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Charge Transport in Dendrimer Melt using Multiscale Modeling Simulation

In this paper we present a theoretical calculation of the charge carrier mobility in two different dendrimeric melt system (Dendritic phenyl azomethine with Triphenyl amine core and Dendritic Carbazole with Cyclic Phenylazomethine as core), which have recently been reported1 to increase the efficiency of Dye-Sensitized solar cells (DSSCs) by interface modification. Our mobility calculation, which is a combination of molecular dynamics simulation, first principles calculation and kinetic Monte Carlo simulation, leads to mobilities that are in quantitative agreement with available experimental data. We also show how the mobility depends on the dendrimer generation. Furthermore, we examine the variation of mobility with external electric field and external reorganization energy. Physical mechanisms behind observed electric field and generation dependencies of mobility are also explored.


INTRODUCTION
Organic semiconductors are the building blocks of the various optoelectronics devices such as solar cells, light emitting diodes (LED), field effect transistor (FET) etc 2 . Charge transport properties of the constituent semiconductor play an important role 3 in the efficiency of the optoelectronic devices. Exploration of new organic semiconducting material with high and tunable charge carrier mobility (for high performance of optoelectronic device), is an active area of research for past several years . The charge carrier mobility of the disordered organic material reported so far usually lies in the range 4-8 of 10 -7 to 10 -10 cm 2 V -1 S -1 . On the other hand the organic semiconducting crystals can give rise to mobility as high 9-10 as 0.1 cm 2 V -1 S -1 .
Although mobility is very high in comparison to the disordered semiconductors the morphology of the organic crystals limits their applicability to a large class of optoelectronic devices. For application to a specific device one needs to have organic semiconductor 11 with specific morphologies. For instance, in some applications like OLEDs, solar cell etc. conduction both parallel and perpendicular to the substrate 11 is required. Organic crystals fail to provide isotropic mobility which is required in this case. In addition, semiconducting crystals are very unrealistic for use in large scale application owing to the difficulty and cost of producing single crystals on a large scale. Cheaper fabrication techniques, such as vacuum sublimation and solution processing, always results in disordered film 2 . Despite the low mobility, the disorder organic semiconducting material is still a promising candidate for the application in large class of optoelectronical devices. The amorphous dendrimer melt system, a disordered organic semiconductor, is a especially good choice for these applications because of its well defined molecular weight [12][13] , batch-tobatch reproducibility and high purity synthesis. Understanding the charge carrier mobility in disordered semiconductors like dendrimers is a step forward in the realization of organic electronics.
The charge carrier mobility in the nitrogen center stilbene dendrimer melt system reported earlier 11,[13][14][15] lies between 10 -3 to 10 -9 cm 2 V -1 S -1 depending on the structure and chemical composition of the respective dendrimer. In most of the theoretical work Gaussian disordered model 11,14,[16][17] was used with empirical fitting parameters. Gaussian disorder model is a macroscopic description of the charge transport phenomena where effect of microscopic details of morphology on mobility is not taken into account. As previously mentioned, the charge carrier mobility in disorder organic semiconductors, like dendrimer, strongly depends on the molecular structure, chemical composition and the packing of the molecules 12,[18][19] in a melt system. In a recent report M. E. Kose et. al 12 have presented a theoretical calculation of mobility in dendrimer using Monte Carlo simulations. In their study, Kose et al. used phenyl-cored four-arm thiophene dendrimers for the charge transport study and reported a charge carrier mobility in the range of 10 -3 cm 2 V -1 S -1 which was two order of magnitude higher than the experimentally measured mobility.
Both theoretical 12 and experimental 15 study of generation and electric field dependence of the mobility in dendrimer has been reported in past. The reduction of mobility with the increase in the electric field strength has been seen in experiment and also in theoretical calculations. In contrast, there are conflicting views as to how the mobility depends on the dendrimer generation. In some oligomeric based conjugated dendrimers, mobility decreases almost one order of magnitude as one goes from G2 to G3 13 , while an increase in mobility is seen in phenyl-cored thiophene dendrimer 12 . However, the physical mechanism behind the observed electric field and generation dependence of the charge carrier mobility is still not well understood.
In a recent report Nakashima et. al 1 have shown how the efficiency of the Dye-Sensitized solar cell (DSSCs) can be increased using the Carbazole dendrimer (with Cyclic Phenylazomethine as core and Dendritic phenyl azomethine with Triphenyl amine core) as the coating material to the TiO2 interface.
Realizing the potential of these dendrimers, it is necessary to have the better understanding of the charge transport properties of these dendrimers.
In this paper we calculate the charge carrier (hole and electron) mobility of the two different dendrimer systems: Dendritic phenyl azomethine with Triphenyl amine core (Dpa-Tpa) and Dendritic Carbazole with Cyclic Phenylazomethine as core (Cpa-Cz). We calculate the charge carrier mobility in both these systems starting from a very realistic morphology which is obtained using fully atomistic molecular dynamics simulation. Our calculation produces the value of mobility which is of the same order of magnitude as measured in experiment 13 .We also study the electric field and the generation dependence of the mobility for both the dendrimer cases. Effect of external reorganization energy on the mobility is also reported. We propose the possible physical mechanism responsible for this kind of dependence and demonstrate this by theoretical calculation.
The paper is organized as follows: In the first section we discuss the details of the molecular dynamics (MD) simulation carried out to obtain a realistic morphology of the system through which charge transport will be simulated. In section II, we discuss the charge transport simulation using semi classical Marcus-Hush formalism. Finally, in section III we summarize the results and end with concluding remarks.

I.A. MODEL AND COMPUTATIONAL DETAILS
The molecular structure of Dendritic Carbazole with Cyclic Phenylazomethine (Cpa-Cz) as core is shown in figure 1 for two different generations: generation 2 (G2) and generation 3 (G3). Figure 2 shows the molecular model of the phenyl azomethine dendrimer with Triphenyl amine as core (Tpa-Dpa). With increase in dendrimer generation, branching increases exponentially. Here we briefly outline the procedure followed to build the initial 3-d model of the dendrimers of both the above mentioned types. First we geometry optimize the core and the branch in Gaussian09 with HF/6-31G(d) as the basis set. During optimization, charges were calculated using electrostatic potential (ESP) method. Then the optimized fragments were added together using Dendrimer builder ToolKit 20  After an initial energy minimization, the system was heated slowly from 0 K to 300 K at a constant pressure of 1 bar. The system was then subjected to simulated annealing of five repeat cycle from 400K to 600K and back, in steps of 50K, allowing 2ps in each step under NPT condition to release the molecules from potential traps or initial biases 23 . Finally, a production run of 100 ns was performed at 300 K and at a pressure of 1 bar. At 300 K the system is in the amorphous melt state. Temperature and pressure were kept fixed using Berendsen weak temperature coupling and pressure coupling method 24 using a temperature coupling constant of 1 ps and pressure coupling constant of 2 ps. During the NPT run, simulation box angles were fixed at 90 0 and length of the simulation box was allowed to very independently in each direction. The bonds involving hydrogen were kept constrained using the SHAKE algorithm. This allowed us to use an integration time step of 2 fs. All calculations were performed with parallel version of PMEMD 21 .

I.B. RESULTS AND DISCUSSION
A representative equilibrated simulation snapshot of the system in case of Cpa-cz,G2 with 64 molecules T=300K is shown in figure 3(c) and 3(d). It shows a completely isotropic assembly of the dendrimers with a density close to ~1 gm/cc.
To extract the information about the structure the dendrimers after the equilibration we calculate the radius of gyration (R g ) of the dendrimer in all four cases. The numerical values of the radius of gyration (averaged over last 10 ns MD run) are given in Table 1. We see that the R g increases with increase in dendrimer generation. This nature is very similar to other classes of dendrimer like PAMAM 25 , PETIM 26 and PPI 27 .

II. A. METHODOLOGY
To study the charge carrier mobility through the amorphous melt phase we follow the semi-classical Marcus-Hush formalism 28 which has been successfully used previously to predict the charge carrier mobility in wide variety of systems including organic semiconductors. Charge transport is described as thermally activated hopping mechanism of charge carriers between charge hopping sites. According to the formalism, the charge transfer rate from th charge hopping site to the th hopping site is given by (1) is the transfer integral, defined as .
(2) 6 Here and is the diabatic wave function localized on th and th site respectively. is the Hamiltonian of the two sites system between which the charge transfer takes place. is the free energy difference between two sites. is the reorganization energy , is the Plank's constant, k is the Boltzmann's constant, and T is the temperature.
The workflow 29-31 of our charge transport simulation is as follows: i) We use fully atomistic MD simulation as described above to get the equilibrium morphology of the system ii) With this morphology we partition the system in charge hopping sites and decide the charge hopping pairs (pairs between which charge hopping is possible) iii) We calculate the transfer integral and all other quantities appearing in the rate expression for all charge hopping pairs. iii) Once the rate is known for all charge hopping pairs, kinetic Monte Carlo [29][30]32 method is used to simulate charge carrier dynamics and calculate the carrier mobility.
The details of molecular dynamics simulation has already been discussed in section I. With the equilibrium morphology in hand we partition the system in charge hopping sites. There is no strict rule in partitioning the system in charge hopping sites. Physical intuition 30 Table 2 below.  to be a dielectric medium. In that case the external reorganization energy can be expressed 35 by the following equation (4) Here is the electronic charge, and are the dielectric constant of the vacuum and the dendrimer melt respectively. R is the effective radius of the charge hopping sites. For a value of R~15 Å and in the range of 2-60 which is the typical value of for the most polymeric systems [36][37] , turns out to be in the range of 0.4 eV to 1 eV. In our calculation we do not calculate external reorganization energy exactly, but we calculate the mobility for different values of in the above range.
The free energy difference appearing in the rate expression is the contribution 12 from the different sources as described below (5) is contribution of external electric field defined defined . Here F is the external electric field and is the displacement vector between the centre of mass of and hopping sites. is the contribution in free energy difference due to different internal energies and can be written as To decide where the charge will hop among N neighbours we choose the biggest for which .
Here is the index of the neighbours of site and is a random number between 0 to 1. The above mentioned condition will ensure that the side is selected with probability . Now the position of the charge is updated and the above process is repeated.
The simulation was repeated with different initial position of charges (as in the snapshots obtained from MD). The charge carrier mobility along x direction is determined [31] from the average charge velocity along that direction by (8) is the average charge velocity of the charge in x direction, is the mobility along x direction and is the applied electric field. Since the system is isotropic the mobility is same in all three directions. In our case average velocity was calculated from the x component of unwrapped charge displacement (since periodic boundary condition was used) divided by total simulation time .

II.B. RESULTS AND DISCUSSION
We calculate the hole and electron mobility for both the dendrimer systems as a function of dendrimer generation (for both the G2 and G3 dendrimers). For the case of Cpa-Cz dendrimer, the hole mobility is of the order of 10 -8 cm 2 V -1 S -1 while electron mobility is in the range of 10 -12 cm 2 V -1 S -1 for an electric field of 10 5 V/cm for both G2 and G3.The magnitude of the electric field strength was chosen to match the experimental field strength 13 .On the other hand for the DPA-TPA dendrimer case, the electron mobility is of the order of 10 -6 cm 2 V -1 S -1 and hole mobility is of the order of 10 -8 cm 2 V -1 S -1 for the case of G2 dendrimer. For G3 dendrimer, both the hole and the electron mobilities are in the range of 10 -13 cm 2 V -1 S -1 .
Our calculated mobility values are in quantitative agreement with the available experimental data 13 . Using time of flight measurement, Lupton et al. 13 reported hole mobility (majority carrier) values of ~5 × 10 -8 cm 2 V -1 S -1 and ~1.5 × 10 -8 cm 2 V -1 S -1 in G2 and G3 conjugated dendrimer system respectively. The point to note from the above discussion is that for CPA-Cz case, hole mobility is higher than the electron mobility, while in DPA-Tpa case electron mobility is higher. To understand the observed difference in hole and electron mobility we calculate the HOMO and LUMO energy of core and branch of the dendrimers separately. The energy values are presented in Table 3. energies would necessarily indicate lower rate of hole (electron) transport between core and branches, since the rate depends exponentially on the energy difference. As a result, the overall mobility of electron will be low in comparison to hole (difference in HOMO energy of dendrimer core and branch is small) in melt system of CPA-cz . For the same reason the hole mobility is lower than the electron mobility in Dpa-Tpa dendrimer.
The point we want to emphasize here is that in case of very low mobility (approximately less than 10 -10 cm 2 V -1 S -1 ) the hopping events are very rare and it is very difficult to get a steady state for the charge transport within reasonable simulation time. So the low mobility values may not be statistically significant. In the next section we will be discussing the electric field and external reorganization energy dependence of the mobility only for the cases where mobility value is of the order of 10 -10 cm 2 V -1 S -1 or higher.
The variation of hole mobility of Cpa-cz dendrimer with respect to electric field is shown in figure 6. At low field the mobility (hole) is higher in G3 than in G2 , but as electric field increases the mobility in G3 decreases rapidly and after a certain value of electric field, mobility in G2 takes over the mobility in G3.
The electric field dependence of hole and electron mobility for G2 Dpa-Tpa is shown in figure 7(a) and 7(b) respectively. The reason for the decrease in mobility with the increase in the electric field strength can be understood as follows: At low electric field, the charge chooses only the most favourable path in hopping process. As the field increases, the charge is forced to occupy also the sites which was previously very less favourable to hop on. As a result the movement of the charge is delayed 12 resulting decrease in mobility. The G3 dendrimer being more branched structure than G2 , allows more paths for the carrier to hop in comparison to G2. As a result under the application of electric field the mobility decreases at a faster rate in G3 than in G2. To have a better microscopic understanding of the electric field dependence of the mobility as described above we present a schematic diagram (Figure 8) representing few specific charge hopping sites and corresponding hopping probabilities for Cpa-cz (G3) dendrimer. At low electric field the charge always hops from site A to C (Figure 8) while at high electric field there is finite probability for the charge to hop to site C as well to site B.
At low electric field when the carrier is supposed to chose only the most favourable path, G3 has higher mobility (figure 6) than G2 in the case of CPa-cz dendrimer while in the case of DPA-Tpa dendrimer, carrier mobility is higher in G2 than in G3. To understand this generation dependence, we look into the possible hopping events transporting the charge from one place to another. Suppose is the average hopping rate between two cores and is the average hopping rate between core and branch of the dendrimer. Now if , then charge transfer will mainly happen through the cores of the dendrimer as shown schematically in figure 9(a). As the dendrimer generation increases, the core-core distance between two dendrimer increases. To verify this we calculate the average distance between the centre of mass (COM) of the cores which are neighbours to each other. The COM distances between the core for all four cases are given in Table 4. We see that the average distance between the neighbouring cores increases with increase in generation. As the core-core distance between two dendrimer increases, decreases, resulting overall decrease in mobility. On the other hand if , then charge will also come to branch and follow the kind of path shown schematically in figure 9(b). With the increase in generation, due to more branching, the more new paths will open up for the charge to chose. Among the new paths, some path will provide faster route and some will provide slower route for the carrier in comparison to the old paths. Since at low electric field the charge only chooses the most probable (fastest) path so as a whole the mobility will go up.
Finally, we study the variation of carrier mobility as a function of the external reorganization energy keeping the value in the range as discussed in section II.A. Figure 10 shows how the mobility changes with the external reorganization energy. In all cases the mobility monotonically decreases with external reorganization energy.

III. SUMMARY AND CONCLUSION
In summary, we have calculated the charge carrier mobility (both electron and hole) in DPA-Tpa and Cpa-Cz dendrimers for both G2 and G3. We have shown that Cpa-cz has larger hole mobility than the electron while in Dpa-Tpa electron mobility is higher. The variation of mobility with respect to external electric field is also reported. We see that in CPA-cz dendrimer hole mobility decreases with the increase of electric field. At low electric field the G3 has higher mobility but after a critical field strength the mobility in G2 becomes higher. We have given physical explanation behind the observed electric field and generation dependence of mobility. Effect of the external reorganization energy on the mobility is also shown by calculating the mobility for various values of the external reorganization energy.
The dendrimers, we have studied have been already reported to increase the efficiency of DSSCs. However we hope that our investigation on the charge transport properties will induce more use of these dendrimers in optoelectronic device as well as DSSCs.   6. Hole mobility as a function of electric field strength in Cpa-Cz dendrimer melt system. At low field the mobility (hole) has higher value in G3 than G2 , but as electric field increases the mobility in G3 falls down rapidly and after a certain electric field mobility in G2 takes over the mobility in G3.  (a) (b) Fig. 9. Two different charge transport pathways in dendrimer system. In first case (a), charge transport mainly happens through the cores of the dendrimers. In the other case (b), the charge hops from core to branch then branch to the branch of other dendrimer and in the end to the core of the latter.