structured material, each nano-particle has been surrounded by a number of next nearest neighbors n. The host lattice is usually disordered but building ordered or partially ordered lattice has become recently attractive. Generally, for each coordination number n > 1 there are a number of possible spatial configurations. For non-overlapping spherical particles and in the planar geometry, we may distinguish each arrangement in spherical coordinates by n azimuth angles (θ 1 , θ 2 , . . . , θ n ). In the case of non-planar geometry we need to determine the polar angle φ too. We always assume that the electron has arrived to the nano-particle from the first neighbor located at θ 1 = 0. Once a particular configuration is picked, we proceed to place the electron traps randomly throughout the particles (r3 model), or on the surface of the particles (r2 ) model. Energy disorder has been implemented by using an exponential distribution for the trap energy Ei , ρ Ei − Ec , (3) g(Ei ) = lv exp k B T0 k B T0 where ρlv is the volume trap density, kB the Boltzman constant, T0 characterizes the mean depth of the trap energy distribution, and Ec is the energy of the extended state through which transport is assumed to occur, i.e., the mobility edge or


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J. Chem. Phys. 141, 094702 (2014) TABLE I. Basic computational parameters.

FIG. 1. Two planar configurations for coordination number n = 2: Left: θ 1 = 0◦ and θ 2 = 90◦ , with traps are distributed in the volume of particles (r3 model). Right: θ 1 = 0◦ and θ 2 = 120◦ , with traps are distributed on the surface of particles (r2 model). The red-marked small circles are the initial locations of the electron.

the conduction band lower bound (Ec > Ei ). The trap crosssection rcut has been assumed to be the same for all traps. We put an electron in a trap which is located from the entrance (θ 1 = 0◦ ) in a distance a little larger than rcut as it has just entered to the nano-particle (see Fig. 1). At each stage, all the traps located within the rcut distance from the current occupied trap are accessible for the next jump. The electron starts a series of random walks until it occupies a trap at a distance slightly larger than rcut located in one of its n exits (including the one it came from). The electron’s waiting time in ith trap is given by E − EC , (4) ti = t0 ln(r) exp − i kT where r is a random number extracted from a uniform deviate, t0 = ν 0 −1 where ν 0 is the attempt to jump frequency, T is the ambient temperature, and Ei is the trap energy (see Fig. 2). This choice of ti corresponds to an exponentially decaying distribution function g(ti ), for a fixed value of Ei .31 For the parameters of Table I the mean trap release time is almost equal to ti = 0.3t0 . A trapped charge carrier can jump to any trap which is located within the cutoff distance (or the trap cross section) rcut of the present trap. Therefore, when an electron enters a nano-particle it encounters many different conduction paths,

d = 19 nm Ec = 0.0 eV T = 300 K T0 = 600 K ν0 = 0.2 × 1013 Hz t0 = ν0−1 = 5.0 × 10−13 s rcut = 2.5 nm ρlv = 1.0 nm−3 ρls = dρlv /6

Nano-particle’s size (diameter) The conduction band energy Ambient temperature Characteristic temperature Attempt to jump frequency Trap-release characteristic time Trap cross section Trap volume density Trap surface density

each of them terminates to one of the n exits (including the nano-particle it starts its journey). Some of pathways are more or less direct and therefore take a small number of number of jumps (short paths), while most of the paths are very tortuous and lengthy. Besides, the trapping time ti is itself a stochastic variable that not only changes from site to site, but for a single trap it changes in a stochastic manner from time to time. The time it takes for the electron flies between traps (via extended states) is usually order of magnitude smaller than the trapping time. As the result, the electron residence time in the particle, τ r , can be estimated by just adding the release times’ of the visited traps, N

τr =

j

ti ,

(5)

i=1

τ r is itself a stochastic variable which even for a specific trap configuration fluctuates randomly in time. Our simulations begin with the selection of spatial configurations of neighbors, with the coordination number n. The mean residence time τ r has been estimated by averaging τ r over 10 000 samples for each configuration. The samples differ on the locations, energies, and release times of the traps. We have studied the behavior of P(τ r ) in different geometries by changing the number of its neighbors and their spatial arrangement. We have also sought the relation between τ r with the size of particles d, when d is much larger than rcut . Unless otherwise mentioned we have used the set of parameters presented in Table I in the simulations. As explained in Sec. I, in r3 model the traps are distributed randomly throughout each nano-particle with a trap density ρlv = 1/nm3 . In r2 model they are scattered on the nano-particles surface with a variable surface density ρls = dρlv /6. This choice of ρ ls ensures the same volume trap density ρlv for all particle sizes. As the result, the total number of traps in each particle of size d varies with d, but it is the same for both model r2 and r3 models. III. RESULTS AND DISCUSSIONS A. Dead ends

FIG. 2. Distribution of trap release time g(ti ) when ti obeys Eq. (4), with the parameters presented in Table I.

A terminating particle or a “dead end” is the special case characterized by n = 1. It is known from percolation theory that the existence of dead ends makes the pathways longer and more tortuous. The effect of terminating particles on electron transport depends on the morphology of the porous material. In an ordered structure, there are little of them. However, in


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FIG. 3. Comparison between (part of) the histograms of P(τ r ) for dead-ends with r2 (circles) and r3 (squares) models.

random packing of nano-particles the number of dead ends increases rapidly with the porosity. According to Ref. 8, the fraction of dead ends in TiO2 films increases from less than 1% for 50% porosity to 31% for 75% porosity. In a terminating nano-particle, the only way for an electron to continue its travel is leaving the particle from the very neighbor it has entered to the particle. There are many from the very neighbor it has entered to the particle. There are many paths for backtracking, each of them correspond to a broad range of resident time. We call a conduction path “short” if it consists of a small number of jumps N j . It is also possible that the electron takes longer and tortuous path characterized by a large value of Nj . Since the trap-release time is not constant and obeys the statistics dictated by Eq. (3), there is not a clear relation between Nj and τ r . However, we anticipate for larger values of Nj the residence time becomes larger on average, compared to residence time of the short paths. The distribution of residence times for terminating nano-particles has been depicted in Fig. 3 assuming a volume distribution of traps (for both r2 and r3 models). As can be seen from the figure, P(τ r ) falls almost exponentially with τ r .

J. Chem. Phys. 141, 094702 (2014)

FIG. 4. The histograms of distribution of residence time, P(τ r ) for n = 3 and r2 model in two different configurations. Case A( squares): the three neighbors are located at θ 1 = 0◦ , θ 2 = 120◦ , and θ 3 = 240◦ . Case B (circles): θ 1 = 0◦ , θ 2 = 60◦ , and θ 3 = 120◦ .

n = 3 we tried 4 different configurations for r3 model: θ = 0◦ , 60◦ , − 60◦ , θ = 0◦ , 60◦ , 120◦ , θ = 0◦ , 120◦ , − 120◦ , and θ = 0◦ , 90◦ , − 90◦ . Our calculations show that in all of these configurations the mean residence time is almost the same and equal to τ r = 205 ± 5t0 (compare it with t i = 0.3t0 ). Not only τ r , but also the mean number of jumps N j are almost the same and fluctuate between 600 and 630 for the tested different configurations of three neighbors. This is not unexpected since, on average, the total moves within a nanoparticle are large enough to expect a simple linear relation between these two values: τ r N j ti .

(6)

Therefore, the equivalence of mean residence time is a consequence of the equality of mean number of jumps. We carried out extensive RW simulations for different n and each n several configurations. We observed that while the electron mean residence time turned out to be independent of the spatial configurations, the coordination number n affects drastically the value of τ r . In Fig. 5, the distribution functions have been compared for two coordination numbers, n = 8

B. Non-terminating particles (n > 1)

In all the studied cases, the distribution of residence time, P(τ r ) shows the general trend like Fig. 3. In Fig. 4 two different P(τ r )’s have been compared for n = 3 and r2 model for the case A where the three neighbors are located at θ = 0, 120◦ , 240◦ and case B where θ = 0, 60◦ , 120◦ . Compared with Fig. 3 we can see how increasing the coordination number makes the distribution broader which is not unexpected because of the introduction of new exits, changes the balance between the old paths and creates new paths at the same time. For the same reasons, the distribution function is slightly affected by the neighbors spatial configurations. From P(τ r ), the mean electron residence time τ r can be estimated for different n and different configurations. Surprisingly, we observed that this value only depends on n and for a specific n, the mean residence time has turned out to be the same for all configurations. For example, for the case

FIG. 5. The resident time distribution functions for the coordination number n = 8 (circles) and n = 3 (squares).


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FIG. 6. The mean residence time τ r as a function of the coordination number n in r2 (circles) and r3 (squares) models.

J. Chem. Phys. 141, 094702 (2014)

These observations suggest that, τ r is a local parameter which can vary remarkably from site to site, unless we are dealing with highly ordered structures. Considering the fact that more compact films have larger mean coordination number,21 this means that τ r is a porosity dependent parameter even if we average it over all the lattice sites. This is an important feature that should not be ignored in coarse-grained simulation models. The case of dead ends is a remarkable example. For the parameters mentioned before and with a surface distribution of traps, it takes on average about 800t0 for an electron to find the route to escape from a dead end. The corresponding value for n = 4, which is the average coordination number for the typical porosity 50% of TiO2 films is about 180t0 . Therefore, the existence of dead ends makes the transport slower at sub-particle scale, by increasing N j , and at the whole material scale, by making the path more torturous. C. The effect of particle size

and n = 2. In the first configuration, 6 particles are lying in the x-y plane, and the other 2 in the ±z direction. In the n = 2 case, neighbors are located on a planar geometry with θ 1 = 0◦ and θ 2 = 60◦ . As can be seen from the figure, the case n = 8 has a smaller characteristic time which is not unexpected, because it provides more exits for the electron to leave the nano-particle. In Fig. 6, the behavior of τ r as a function of coordination number has been depicted. The decrease in τ r with the coordination number n is related to the increase of the exits in higher coordination number. As mentioned before, this is a reflection of the average path’s length measured by the average total jumps the electron makes from the entrance to the exit. Figure 7 shows how N j decreases with n. We also carried out simulations for smaller values of rcut and observed that although the residence time (and also its fluctuations around the mean value) increases for smaller values of rcut , but still we can consider it to be independent of the local geometry, of course with less precision. For example, for the case n = 4, we estimated τ r = 450 ± 30t0 in r2 model with rcut = 0.75 nm, and τ r = 870 ± 40t0 in r3 model withrcut = 1.0 nm.

FIG. 7. The mean number of jumps N j as a function of the coordination number n for r2 (circles) and r3 (squares) models.

The size of nano-particles d is an important geometrical factor which can be changed by controlling material’s preparation conditions. By increasing the size of nano-particle the number of possible pathways within the particle grows as well as the relative number of longer paths. In Fig. 8, we have compared P(τ r ) for the case n = 2 with a volume distribution of traps for d = 21 nm (circles) and d = 11 nm (squares). One can see how the addition of new longer paths in the larger particle affects the distribution of the statistics of the resident times and makes it broader. Consequently, both the average of total jumps and mean residence time increase with d in both r3 and r2 models, as can be seen from Figs. 9 and 10, respectively. We can use these results to estimate the dependence of the diffusion coefficient on particle size. In ordered lattices or in compact clusters where particle concentration is above percolation threshold we have Eq. (2) to estimate the diffusion coefficient. For a fixed spatial arrangement of nano-particles Mj is constant and does not depend on particles’ size d, but R2 scales as d2 . Therefore, with τt = τ r Mj ∝ d, the diffusion

FIG. 8. Distribution of residence times for coordination number n = 2, and with a volume distribution of traps for d = 21 nm (circles) and d = 11 nm (squares).


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FIG. 9. The mean residence time τ r as a function of particle size d for the coordination numbers n = 2 (circles), n = 4 (squares), and n = 6 (triangles) for r3 model.

FIG. 11. The behavior of

d2 τr

as a function of particle size d for the coordi-

d2 τr

as a function of particle size d for the coordi-

d2 τr

versus particle size for n = 4 and r2 model.

nation numbers n = 2 (circles), n = 4 (squares), and n = 6 (triangles) for r3 model.

coefficient varies with the particle size as Dc =

R2 d2 ∝ . 6τt τr

(7)

The above value have been calculated for different coordination numbers and depicted in Figs. 11 and 12 for r3 and r2 models. From these figures we can predict that for r3 model the diffusion coefficient slightly decreases with particle size specially for a more compact layer (larger values of n). On the contrary, for r2 model, the results indicate that Dc grows with d. In both cases the trap volume density has been the same. However, the same behavior has been observed for a constant surface density of traps (Fig. 13). It is worthwhile to mention that the same behavior for Dc versus d has been reported as a result of a full-scale simulation over 50 nanostructure samples of size 100 nm × 100 nm × 100 nm13 with a fixed porosity equal to 55%. The increase in electron diffusion coefficient with increasing the TiO2 particle size has been observed experimentally among the samples prepared from the same starting materials32 and samples with almost the same porosity.33

FIG. 10. The mean residence time τ r as a function of particle size d for the coordination numbers n = 2 (circles), n = 4 (squares), and n = 6 (triangles) for r2 model.


nation numbers n = 2 (circles), n = 4 (squares), and n = 6 (triangles) for r2 model.


Full and open circles stand for a fixed volume density (1 nm−3 ) and a fixed surface density (19/6 nm−2 ) of traps, respectively.


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D. Where is the exit?

An interesting question arises here: Which neighbor is the next host of the electron? To answer this question, we have calculated for each neighbor the probability of being the exit site for the diffusive electron by calculating the fraction of paths ends to it. Our results show that despite the common assumption, this probability is not the same for all neighbors, even for highly symmetric configurations. For example, in the case of three nearest neighbors forming a triangular configuration we have estimated the chance of exit from θ 1 = 0◦ , θ 2 = 120◦ , and θ 3 = −120◦ to be 43%, 28.5%, and 28.5%, respectively, for the r2 model, and 46%, 27%, and 27% for r3 model. The most striking point here is that the probability of exiting from the entrance is almost greater than that of other exits. This effect is a consequence of the non-zero trap cross-section rcut which makes the paths crossing the entrance (θ 1 = 0◦ ) more accessible. From this observation we can conclude that the electron pathways are even more tortuous than we used to think before, because in addition to the ordinary percolation mechanism, the excessive chance of backtracking to the θ 1 = 0◦ neighbors makes the average path’s length larger, and as the results this is another factor that makes the electron transport slower than before. This effect softens by decreasing porosity. For example, for n = 4 with a planar geometry we found the chance of exit from θ 1 = 0, θ 2 = 90, θ 3 = 180, and θ 4 = 270 to be 34%, 23%, 20%, and 23%, respectively for r2 model and 39%, 21.5%, 21.5%, and 18% for r3 model. This feature is another factor that makes the more compact structures favorable for faster charge transport.

IV. CONCLUSION

The concept of electron residence time is crucial for devising a robust particle-level model of transport in nanostructured semiconductors. One of the most important goal of a coarse-grained model is to separate thoroughly the role of the morphology on charge carrier transport in nano-structured materials. We carried out fine-scale RW simulation of traplimited electron transport and studied the effect of particle size, its coordination number, and the spatial arrangement of the next neighbors on the time interval an electron spends in a nano-particle. Both r2 and r3 models were examined. We observed that although the spatial configurations affect the statistics of τ r , but the average of residence time only depends the coordination number n. Since the mean coordination number is a function of the porosity, we conclude that the mean residence time is also porosity dependence quantity. We also investigated the effect of particle’s size on the value of τ r and observed that for r3 model and with a constant volume trap density, this parameter scales as d2 . This leads to a diffusion coefficient which decreases slightly with nano-particle size for compact nano-structures. On the other hand, the r2 model predicts the diffusion coefficient increases with particle size d, for constant volume or surface trap density. Compared to the exact, but time consuming fine-scale models, the coarse-grained models are much faster and

J. Chem. Phys. 141, 094702 (2014)

straightforward. It is worth to mention that estimation of the diffusion coefficient via “fine-level” simulation models like Anta and Morales-Florez13 and Ansari-Rad et al.29 is exact but very time consuming. The reason is that a realistic sample consists of many nano-particles each of them contains many traps. During the diffusion process the charge carrier visits each of the nano-particles too many times. And this happens for all the samples needed for evaluation the statistical average of diffusion coefficient! The implementation of the periodic boundary conditions helps to reduce the size of samples and therefore the number of particles, but to avoid the sizeeffects and for accurate estimation of the diffusion coefficient one cannot use very small samples.

ACKNOWLEDGMENTS

We are thankful to H. Farsi who introduced us to the exciting field of dye sensitized solar cells. This work has been partially supported by University of Birjand under project No. 14839/dal/1392. 1 B.

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