First Principle Analysis of Charge Dissociation and Charge

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The Journal of Physical Chemistry

First Principle Analysis of Charge Dissociation and Charge Recombination Processes in Organic Solar Cells Amalia Velardo,† Raffaele Borrelli,‡ Amedeo Capobianco,† Mario Vincenzo La Rocca,† and Andrea Peluso∗,† †Dipartimento di Chimica e Biologia, Università di Salerno, Via G. Paolo II, I-84084 Fisciano (SA), Italy ‡Dipartimento di Scienze Agrarie, Forestali e Alimentari, Università di Torino, Largo Paolo Braccini 2, I-10095 Grugliasco, Italy E-mail: [email protected]

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Abstract The rates of charge separation and charge recombination processes for a class of small organic molecules employed as donors in bulk heterojunction solar cells have been obtained from first principles. The Fermi Golden Rule and Kubo’s generating function method for evaluating the Franck-Condon weighted densities of states have been employed, with equilibrium geometries, vibrational frequencies, and normal modes computed at density functional theory level. The comparative analysis shows that the most performing donor dye exhibits the highest rate of the photoinduced electron transfer step.

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Introduction Photoinduced charge separation is the main mechanism through which living systems efficiently use solar energy for triggering the synthesis of chemical species of high free energy content. 1 In photosynthetic reaction centers, a rigid supramolecular assembly, consisting of one electron donor (D) and several electron acceptors (A) forming a supramolecular electron transport chain, enables a highly efficient photoinduced charge separation over long distances. 2 Since each reaction center possesses only one electron donor and the electron transport chain can carry only one electron at a time, there is a strict kinetic control over the spin of the transferred electron: charge recombination via the donor triplet state (CRT), detected in reaction centers that are prevented from completing electron transfer (ET) processes, 3,4 is spin forbidden and therefore much slower than charge separation. In bulk heterojunction (BHJ) solar cells the situation is very different; 5–10 BHJs are characterized by highly disordered solid state structures, in which domains of the acceptor molecules and domains of the donor molecules interpenetrate each other to increase the contact surface. As long as charge separation takes place, it is well possible that a nongeminate electron-hole encounter, i.e. an electron and a hole generated by two different photoinduced ET processes, leads to the formation of a charge transfer (CT) state, whose spin state can be either singlet or triplet. From a triplet CT state, CRT is no longer a spin forbidden process and can efficiently compete with charge separation. Formation of triplet CT excitons on nanosecond timescales have been recently observed in thin films consisting of blends of polymer donors and fullerene derivative acceptors, 11–14 and their low conversion efficiencies have been tentatively attributed to the existence of efficient CRT pathways. Herein, we present a comparative analysis of the rates of charge separation and charge recombination processes, including CRT, for a class of small organic push-pull molecules, successfully employed as donors in BHJ solar cells. 15 Although many other small organic molecules with higher power conversion efficiency have been reported in the literature, 9,16–23 the class of molecules considered here offers a well suited set of experimental data for a 3



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theoretical investigation of the structure-property relationships in BHJ devices. Indeed, one of those molecules, 2-2-[5-(Butyl-ethyl-amino)-thiophen-2-yl-methylene]-3-oxo-indan-1ylidene-malononitrile (here denoted as HB366, as in the original paper), 15 see Scheme 1, exhibits a remarkably higher power conversion efficiency than all the other molecules of the same class – 3.0% in blend with [6,6]-phenyl C61 butyric acid methyl ester (PCBM) (55% in weight) under standard conditions, increasing up to 4.5% after optimization of the solar cell. On the basis of the available X-ray data, W¨ urthner and coworkers suggested that the absence of bulky substituents is one of the distinguishing structural features of HB366. 15 That is certainly an important point, but energetic and spectral factors must be considered as well, for a deeper understanding of the different power conversion efficiencies observed within that class of small organic donors. Thus we have selected the three dyes shown in Scheme 1 for a comparative analysis: HB366, the top performing donor within W¨ urthner’s class of donor dyes, HB238 and MD333, which, even in the absence of bulky ring substituents, exhibit in comparable conditions a significantly lower power conversion efficiency, ≈ 1% in PCBM. The comparative analysis presented here will show that another distinguishing feature of the most performing dye is its higher rate of the photoinduced electron transfer (PET) step; the possible presence of CRT decay pathways does not appear to severely limit device performances.

Computational details Equilibrium geometries, vibrational frequencies, and normal modes of vibration have been computed at DFT level of theory, employing the M05-2X functional in conjunction with the 6-31+G(d,p) basis set. The M05-2X functional has been adopted because, holding a high percentage of Hartree-Fock exchange, it yields more accurate results than standard hybrid functionals for donor-acceptor conjugate dyes, reaching the quality of explicitly correlated methods. 24–26 Effects due to the polarization of solvent were included by using a polarizable

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CN S

MD333

O

N

O

N

Bu

Et

Bu Bu NC

CN HB366

S O Bu

N Et t-Bu

HB238

S

Bu

S

N Bu

NC

N CN

Scheme 1 continuum model (PCM); 27 dichloromethane has been chosen both because experimental results are available in that solvent and because of its moderately low dielectric constant, suitable for mimicking the environment in solid state devices. The unrestricted formalism was adopted for open shell systems. For all dyes, geometry optimizations started from the lowest-energy conformations obtained by molecular mechanics computations, carried out by using the MMFF force field as implemented in Spartan. 28 Long alkyl substituents have been replaced by methyl groups in computations. Rate constants have been evaluated by using the Fermi Golden Rule (FGR) expression of the rate of a radiationless transition between two electronic states,

k=

2π 2 |V | F (∆E, T ) ~ 5


(1)


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where V is the electronic coupling element, F (∆E, T ) is the Franck-Condon weighted density of states, and ∆E is the energy difference between the two electronic states. In order to include in calculations all normal modes of vibration, F (∆E, T ) has been computed employing Kubo’s generating function approach, in which F (∆E, T ) is obtained as the inverse Fourier transform of a time correlation function f (τ ): 29,30 1 F (∆E, T ) = 2π

Z

+∞

eiτ ∆E f (τ ) dτ,

(2)

−∞

with f (τ ) = Tr{e−iτ Hb e−(β−iτ )Ha }/Tr{e−βHa };

(3)

where Ha and Hb are the vibrational Hamiltonians of the two electronic states and β = 1/kB T 29–35 (see the Supporting Information for more details). The Duschinsky matrix J and the displacement vector K, necessary to carry out the trace operation of eq. 3, have been computed using the curvilinear coordinate representation of the normal modes as implemented in a locally modified version of the MolFC software. 36,37 The use of internal coordinates prevents unphysical large shifts of the involved bond distances caused by large displacements of angular coordinates. 35,38–43 Rate constants have been computed by using an average value of F (∆E, T ), taken over a range of ±0.05 eV around the corresponding ∆E values. In the computation of F (∆E, T ), PCBM has been modeled by the C60 molecule, in order to take advantage of the high symmetry of the latter to speed up calculations. Icosahedral geometry has been assumed for the ground state, whereas for the one electron reduced species the D2h symmetry has been adopted, since it yields the lowest energy structure. All electronic wavefunction calculations have been performed by using the Gaussian 09 package, 44 but for C60 in both neutral and anionic form for which the Turbomole suite of programs has been employed. 45

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Results and Discussion Full geometry optimizations of the ground states and of the first excited singlet (S1 ) and triplet (T1 ) states of the three dyes in solvent of moderate polarity (dichloromethane ǫ = 8.93) lead to the results schematically reported in Figure 1 and summarized in Table 1. The adopted computational approach predicts oxidation free energies in very good agreement with the experimental data, see Table 1. The energies of the CT states have been obtained by subtracting the experimental reduction free energy of PCBM, ≈ −4.1 eV, 46,47 from the computed oxidation free energies of the dyes, obtained by full optimizations of the ground states (D+ 0 ) of the one electron oxidized dyes. This choice has been dictated by the fact that, as well known, it is difficult to reach a high degree of accuracy for the computed reduction free energies of neutral species. DFT/PCM computations yield for PCBM in dichloromethane ∆Gred = −3.6 eV; the agreement with the experimental value slightly improves by performing single point computations with a triple zeta basis set, with diffusion functions on both light and heavy atoms, which yield ∆Gred = −3.7 eV. Noteworthy, the equilibrium geometry of C− 60 , necessary for the calculation of F (∆E, T ) for the PET step, is well reproduced by calculations, as testified by the simulated vibrationally resolved photoelectron spectrum of cold C60 anion, reported in the Supporting Information. Computed vertical excited state energies are slightly overestimated (≈ 0.2 eV) if compared with the peaks of the corresponding absorption bands. 15 Table 1: Computed energies (eV) of the first excited singlet, S1 , and triplet, T1 , states, oxidation potentials (Volt), and dipole moment changes upon photo-excitation (Debye) of the three dyes of Scheme 1. All energies refer to the ground state of each dye.

HB238 HB366 MD333 a From

S1

CT

T1

2.19 2.38 2.61

1.49 1.62 1.64

1.43 1.72 1.84

Eox Theor Expa 5.52 5.52 5.65 5.69 5.67 5.65

∆µ Theorb Expc 2.2 2.5 3.0 4.0 2.0 1.7

voltammetric measurements in dichloromethane; b from gas-phase computations; to the gas-phase by using the Onsager model, see ref. 15.

c extrapolated

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Figure 1: Computed energies (eV) of the first excited singlet (S1 ), triplet (T1 ), and charge transfer (CT) states of the three dyes of Scheme 1. The zero of energy corresponds to the ground state of each dye.

Figure 1 shows that the triplet state of HB238 is predicted to be slightly below the CT state, so that at first sight the lower power conversion efficiency of HB238 with respect to HB366 could be explained by the presence of an efficient exoergic charge recombination pathway via the triplet state of the donor. To put this argument on a more quantitative ground and to better understand the reasons of the low efficiency of MD333, whose triplet state energy is well above that of the CT state, we have performed a comparative theoretical analysis of the rates of charge separation and charge recombination processes, using a full quantum mechanics approach, which has been successfully employed in the past to reliably reproduce the temperature dependence of electron transfer between pheophytin and primary ubiquinone in bacterial reaction centers. 32,48 The F (∆E, T )’s for photoinduced electron transfer (PET) from the donors to the PCBM acceptor at T = 298 K are reported in Figure 2, as a function of the energy difference (∆E) between S1 and the CT state. The rate constants corresponding to the ∆E’s of Table 1 are listed in Table 2. All values in Table 2 refer to |V |2 = 1 cm−1 ; that is a convenient choice for comparative purposes, because the three selected molecules have similar π backbones and therefore their weak interactions with PCBM, which determine the electronic coupling elements V , are likely to be similar. In real materials different geometrical arrangements at the A/D interface must be expected and the electronic coupling element V is therefore characterized by a statistical distribution V (R), which is a function of the intermolecular 8


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coordinates R. It has been recently shown that even though a molecule shows strong deviations from planarity, its electronic transport property could not be compromised, inasmuch as short contacts across different directions can significantly enhance exciton diffusion. 49 Because of that, here we make the reasonable assumption that the V (R)’s of the three dyes are similar to each other, because the molecules have similar shapes and sizes. Vice versa the intramolecular vibrational states which determine F (∆E, T ) are preserved at A/D interfaces, i.e. they do not significantly depend on geometrical arrangements in the solid state, because electronic coupling elements are much smaller than the strengths of chemical bonds. Thus, within a class of similar molecules, F (∆E, T ) represents a physically well sound quantity for determining the chemico-physical factors which control the rates of charge separation and charge recombination processes. Inspection of Figure 2 shows that for MD333 and HB238 PET occurs in an energy region in which the corresponding F (∆E, T )s rapidly decay as |∆E| increases, i.e. the inverted Marcus region. Vice versa, for HB366 the computed ∆E for PET matches the energy region in which F(∆E, T ) is maximum, so that HB366 exhibits higher PET rates, ca. one order of magnitude, than the other two donor dyes. Remarkably, the differences in predicted PET rates among HB366 and the other two dyes will increase as the Coulomb binding energy of the CT state, not considered in Table 1, increases the |∆E| for PETs. Table 2: Energy differences (∆E, eV) and computed rate constants (k,s−1 ) for PET (kPET ), hole hopping (kh ), formation of CT state (kfCT ) and its backward dissociation (kbCT ), charge recombination via donor triplet state (kfCRT ), and charge dissociation from the triplet state of the donor (kbCRT ) at T = 298 K; all rate constants refer to |V |2 = 1 cm−1 .

HB238 HB366 MD333

PET ∆E k PET -0.70 3.1 ·107 -0.76 1.6 ·108 -0.97 9.3 ·106 a Including

hole hopping kh 2.9 ·108 1.9 ·107 3.7 ·107

CT ∆E a -0.13 -0.13 -0.13

kfCT

3.0 7.9 ·107 2.7 ·108 ·108

CRT kbCT

1.9 5.0 ·105 1.7·106 ·106

∆E a 0.07 0.23 0.33

kfCRT

1.7 3.7 ·104 7.8 ·102 ·107

kbCRT 2.6 ·108 2.8 ·108 3.0·108

the contribution of Coulomb Binding energy, Eb = 0.13 eV, see text.

The shapes of the F (∆E, T )’s for hole hopping, the process in which an electron hole 9



3 × 10

-4

HB238 HB366 MD333

2 F (DE,T )

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1

0 -1.5

-1.0

-0.5

0

DE / eV

A

D*

k PET

-

A

Eb

D+

Figure 2: Franck-Condon weighted densities of states for photoinduced electron transfer from the excited donor to PCBM acceptor as a function of the energy difference between S1 and CT electronic states (∆E) at T = 298 K.

hops from one donor molecule to a neighboring one in quasi-resonance conditions, are very similar to those computed for PET, see the Supporting Information, so that at ∆E ≈ 0 HB238 and MD333 exhibit higher vibrational contributions to hopping rates than the most performing HB366. This result is in line with the expectation that charge transport within donor domains is favored in compounds possessing electronic structures close to the cyanine limit, because of the larger delocalization of the frontier orbitals, which implies smaller nuclear relaxation and therefore smaller reorganization energies, see Table 3. Indeed, both the changes of dipole moments upon photoexcitation as well as the absorption bandwidths indicate that HB238 and MD333 possess an electronic structure close to the cyanine limit, 15 with HOMO and LUMO delocalized over the whole conjugated π path, whereas HB366 is a moderate push-pull molecule, undergoing larger structural changes than HB238 and MD333 upon oxidation, as confirmed by the computed reorganization energies reported in Table 3 and by the shapes of HOMO and LUMO reported in Figure 3. Hole hopping can also lead to an electron-hole encounter at A/D interfaces, forming a CT

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Figure 3: Isosurface contour plots of the HOMO (left) and LUMO (right) of the investigated dyes. state which experiences a Coulomb stabilization energy (Eb ). 50 The computed rate constants for an electron-hole encounter (kfCT ) and for backward charge dissociation (kbCT ) are reported in Table 2, where the Coulomb binding energy for CT state (Eb ) has been set to 0.13 eV, as measurements of electric field induced quenching of photoluminescence would suggest. 51 Even in the case of such a weak Eb , the computed rate constants show that electron-hole encounter at A/D interfaces represents the most efficient trap for charge transport in organic materials. 7,51–53 Those results evidence the double role of CT states at A/D heterojunctions: they promote electron hole dissociation, but at the same time limit charge diffusion. 10,51,54,55 There has been much discussion in the literature about the mechanism by which efficient electron-hole dissociation occurs at the donor/acceptor interface, overcoming energy barriers largely exceeding the thermal quantum. 7,11,52,56–62 Figure 4, right panel, shows that for all the three donors ln kbCT scales linearly with Eb , so that as Eb increases dissociation into separated charges becomes rapidly exceedingly slow, no longer competitive with decay to the ground state. From nongeminate electron-hole encounter, CT states can be formed either in the singlet

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HB238 HB366 MD333

10

7

10

6

10

8

10

5

CT

CT kb 10

kf

4

10

3

7

10

0

10

HB238 HB366 MD333 0.1

0.2 E b / eV

-

0.3

k CT f e-

+

D

1

10

0.4

D

A

2

10

0

0.1

-

A

k CT b

Eb

0.2 E b / eV

0.3

0.4

D+ D

Figure 4: Computed rate constants at T = 298 K for formation (left) and backward dissociation (right) of an electron-hole pair at A/D interface as a function of the Coulomb binding energy Eb . Table 3: Reorganization energies (eV) of the donor dyes for photo-excitation, and for photoinduced charge separation, charge recombination via triplet, and hole hopping half-reactions. HB238 HB366 MD333

S1 ← S0 0.044 0.084 0.049

D+ 0 ← S1 0.115 0.171 0.138

D+ 0 ← T1 0.149 0.133 0.170

D+ 0 ← S0 0.088 0.114 0.096

or in the triplet state. In the latter case CRT can efficiently occur, since the process is not spin forbidden and the energy of T1 is closer to that of 3 CT (Figure 5). The computed vibrational contributions to rate constants for CRT and for the backward charge dissociation from T1 are reported in Table 2, where Eb = 0.13 eV has been included in the reported ∆E’s. The triplet states of HB366 and MD333 are predicted to be at higher energies than the CT state so that CRT plays a very marginal role for those dyes, inasmuch as kbCRT is always much greater than kfCRT , see Table 2 and Figure 5. For HB238 the triplet state is predicted slightly below the CT state (0.06 eV) for Eb = 0, so that CRT is kinetically competitive with hole hopping, but only in the region of small Eb , see Figure 5. However, charge dissociation from T1 is also predicted to be fast enough to effectively compete 12


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with the spin forbidden decay to the ground state, thus suggesting that also for HB238 T1 does not represent an insurmountable problem for charge transport. This conclusion is in line with time dependent spectroscopic measurements, which have shown that in blends of poly(indacenodithiophene-co-phenantro[9.10-b]quinoxaline) (PIDT-PhanQ) with PCBM, where CRT occurs on nanosecond timescales at T < 240 K, a thermally activated process, tentatively assigned to the backward dissociation of CRT to free charges, effectively competes with relaxation of T1 to the ground state. 13,14 8

10

6

10

8

10 4

10

k CRT b

k fCRT 2

10

0

10

-2

10

0

HB238 HB366 MD333

HB238 HB366 MD333

7

10

0.1

0.2 E b / eV

0.3

0.4

0

0.1

0.2 E b / eV

0.3

0.4

eEb -

A

3

k CRT f

D+

k CRT b

T1

A

3

D

CT

Figure 5: Computed rate constants for charge recombination via triplet (left) and for the backward charge dissociation (right) of a CT state as a function of the Coulomb binding energy; T = 298 K.

Conclusions We have presented a comparative analysis of the nuclear contributions to the rates of charge separation and charge recombination processes for a class of three donor dyes, used in BHJ solar cells in blends with PCBM acceptor. Our analysis shows that the evaluation from

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first principles of the Franck-Condon weighted density of states as a function of the energy difference between the electronic states involved in the non radiative transition provides important guidelines for a rational design of a donor dye and for finding out its best operational conditions. Indeed, we have shown that by properly matching the driving energy for photoinduced charge separation with the value at which the Franck-Condon weighted density of states exhibits a maximum, the rate constants for photoinduced ET can be increased up to one order of magnitude. From the comparative analysis over a small sample of donor dyes sharing similar chemical structures for size and shape, it comes out that for dyes possessing an electronic structure close to the cyanine limit, characterized by narrow absorption bands and small dipole moment changes upon photoexcitation, the energy difference between the first excited singlet of the donor and the CT state must be small, within 0.3 eV, in line with the values proposed by Brabec et al., on the basis of an exhaustive statistical analysis of the most performing dyes in BHJ solar cells. 5 In the case of dyes exhibiting a more pronounced push-pull character, evidenced by broad absorption bands and larger dipole moment changes upon excitation, the energy difference for photoinduced ET should increase, the optimum −∆E for HB366, one of the dyes analyzed here, being in the range 0.4-0.7 eV. Obviously, other factors than those considered here can also play an important role, as for instance the presence of bulky substituents, which can prevent from the formation of well stacked arrangements in the solid state, decreasing electronic couplings for charge transport. Finally, charge recombination paths via a low lying triplet state of the donor do not appear to severely limit device performances, as strongly bound CT states do.

Acknowledgement The financial supports of PON2007-2014 (Relight project) and of the University of Salerno are gratefully acknowledged.

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Supporting Information Available Additional computational and theoretical details. Computed F (∆E, T ) for the S1 ← S0 transition (Figure S1). Computed photoelectron spectrum of C− 60 fullerene at T = 298 K (Figure S2). Figure S3: Computed F (∆E, T ) for hole hopping (Figure S3). This material is available free of charge via the Internet at http://pubs.acs.org/.

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First Principle Analysis of Charge Dissociation and Charge

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