Insights into the Charge-Transfer Mechanism of Organic Photovoltaics: Eﬀect of Domain Size

A great eﬀort has been devoted into understanding the mechanisms of charge generation and charge separation processes in bulk heterojunction solar cells, with the aim of improving their performance. Theoretical methods, such as DFT, have been used to shed light into these complex processes, but the computational cost associated with the simulations limits the model size, and thus its accuracy with respect to real hetero-junctions. To overcome this limitation, a linear-scaling reformulation of (TD)DFT is employed, allowing to move beyond the simple polymer-fullerene models and to consider larger complexes composed of more than a single oligomer chain and numerous fullerene molecules. In this work the interaction between an analogue of PBTZT-stat-BDTT-8, a high-performance D-A statistical copolymer developed by Merck, and PCBM is explored, with a focus on: (i) the eﬀect of the size of the polymer’s acceptor (A) blocks, and (ii) the eﬀect of the domain size. Results suggest that large acceptor blocks enhance the probability of a charge transfer (CT) to occur, and that CT states are more signiﬁcantly aﬀected by the size of the polymer rather than the fullerene phase. Evidence of long-range CT states in the low energy part of the excited state manifold is also observed.


Introduction
Organic photovoltaic (OPV) devices have attracted considerable interest in the last 15-20 years, mainly due to their relatively low manufacturing costs, high flexibility, and the possibility of roll-to-roll processing, which enables high and cheap throughput device production; moreover, OPVs also work in both diffuse light and indoor, making them extremely promising as a light harvesting technology. 1,2 Schematically, a typical OPV module consists of an active layer sandwiched between an anode and a cathode, and composed of a blend of electron donor and electron acceptor materials. This kind of device architecture, which at present is the state of the art in OPVs, is iv). Charge separation may also happen via coupling of excited CT n states with the CS state, although at present it is still not clear whether these 'hot' states, as well as the excess energy (i.e., the energy dissipated during the S 1 → CT 1 relaxation process), are important for charge generation: on the one hand, it has been argued that 'hot' CT states, being in general more delocalized, are useful for charge separation because of the significant decrease in the Coulomb interaction between the electron and the hole, and that a large excess energy is therefore beneficial for an efficient charge dissociation; 4-7 on the other hand, this view has been challenged by experiments that showed that the efficiency of the charge generation step does not depend on the population of high energy CT n states, but only on the lowest lying one (CT 1 ), and therefore, excess energy is regarded as being wasted. [8][9][10] The overall power conversion efficiency (PCE) of the device largely depends on the efficiency of steps (iii) and (iv), but due to the morphology of the active layer, an interpenetrating network of polymer and fullenere domains, it is not surprising for many energy loss mechanisms to take place: for instance, the electron-hole pair can undergo geminate recombination even before it is able to diffuse to the donor-acceptor interface, and even assuming the CT to happen, there may occur non-geminate recombination between electrons and holes generated from different excitons. Nevertheless, devices with PCEs higher than 10% have been widely reported in the literature. [11][12][13] Being able to model the formation of charge transfer states in the polymer-fullerene interface would be extremely helpful in order to elucidate the mechanisms of charge generation and separation in OPVs. However, constructing such theoretical models is a rather challenging task, due to the complexity of the system: for instance, although atomistic models based on first-principles calculations (such as density functional theory) would provide some of the necessary information useful to shed light into these highly complex problems, it is not straightforward to correctly represent a BHJ, one of the main reasons being the computational cost associated with the size of the system in study. For this reason, BHJs have often been modelled with DFT mainly as a single small chain oligomer complex with a single fullerene. DFT, in its time-dependent extension (TDDFT), has been used to study excited state properties of this type of materials: for instance, Few et al. 14 investigated the influence of the chemical structure on the properties of CT states of a series of polymer-fullerene complexes, while Niedzialek et al. 15 tried to correlate the device efficiencies with the excited state properties of three different polymer-fullerene blends, with a focus on the effect of the excess energy; TDDFT has also been used in conjunction with Ehrenfest dynamics for the study of the charge transfer dynamics of a P3HT:PCBM blend. 16 Recently, Fazzi et al. 17 explored the influence of the P3HT-PCBM relative orientation (on-top and on-edge) on the excited state manifold of this class of materials, and showed that this is able to affect the charge generation mechanism of OPV blends. However, TDDFT calculations become extremely prohibitive when performed on large-size systems, and this limitation might lead to an oversimplification of the BHJ model. The aim of this work is to move beyond the simple small oligomer and fullerene pair model by increasing the complexity of the system (i.e., by significantly increasing the size of both the oligomer and the fullerene domains), while remaining within the fully ab initio (TD)DFT framework: this is achieved by the use of a linear-scaling reformulation of (TD)DFT, which enables to investigate the ground and excited state properties (with a focus on the charge transfer states) of BHJs on a far larger scale than possible before. We chose to model the electron acceptor material as PCBM and the electron donor material as an analogue of PBTZT-stat-BDTT-8, a high-performance polymer developed by Merck that was used in the solar tree installation at the German pavilion, as part of the Universal Exhibition in Milan, 2015. 18 PBTZT-stat-BDTT-8 is a statistical push-pull block copolymer, and this means: (i) it is itself composed of electron-deficient (acceptor, A) and electron-rich (donor, D) blocks, 2,1,3-benzothiadiazole-thiophene (BTZ-T) and benzo-[1,2b:4,5-b']dithiophene-thiophene (BDT-T), respectively; (ii) the length of the A and D blocks is not likely to be constant throughout the polymer chain, as this material presents different properties when compared to its regioregular analogues. 19 Moreover, given its complex com-position, it is possible that the size of the blocks may somehow influence the excited state properties of this kind of blend.
In this work we first explore the effect of the polymer's block size by coupling three oligomers, derived from PBTZT-stat-BDTT-8, with a different A block length together with a pristine PCBM supercell, focusing on the charge transfer excitations of the resulting complexes. We then multiply in turn the size of the fullerene and the polymer domains (resulting in models with as many as 2360 and 2048 atoms, respectively), in order to investigate how far the charge transfer states can delocalize in both domains, how (and if) CT states are influenced by the domain size, and to elucidate whether long-range charge transfer (i.e., charge transfer states with an increased electron-hole distance) might occur in OPV devices, as the work of Ma et al. 20 seems to suggest.
This study has been carried out within the density functional theory framework by employing linear-scaling DFT and linear-scaling TDDFT with the ONETEP code, and it is organized as follows: in Section 2 (Methods) the concepts behind the ONETEP program,

The ONETEP program
All the simulations were performed at the density functional theory level by using the ONETEP (Order-N Electronic Total Energy Package) code for linear-scaling DFT. 21 ONETEP is based on a reformulation of the single-particle density matrix ρ(r, r'), which in conventional Kohn-Sham DFT is defined as where f i is the occupancy of the state ψ i (r), which is a Kohn-Sham orbital function. In the ONETEP formalism, the density matrix is expressed in a separable form, equivalent to the expression in Equation 1: where φ α (r) are localized atom-centered orbital functions known as non-orthogonal generalised Wannier functions (NGWFs), 22 and K is known as the density kernel. The NGWFs are expanded in a basis of periodic sinc (psinc) functions, 23 which correspond to plane waves through a Fourier transform. The peculiarity of ONETEP is that it is able to achieve linearscaling behavior while maintaining at the same time near-complete basis set accuracy: linear scaling is obtained by enforcing strict localization of the NGWFs within a set radius and by truncation of the density kernel, K, imposing a spatial cutoff, which makes ρ(r, r') sparse; during a calculation both the density kernel and the NGWFs are optimized in a self-consistent way, and this allows plane wave accuracy by using a minimal set of orbital functions only.

Simulation Details
In this work the ground and excited state properties of a donor-acceptor model complex have been studied (see Figure 2 for the structures of the materials); in order to avoid confusion with the electron rich and electron poor units of the polymer itself, from this point onwards the donor material will be simply called polymer (P) and the acceptor material fullerene (F). Also note that, although the name PBTZT-stat-BDTT-8 will still be used, we here emphasize that the structures that were used to carry out this study were all analogues of this polymer. In this study the fullerene domain was modelled as twelve close-packed PCBM molecules (effectively a 3x1x1 supercell) whose atomic positions were taken from a pristine x being the BDT-T block (donor, D) and y the BTZ-T block (acceptor, A). In this work, the R 1 and R 2 functional groups of the polymer are CH 3 and OCH 3 , respectively. PCBM crystal experimentally determined from XRD by Paternò et al. 24 We first focused on the effect of the A block length on the properties of the resulting polymer:fullerene complex, since, as previously mentioned, this might or might not be important for polymer design considerations. It is worth noting that our previous studies suggested different electronic and optical properties of the polymer's D and A units when blended with PCBM (see Supporting Information). To explore the effect of the A block length, the D and A units of PBTZT-stat-BDTT-8 were combined so that three different oligomers were obtained: (i) DDAAAAADD, (ii) ADDAAADDA, and (iii) AADDADDAA; for comparison with the regioregular copolymer, a fourth oligomer was also constructed, the alternating ADADADADA. The oligomers were called 5A, 3A, 1A, and AD, respectively (see Figure 3 for a schematic representation of these structures). It is worth noting that the oligomers mentioned above are all isomers. The PCBM supercell was placed at the center of each oligomer chain, above the central A block, at an initial intermolecular distance of ∼ 3.5 Å, in line with other theoretical works in which the distance was set between 3.2 and 3.5 Å. [14][15][16]25 The size of the A block was gradually reduced by taking away at each step two A units from the central block and by placing them at the edge of the chain: in this way the number of A and D blocks, the length of the chains, the number of atoms and double bonds were kept constant for each simulation, assuring the results to be comparable; the assumption made behind these experiments is that the external A units are too far away from the central fullerene domain to be able to significantly interact with it. Each of the complexes consists of a total of 1304 atoms.
For the final, and main part of this study we focused on the 5A complex and we increased the complexity of the system: we multiplied in turn the size of the fullerene domain (from a 3x1x1 to a 3x1x2 supercell) and the size of the polymer domain (two, three, four stacked oligomer chains), in order to investigate how far the charge transfer states can extend in the PCBM crystal and in the polymer domain, respectively, and if this has any effect on the interface energetics. The geometries of the stacked oligomers were first fully optimized without the presence of the fullerene phase, and we set the interchain distance at a value of ∼ 3.5 Å. The chains were stacked in such a way that each oligomer was flipped by 180 • with respect to the closest ones above and below, with the functional groups pointing in opposite directions, in order to avoid steric hindrance. We then placed the PCBM phase above the central part of the polymer domain, and we fully optimized each one of the complexes.
The geometries of all oligomers, fullerenes, and polymer-fullerene complexes were optimized in vacuum by using the Perdew-Burke-Ernzerhof exchange-correlation functional (PBE) 26 in conjunction with the projector augmented wave (PAW) method; 27 to take into account dispersion effects, each simulation was performed also by including the D2 correction by Grimme. Describing charge transfer states with TDDFT is a rather challenging task, since this is an area in which this method has well known limitations due to both the DFT formalism itself (i.e., the missing derivative discontinuity of the exchange-correlation functional), and the incorrect 1/R behavior of the potential energy curves of such states, with R being the intermolecular separation distance. These both lead to one major drawback: an underestimate of the energy of the CT transitions, and consequently the loss of a direct quantitative comparison with experimental values. This is in general a significant issue when using a GGA functional, such as PBE, and it might in part be corrected (but not necessarily fully solved) by employing functionals which include a fraction of Hartree-Fock exchange, such as hybrids. Range-separated hybrids instead seem to work particularly well for describing CT states, however, they are more computationally expensive, and they introduce an empirical parameter that significantly affects the excitation energies, and that therefore needs to be tuned. Taking these into account, we note that qualitative results that can be useful to shed light into some of the properties of polymer-fullerene complexes can still be obtained despite the use of a GGA functional, and moreover, to the best of our knowledge, DFT and TDDFT have never been applied so far to such large systems which are relevant for OPV functionality. To assess TDDFT accuracy, this methodology was first tested by comparing results obtained with ONETEP against published theoretical and experimental data. A number of eight polymer-fullerene blends were considered, of which the experimental CT peak emissions obtained with electroluminescence (EL) were taken from the work of Yao et al. 34 Amongst these eight blends, five of them were also modelled with TDDFT by Few et al.: 14 as previously mentioned, in their work they performed TDDFT to model the first CT excitation energies of different blends at the B3LYP/6-31G* level (for further details the reader is referred to their paper). In this paper the blends were modelled by considering oligomers consisting of 3-4 repeat units (12 in the case of P3HT, due to the very small size of its repeat unit), and by placing the fullerene (either PC 61 BM or PC 71 BM) above the acceptor unit of the oligomer, with the only exception of P3HT, the latter being a homopolymer. Note that for every complex, the fullerene was placed above the central A unit, where present, of the oligomer chain. Results are shown in Figure 4: as can be observed, and as expected, a GGA functional like PBE underestimates the energies of the CT states, while the hybrid functional B3LYP has the opposite effect; as Few et al. pointed out in their work, this overestimate with the B3LYP functional is most probably due to the lack of polarizable medium and to the use of a rather limited basis set. Despite the expected systematical underestimate of the frequency of CT 1 states, results seem to suggest that PBE is able to at least qualitatively represent the overall experimental trend, with the only exception of MDMO-PPV:PC 61 BM; however, as it can be observed, also B3LYP seems to fail for this particular blend. Therefore, it seems that by using a less computationally expensive functional such as PBE, it is possible to compute charge transfer excitation energies that overall agree qualitatively with experiments, at least with this class of systems.
where E PF is the energy of the whole complex, and E P and E F correspond to the energy of the polymer and the fullerene phase, respectively. E CT B was estimated according to the following expression: 36 where E CS is the energy of the charge separated state, and E CT 1 is the energy of the first charge transfer state; as shown by Brédas et al., 37 the energy of the CS state can be estimated as the difference between the ionization potential (IP) and the electron affinity (EA) of the complex.   a Note that every transition has a charge transfer character, and therefore every electron-donating orbital is located on the oligomer while every electron-accepting orbital is located on the fullerene domain. We then explored the excited state properties by performing TDDFT on each complex, and requiring the calculation of the first ten excited states; results are shown in Figure 6, lower plot, and a breakdown of each transition is presented in Table 1. Despite involving similar orbital transitions, the shape of the excited state spectra is different for each complex (see Figure 6, lower plot), and, interestingly, we observe an average decreasing trend in the oscillator strengths with the decrease of the A block size, with the only exception of the first CT transition of PCBM:3A, which is, notably, the brightest one;

Results and Discussion
interestingly, we also observe low oscillator strengths in PCBM:AD. This gradual decrease in the oscillator strength may depend on the amount of charge density over the A block of the oligomers: it has been observed that push-pull copolymers undergo intramolecular charge transfer upon photoabsorption, i.e., the charge gets transferred from the D block to the A block within the polymer. 41,42 Since we decreased the size of the A block (5A > 3A > 1A -AD), it is reasonable to believe that the amount of charge located in the A block will also decrease, effectively lowering the probability of an electronic transition to take place, and consequently switching off the oscillator strength. In order to gain more insight into this matter, we focused on the oligomers alone, and we studied their excited state properties when decoupled from the fullerene domain.
We performed TDDFT on 1A, 3A, 5A, and AD oligomers, requiring the calculation of the S 1 state only. The hole-electron density plots of each oligomer obtained with TDDFT are shown in Figure 8. Results now confirm the previous hypothesis: it is clear that the extent of the electron density correlates with the length of the central A block of the oligomers, as can be observed. For oligomers 3A and 5A the electron density is localized onto the central A block and the hole density is found onto the D units, while, interestingly, despite the presence of a single A unit at the center of the 1A oligomer chain, the electron density is localized exclusively at the edge, above the external A block composed of two A units.
The hole density extends over the center of the chain, that is the part of the oligomer that in our model will mainly interact with the PCBM domain. For oligomer AD, as expected, the hole density is found on the D units, while the electron density on the A units; we note that most probably it is mainly the central A unit that is able to significantly interact with the fullerene domain, and this explains why PCBM:AD and PCBM:1A transitions have a comparable intensity. This finding seems to suggest that the relative size of different blocks in the polymer may be an important factor, since, assuming the size of the blocks not being constant in the real polymer, when the material undergoes photoexcitation the charge seems to get transferred mainly to the A blocks whose size is bigger or comparable to the D blocks.
This is relevant because in the real blend the polymer most likely contains long A blocks rather than long D blocks. We also note that the energy of the S 1 transition is the same for both oligomers 3A and 5A (0.93-0.92 eV), while for 1A the energy is slightly lower (0.86 eV), and, on the contrary, oligomer AD has a slightly higher value (0.98 eV). Overall, these results suggest that the size of the A block does indeed have an influence on the excited state properties of both the polymer itself and the PCBM complex: the length of the blocks in the polymer chain determines where the charge is more likely to get transferred to, and if the PCBM domain does not overlap with the portion of the polymer chain where there is a high charge concentration, this will decrease the probability of a charge transfer transition to be observed. We stress here that even if low oscillator strengths correlate with a low probability of electronic transitions to occur, this does not necessary mean that in the actual device a low efficiency will be observed, since this property does not depend exclusively on the energetics of the molecules involved, but for instance both the morphology and the device architecture are crucial. Nevertheless, these results can provide some guidance on how to enhance the charge transfer rate of push-pull polymer-fullerene blends.

Effect of the size of the domain
Having demonstrated in the previous section that large A blocks are beneficial for charge transfer processes, we chose PCBM:5A complex as the starting model for this final and most relevant part of the paper. From this point onwards the starting model will be called simply P:F (polymer:fullerene), the complexes composed of n oligomer chains (with n = 2, 3, 4) will be called nP:F, and the complex with twice the number of PCBM molecules will be called P:2F (the structures of the five complexes are shown in Figure 9) complexes. P:2F is particularly interesting because the electron density of its CT 1 state extends mostly over the fullerenes which are located far away from the oligomer chain. We estimated the hole-electron separation of this complex by computing the distance between the centroids of the densities, following the procedure described by Le Bahers et al. 43 We obtained a value of ∼ 2.8 nm, much higher than ∼ 1 nm, as estimated by Zhang et al. 44 for short-range CT states. This confirms that CT 1 is a long-range charge transfer state, found in the lowest part of the CT states manifold. This is remarkable, since it is generally believed that CT states with a large electron and hole spatial separation occur at higher energies, and therefore should be regarded as 'hot'. The results here obtained suggest that these states can also have a 'cold' nature. However, the spacing between the excited states of the complexes is really small: for instance, the first three CT states of P:2F all lie within 0.02 eV, a value smaller than the thermal energy at room temperature (0.025 eV), which makes these states equally thermally accessible. Since the electron densities of these three CT states are located on the PCBM molecules which are both close to and far away from the oligomer, this is further confirmation of the delocalized nature of the electron density, and of the possibility of long-range charge transfer processes to occur at lower energies.
Similarly, also the electron densities of the 2P:F, 3P:F, and 4P:F complexes are delocalized over the whole fullerene domain, as well as the hole densities over the polymer domain.
However, it appears that increasing the number of chains leads to a more localized hole over the inner oligomers of the phase. Overall, these results seem to suggest that: (i) in a bulk heterojunction the charge transfer process does not involve only a single polymer chain and a single PCBM molecule; (ii) long-range charge transfer is possible, and it may occur in the lowest part of the CT state spectrum, in competition with short-range charge transfer; (iii) excited states are generally delocalized over the whole fullerene domain, and over more than a single polymer chain. the energies and the intensities of the transitions, as can be observed in Figure 10, lower plot.
The decrease of the transition energies reflects the decrease of the band gap of the complex, as previously discussed, while the decrease of the oscillator strengths can be explained with a reduced coupling between the densities of the fullerene and the polymer domain, the latter progressively moving onto the inner chains of the phase, this causing a poor overlap with the electron density. We also note here that low oscillator strengths are beneficial for the population of CT states, since this process is more likely to happen via internal conversion Although increasing the size of both phases seems to be beneficial, we do note that E CT B presents a stronger decrease when the polymer domain is multiplied, suggesting that a large polymer phase may significantly facilitate the hole-electron splitting rather than a large fullerene phase.

Conclusion
There is still some lack of understanding of the mechanisms that govern the photoinduced charge-carrier formation in bulk heterojunction devices. Due to their complex morphologies, it is not trivial to construct theoretical models that are representative of a real device, one of the main problems being the computational cost associated with density functional calculations on large systems such as polymer-fullerene complexes. By using a linear-scaling reformulation of DFT within the ONETEP code, we were able to overcome the size issue and perform fully ab initio calculations on a number of large-scale polymer-fullerene blends, with the aim of exploring some of the ground-and excited-state properties relevant for OPV functionality. Ground-state properties were investigated by performing DFT calculations, while to study excited-state properties we used time-dependent DFT (TDDFT).
We first explored the effect of different block lengths of the polymer chain when interacting with the fullerene domain, with a focus on the size of the A block. Despite observing no significant changes in the ground-state properties, we did note some differences in the excitedstate spectra of these complexes: we found that the size of the oligomers' A block correlates with the average intensity of the charge transfer states, that is, the larger the extent of the A block, the stronger the oscillator strength. By decoupling the oligomer from the PCBM phase, we were able to observe that the electron density of the oligomers S 1 transition is found exclusively above the A blocks: therefore, large A blocks enhance the coupling between the oligomer electron density and the fullerene crystal, increasing the probability of a charge transfer to occur. By treating the fullerene phase as a close-packed crystal, we were also able to observe a delocalized electron density, suggesting that during the charge transfer more than a single PCBM molecule is able to interact at the same time with the polymer. We also computed the charge transfer exciton binding energies, and we observed that the complex composed of the oligomer with the shortest A block had the highest E CT B (0.71 eV), while the lowest value (0.61 eV) was found for the complex composed of the alternating A-D oligomer; both complexes, however, were characterized by similar and low oscillator strengths. These results suggest that the size of the polymer A block does influence the excited state manifold of a BHJ blend, and that, in order to maximize the probability of a CT transition to occur, a polymer with large acceptor blocks may be beneficial.
We then examined the effect of both the fullerene and the polymer domains size, by selectively increasing the size of one phase with respect to the other. When we doubled the fullerene phase (from 12 to 24 PCBM molecules) we could not observe any significant difference in the energy range of the CT states with respect to our starting model; we did observe, however, an increase in the number of CT states, due to the higher number of molecules in the complex. The electron densities of the CT transitions all have a strongly delocalized character, involving fullerenes both close to and far away from the oligomer chain; some of the lowest states (for instance, the first one, CT 1 ) are characterized by an electron density found almost exclusively on the PCBM molecules positioned at the upper edge of the complex, with a hole-electron separation of ∼ 2.8 nm; this suggests that long-range charge transfer states are possible, and that they can be observed even at the lowest energies.
These long-range CT states most probably coexist with short-range states, since the energy difference of both CT 2 and CT 3 with respect to CT 1 is less than the thermal energy at room temperature. We then progressively increased the number of oligomer chains (from 1 to 4) while keeping the fullerene phase constant, and we observed a decrease of the CT state energies, which correlates with the number of oligomer chains. Lower oscillator strengths also correlate with a higher number of chains, and this can be explained by examining the hole densities of the complexes, which, like the electron densities, show a delocalized character over more than a single polymer: the higher the number of chains and the more the hole density moves into the inner chains of the polymer phase, reducing the coupling with the fullerene phase. We finally computed the charge transfer exciton binding energies and we observed that increasing the polymer phase significantly lowers the value of E CT B , while increasing the PCBM domain has a less strong effect on E CT B , despite also causing a lowering of the value.
Clearly, given the complexity of the problem, only a limited number of questions were addressed here; nevertheless, we expect the results obtained in this paper to be helpful for the design of high-performance OPV devices, as we were able to shed light into some of the charge generation mechanisms of a polymer-fullerene BHJ.