Density Functional Study of Stacking Structures and Electronic

Article pubs.acs.org/JPCB

Density Functional Study of Stacking Structures and Electronic Behaviors of AnE-PV Copolymer Chuan-Ding Dong and Wichard J. D. Beenken* Institut für Physik and Institut für Mikro- und Nanotechnologie, Technische Universität Ilmenau, 98693 Ilmenau, Germany ABSTRACT: In this work, we report an in-depth investigation on the π-stacking and interdigitating structures of poly(p-anthracene-ethynylene)-alt-poly(p-phenylene-vinylene) copolymer with octyl and ethylhexyl side chains and the resulting electronic band structures using density functional theory calculations. We found that in the π-stacking direction, the preferred stacking structure, determined by the steric effect of the branched ethyl-hexyl side chains, is featured by the anthracene-ethynylene units stacking on the phenylene-vinylene units of the neighboring chains and vice versa. This stacking structure, combined with the interdigitating structure where the branched side chains of the laterally neighboring chains are isolated, defines the energetically favorable structure of the ordered copolymer phase, which provides a good compromise between light absorption and charge-carrier transport.

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substituted to the remaining phenylenes.11 This structure shown in Figure 1 has been named AnE-PV ab.9 For the reverse substitution, that is, branched 2-ethyl-hexyl side chains substituted to the phenylenes next to the anthracene units and linear otcyl side chains substituted to the remaining phenylenes, which is named AnE-PV ba, only amorphous phases have been found.9 Being freely intermixable with each other and being

INTRODUCTION In organic photovoltaic cells based on conjugated polymers, the structural order of the polymer phase plays an important role.1−4 It has been found in a series of studies that the formation of an ordered or semicrystalline phase of the conjugated polymer significantly improves the charge generation yield. For example, Herrmann et al.1 observed that photoexcitation results in a higher charge generation rate in the ordered subphases of poly(3-hexylthiophene)(P3HT) than in the amorphous P3HT. Similarly, Howard et al.5 claimed that the key to understanding the ultrafast free charge carrier generation in P3HT:fullerene blends is the ordered structure of annealed rr-P3HT rather than the excess energy of the exciton. The enhancement of charge-carrier generation in ordered polymer phases is generally attributed to charge delocalization. Wang et al.6 demonstrated that the delocalization of exciton in ordered P3HT phases facilitates direct dissociation at the interface. Schwarz et al.3,7 found in an investigation of poly(pphenylene):C60 system that the electric field required to sustain the charge-carrier generation decreases with an increasing conjugation length of the chromophore. Moreover, it has been proposed that charge carriers are stabilized against recombination by delocalization.4,8 Although an annealing process often improves crystallinity in polymer thin films,5 the formation of ordered phases relies mainly on the structure of the polymer. To this end, the configurations of side chains substituted to the polymer backbone play a crucial role. A good model for this dependency in the framework of conjugated polymers, which may be used for solar cells, is the recently emerging poly(panthracene-ethynylene)-alt-poly(p-phenylene-vinylene) (AnEPV) copolymer,9−12 which forms a quite ordered phase for linear otcyl side chains substituted to the phenylenes next to the anthracene and branched 2-ethyl-hexyl side chains © 2016 American Chemical Society

Figure 1. Atomic structure of AnE-PV ab copolymer. The dashed frame marks the phenylene unit with the branched side chains b. Red balls stand for O, green for H, and blue for C atoms. Received: July 29, 2016 Revised: September 28, 2016 Published: September 28, 2016 10854

DOI: 10.1021/acs.jpcb.6b07657 J. Phys. Chem. B 2016, 120, 10854−10859

Article

The Journal of Physical Chemistry B able to form blends with fullerene derivates like PCBM, which represents quite efficient bulk-heterojunctions as active layers,10,12 the AnE-PV copolymer is an ideal model system for studying order/disorder issues in organic solar cells. It has been shown that the absorption and the electroluminescence spectra depend significantly on the ratio between AnE-PV ab and AnE-PV ba content.10 However, in a previous density functional study13 on single AnE-PV ab and AnE-PV ba chains, we showed that this is not an effect of the intrachain order, that is, a good planarity of the conjugated AnE-PV ab backbone enhancing the conjugation length and the exciton delocalization, but that this must be an interchain effect, that is, the formation of stacks of AnE-PV ab chains by means of π−πstacking. For AnE-PV ba, such stacking is hindered because of its lower planarity. The existence of AnE-PV ab stacks has been proved by grazing incidence wide-angle X-ray scattering (GIWAXS) experiments,11,12 which have shown that the stacks of AnEPV ab consist of layers with approximately 3.8 Å interlayer distance, and the lateral distance, that is, perpendicular to the stacking and backbone directions, between the conjugated backbones is around 18.0 Å.11 However, these experiments could not reveal the detailed structure, which concerns a shearing between the layers in the direction of the conjugated backbones as well as the details of the interdigitation of the side chains within the layers. Therefore, it is interesting to study these aspects by density functional theory (DFT) calculations. The fundamental importance of shearing and interdigitation for the electronic band structure of conjugated polymer crystallites has been already shown for other polymers, for example, P3HT14−17and PTB7,18 as well as for conjugated systems in general.19,20 In our case, besides the known distance between the layers, it is in particular the relative positions of the anthracene units on neighboring chains, that is, stacking just above each other or on ethynylene, phenylene, or vinylene units, that determine the overlap between the π-electron systems and, consequently, the band structures, the optical transitions, and the interchain charge transport properties along the π-stacking direction. How the conformation and the interdigitation of the side chains influence the shearing and the interlayer distance will be the subject of the first part of this study, which is followed by the calculation and discussion of the resulting band structures and optical spectra for some local minimum stacking structures.

Figure 2. Side view of two monoclinic unit cells used to represent the stacking of single AnE-PVab chains. The π-stacking distance d is fixed to 3.8 Å, and the different stacking positions are characterized by the shear between neighboring chains.

= 27.0 Å. On the other hand, the shearing in Y direction tends to render the backbone stacking on the side chains of the next layer and vice versa and is in contradiction to the X-ray diffraction (XRD) measurement indication that backbones are stacking on each other (Figure 3 of ref 11). Moreover, with a Y direction shear of around 2.0 Å, the partially stacking backbones cause significantly higher energies according to our calculations. Therefore, in this work, we confined ourselves in the discussion of X direction shearing. All calculations were performed by the VASP package21 using the plane-wave and PAW potential method with the PBE functional to approximate the exchangecorrelation22 and Grimme’s D2 correction for the van der Waals interaction.23 A large energy cutoff of 520 eV was used to eliminate the Pulay stress caused by the varying unit cell geometry. Given the dimension of the unit cell described earlier, a 2 × 2 × 8 k-point grid was used to homogeneously sample the Brillouin zone. Figure 4 shows the variation of the energies with shear. Most prominent is that the face-to-face conformation, that is, the

Figure 3. Two different orientations for the ethyl branches (ball-andstick model) of the ethyl-hexyl side chains.

orthorhombic unit cell with β = 90° and c = d = 3.8 Å, is unstable anyway. This instability is because that carbon atoms in aromats prefer not stacking uprightly on each other but sitting either above the center of an aromatic hexagon or above the position between two carbons of the lower layer, which is well-known from, for example, the structure of graphite. For the same reason, the total energy of the stack is oscillating with the shear, which is clearly seen for the AnE-PV without side chains. Including the side chains, for AnE-PV ab, this oscillation is persistent but is superimposed by the effect of the side chains, which results from the fact that the ethyl branches of the ethylhexyl side chains even in the in-plane conformation (see Figure 3) have to find gaps between the side chain row of the overlaid copolymer chain. At the shear of ±13.0 Å, the phenylene unit with the ethyl-hexyl side chains lie just below the anthracene unit and have enough space for steric relaxation, which leads to the two deep minima in energy. The shear −13.5 Å is equivalent with 13.5 Å since a = 27.0 Å. See also Figure 5D and Figure 7 for the structural details of 13.0 Å shear. In a second run for the interdigitating structures, we considered two different patterns of interdigitation (see Figure 5). (1) The first pattern is the interdigitation where the

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STACKING STRUCTURE CALCULATIONS In a first run, for the determination of the shear between the stacking layers, we defined a monoclinic unit cell with a rectangular base of a = 27.0 Å, b = 35.0 Å, and a height d = 3.8 Å. Note that only the height d was fixed while the lattice constant c and the angle β of the monoclinic lattice (α = γ = 90°) were varied for the shearing (see Figure 2). The value for d was taken from the crystallographic data11,12 and the value of a from the length of the repetitive AnE-PV unit. For the faceto-face structure, the shear is 0.0 Å, and we get a unit cell with c = d and β = 90°. In this first run, the value of b was taken much larger than the experimental interchain distance 18.0 Å in order to avoid interdigitation between the side chains. Thus, keeping a, b, and d constant, we sheared the unit cell along the X direction from −13.5 Å to 13.5 Å with a step of 0.5 Å and performed DFT calculations for AnE-PV chains with and without ab side chain substitution at each step. The range from −13.5 Å to 13.5 covers the whole length of the AnE-PV unit a 10855


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The Journal of Physical Chemistry B

interdigitation. For this pattern, we used a monoclinic unit cell (α = γ = 90°) with a = 27.0 Å and b = 18.0 Å, the latter corresponding to the crystallographic data11,12 of the lateral interchain distance. Note that AnE-to-AnE interdigitation is not sterically compatible with the in-plane orientation of the ethyl side chain branches. (2) The second pattern is the interdigitation where the anthracene units are neighbored by vinylene units of the next chain, and the branched side chains from neighboring chains are isolated from each other, which we called AnE-to-PV interdigitation. For this pattern, a triclinic unit cell with a = 27.0 Å, b = 19.1 Å, α = 70°, and γ = 71° is used, which also guarantees the lateral interchain distance 18.0 Å within the layer. For both of these unit cells, the values of c and β depend on the interlayer shearing for which we considered only three cases: 0.0 (face-to-face stacking), 3.5, and 13.0 Å. Within these unit cells, we optimized the π-stacking distance d. In the optimization for each stacking and interdigitating structure, the orientations of the side chain branches which are sterically not allowed or compatible were ruled out. This results in the four stacking and intercalating structures that are shown in Figure 5 in 2 × 2 × 2 supercells. The results of the optimization using one unit cell are shown in Figure 6. The calculations showed that only those structures with AnEto-PV interdigitation have energetic minima with respect to the interlayer distance d around 3.8 Å. The face-to-face structure

Figure 4. Total energies of optimized stacking single AnE-PV chains with and without side chains as functions of the shear between neighboring chains. The π-stacking distance is fixed to 3.8 Å. For the complete AnE-PV ab copolymer, the ethyl side chain branches are oriented in-plane. The dashed lines mark the stacking positions for which electronic properties are studied (Figure 8 and Figure 9).

anthracene units are neighboring anthracene units of the next chain, and the branched side chains from neighboring chains are next to each other, which we called AnE-to-AnE

Figure 5. Illustration of the stacking and interdigitating structures of AnE-PVab using 2 × 2 × 2 supercells and stick model. The red sticks stand for O atoms and blue for C. H atoms are not shown for clarity. Thick sticks stand for the upper layer and thin sticks for the lower layer. The detailed features of the four structures are given in the table. 10856


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Figure 7. Backbones of neighboring AnE-PV chains in different stacking positions. Blue stands for the backbone above and gray for the backbone below.

show only the calculated band structures and the dielectric matrices for the positive shears mentioned earlier. As shown in the first panel of Figure 8, for face-to-face stacking (0.0 Å shear), the valence band maximum is located at Γ point and the conduction band minimum at Z point, which renders an indirect band gap of 0.28 eV, whereas the vertical energy gaps at Γ and Z points are 1.08 and 1.00 eV, respectively. The calculated absorption spectrum of face-to-face stack (see Figure 9) is quite similar to that of a single chain except for one weak aggregation band at 1.3 eV. Because the joint density of states is quite high for the parallel conduction and valence bands, it is the approximate 2/m symmetry of the face-to-face stack with its inversion center that causes the quenching of this optical aggregation band. Notably, both the valence bands and the conduction bands exhibit steep slopes in the π-stacking direction, that is, from Γ to Z as well as from N to X. From their curvatures, one may deduce a low effective mass, which is beneficial for charge-carrier mobility. On a first sight, the band structures for stacks with shear of 1.5, 3.5, and 8.0 Å are quite similar to each other but dramatically different from that of the face-to-face stack. In detail, however, they reveal an interesting tendency in respect to their band gaps. For the 1.5 Å shear, the minimum band gap of 0.85 eV is located at the Z point, which, together with the band gap at Γ point, results in the double-peak structure at around 1.0 eV in the absorption spectrum (see Figure 9). For 8.0 Å shear, the minimum band gap of 0.97 eV is now located at Γ point and is quite close to the gap of 1.08 eV at Z point. The band structure for 3.5 Å shear represents the intermediate case with even more close band gaps of 1.04 eV at the Γ point and 1.01 eV at the Z point. For these two spectra, the peaks around 1.0 eV are actually two peaks merging with each other. Notably, in this case, the dispersion of the conduction bands and valence bands along the π-stacking direction, especially from Γ to Z, is quite weak, which indicates high effective masses in this direction and, consequently, low mobilities for both charge carriers, electrons, and holes. On the other hand, the parallel conduction and valence band result in a peaking joint density of states, which gives rise to the peak at 1.0 eV in the absorption spectra as shown in Figure 9. In spite of keeping the tendency already seen for shear from 1.5 to8.0 Å, the band structure of the energetically most favorable stack with 13.0 Å shear is again different from those mentioned earlier. With the band structure of face-to-face stack, it shares the indirect band gap between Γ and Z point (0.65 eV) and the comparably steep dispersions of valence band and conduction band between these points, indicating low effective masses of charge carriers along the π-stacking direction. Nevertheless, like the other three sheared stacks, the

Figure 6. Calculated total energies as functions of the π-stacking distance d for the structures described in Figure 5 but only for one unit cell. Structure B’ (not shown in Figure 5) is the same as structure B but with out-of-plane side chain branches.

(A) has a shallow minimum between d = 3.8 and 4.0 Å. Stacks with 3.5 Å shear, as shown in Figure 6, can be formed for ethyl branches either in-plane (B) or out-of-plane (B’), both having the minima at d = 3.8 Å, though the in-plane side chain branch is more preferred than the out-of-plane branch. For comparison, we have also shown the energy variation for the AnE-to-AnE interdigitation with 3.5 Å shear (Figure 5 C), which is unstable and which does not form well-defined stacks. Importantly, the most stable stacking structure is that with AnE-to-PV interdigitation, in-plane ethyl branches, 13.0 Å shear, and the optimized interlayer distance of d = 3.8 Å (Figure 5 D). This could be the main structure of the semicrystalline phase of AnE-PV ab, although it cannot be excluded that some layers are stacked according to the structures with different shear like B or B’, which all agree in interlayer distance of 3.8 Å with the crystallographic data.11,12 Without interdigitation, the optimum interlayer distance shrinks to d = 3.4 Å, which is significantly smaller than the experimental value.

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BAND STRUCTURE CALCULATIONS To demonstrate the effect of different stacking structures on the electronic properties of AnE-PV ab crystals, we have calculated the band structures and the dielectric matrices for the structures with shear 0.0, 1.5, 3.5, 8.0, and 13.0 Å. These sheared stacks are the minimum positions in the optimization calculations with fixed d = 3.8 Å (Figure 4). The dielectric matrices, the imaginary part of which, ε(2), represents the absorption spectrum without excitonic effects, were calculated without including local field effect.24 Because the side chains have only little effect on the electronic properties of the copolymer chain and because the interdigitation does not cause electronic coupling between the chain neighboring in the Y direction, for the electronic structure calculations in this section, the side chains are cut, as shown in Figure 7, to reduce the computational effort. This structural approximation is justified by both our test calculations and literature.25,26 Moreover, our calculations show that the minimum structures with respect to the shearing in the opposite direction, such as shear of −8.0 Å and −13.0 Å, have very similar electronic behaviors as their counterparts with positive shears. Therefore, in this paper, we 10857


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Figure 8. Band structures of stacked AnE-PV backbones with face-to-face stacking and the stacking structures shown in Figure 7. Blue marks the two highest valence bands and red marks the two lowest conduction bands. The dashed lines mark the positions of the high-symmetry k points used. N = (π/a, 0, π/c); X = (π/a, 0, 0); Γ = (0, 0, 0); Z = (0, 0, π/c).

stacking structure the central anthracene ring is exactly sandwiched between the phenyl rings in the neighboring layers (see Figure 7 13.0 Å). Nevertheless, this stacking rule still shows its effect, especially for some configurations representing energetic local minima and may result in stacking failures. Notably, only the band structure of the optimal stacking structure promises a reasonable compromise between sufficient light absorption in the relevant range of the sun spectrum and high charge carrier mobility along the π-stacking direction, which may explain the better performance of AnE-PVab than that of the amorphous AnE-PV ba copolymer in solar cells.9 For the subtle interplay of π-stacking and interdigitation, we may predict that the analogous poly(p-pentacene-ethynylene)alt-poly(p-phenylene-vinylene) copolymer with octyl and ethylhexyl side chains in the ab substitution scheme may fail for efficient solar cells, since stacking is hindered by the ethybranches of the side chains, which cannot find a gap next to the space-filling pentacene units in the layer above or below.

Figure 9. Natural logarithm of the calculated imaginary part of dielectric matrices of stacked AnE-PV backbones with face-to-face stacking and the stacking structures shown in Figure 7. Local field effects are neglected.

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corresponding calculated imaginary parts of the dielectric matrices (Figure 9) possess a lower but still significant peak at 1.1 eV. This absorption is in the relevant spectral range for efficient photovoltaic devices.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49 3677 69 3258. Fax: +49 3677 69 3271.

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Notes

The authors declare no competing financial interest.

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CONCLUSION We have shown that the AnE-PV ab copolymer forms stacks favorably in such a way, as shown in Figure 5 D, that the anthracene units are stacked over the phenylene units and vice versa in the π-stacking direction and are neighbors also to the phenylene units in interdigitation, which are denoted by 13.0 Å shear and AnE-to-PV interdigitation in this work. The reason is that the ethyl-hexyl side chains, even in the in-plane conformation, can find the most space for steric relaxation only beside the anthracenes which have no side chains attached. The π-stacking rule that carbon atoms of the aromats should sit on top of the interstitial sites of the neighboring layers is overwhelmed by this steric hindrance, since in the optimal

ACKNOWLEDGMENTS This work is financially supported by the DFG research grant “PhotogenOrder”. We would also like to show our gratitude to H. Schwanbeck from the computing center of the Technische Universität Ilmenau for quick and professional IT support.

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