Modeling Photoinduced Charge Transfer Across π-Conjugated

Article pubs.acs.org/JPCC

Modeling Photoinduced Charge Transfer Across π‑Conjugated Heterojunctions Yongwoo Shin and Xi Lin* Department of Mechanical Engineering and Division of Materials Science and Engineering, Boston University, Boston, Massachusetts 02215, United States ABSTRACT: The adapted Su−Schrieffer−Heeger (aSSH) model Hamiltonian is extended in this work to incorporate the interchain π−π stacking and dynamical electron−phonon coupling effects so that the photoinduced charge-transfer mechanism can be directly probed at the π-conjugated heterojunction interfaces. It is found that excitons generated in the bulk poly(p-phenylene vinylene) (PPV) phase require an activation energy of 0.23 eV to reach the heterojunction interfaces before getting their charges separated. Electron transfers from the D1* state of PPV to the t1u * state of C60 follow the nonadiabatic mechanism, which is accelerated by three major factors including the 0.95 eV energy drop between the two states, close vicinity of the electron-donating D*1 state to the C60 phase, and suppressed inversion symmetry of C60 at the interfaces. The irreversible phonon relaxation energy associated with the nonadiabatic electron transfer is estimated to be 0.3 eV, which explains the widely accepted empirical energy offsets between the measured open-circuit voltage and the theoretical built-in potential.

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INTRODUCTION Since the discovery of conducting polymers in the late 1970s,1 it has been well recognized that the power conversion efficiency (PCE) of photovoltaic cells made of pure conducting polymers were too low for practical applications, in the order of 10−5 or less due to fast recombination of excitons.2 Therefore, photoinduced electron transfer3 using composite conducting polymers as electron donors and fullerenes as electron acceptors was introduced to stabilize the spatially separated electrons and holes. Such photoinduced electron-transfer processes occur effectively only in the vicinity of the donor− acceptor interfaces, since any excitons created in either of the two bulk phases will be subject to recombination and decay before reaching the charge-separating interface. In single-phase conducting polymers, the exciton diffusion length is about 10 nm, which in turn sets the dimensional limit of a double-layer thin film.4 However, a 20 nm double layer of such thin films is essentially transparent to visible lights, allowing most photons to pass through freely. Therefore, in practice, by blending conducting polymers and fullerenes, one anticipates formation of the bicontinuous interpenetrating composite materials so that excitons can be generated in the vicinity of the interfaces and the spatially separated charge carriers can migrate to the collecting electrodes inside of their own semiconducting phase. Recent studies further suggested that hierarchical structures with multiple length scales from nanometers to submicrometers are present in these polymer/fullerene systems.5,6 Therefore, despite the fact that PCE of 3−9% can now be routinely achieved 4,7 using these bulk heterojunction structures, fundamentally challenging problems still remained such as designing novel low bandgap polymers, enhancing the exciton diffusion lengths, improving morphology of these composites, and many others. These practically important problems require © XXXX American Chemical Society

comprehensive understanding of the electronic structures at these π-conjugated heterojunction interfaces. As one of the most widely used electronic structure computational techniques, the first-principles density functional theory (DFT) with standard functionals is known to fail in predicting many electron localizations in solids.8 The timedependent DFT (TDDFT) with the adiabatic local density approximation does not perform systematically better9 and in most cases is only limited to single-point energy calculations at given fixed geometries. In particular, Kanai et al. applied DFT with standard generalized gradient-corrected functionals and found that the electron state of exciton is delocalized across the donor−acceptor interface.10 Tamura et al. computed the absorption spectra of heterojunctions using TDDFT and found that the photoabsorption energy is highly sensitive to the presumed π−π stacking orientations.11 Generally speaking, these DFT-based approaches have three notable limitations. First, the optical bandgaps of conducting polymers are significantly underestimated12 which makes the exciton wave functions delocalize across heterojunction interface. Second, the DFT calculations could not dynamically follow the nonlinear electron−phonon couplings that are intrinsic to the quasi-onedimensional conducting polymers13,14 and bulk heterojunctions.15 These nonlinear coupling effects are essential in creating the self-localized solitons and polarons, which are far beyond the harmonic expansion around ground states.16,17 Third, due to high demands in their computational costs, these DFT-based calculations10,11 routinely model the heterojunctions as one single polymer chain coupled to one C60 molecule, Received: March 1, 2013 Revised: April 24, 2013

A

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neglecting π−π stacking effects in either of the two phases. Therefore, despite numerous experimental and theoretical investigations,18−20 the electronic structures and photoinduced electron-transfer mechanisms of these polymer and fullerene heterostructures have not yet been clearly presented.18 In this work, we develop a computationally efficient model Hamiltonian that can reliably predict the dynamical π−π stacking structures, photoabsorption spectra, and photoinduced absorption (PIA) changes for realistic π-conjugated heterojunction structures containing a few thousand atoms. These predicted results agree quantitatively with the corresponding experimental measurements, with new insights on the photoinduced charge-transfer mechanisms across π-conjugated heterojunctions discussed in detail.

Hph =

2

+

∑

(1)

γ α(bij − bij0)](ci†cj + c †j ci)

i

⎡ 2⎣

ij

4 γα⎤ ⎥ πK ⎦

2

0 2 K iT(φijkl − φijkl )

(3)

n=0

where Mi is the mass of the ith atomic site and the force constants KB, KiT, and KpT specify the strengths in the bond bending, improper torsional, and proper torsional DOF, respectively. These force constants and their corresponding equilibrium angles θ0ijk and φ0ijkl are obtained by fitting to the adiabatic potential energy surface (PES) computed using the second-order truncated Møller−Plesset correlation energy correction (MP2) method27 with the 6-31G basis set.28 As shown in Figure 1a−e, the aSSH PES (solid lines) precisely follow the MP2 results (red symbols) over a large temperature range. It is important to note that proper torsions at the bridge bonds between the phenyl and dimer units in PPV require four extra high-frequency terms to fully account for the nonplanar double-well PES shown in Figure 1e, where a kinetic energy of 300 K may twist φijkl = ± 46° (shaded orange area of Figure 1d). Upon photoexcitation, self-localized electron and hole states reverse the local bond length alternation pattern and therefore the soft bridge bonds are greatly strengthened.29,30 To account for such a strong coupling between torsions and self-localized polarons and excitons,29,30 we introduce an analytical switching function in eq 3 so that the local torsional strength dynamically follows the local bond-alternation order parameter17,22 of

ij

∑ K ⎢(bij − bij0) −

1 − ηξ

2

∑ ηξ ∑ K pT;n cos 2n(φijkl − φijkl0 ) ijkl

where Hπ−ph12,21 describes the π-electron hopping along conjugated backbones with the universal dimensionless electron−phonon coupling constant λ inherited from the original SSH Hamiltonian;22,23 Hph is a classical potential specifying the bending and torsional degrees of freedom (DOF) for saturated σ bonds; Hπ−π takes care of the interchain π−π stacking interactions; and HR accounts for the steric effects and the van der Waals interactions of alkyl (−R) side chains. The intrachain π-electron Hamiltonian12,21 contains

− ε ∑ ci+ci +

ijk

4

+

CONSTRUCTION OF ASSH HAMILTONIAN The three-dimensional (3D) aSSH Hamiltonian can be written as

Hπ− ph = −∑ [γt0 −

i

ijkl

■

HaSSH = Hπ − ph + Hph + Hπ − π + HR

∑ 1 Mi r i̇2 + ∑ 1 KB(θijk − θijk0 )2

ηξ (bij) =

2

⎡ bij − (b− + b=)/2 ⎤ 1 1 ⎥+ tanh⎢ ⎢⎣ ⎥⎦ (b− − b=) 2 2

(4)

where the single-bond and double-bond reference lengths are b− = 1.44 Å and b= = 1.36 Å, respectively.31,32 The smoothly switching PES using the aSSH Hamiltonian is plotted in Figure 1f, showing the shallow double-well energy landscape for long bridge bonds and steep single-well energy landscape for short bridge bonds. The interchain π−π stacking33−36 and the alkyl side chain interactions are modeled in this work as

(2)

Here the double summations run over all the chemically bonded π-conjugated pairs ⟨i,j⟩. The bond length bij = |rij| = |rj − ri| between the ith and jth conjugated sites takes the dummy variable b0ij as its reference,12 in which ri is the 3D position vector of the ith conjugated site. Operators c†i and ci create and annihilate a π-electron on the ith conjugated site, respectively. The standard SSH parameters consist of the reference hopping integral t0 = 2.5 eV, the linear electron−phonon coupling strength α = 4.1 eV/Å, and the elastic constant K = 21.0 eV/ Å2.22,23 The linear ((4√γα)/(πK)) term compensates the contractive force of the hopping integral.24,32 The dimensionless aSSH parameter γ = 1.11 scales the electron−phonon coupling strength in the aromatic units of PPV,21 which is consistent with the widely accepted hopping integral t = 2.8 eV, or γ = t/t0 = 1.12, for graphene and graphene nanoribbons.25 Since fullerene is a wrapped graphene sheet with six disclinations of s = π/3,26 all carbon sites on C60 are treated equally using the identical γ = 1.12 as graphene. Finally, the onsite aSSH parameter ε = 0 is taken throughout this work, since there are no heterogeneous N, S, or O sites in PPV or fullerene.12,21 The classical phonon Hamiltonian includes the following contributions

⎡ α ⎤ Hπ − π + HR = −∑ tππ exp⎢ − ππ (Rδζ − 21/6σ )⎥ ⎣ tππ ⎦ δζ (cδ†cζ

+

cδcζ†)

12 ⎡⎛ ⎛ σ ⎞6 ⎤ σ ⎞ ⎢ ⎟⎟ ⎥ ⎟ − ⎜⎜ + ∑ 4ε ⎜⎜ ⎢⎝ Rδζ ⎟⎠ R ⎠ ⎥⎦ ⎝ δζ δζ ⎣

(5)

Here Rδζ = |rδζ| = |rζ − rδ|. The first double summation runs through all the conjugated carbon pairs located on two different polymer chains, where the interchain π−π hopping integral tππ = 0.1t0 = 0.25 eV37 and the corresponding electron−phonon coupling constant αππ = 0.36 eV/Å.38 In the case of C60, Hπ−π also apply to all the carbon pairs beyond the second-nearest neighbors, accounting for closely stacked carbon atoms on the same fullerene sphere. These long-range electron−phonon couplings effectively play the same role as the electron− B

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Figure 2. PES of a π−π stacking benzene dimer computed by the aSSH Hamiltonian and MP2/6-31G. (a) The separation distance R and orientational angles ϕ1 and ϕ2 of a benzene dimer. (b) PES along the separation distance R direction for the perpendicular configuration, where the minimum at R = 5.1 Å is the global minimum configuration of the dimer. (c) PES projected onto the orientational-angle subspace at fixed R = 5.1 Å computed by the (d) aSSH Hamiltonian and (e) MP2/6-31G. In this case, the local minima are located at |ϕ1 − ϕ2| = π/2. (f) PES projected onto the orientational-angle subspace at fixed R = 3.9 Å computed by the (g) aSSH Hamiltonian and (h) MP2/6-31G. In this case, the local minimum is located at ϕ1 = 0 and ϕ2 = 0. Dashed contour lines in (d, e, g, and h) correspond to the kinetic energy at 300 K. Figure 1. PES computed by the aSSH Hamiltonian (solid lines) and ab initio MP2/6-31G method (red symbols) for the normal modes of (a) bending in aromatic units where KB = 27.9 eV/rad2 and θ0ijk = 120°; (b) bending between the bridge bond and aromatic unit where KB = 10.6 eV/rad2 and θ0ijk = 122° (blue curve) and bending between the bridge and dimer bonds where KB = 8.8 eV/rad2 and θ0klm = 125° (green curve); (c) improper torsion in aromatic units where KiT = 3.6 eV/rad2 and φ0ijkl = 0; and (d) improper torsion at the dimer bond where KiT = 1.8 eV/rad2 and φ0jklm = 0 (green curve) and proper torsion around the bridge bond (blue curve, zoomed out in (e)) where φ0ijkl = 24° and KpT;n = (50.00,−68.54,25.52,0.99,0.07) meV for n = 0, 1, 2, 3, and 4, respectively. Shaded orange areas highlight variation in angles with the kinetic energy at 300 K over the corresponding local minima. (f) Torsion around the bridge bond dynamically switches between the proper form for the neutral state and the improper form for the exciton state, depending on the instantaneous length of the bridge bond.

ence terms in eq 5, the aSSH Hamiltonian of eq 1 can capture both the distance dependence (Figure 2b) and the orientational-angle dependence (Figure 2d,g) for the π−π stacked benzene dimer, as compared to the MP2/6-31G results (Figure 2b,e,h).

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RESULTS AND DISCUSSION Figure 3 shows our computed polarized absorption spectra for monoclinic PPV (Figure 3a) and absorption spectra for facecentered cubic (fcc) C60 (Figure 3c) and PPV/C60 heterojunction (Figure 3e). The oscillator strength45 from the ith state ψi of energy Ei to the jth state ψj of energy Ej can be written as 2me (Ej − Ei)⟨ψi|R|ψj⟩2 fij = (6) 3ℏ2 where me is mass of electron, ℏ is reduced Planck constant, and R is 3-dimensional position operator. The computed polarized absorption spectrum parallel to the PPV chain axis, red solid line in Figure 3a, has two strong peaks of similar oscillator strengths. The lower-energy peak at 2.8 eV comes from the D1 → D1* transition, aligned well with the experimental value of 2.9 eV for oriented poly[2-butyl-5-(2′ethyl)hexyl-p-phenylene vinylene] (BuEH-PPV) films.42 This D1 → D*1 peak has a shoulder at 3.4 eV, as compared to the 3.7 eV shoulder in experiment.42 This shoulder becomes a more pronounced peak in oriented poly[(2-methoxy,5-(2′-ethyl)-

electron correlations on the C60 absorption spectrum39,40 as to be discussed below. The double summation in the last Lennard-Jones (LJ) term runs through the conjugated sites on different chains and all the nonconjugated alkyl side groups. Standard van der Waals radius values σ = 1.85 Å for C−C and C−H pairs and σ = 1.00 Å for H−H pairs are taken directly from the AMBER force field.41 The LJ energy depth ε are fitted to the benzene dimer (Figure 2a) dissociation path (Figure 2b) computed by MP2/6-31G,28 which gives ε = 2.0 and 0.26 meV for C and H atoms, respectively. Even though there are no explicit angle-dependC

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Different from PPV, the valence band maximum to the conduction band minimum transition in C60, hu → t*1u, has a tiny peak due to the same u-symmetry.39 Therefore, the lowest notable absorption peaks appear at 3.7−4.1 eV coming from hu → t*1g (Figure 3c). Strong absorptions near 5 eV come from gg,hg → t*1u. As mentioned earlier, the long-range π−π stacking terms in eq 5 are necessary to obtain the photoabsorption spectrum in Figure 3c, which is comparable to the random phase approximation39 and effective Coulomb interaction40 results. The black dashed line in Figure 3c is from experimental measurements.43 While the density of states (DOS) of the PPV/C 60 heterojunction (Figure 3f) contains many nontrivial features, the photoabsorption spectrum (Figure 3e) resembles the two spectra of Figure 3a and Figure 3c. The lowest transition D1 → D1* at 2.8 eV and the strongest L → L* absorption at 6.0 eV are both from PPV. The computed D1 → D1* peak is 0.5 eV higher than the experimental values, presumably due to the difference between the PPV and MEH-PPV samples. Experimentally, the D1 → D1* peak is 2.9 eV for BuFH-PPV42 and 2.5 eV for MEHPPV,46 a 0.4 eV difference. Such an energy difference is due to the presence of the methoxy and ethylhexyloxy side groups that are conjugated to the MEH-PPV polymer backbone. In contrast to previous DFT calculations10 where electronic states are found to be delocalized across the polythiophene/C60 heterojunction interface, no mixed transitions between the PPV and C60 are found in Figure 3e within the experimentally relevant energy range (6 eV and below). However, if higherenergy excitations are considered, mixed states containing D*2 of PPV and hg* and t2u * of C60 will be populated and direct transitions across heterojunction interface may occur. Moreover, due to the lack of inversion symmetry at the heterojunction interface, the energy splitting between t1u * and * of C60 is suppressed in Figure 3f as compared to Figure 3d. t1g This effectively doubles the number of the electron-accepting states in the C60 phase, favoring nonadiabatic electron transfers across the heterojunction interface. Besides the photoabsorption spectra for the ground-state heterojunction structures (Figure 3e), the aSSH Hamiltonian of eq 1 can be used to probe the photoinduced charge-transfer mechanism and to compute the PIA changes. Since the oscillator strength between the D1 state of PPV and the t1u * state of C60 is zero, the lowest allowed transition is D1 → D*1 in the PPV phase. This gives rise to the interfacial exciton shown in Figure 4a with a formation energy of 2.58 eV, which consists of vertical Franck−Condon transition of 2.61 eV (green arrow) and small phonon relaxation energy of 0.03 eV. As a direct comparison, the exciton formation energy in bulk PPV (Figure 3a) is 2.35 eV. Therefore, the minimal diffusion barrier for excitons generated in bulk PPV to reach heterojunction interfaces is 0.23 eV. Unlike excitons in bulk PPV where the electron and hole states are energetically favorable to be centered at the same location, the interfacial exciton of Figure 4a shows clear charge separation, with the electron D1* state leaning toward heterojunction interface. However, both the electron D*1 and the hole D1 states of the interfacial exciton are localized in the PPV phase. Their wave functions do not overlap with the C60 wave functions. This implies only nonadiabatic electron transfers are allowed. * are Nonadiabatic electron transfers from D1* to t1u accelerated by the large energy drop of 0.95 eV, sufficient to overcome the 0.23 eV bulk exciton diffusion barrier. In

Figure 3. Absorption spectra (colored lines in a, c, and e) and DOS (b, d, and f) computed by the aSSH Hamiltonian of eq 1 for monoclinic PPV, FCC C60, and PPV/C60 heterojunction, as compared to the polarized absorption spectra parallel (black line in (a)) and perpendicular (black dashed line in (a)) to the PPV chain axis in highly oriented PPV films,42 absorption spectrum (black dashed line in (c)) of C60,43 and two absorption spectra (black solid and dashed lines in (e)) of MEH-PPV/C60 blends.44 In the DOS plots, all peaks are marked by arrows and the widths of those broad energy bands are indicated by shaded blocks. Optimized each geometries are presented with different angles.

heyloxy)-p-phenylene vinylene] (MEH-PPV) films,46 because the latter has stronger charge-conjugation symmetry-breaking effects.47 The higher-energy peak at 5.7 eV, versus the experimental value of 5.8 eV,46 is due to the excitation between the two localized bands L → L*, whose wave functions contain nodes precisely at the bridge sites.12 Their cross transitions D1 → L* and L → D*1 contribute to the 4.4 eV absorption peak in the polarized absorption perpendicular to the PPV chain axis, red dashed line in Figure 3a. The experimental value is 4.8 eV.42 As shown in Figure 3a, both the absorption peak energies and their relative oscillator strengths agree well with the corresponding experimental measurements.42 One notable difference is the vanishing contribution of D1 → D1* to the computed perpendicular absorption spectrum, implying that PPV chains in the oriented films42 are not completely aligned. We note that in our calculation all the intra- and interchain degrees of freedom are fully relaxed via self-consistent iterations, which is particularly important to compute the PIA spectra (see below) in which excitons and charged polarons are strongly coupled to optical phonons in conducting polymers. Moreover, the adsorption spectra shown in Figure 3 are computed for solids with explicit interchain π−π stackings, as contrast to previous calculations where only one single PPV chain was used.48,49 The capability of treating bulk solids is crucial to probe the electron transfer occurring near the heterojunction interface to be discussed below. D

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Figure 4d shows our computed PIA change (black solid line) of PPV/C60 heterojunction, which is the difference in the photoabsorption spectra before and after charge transfers, plotted in Figure 4c as grey and black solid lines, respectively. When red-shifted by the same 0.5 eV which aligns the D1 → D*1 peak for PPV and MEH-PPV films (Figure 3e), our computed PIA change (black line) for PPV/C60 matches exactly the experimental curve (gray circles) for MEH-PPV/ C60,3,44 showing a characteristic peak from the polaron band, followed by a plateau and then a photobleaching dip.

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CONCLUSIONS In summary, we introduce a new model Hamiltonian that quantitatively captures the photoinduced charge-transfer processes occurring at PPV/C60 heterojunction. We identify a few key features that ensure the efficient nonadiabatical electron-transfer process. First, there is a large energy drop between the electron-donating D*1 state and the electronaccepting t*1u state. Second, the D*1 electron state is physically located at the interface, ready for nonadiabatic electron transfers to t1u * across the heterojunction. Third, the reduced inversion symmetry near the heterojunction interface doubles the electron-accepting states in the C60 phase. Fourth, after nonadiabatic charge transfer, the D1 hole state is screened by optical phonons in PPV and becomes a highly localized hole polaron P+ moving away from the interface. The irreversible phonon relaxation energy is 0.3 eV, exactly matching the empirical energy offset of VBI − Voc. In addition, our results indicate that excitons generated in bulk PPV require a minimal energy lift of 0.23 eV to reach heterojunction interface.

Figure 4. (a) For PPV/C60 heterojunctions, the lowest-energy transition D1 → D*1 occurs at 2.61 eV. The 0.95 eV energy drop * in C60 accelerates nonadiabatic electron from D1* in PPV to t1u transfers. After electrons being transferred, the D1 hole states are strongly coupled to optical phonons in PPV, leading to self-localized polarons P+, which moves away from the heterojunction interface with * )− to P+ are forbidden. (b) P0. Direct radiative decays from (t1u Photoinduced electron transfer leads to the formation of new P+ states in the optical bandgap. (c) Absorption spectra before (purple) and after (black) photoinduced electron transfers, with their difference plotted in (d). With a 0.5 eV red shift to compensate the D1 → D1* peak offset (Figure 3e) between the PPV and MEH-PPV films, our computed PIA change (black solid line) for PPV/C60 matches exactly the experimental measurement3 (gray circles) for PPV-MEH/C60.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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addition, the close vicinity of D*1 to the C60 phase and the suppressed energy gap between t1u * and t1g * greatly enhance the nonadiabatic electron-transfer processes. On the other hand, adiabatic electron transfers are only allowed at much higher photoexcitation energies, when the D*2 band is populated. Our results indicate that the D2* wave functions (not shown) are well mixed to the C60 states. After nonadiabatic electron transfers, the hole D1 states left in the PPV phase are subject to strong screening by the optical phonons in PPV and become self-localized hole polarons P+. As shown in Figure 4a, P+ are much more localized and move further away from heterojunction interface as compared to D1. At the same time, the empty D*1 states are coupled to phonons and form self-localized polarons P0, moving away from heterojunction interface together with P+. As indicated in Figure 4a, the total phonon relaxation energy accompanied with the nonadiabatic electron transfer is 0.30 eV, consisting of 0.18 eV in forming P0, 0.11 eV for P+, and 0.01 eV for (t1u * )−. Such an irreversible phonon relaxation energy exactly matches the widely accepted empirical energy offset of 0.3 eV between the measured open-circuit voltage Voc and the theoretical built-in potential VBI.50 Since the oscillator strength between P+ and (t1u * )− is zero, direct radiative decay between them is forbidden. E

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The Journal of Physical Chemistry C

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