Hopping-Based Charge Transfer in Diketopyrrolopyrrole-Based Donor

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Hopping Based Charge Transfer in Diketopyrrolopyrrole Based Donor-Acceptor Polymers: A Theoretical Study Florian Steffen Günther, Sibylle Gemming, and Gotthard Seifert J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b01310 • Publication Date (Web): 19 Apr 2016 Downloaded from http://pubs.acs.org on April 23, 2016

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The Journal of Physical Chemistry

Hopping Based Charge Transfer in Diketopyrrolopyrrole Based Donor-Acceptor Polymers: A Theoretical Study Florian Günther,†,‡,¶ Sibylle Gemming,†,§,¶ and Gotthard Seifert∗,‡,¶ †Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01314 Dresden, Germany ‡Department Chemie, Technical University Dresden, 01062 Dresden, Germany ¶Center for Advancing Electronics Dresden, Technical University Dresden, 01062 Dresden, Germany §Faculty of Natural Sciences, Institute of Physics, Technical University Chemnitz, 09126 Chemnitz, Germany E-mail: [email protected] Phone: (+49) (351) 463 37637

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Abstract Diketopyrrolopyrrole (DPP) based materials have recently been considered as promising candidates for novel organic electronics. In this article, we report an investigation on intermolecular charge transfer between DPP based polymers. We use Marcus transfer theory and evaluate the required quantities, the reorganization energy and the coupling, by density functional-based tight binding (DFTB) calculations. Since the coupling is dependent on the stacking geometry we employ an energy-weighted statistical approach to derive a single quantity, which can been entered in the Marcus formula. This value contains the variation of the coupling when the stacking conformation is changed. The application of this method, as we implement it in this study, does not require a detailed analysis of the energy landscape, but samples over large number of stacking possibilities on a regular, but very dense grid into account. These average values can been used to analyze isomeric effects such as the orientation of units, the influence of the molecular structure as functionalization, or the importance of stacking properties as parallel and anti-parallel stacking. The obtained results show that enhanced charge carrier mobilities can be achieved when specific molecular configurations are considered rather than by working with a set of random orientations.

Abbreviations DFT,DFTB,CT, DA, DPP

Introduction Diketopyrrolopyrrole (DPP) and its derivatives have been known for several decades, mainly as pigments and dyes. 1,2 Recently these materials have attracted a lot of attention, since


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both organic crystals 3 as well as polymers 4,5 based on this compound are highly promising candidates for novel materials for organic electronics, e.g. as solar cells 3 (OSCs) or fieldeffect transistors 6 (OFETs). Polymeric systems yield hole mobilities up to 10 cm2 V−1 s−1 and electron mobilities up to 6 cm2 V−1 s−1 , which are the current record values for organic systems. 6–8 These values are still up to three orders of magnitude lower than for common rigid bulk semiconductors. Nevertheless, these systems provide the possibility of new applications as e-papers or transparent and flexible devices. In this study, the charge transport processes in DPP based polymers are investigated on an atomistic scale. The general molecular structure is depicted in Fig. 1a) and Fig. 1b). Two thiophene rings are symmetrically attached to the DPP unit. In the following, this TDPP-T part of the molecule is referred to as acceptor unit according to its electron-deficient nature. This acceptor unit is attached to an electron-rich unit, the donor unit, for which we consider different thiophene based compounds as shown in Fig. 1c). As they are consist of electron-deficient and electron-rich regions, these systems are called donor-acceptor (DA) polymers. Note, that for all considered structures there are two planar skeletal isomers concerning the orientation of the thiophene rings of the acceptor unit: The sulphur atom of the thiophene ring can point towards the nitrogen atom of the DPP unit, Fig. 1a), or away from it, Fig. 1b). The former one we refer to as cis configuration and the latter one is called trans configuration, respectively.



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π-acceptor

π-acceptor R

R

O

O

N

N

S

S N

O

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R

S

Ar

π-donor

O

b)

TT

c)

N

R

n

a)

S

Ar

π-donor n

T-T

T-E-T

T

Figure 1: (a) General structure in cis conformation: DPP with two thiophene rings symmetrically attached form the acceptor unit. The donor unit Ar connects these thiophene rings of successive units. (b) trans conformation of the thiophene rings. (c) Systems considered for the donor unit: thienothiophene (TT), bithiophene (T-T), 1,2di(2-thienyl)ethylene (T-E-T), and single thiophene (T).

Commonly, long alkyl side chains are attached to the nitrogen atoms of the DPP unit which make the polymer soluble in organic solvents. Those chains, however, do not affect the electronic structure properties we are interested in. Thus, for simplicity methyl groups are substituted for this part. A thin film consisting of DA polymers tends to form so-called π-π stacking regions, where adjacent chains are aligned in a parallel manner. 5,6 In π-conjugated systems, the transport along the polymer backbone is about 10 times larger than the intermolecular charge transport. 9,10 For the charge transport through the whole device, the latter one is of crucial importance to overcome impurities in the π-conjugation which appear due to the finiteness of the chains. Thus, the CT between adjacent chains is the bottleneck for the charge transport through the whole system. For this reason, we only focus on the hopping based charge transfer (CT) between adjacent polymer chains. To this end, we are using Marcus theory of 4 ACS Paragon Plus Environment

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CT 11 where the hopping rate between equivalent sites is given by

tAB

|HAB |2 = ~

r

π λ exp − . λkB T 4kB T

(1)

Here, λ is the reorganization energy and HAB the electronic coupling element. The letters A and B represent the interacting sites. The Plank and the Boltzmann constants and the temperature are denoted as ~, kB , and T , respectively. In order to determine the hopping rate using the Marcus formula, the two quantities λ and HAB need to be calculated. The reorganization energy λ is a physical quantity which indicates the structural differences between neutral and charged states. Since it arises from physical observables it can simply be calculated via simulation methods that provides the total energy of a given structure, for example density functional theory (DFT). The coupling element HAB , however, as it appears in Fermi’s golden rule 12 is based on the quantum mechanical wave function. The physical meaning and the reliability of the single-particle wave functions which arise from DFT based methods, the Kohn-Sham wave functions, are often controversially discussed. For some recent discussions, see Ref. 13. A. Kubas et al., 14,15 however, have benchmarked the use of different DFT methods for the calculation of HAB against high-level ab initio calculations. Their study includes dimers of organic systems such as thiophene or pyrrol which are the compounds of the DA polymers discussed here. They have shown that the considered methods provide good estimates of the coupling HAB and give the correct trend for these organic molecules. Since the coupling HAB depends on the two sites A and B, it also depends on their relative arrangement. Due to this, several relevant orientations should be considered rather than just a single structure. The associated energy landscape of conjugated polymer chains which is governed by the π-π interaction, however, may be characterized by several local minima. Thus, we suggest a statistical approach averaging over a large number of configurations to calculate a more reliable average value for HAB . This, however, implies the need for 5 ACS Paragon Plus Environment


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a computationally efficient method to calculate HAB . The density functional based tight binding (DFTB) method is well suited for this purpose. 14,16,17 This article is organized as follows: First, the theoretical derivation of the parameters for the Marcus approach and the computational details are briefly introduced. Second, the transport properties obtained are presented and discussed relating them to the molecular structure. Here, we also focus on different stacking possibilities when the coupling HAB is evaluated. Finally, we close with a summary highlighting the most important conclusions.

Methods Theoretical Background In fabricated OSCs and OFETs, the charge transport in semiconducting layers will be influenced a lot by effects as fabrication process or device layout. In this study, the main influence factors are considered, which intuitively affect the parts directly involved in the CT. Neglecting the influence of the adjacent chains and the remaining environment, the reorganization energy λ can be determined by considering the repeat unit. The reorganization energy is given via h i h i ~ c ) − E n (R ~ n ) + Ec (R ~ n ) − Ec (R ~ c ) = λn + λc , λ = En (R

(2)

~ n and R ~ c are the equilibrium geometries for the neutral and the charged sites, where R respectively. The energies En and Ec refer to the total energy in these specific geometries when the system is either neutral or charged. To this end, a positive or a negative charge is considered for hole or electron transport, respectively. Note that λ consists of two terms: λn denotes the energy for bringing the neutral system to the geometry of the charged one and λc describing the opposite case. Since for the CT both processes take place simultaneously, 6 ACS Paragon Plus Environment

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it is intuitive that λ is the sum of these two terms. In contrast to the evaluation of λ, the coupling element HAB cannot be derived from the calculation of single sites. Its value is dependent on the stacking geometry of neighboring sites. For a given orientation, HAB can be calculated via D E ˆ AB |Ψ(B) , HAB = Ψ(A) |H

(3)

where Ψ(A) and Ψ(B) are the many-body wave functions where the charge, which is ˆ AB is the transported through the system, is either localized at A or B, respectively, and H Hamiltonian describing the electronic coupling between these two states. Applying a fragment orbital (FO) approach where Ψ(A) and Ψ(B) are expressed by Slater E E (B) (A) of the isolated fragments A and and φj determinants composed of the orbitals φi B, and applying Slater-Condon-rules 18,19 and orthogonality arguments for the fragment orbitals, 14 Eq. (3) can be transformed for hole transfer to

hole HAB

=

D

(A) ˆ AB |φ(B) φHOMO |H HOMO

E

,

(4)

and for electron transfer to

electron HAB

=

D

(A) ˆ AB |φ(B) φLUMO |H LUMO

E

,

(5)

depending on the charge considered. These orthogonality arguments, however, correspond to the approximations when utilizing an orthogonalized basis. Alternatively, the value approximated in a non-orthogonal basis can be corrected via a Löwdin-like transformation: 20 ˜ AB = HAB − SAB HAA + HBB . H 2

(6)

In this equation, SAB = φ(A) |φ(B) is the overlap of the corresponding orbitals and HAA 7 ACS Paragon Plus Environment


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and HBB are the associated on-site energies. As Eqs. (4) and (5) show, HAB depends on the spatial form of the highest occupied molecular orbitals (HOMOs) and lowest unoccupied molecular orbitals (LUMOs), and the orientation of the sites relative to each other. In this work, we introduce a Boltzmann-type statistical approach for averaging over all significant configurations. This enables the calculation of a thermodynamically averaged value of HAB which is much more representative for the system than the value of HAB for an arbitrary configuration. Since the coupling enters the Marcus formula in squared form, 2 : we evaluate the average value of HAB

R 2 HAB =

~ ~ E(R)−E loc (R) ˜ 2 (R)exp ~ ~ H − dR AB kB T , R ~ ~ E(R)−E loc (R) ~ exp − dR kB T

(7)

~ stands for an arbitrary stacking geometry and Eloc (R) ~ is the energy of In this equation, R the “next local minimum” of the corresponding energy landscape. In this context, we refer to such a minimum where a steepest descent optimization ends as “next local minimum”. Compared to the standard Boltzmann distribution, we substitute Eloc for the energy of the global minimum. This ansatz can be justified for the following reasons: The interaction energy of stacked chains exhibits several local minima separated by barriers > 0.3 eV per interacting unit. While fabricating OSCs or OFETs, it is rather unlikely, that all polymers will align in the most stable conformation. Overcoming the barriers by thermal diffusion is improbable, since conjugation lengths of several hundred repeat units per polymer are common. Furthermore, the long alkyl side chains may affect the diffusion as well. So, the individual chains will equilibrate to the next rather than to the global minimum. Thus, the approach reflects a fully percolated polymer film composed of subregions that are in local thermodynamic equilibrium. The statistical approach in Eq. (7) provides two sampling strategies: On the one hand, only 8 ACS Paragon Plus Environment

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configurations next to local minima can be considered, since those contribute significantly to the average value. On the other hand, a large number of configurations can be calculated including those which will not affect the average value much. The first one requires less computing effort, given that all stable configurations are already known. If this is not the case, however, they need to be determined at first followed by the calculation of the couplings for configurations next to this local minima. The energy landscape, however, may be intricate, such that for its evaluation a lot of configurations have to be considered. For this, a sampling on a regular but very dense grid is the intuitive choice to ensure the consideration of all important regions. In case of the second strategy, the couplings HAB are simultaneously calculated for all configurations. Since the contribution of configurations with high energy compared to their next local minimum is small, the result is almost the same as in the first case, but the sampling is more complete. The advantage of the second approach is that a detailed analysis of the exact positions of the local minima is not required. Thus, this version is much easier to apply to an arbitrary system. The additional computational effort can be compensated by performing the calculations in a parallel manner which can easily be achieved, since the evaluation of one configuration is independent from all the others. Thus, we focus on the second approach in this study.

Computational Details In our study, we performed self-consistent charge DFTB simulations 16,17 for the structural properties and the electronic structure utilizing the program package DFTB+. 21 The calculation of the coupling using Eqs. (4) and (5) as well as the determination of the average value following Eq. (7) have been performed by our own subroutines. Our DFTB calculations have been performed using the following parameter set. Mainly, the standard parameters were applied, which were obtained using a contraction basis. 17,22 Only ˆ AB a different parameter set was for the evaluation of the coupling Hamiltonian matrix H 9 ACS Paragon Plus Environment


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used. It was derived from a basis set of uncontracted atomic orbitals, reflecting better the long range part of the wave function. This ensures that the interaction of the two neighboring fragments is fully covered. The geometry has been obtained by starting with optimizing the monomeric systems saturated by hydrogen atoms. Then, periodic boundary conditions were introduced to describe infinitely long polymers. In order to determine the optimal cell size, optimizations for different unit lengths have been performed. For the evaluation of the reorganization energy λ we considered the monomeric systems instead of the infinite polymer. Firstly, λ is a local quantity such that the influence of adjacent units is small. Secondly, the delocalization effect which arises as an artefact of Kohn-Sham approaches is partially suppressed when the finite system is considered. Lastly, calculations of charged systems with periodic boundary conditions are in principle impossible without computational artefacts. Thus, the monomer model should be sufficient for our purpose. To calculate HAB the following strategy was used: For all systems, two repeat units per periodic cell have been considered, such that the number of nodal planes along the backbone is even. Doing so, the parity of wave functions of the frontier orbitals are the same at the boundary of the unit cell. This allows considering the Γ-point only. First, the HOMO and the LUMO wave functions, |φHOMO i and |φLUMO i, are derived for isolated chains. Second, the ˆ AB is calculated for a pair of two parallel π-π stacked molecules. For this coupling matrix H calculation the parameter set derived from the uncontracted basis has been used. Third, the interaction energy has again been calculated using the standard parameters. 17 Furthermore, we used the universal force field 23,24 for dispersion correction. With this, we mainly restrict the interaction energy to the van-der-Waals interaction. Since we would like to describe the polymer in a regular stacking, we utilize a periodic [AB] stacking instead of two isolated chains. In doing this, we ensure a more correct energy weight because the interaction to


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both neighboring chains is included. As already mentioned, we utilize Eq. (7) using a large number of stacking configurations. Since the geometry of the individual chains is considered to be constant and only parallel π-π stacked chains are considered, three degrees of freedom remain: first, the shift along the backbone which we refer to as x direction, second, the shift perpendicular to x but with constant π-π stacking distance (y direction), and last, the direction of the π stacking (z direction). Because of the symmetry of the considered systems, only a relative shift up to the half length of the repeat unit in x direction needs to be considered. In our study, we used a grid of 51 equidistant points in x direction in the range of 0% to 50% of the unit length. In y and z direction, a grid was considered which contains 41 equidistant points in each of both directions. The distance of the two chains was chosen in the range between 3.2 Å and 3.7 Å. The shift in y direction was varied in the range of ±10 Å. These values turned out to reproduce a smooth landscape of the interaction energy of the two chains. The boundaries were chosen to ensure that no local minimum is out of this range. In total, over 85000 different stacking configurations are considered for each system.

Results and Discussion In Table 1, the results for all considered systems are listed. Substituting thiophene (T), thienothiophene (TT), bithiophene (T-T) or 1,2-di(2-thienyl)ethylene (T-E-T) for the donor unit leads to different lengths of the unit cell. The columns in Table 1 are ordered according to increasing length of the units from left to right. The influence of the variation of the donor unit on the reorganization energy is discussed first. Comparing the values for the cis and trans configurations, the difference is always smaller than 5 meV per repeat unit for all the considered systems. Since this is within the error range of the applied methods, we conclude that there is no strong dependence of λ on



q 2 Table 1: Reorganization energy λ and average value HAB as well as resulting charge transfer rate t and charge carrier mobility for the different DPP based DA polymers considering thiophene (T), thienothiophene (TT), bithiophene (TT) or 1,2-Di(2-thienyl)ethylene (T-E-T) for the donor unit. The columns are ordered according to the length of the individual repeat units. For each system the results for the cis and trans isomers in parallel (P) as well as in anti-parallel (A) stacking conformations are listed. In each row, the largest and smallest values for hole and electron transport are highlighted in bold. aryl group

t (1013 s−1 )

q

2 HAB (meV)

λ (meV)

charge type

T

TT

T-T

T-E-T

hole

electron

hole

electron

hole

electron

hole

electron

trans

207

177

189

170

174

163

160

161

cis

209

180

192

175

177

168

163

166

119

55.1

133

107

108

59.7

132

64.3 97.3

75.5

92.3

81.5

102

72.3

97.8

92.3

127

77.8

78.5

106

74.6

113

122

86.8

96.1

103

8.7

2.4

13.1

9.8

10.4

3.1

5.8

4.5

6.4

5.7

9.3

4.6

5.7

6.3

11.6

4.9

5.3

9.3

3.3

9.5

10.7

6.1

7.9

8.8

4.4

1.2

6.6

4.9

5.2

1.6

2.9

2.2

3.2

2.9

4.7

2.3

2.9

3.2

5.8

2.5

2.7

4.6

1.7

4.8

5.4

3.1

4.0

4.4

2.9

2.8

5.2

3.3

4.1

3.2

trans/P trans/A cis/P 75.0

90.3

cis/A trans/P 8.7

3.0

trans/A cis/P 2.7

5.6

cis/A µ (cm2 V−1 s−1 )

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trans/P 4.4

1.5

trans/A cis/P 1.4

2.8

cis/A

µ ¯ (cm2 V−1 s−1 )

2.9

2.2

this conformational aspect. Comparing the values for different donor units, however, one sees a decrease for both hole and electron transfer with increasing length of the donor unit. The difference for hole transport is in the range of 14 to 18 meV, while for electron transport it is with 2 to 5 meV up to a factor of seven smaller. This is in line with the fact, that the donor unit was varied which mainly affects the transport properties of holes. The acceptor unit, however, stays the same for all systems. Thus, λ for electron transfer stays rather constant,


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as well. The decreasing trend may arise from the delocalization of the charge density within DFT based methods. Next, the results of the coupling HAB will be presented. As already mentioned, HAB strongly depends on the stacking conformation. Since the considered polymers have no mirror plane perpendicular to the molecule plane, a parallel (P) and an anti-parallel (A) stacking need to be considered. Thus, for each system four different stacking conformations were studied: cis/P, cis/A, trans/P, and trans/A. The variation of HAB for the T-DPP-T-TT system in cis/P configuration and a distance of hole electron 3.5 Å, is depicted in Fig. 2a). The colored curves show the behavior of HAB and HAB

when the two chains are shifted along the polymer backbone starting from a direct π-π stack. Both show an odd number of oscillations with maximal values at the edges and an inversion symmetry at a relative shift of 50% of the unit length. The form of these curves can be related to the forms of the HOMO and LUMO wave functions, respectively, which are shown in Figure 3. For a relative shift of zero, the nodal planes of the wave functions are on top of each other which means that the overlap of the frontier orbitals obtains its maximum. Thus, the coupling HAB yields maximal values, too. When the polymers are displaced, the overlap becomes smaller, and thus HAB decreases. For stackings, where the nodal planes coincide with the maximum values of the orbital of the other site, the overlap and thus the coupling becomes zero. Further displacement increases the overlap again, but since the sign changes the coupling becomes negative. Due to these facts, the number of oscillations of HAB in Fig. 2a) is equal to the number of nodal surfaces along the polymer backbone. Since these nodal planes are nearly perpendicular to the chain direction for the HOMO, see Fig. 3a), the hole electron oscillations of HAB have a higher amplitude compared to HAB . As shown in Fig. 3b),

the nodal planes of the LUMO wave function are rather diagonal to the polymer backbone. The oscillating behavior and the correlation with the frontier orbitals have been observed for the remaining systems, as well. The maximal amplitudes show the same order of magnitude



electron HAB

400

∆Estack

-3.5 -4

200

-4.5 -5

0

-5.5 -200

-6

hole HAB

-6.5

-400 20

0

40 60 80 relative shift (%)

stacking energy ∆Estack (eV)

of about 500 meV. Thus, they are not reported here in detail.

coupling element HAB (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

100 b)

a)

Figure 2: (a) Coupling HAB for electrons (red, left scale) and holes (green, left scale) as well as interaction energy (black, right scale) of ABA stacked chains for the T-DPP-T-TT polymers in cis/P configuration. (b) Schematic plot of the cis/P configuration of the TDPP-T-TT system. The colored planes represent the molecular planes. For the evaluation of HAB only two chains have been considered. For the evaluation of the energy an ABA stack was utilized.

b)

a)

Figure 3: Surface plots of the HOMO and LUMO wave functions of T-DPP-T-TT polymers in cis conformation. The nodal planes of the HOMO (a) are preferably perpendicular to the polymer backbone, the nodal planes of the LUMO (b), however, are predominantly diagonal.

The black curve in Figure 2a) refers to the interaction energy of a periodic [AB] stacking as depicted Fig. 2b). When the shifts perpendicular to the polymer direction are fixed, the 14 ACS Paragon Plus Environment

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most stable conformation is reached by a shift of roughly 30% of the unit length. For shifts of roughly 15%, 40%, and 50% (see arrows in Fig. 2a)) meta stable stackings arise. Due to the symmetry of the systems the same configurations are obtained by shifts of 60%, 70%, and 85% (see arrows in Fig. 2a)). These geometrically stable configurations, however, do not coincide with the maxima of HAB . To estimate a single quantity for each system, the statistical approach (Eq. (7)) is used assuming a temperature of 300 K. Now, stacking conformations including shifts perpendicular to the polymer backbone were considered as well. The results for all considered systems are shown in the fourth row of Table 1. As a single thiophene donor unit does not lead to A/P isomerism, the parallel and anti-parallel stackings are considered at once. The values vary over a broad range from 74.6 up to 133 meV for hole and from 55.1 to 113 meV for electron transport. Already for one system, the values differ very much between the four different hole stacking possibilities. For instance, on the one hand, HAB of the T-DPP-T-T system in

the cis conformation yields one of the lowest values, but one of the highest in the trans conformation. On the other hand, the T-DPP-T-TT system in trans/P conformation yields electron , but the highest value in the cis/A. Thus, we conclude that the lowest values for HAB

the dependency of the coupling HAB on the molecular structure is rather complex. Nevertheless, these results may be helpful to design novel devices with enhanced transport properties. The fifth row in Table 1 show the hopping rates obtained for the different stacking configurations where again a temperature of 300 K was applied. Using a simple Einstein diffusion model, 25 those rates can be translated into charge carrier mobilities via ed2 k . µ= kB T

(8)

In this equation e is the elementary charge and d the hopping distance, for which we use a value of 3.6 Å since it is the average π − π stacking distance, which has also been observed in experiments. 6 The highest hole and electron mobilities are obtained by T-DPP-T-T-T in 15 ACS Paragon Plus Environment


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trans/P configuration and by T-DPP-T-T-E-T in cis/P configuration, respectively. We note that these values are about one order of magnitude lower than the ones already observed in experiments. Since our model considers ideal stacks of chains, the values should be larger than the experimental ones. This deviation between calculated and measured values arises from the following reasons: First, it was shown that the FO approach underestimates the electronic coupling. 14,15 Second, we applied the Einstein model as a simple approach to obtain mobility values based on the calculated hopping rates. In doing so, the consideration of a fixed hopping distance leads to a further underestimation, since d will be affected by the thermal effects as well. Third, the Einstein model assumes an isotropic diffusion. Since the CT along the backbone is superior to the intermolecular CT, 9,10 the charge carrier mobility of a system in which both mechanisms occur will be larger. Thus, our results provide a lower boundary for ideal stacks rather than an upper one. In the last row of Table 1, the averages over all the configurations are listed. Comparing the average values for each system shows that the hole mobility is always a bit larger than the electron mobility. This is in line with the fact, that these systems are mainly intrinsic hole conductors. The highest hole mobility was found for the T-DPP-T-T-T system while the lowest one was achieved by the T-DPP-T-T system.

Conclusions In this article, the charge transfer of donor-acceptor polymers consisting of a DPP acceptor unit and different thiophene based donor units was investigated. For this, Marcus theory has been applied to estimate intermolecular hopping rates. The two required quantities, λ and HAB , were calculated using DFTB based methods. We discussed two kinds of structural properties: On the one hand we considered different 16 ACS Paragon Plus Environment

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isomeric conformations, cis and trans concerning the orientation of the thiophene units related to the DPP unit, combined with the two stacking orientations parallel and antiparallel. This results in four groups of orientation: cis/P, cis/A, trans/P, and trans/A. On the other hand, two chains in one of these orientations were shifted with respect to each other. At first, we discussed the dependency of the reorganization energy λ. For this, we considered the monomer model, such that from the structural properties only the cis-trans isomerism remained as degree of freedom. Here, we found that the elongation of the repeat unit, which coincides with the variation of the donor part, leads to a decrease of the reorganization energy. This change was much larger for holes than for electrons which matches the fact that the donor unit affects transport of holes more strongly than of the one of electrons. Due to the fact, that the acceptor unit was unmodified, λ stays rather constant when electron transfer is considered. A dependence on the orientation of the thiophene rings, however, was not observed. Second, we calculated the coupling HAB of two chains for shifts with respect to each other. Here, the shift along the polymer backbone was mainly discussed. For this, we observe a clear correlation between the number and position of node planes calculated for the frontier orbitals and the corresponding values for HAB . This results match the fact that HAB behaves like the overlap between the wave functions of the individual fragments. Furthermore, we found several local minima in the interaction energy of twoπ-π stacked chains which are separated by barriers of more than 0.3 eV per interacting unit. To estimate a single quantity that can be entered to the Marcus formula, we introduce a 2 averaging over a large number Boltzmann-type statistical approach to calculate a value HAB

of stacking alignments. This approach enabled an investigation of the transport properties for each of the four groups of orientation separately. The obtained results range from 55.1 up to 133 meV. Even, when comparing the different possibilities for a certain donor, a broad variation was observed. Among all studied systems, the highest hole mobility arises from



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the T-DPP-T-T-T system in trans/P configuration while the highest electron mobility was found for T-DPP-T-T-E-T in cis/P configuration. Other orientations of the same system, however, result in values much smaller, such that the mean value of all orientations does not produce clear trends. We therefore conclude that the electronic transport properties are not pronouncedly influenced by the molecular structure of the studied compounds but by the isomeric conformation and the stacking orientation. Based on this, we suggest to target these specific molecular configurations (rather than to work with a set of random orientations) as a way to improve the transport properties of organic devices. q 2 Finally, we note that for the largest obtained coupling elements, the ration of λ and HAB exceeds the critical value for the applicability of the Marcus theory. For these cases, a delocalised transport mechanism might be applied, see Refs. 26, 27, and 28.

Acknowledgement The authors thank T. Erdmann and Dr. A. Kiriy for valuable discussions and experimental insights. F.G. thanks K. Vietze for productive and helpful IT support. The research was funded by the cluster of excellence Center for Advancing Electronics Dresden (cfAED) and supported by the International Helmhotz Research School (IHRS) NanoNet. Many colleagues have provided feedback on the methodology and the manuscript, which is highly acknowledged.

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(28) Troisi, A. The Speed Limit for Sequential Charge Hopping in Molecular Materials. Org. Electron. 2011, 12, 1988–1991.


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Hopping-Based Charge Transfer in Diketopyrrolopyrrole-Based Donor

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