Different Dissociation Rates of Singlet and Triplet Excitons of

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C: Surfaces, Interfaces, Porous Materials, and Catalysis

Different Dissociation Rate of Singlet and Triplet Excitons of Pentacene at the Interface in Solar Cells Min Wei, Fan Jin, Tingwei Chen, Huizhong Ma, Chengbu Liu, and Yuchen Ma J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b10352 • Publication Date (Web): 18 Jan 2019 Downloaded from http://pubs.acs.org on January 20, 2019

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The Journal of Physical Chemistry

Different Dissociation Rate of Singlet and Triplet Excitons of Pentacene at the Interface in Solar Cells Min Wei, Fan Jin, Tingwei Chen, Huizhong Ma, Chengbu Liu and Yuchen Ma* School of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, China *Corresponding author: [email protected]

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Abstract Fission of the lowest-energy singlet exciton (S1) to two lowest-energy triplet excitons (T1) in pentacene has been expected to be a promising means for increasing the quantum efficiency of solar cells. Experiments find that S1 and T1 dissociate at quite different time scales at the donor/acceptor interface. Using the pentacene/TiO2 heterojunction as the model, we investigate the dissociation of pentacene excitons by a combination of many-body Green’s function theory and time-dependent Schrödinger equation. Singlet and higher-energy triplet (Tn) excitons of pentacene could dissociate at the same time scale of ~100 fs, benefiting from their capability to scatter into charge-transfer (CT) states and weakly bound charge-separated (CS) states across the interface. Resonance of pentacene excitons with CS states could facilitate the creation of free charge carriers. However, dissociation of T1 is hampered due to its poor density of states projected onto the interfacial states which preventing its scattering into CT and CS states. According to this phenomenon, we suspect that the electron transfer from T1 to acceptor as observed in experiments might undergo two successive processes, promotion of T1 to Tn by visible light and dissociation of Tn via scattering. Involvement of the additional light absorption process might result in the low dissociate rate of T1.

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1 Introduction Organic solar cells have attracted broad interest due to their low-cost production and easy fabrication.1-4 In organic solar cells, donor and acceptor form a type-II heterojunction which could promote separation of charge carriers (electron and hole) from photogenerated excitons. The solar-to-electric power conversion efficiency of organic solar cells is still low, e.g. 11.5% at most for the typical ones based on fullerene derivative acceptors,5 compared to the commercial inorganic solar cells based on silicon and the recently emerged perovskite solar cell. Charge separation at the donor/acceptor interface is one of the key factors that determine the efficiency of solar cells. Understanding the mechanism of charge separation is therefore essential for raising the efficiency. This topic has been discussed widely but still remains elusive. Several models have been proposed to account for the charge separation process according to experimental and theoretical studies.6-14 In the most common model, the exciton created in the donor bulk diffuses to the interface where it decays nonadiabatically into strongly bound interfacial charge-transfer (CT) states. To realize charge separation, excess energy, such as thermal energy and electric field, is required to lift the system up into high-energy long-range charge-separated (CS) states whose electron and hole are weakly bound. In this model, charge separation is considered to be a slower process happening at the picosecond time scale. This is supported by many experimental and theoretical studies.15-27 For example, the experimental measurements by Vithanage et al. propose that it takes ~10 ps for the separation between charges to reach ~3 nm in the bulk P3HT:PCBM heterojunction.22 However, some experiments raise an opposite opinion and suggest that charge separation at the interface is very fast, typically within 100 fs, and is therefore independent of temperature and electric field.28-33 For example, Jailaubekov et al.32 and Grancini et al.31 measure that conversion of singlet excitons in donor to interfacial CT ones can be realized at the time scale of ~80 fs and ~50 fs at the CuPc/C60 and polymer/PCBM interfaces respectively. This femtosecond-scale process excludes the assistance of 3



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nuclear motions in charge separation. To account for this ultrafast phenomenon, some researchers, such as Ma and Troisi, Janković and Vukmirović, put forward another mechanism in theory. They suggest that separated charge carriers might be generated directly by photo-excitation of the high-energy long-range CS states.11,

13, 17, 26, 34

Although the oscillator strengths of CS states are very small, their density of states are high so that their overall optical absorption intensity is significant, as argued by Ma and Troisi. Recently, Masugata et al. propose that scattering of donor excitons by the interface might result in the exciton dissociation.14 One approach to increase the efficiency of organic solar cell and to exceed the Shockley-Queisser limit is to employ donor materials in which one photon could yield multiple electron-hole pairs and thus more free charge carriers.35 The lowest-energy singlet exciton (S1) of pentacene can split into two lowest-energy triplet excitons (T1). Pentacene and its derivatives have been intensively studied in recent years for both the mechanism of singlet fission and the exciton dissociation at the donor/acceptor interface.6,

35-44

Recently there are also some attempts to utilize singlet-fission

molecular materials as donor in dye-sensitized solar cells (DSSC) to improve the performance of DSSC. For example, Kunzmann et al. and Pace et al. study respectively the electron injection into ZnO and TiO2 acceptors from the triplet excited states formed after singlet fission in the carboxylated pentacene and tetracene.43, 45 Distinct from the efficient dissociation of S1, electron transfer from T1 to acceptor is significantly slow or even cannot be achieved at all.35, 43 For example, the time constant of T1 dissociation is measured to be about 5 ps at the interface of the pentacene/C60 bilayer, which is more than one order of magnitude longer than the singlet exciton;35 no free electron originating from T1 dissociation is detected at the interfaces of pentacene/TiO2 and tetracene/TiO2.43 People usually compare the excitation energy of donor T1 with the energy gap between the electron affinity of acceptor bulk and the ionization potential of donor bulk to predict whether T1 can relax into the long-range CS states to realize charge separation. From this point, at least for tetracene/TiO2 the dissociation of T1 is energetically favorable, just like pentacene/C60.35,

43, 46

Therefore, there might be some other factors that affect T1 4


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dissociation. Theoretical investigations are required to gain insights into the dissociation mechanism of singlet and triplet excitons of pentacene at the interfaces of solar cells. In the present work, we employ a combination of many-body Green’s function theory (MBGFT)47-49 and time-dependent Schrödinger equation50 to research the dissociation dynamical processes of pentacene excitons at the interface of solar cell, attempting to give some explanations for how T1 evolves into the weakly bound CS states and why T1 is slower to dissociate than S1. The work in this article is mainly carried out with rutile TiO2 as the acceptor material. The pentacene/TiO2 interface attracts some attentions recently.51-53 The ordered pentacene film has been deposited successfully onto the rutile TiO2(110) surface by Lanzilotto et al., with well-matched lattice between the pentacene film and TiO2(110).54 The dissociation dynamics of singlet and triplet excitons at the pentacene/TiO2 interface has also been studied by Pace et al. from experiment.43 For the pentacene/TiO2 heterojunction, theoretical study can be carried out with a moderate-size supercell as shown in Figure 1. Based on our results for the pentacene/TiO2 interface, we further predict the possible behavior of T1 at the pentacene/C60 interface according to our calculated excitonic level alignment between T1 and the lowest interfacial CT state with an approximate interface model. MBGFT contains two parts, the GW method which is used to evaluate the single-particle (electron or hole) energy levels, and the Bethe-Salpeter equation (BSE) which is specially for the description of electron-hole pair.55-57 The long-range Coulomb interaction between electron and hole can be modeled with high accuracy in BSE. MBGFT is especially suitable for the investigation of CT and CS states, as have been demonstrated in previous studies on organic molecules and organic molecular crystals.58-64

2 Computational details 2.1 Interface models The pentacene/TiO2 interface supercell is constructed by a pentacene molecule and the rutile TiO2(110) surface, where TiO2(110) is modeled by a periodic orthorhombic 5



(6 1) slab constituted by three O-Ti-O trilayers (Figure 1). Dimension of the supercell is set to 17.754 Å and 6.497 Å along the [001] and [1 ̅ 0] directions respectively according to the experimental lattice constants65 of rutile TiO2. Periodic boundary condition is applied along the surface directions. The bottom O-Ti-O trilayer of TiO2 substrate is fixed at its bulk position, together with the under-coordinated Ti (5-fold) and O (2-fold) atoms in this trilayer being saturated by pseudohydrogen atoms with nuclear charges of +4/3 for Ti and +2/3 for O. This protocol can produce a TiO2 bulk-like environment at the bottom side with few O-Ti-O trilayers as demonstrated by Kowalski et al.,66

and has been employed in the

studies of defects in TiO2(110)67-69 and the behavior of holes at the CH3OH/TiO2(110) and water/TiO2(110) interfaces70-71. Geometries of the top two O-Ti-O trilayers of the TiO2 substrate and the pentacene molecule adsorbed on it are relaxed by the density-functional theory (DFT) within the generalized gradient approximation of Perdew-Burke-Ernzerhof (PBE)72 using the plane-wave-based Vienna ab initio Simulation Package (VASP) code.73-74 Projector-augmented wave potential is used to separate valence electrons from core ion in VASP. To describe the van der Waals (vdW) interaction between pentacene and TiO2, the vdW-D2 correction to PBE proposed by Grimme is applied.75 Ionic relaxation is stopped until the force on each atom is less than 0.05 eV/Å. An energy cutoff of 450 eV is employed for plane-wave expansion of electronic wave functions. Ti 3s23p63d24s2, O 2s22p4 and C 2s22p2 are treated as valence in the pseudopotential. 2.2 GW method and Bethe-Salpeter equation GW and BSE are state-of-the-art first-principles approaches for studying electronic and excitonic properties of materials. They are especially applicable to materials composed by ingredients having substantially distinct dielectric properties,76-77 such as the interfaces discussed in this article. Our GW+BSE calulations are carried out by a Gaussian orbital based code.55-57 We use atom-centered Gaussian orbitals with decay constants (in atomic unit) of 0.16, 0.51, 1.64 and 5.24 for Ti, 0.30, 0.89, 2.68 and 8.04 for O, 0.2, 0.71 and 2.5 for C, 0.1, 0.4, and 1.5 for H. GW runs are performed at the level of G0W0, with the physical quantities in GW, including Green’s function and 6


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polarizability, constructed by single-particle eigenvalues and eigenwave functions from DFT within the local density approximation (LDA). Electronic screening is described by the random-phase approximation and the plasmon-pole model proposed by von der Linden and Horsch.56, 78 GW calculation is extremely time-consuming, especially for our pentacene/TiO2 interface model which contains 156 atoms, since it involves band summations over a great number of unoccupied states in the evaluation of electronic screening and self-energy. To reduce the computational cost, in our work band summation is truncated at the unoccupied state which is 50 eV (130 eV) above the conduction band minimum for electronic screening (self-energy). By comparing with the case that all unoccupied states are included in the band summations, these band truncations affect little the band gaps (< 0.02 eV). Figure S1 and Figure S2 present some information on the convergence tests of these two truncations. Under these truncations our calculated GW band gap of rutile TiO2 bulk is 3.33 eV, agreeing with previous GW results (e.g. 3.38 eV by Kang and Hybertsen,79 3.32 eV by Migani et al.80) and experiments (3.3±0.5 eV).81 A Γ-point and 3×9×1 Monkhorst-Pack k-point are sampled in the first Brillouin zone to construct the dielectric matrix and self-energy respectively, which are tested to converge GW energies within 0.08 eV. A 30 Å vacuum gap above the TiO2 surface is included in the supercell, which we have tested that GW band energies can converge within 0.1 eV (see Figure S3). In BSE, the wave function of the exciton S is expressed as 𝜒 (𝐫 , 𝐫 ) = ∑ 𝐴

𝜓 (𝐫 )𝜓 (𝐫 )

( )

where 𝑐 (𝑣) labels the conduction (valence) band,

is the k-point in the first

Brillouin zone, 𝐫 (𝐫 ) labels the position of the electron (hole) in this exciton. In this representation, the BSE eigenvalue equation is ∑ 𝐻

with 𝐻

𝐴

,

,

= (𝜀

=Ω 𝐴 − 𝜀 )𝛿

(2)

𝛿

𝛿

+𝐾

,

. 𝜀 is the single-particle

energy level, 𝐾 is the BSE electron-hole interaction kernel. In our work, 𝜀 uses the 7



quasiparticle energies calculated from GW. Exciton binding energy (Eb) for the state S in this work is defined as 𝐸 =

∑

𝐴

𝐴

𝐾

(3)

,

,

In BSE calculations, 21 valence bands and 40 conduction bands, which compose the electronic density of states (DOS) in Figure 2, and a 5×8×1 k-point grid are employed to represent the excited states of our pentacene/TiO2 interface model (see Figure S4 for the convergence test). More detailed description of the GW and BSE methods is provided in the supporting information. 2.3 Time-dependent dynamics of excitons In real solar cells, excitons are initially created in the region of pentacene crystal which is distant from the interface and are thus eigenstates of the pentacene crystal itself. As they approach the interface, their states will be scattered since they are usually not in the eigenstates of the interfacial region. We will see how the pentacene excitons evolve in space and time as the result of scattering via the time-dependent Schrödinger equation50 𝑖ℏ

∂ |Ψ(𝑡)〉 = 𝐻𝑖𝑛𝑡 |Ψ(𝑡)〉 ∂𝑡

(4)

where Ψ(𝑡) is the excitonic wave function, 𝐻𝑖𝑛𝑡 the two-particle BSE Hamiltonian of the pentacene/TiO2 interface (int). We assume that nuclear positions do not change in this process, so does not 𝐻𝑖𝑛𝑡 . The initial state Ψ(𝑡 = 0) is set to the eigen excitation of the isolated monolayer pentacene film (PF). Ψ(𝑡 = 0) is not an eigenstate of 𝐻𝑖𝑛𝑡 . If expanding Ψ(𝑡 = 0) in all the eigenstates |𝑆〉 of 𝐻𝑖𝑛𝑡 , we can get 𝑖 |Ψ(𝑡)〉 = ∑⟨𝑆|Ψ(𝑡 = 0)⟩exp (− Ω𝑠 𝑡) |𝑆〉 ℏ

(5)

where Ω𝑠 is the eigenvalue of the state |𝑆〉. For comparison, time propagation of the single-particle state |𝜓k (𝑡)〉 at the wave vector k for the isolated photogenerated electron in PF is calculated as well through the time-dependent Schrödinger equation within DFT-LDA. |𝜓k (𝑡)〉 is obtained via 8


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𝑖 𝑖𝑛𝑡 𝑖𝑛𝑡 𝑖𝑛𝑡 〉 |𝜓k (𝑡)〉 = ∑⟨𝜓𝑛k |𝜓k (𝑡 = 0)⟩exp (− 𝐸𝑛k 𝑡) |𝜓𝑛k ℏ

(6)

𝑛

𝑖𝑛𝑡 𝑖𝑛𝑡 〉 are the eigenvalue and eigenwave function for the where 𝐸𝑛k and |𝜓𝑛k

single-particle state |𝑛k〉 of the interface.

Figure 1. Structure of the periodic pentacene/TiO2 interface. (a): front view; (b): side view. In (b) the supercell is repeated doubly along the [1 ̅ 0] direction. Blue, red, brown, and white balls represent Ti, O, C and H atoms, respectively.

3 Results and discussion 3.1 Geometry of pentacene/TiO2 Figure 1 shows the stable configuration of the pentacene/TiO2 interface after optimization by VASP. Pentacene lies on TiO2(110) with an angle of 23°, which is in accordance with the experimental value of 25°.54 The vertical distance between the topmost surface of TiO2 and the center of mass of pentacene is 3.36 Å. The nearest distance between adjacent pentacene molecules along the [001] direction is 3.60 Å, while the interplanar distance between pentacene molecules along the [1 ̅ 0] direction is 2.50 Å. Our arrangement of pentacene is consistent with the structure of pentacene monolayer deposited on TiO2(110) in the experiment by Lanzilotto et al.,54 e.g. (i) pentacene molecules align head-to-head (side-by-side) along the [001] ([1 ̅ 0]) direction, (ii) the density of pentacene is 0.910.01 molecule nm-2 (0.87 in our model), (iii) the spacing between pentacene molecules along the [1 ̅ 0] direction is nearly the same as ours. 3.2 Electronic level alignment at pentacene/TiO2 Figure 2 displays the electronic DOS of pentacene/TiO2. In LDA, both the highest occupied molecular orbital (HOMO) and HOMO-1 of PF sit within the band gap of 9



TiO2, with the gap between PF HOMO and the conduction band minimum (CBM) of TiO2 being 0.2 eV. In GW, PF HOMO still stays within the TiO2 band gap, and the gap between PF HOMO and TiO2 CBM increases to 2.5 eV. The GW DOS peak of PF LUMO is ~0.6 eV above TiO2 CBM. The highest occupied and the lowest unoccupied orbitals belong to PF and TiO2 respectively (Figure S5). Therefore, the pentacene/TiO2 interface is a type-II heterojunction.

Figure 2. Total (cyan) and projected (orange and blue) electronic DOS of the pentacene/TiO2 interface calculated by DFT-LDA (a) and GW (b). Energy is aligned with respect to the valence band maximum of TiO2 in each panel. A Gaussian broadening of 0.05 eV is employed. 3.3 Excitons at pentacene/TiO2 Figure 3(a) presents the energies and optical absorption spectra of singlet excitons for the isolated PF. The lowest singlet excitation of PF (state 𝑆1PF−F in Figure 3) is at 2.23 eV, composed by the HOMOLUMO transition. It is a Frenkel-type exciton (Figure S6) with Eb of 1.52 eV. In this article, the Frenkel-type exciton within PF refers to the exciton whose electron and hole are predominantly confined within one pentacene molecule. Following 𝑆1PF−F are two CT excitations at 2.57 eV (state 𝑆2PF−CT) and 2.73 eV (state 𝑆3PF−CT) with Eb of 0.95 eV and 0.76 eV respectively. States 𝑆2PF−CT and 𝑆3PF−CT arise from the HOMOLUMO transition too, with the electron promoted to two adjacent molecules along the [1 ̅ 0] direction (Figure S6). The fourth state (state 𝑆4PF−F at 2.84 eV) is a Frenkel-type exciton again (Figure S6) with Eb of 1.63 eV. Composition of 𝑆4PF−F is complicated, with nearly equal contribution from the transitions HOMO-1LUMO, HOMOLUMO and

10


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Figure 3. Singlet [(a)~(c)] and triplet [(d)~(f)] excitonic levels (colored vertical lines) of the isolated PF (black), TiO2 substrate (red) and pentacene/TiO2 interface (orange). The singlet optical absorption spectra are given in (a)~(c) for light irradiation along the [110] direction. The insets of (a)~(c) present the absorption spectra in a larger energy range. A Gaussian broadening of 0.1 eV is employed in the spectra. HOMOLUMO+1. Figure 3(d) indicates the energies of triplet excitons for the isolated PF. Below 3.0 eV there are eight triplet excitations (𝑇1PF−F~𝑇8PF−F in Table S1, see also Figure 3). The lowest triplet state 𝑇1PF−F locates at 0.96 eV, resulting from the HOMOLUMO transition. The second lowest triplet excitation of PF (state 𝑇2PF−F) is at 2.01 eV, composed by the HOMO-1LUMO and HOMOLUMO+1 transitions. Both triplets 𝑇1PF−F and 𝑇2PF−F are highly localized Frenkel-type 11



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excitons (Figure S6), with Eb of 2.63 eV and 2.94 eV respectively. Above 𝑇2PF−F are two CT states (𝑇3PF−CT at 2.55 eV and 𝑇4PF−CT at 2.73 eV in Figure 3) which originate from the HOMOLUMO transition (Figure S6). Their Eb are 0.93 eV and 0.75 eV, respectively, similar to those of 𝑆2PF−CT and 𝑆3PF−CT. Starting from 𝑇5PF−F, deeper occupied and unoccupied orbitals are involved in the excitations (Table S1). For example, the HOMO-2LUMO transition dominates 𝑇5PF−F. Both the lowest singlet and triplet excitations of the TiO2 substrate are calculated to be around 3.1 eV (Figure 3(b) and Figure 3(e)). This is consistent with the experimental absorption edge of the rutile TiO2.82 When PF and TiO2 are coupled together, new excitations of the CT character emerge. For singlets, excitations below 2.4 eV are pure CT states with the electron promoted from PF HOMO to the conduction bands of TiO2. The lowest one of them appears at 2.0 eV (state 𝑆1𝑖𝑛𝑡−CT in Figure 3 and Figure S7), with Eb of 0.65 eV. Mixing between CT and Frenkel-type excitons begins to appear at 2.43 eV (Figure S7). At the interface there is no exciton that is completely localized on PF. Distinct from the singlets, the lowest triplet excitation of pentacene/TiO2 (state 𝑇1𝑖𝑛𝑡−F in Figure 3) belongs to the Frenkel-type exciton localized on PF (Figure S7). However, its energy is 1.62 eV, 0.66 eV higher than the lowest triplet exciton 𝑇1PF−F of the isolated PF. Spatial distribution of 𝑇1𝑖𝑛𝑡−F is highly similar to that of 𝑇1PF−F of PF. In BSE, the exciton energy can be expressed as Ω = 𝐸

− 𝐸 , where 𝐸

is

the free transition energy from valence to conduction bands, i.e. the term 𝜀 − 𝜀 in the BSE Hamiltonian as shown in Eq. (2), while 𝐸 is the electron-hole binding energy as given in Eq. (3). 𝐸

and 𝐸

reduce by 0.46 eV and 1.12 eV

respectively from 𝑇1PF−F to 𝑇1𝑖𝑛𝑡−F. It seems that the higher energy of 𝑇1𝑖𝑛𝑡−F than 𝑇1PF−F arises mainly from the much smaller Eb in the former. Eb is proportional to the electron-hole interaction kernel 𝐾 (see Eq. (3)), while the attractive term of 𝐾 is inversely proportional to the dielectric function, or the electronic screening, of the environment (see Eqs. (S3), (S6) and (S8) in the supporting information). Compared to the 𝑇1PF−F exciton, the 𝑇1𝑖𝑛𝑡−F exciton feels additional electronic screening from the TiO2 substrate. Dielectric constants of the pentacene crystal and the rutile TiO2 12


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crystal are calculated to be 3.1 and 8.0 by our GW respectively. Therefore, the electronic screening felt by 𝑇1𝑖𝑛𝑡−F is stronger than that felt by 𝑇1PF−F , making 𝐸 of 𝑇1𝑖𝑛𝑡−F weaker than that of 𝑇1PF−F . Pronounced reduction of the electron-hole binding energy and the HOMO-LUMO gap of adsorbates by the electronic screening from the substrate has been observed and discussed in previous MBGFT studies for molecules and carbon nanotubes adsorbed on metallic and semiconducting surfaces.83-86 The first triplet state of the CT character (state 𝑇2𝑖𝑛𝑡−CT in Figure 3) emerges at 1.98 eV, with the electron excited from PF to TiO2 (Figure S7). States 𝑇2𝑖𝑛𝑡−CT and 𝑆1𝑖𝑛𝑡−CT are both caused by the transition from PF HOMO to the conduction band bottom of TiO2. They have nearly the same Eb. From 𝑇2𝑖𝑛𝑡−CT until 2.4 eV, triplet states of pentacene/TiO2 are all pure CT ones. Lower-energy interfacial singlet and triplet CT states are obstacles for charge separation since electron-hole pairs trapped in them might recombine with a high rate. Much discussion has proposed that CS states are the precursors of free carriers. In this case, the incoming excitonic state from donor, at least some portion of it, need to be resonant with CS states to realize dissociation. Traditionally, CS states refer to CT ones whose exciton binding energies are smaller than the room-temperature thermal energy (25 meV).19, 87 Recent studies extend the range of CS states and show that electron-hole pairs with binding energies smaller than ~0.1 eV are all possible to dissociate with the help of entropy and disorder.18-19 In this article, we still use the traditional definition, assigning CT states with Eb smaller than the room-temperature thermal energy to CS ones. A group of CS states appears between 3.5 eV and 3.8 eV for both singlet and triplet excitations of the pentacene/TiO2 interface (Figure 4). In these CS states, both the electron and hole are highly delocalized to an extent of 8 nm at least along the lateral directions parallel to the interface. Figure 4(c) and Figure 4(d) illustrate one of them, whose electron is constrained within the bottom O-Ti-O trilayer of TiO2 substrate. In comparison, electrons of CT states with large Eb are closer to the interface (Figure S7). The vertical average distance between the bottom O-Ti-O trilayer of TiO2 substrate and PF is 11 Å. It seems that free carriers can be created from CS states whose electron and hole are not too far away from the interface. At 13



Figure 4. Exciton binding energy for each singlet (a) and triplet (b) excitonic level of the pentacene/TiO2 interface. The insets of (a) and (b) display the enlarged views for the energy range of 3.4~3.8 eV. (c) and (d): spatial distributions (up: side view; down: top view) of electron and hole respectively for the singlet exciton at 3.49 eV which has a binding energy of 0.4 meV. Side view of the pentacene/TiO2 interface is given between (c) and (d) to facilitate the identification of electron and hole positions. A length ruler is given in (c). When we calculate the electron (hole) distribution, its associated hole (electron) is fixed at atoms within the central supercell as marked by the pink rectangle in (d) as the example.

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Figure 5. Mechanism for exciton dissociation through scattering (a) and mechanism for dissociation of the lowest triplet exciton (𝑇1 ) of pentacene (b). In (a), the donor exciton is scattered into interfacial charge-transfer (CT) and charge-separation (CS) excited states. PDOS: projected density of states of the donor exciton onto CT and CS states. GS: ground state. In (b), 𝑇1 needs to be promoted by light to 𝑇𝑛 to realize dissociation. least in our pentacene/TiO2 model in which the donor, acceptor and interface are all regularly crystallized, there are some CS states within 1 nm to the interface. If there are some disorders or defects around the interface domain which make the delocalized “bandlike” states unavailable there, the electron-hole separation distance in CS states should be larger. Lower conduction bands of the TiO2 substrate belong to the states close to the interface. Photogenerated electrons occupying these states form CT states with large Eb. In contrast, CS states in the range of 3.5~3.8 eV stem from transitions between PF HOMO and higher-energy bulklike conduction bands of TiO2 which are away from the interface. Since there are only a pentacene monolayer and three O-Ti-O trilayer in our interface model, the number of CS states is very limited. When the thicknesses of donor and acceptor increase, first, the number of CS states increases rapidly; second, CS states might extend further into lower-energy region since the interfacial band bending effect would attenuate gradually with the distance from the interface. This elevates the dissociation efficiency of exciton. According to the discussion of Ma and Troisi, the number of CS states scales as 𝑚

𝑛, where 𝑚

and 𝑛 are the thickness of donor and acceptor respectively.11 The band bending and 15



interfacial dipole created by charge transfer between donor and acceptor may provide additional driving forces to promote the separation of charge carriers.88-90 Hence, besides the CS states with Eb at the level of thermal energy, CT states with smaller Eb might be precursors of charge separation. Although, limited by the huge computational cost of the GW+BSE method, the size of our pentacene/TiO2 interface model is too small to give very accurate energy spectrum for CT and CS states and to very reliably distinguish between them, the spectrum we get based on this small model should qualitatively resemble the true interface spectrum and can help us to explore the dissociation dynamics of excitons at the interface. 3.4 Time evolution of excitons at pentacene/TiO2 In this section we discuss the possibility and time scale for the dissociation of PF excitons (labelled by |Φ⟩) through scattering at the pentacene/TiO2 interface. We first take the lowest four singlet excitons of PF, whose energy are smaller than 3.0 eV and within the visible light region, as the initial state to examine their time-dependent propagation, as displayed in Figure 6. Initially (𝑡 = 0), the electron and hole are localized within PF. When the exciton approaches the interface with the kinetic energy acquired from the energy gradient or the decay from higher excited states, its wave function |Φ⟩ will resonate with eigenstates |𝑆⟩ of the interfacial BSE Hamiltonian (Figure 5). This can be measured through the projected density of excitonic states (PDOS)50: Γ(𝐸) = ∑|⟨𝑆|Φ⟩|2 𝛿(𝐸 − Ω )

(7)

Figure S8 gives the PDOS for the lowest four singlet excitons of PF. Distributions of PDOS are very broad for them. Their |Φ⟩ become resonant with a large amount of CT and CS states. Figure 6 and Figure S9 present time evolution of the photoelectron population within PF. With time, the electron migrates into TiO2, while the hole remains in PF. 90% of the electron is transmitted into TiO2 after 200 fs. After fitting the time-evolution curve of photoelectron population within PF to an exponential

16


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Figure 6. Time evolution of the photoelectron population on PF at the pentacene/TiO2 interface with the lowest four singlet excitons 𝑆1PF−F (a), 𝑆2PF−CT (b), 𝑆3PF−CT (c) and 𝑆4PF−F (d) of isolated PF as the initial state. The red straight line denotes a fit to the exponential decay as a guide to eye. (e) and (f): spatial distributions (up: side view; down: top view) of electron and hole respectively at t=0 for 𝑆1PF−F . (g) and (h): spatial distributions (up: side view; down: top view) of

electron and hole respectively at t=200 fs for 𝑆1PF−F . Side view of the pentacene/TiO2 interface is provided between (e) and (f) to facilitate the identification of electron and hole positions. A length ruler is given in (e). When we calculate the electron (hole) distribution, its associated hole (electron) is fixed at atoms within the central supercell as marked by the pink rectangle in (e) as the example.

17



function, we get time constants of 85 fs, 87 fs, 101 fs and 156 fs for these four singlet states respectively. This coincides with the fast charge separation (< 100 fs) reported in some experiments. Electron and hole spread laterally along directions parallel to the interface as well (Figure 6 and Figure S11). For example, the electron and hole stemming from the lowest singlet exciton expand laterally by one order of magnitude after 200 fs. If increasing the thickness of donor and acceptor materials, the electron and hole will penetrate into the depth of TiO2 and pentacene crystals further, enhancing the dissociation of exciton. Different from singlet excitons, some triplet excitons are hard to dissociate. We choose the lowest four triplets 𝑇1PF−F, 𝑇2PF−F, 𝑇3PF−CT and 𝑇4PF−CT as denoted in Figure 3 as the examples to discuss their time propagation (Figure 7 and Figure S10). For 𝑇1PF−F, nearly no electron can be transferred into rutile. From Figure S8, the PDOS peak of 𝑇1PF−F is overwhelmingly dominated by the contribution from the interfacial triplet state 𝑇1𝑖𝑛𝑡−F, with very few contribution from CT and CS states. In other words, 𝑇1PF−F can only resonate with 𝑇1𝑖𝑛𝑡−F, and there are no CT and CS manifolds available for 𝑇1PF−F to be scattered into. In our pentacene/TiO2 system, 𝑇1𝑖𝑛𝑡−F cannot relax into CT states nonadiabatically since its energy is much lower than CT. In some systems this decay process may happen, however, sufficient excess energy might then be required to drive the system out of the potential well of CT state. 90% of the photoelectrons in 𝑇2PF−F , 𝑇3PF−CT and 𝑇4PF−CT could relax into TiO2 after 200 fs with the time constants of 76 fs, 110 fs and 97 fs respectively. Similar to singlet excitons, the photoelectron and hole of 𝑇2PF−F, 𝑇3PF−CT and 𝑇4PF−CT are far more delocalized after their separation (Figure 7 and Figure S12). From Figure 3, the energy of 𝑆1PF−F is above that of 𝑆1𝑖𝑛𝑡−CT and close to those of many interfacial CT states, while 𝑇1PF−F is well separated from interfacial CT states in energy. This might simply account for the hardness of 𝑇1PF−F to dissociate compared to 𝑆1PF−F . The role of exciton-phonon coupling in exciton dissociation at the donor/acceptor interface has been extensively discussed theoretically, e.g. in the recent work by Janković and Vukmirović and that by Yao et al.12, 26 Since our MBGFT code does not possess the capability to evaluate accurate excited-state forces, in fact no formula has 18


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Figure 7. Time evolution of the photoelectron population on PF at the pentacene/TiO2 interface with the lowest four triplet excitons 𝑇1PF−F (a), 𝑇2PF−F (b), 𝑇3PF−CT (c) and 𝑇4PF−CT (d) of isolated PF as the initial state. The red straight line denotes a fit to the exponential decay as a guide to eye. (e) and (f): spatial distributions (up: side view; down: top view) of electron and hole respectively at t=0 for 𝑇2PF−F. (g) and (h): spatial distributions (up: side view; down: top view) of electron and hole respectively at t=200 fs for 𝑇2PF−F. Side view of the pentacene/TiO2 interface is provided between (e) and (f) to facilitate the identification of electron and hole positions. A length ruler is given in (e). When we calculate the electron (hole) distribution, its associated hole (electron) is fixed at atoms within the central supercell as marked by the pink rectangle in (e) as the example. 19



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been set up yet to evaluate accurate excited-state forces analytically within MBGFT, atomic motion in the exciton dissociation process is not taken into account in our studies. Even if atomic motion in the excited state can be modeled approximately by some other first-principles approaches, such as the constrained DFT which has been applied in some periodic systems,91-94 it is impractical to perform GW+BSE calculations to update interfacial exciton eigenstates at each time step owing to the huge computational demand of GW+BSE. In the first-principles study on the nonadiabatic decay dynamics of excitons at the pentacene/C60 heterojunction by Akimov and Prezhdo, in which the phonon effect and atomic motion are considered, the lowest singlet excited state of pentacene is found to decay into the interfacial CT state within 40 fs.8 This time scale is similar to our results in which the exciton-phonon coupling is neglected. Certainly, the effect of exciton-phonon coupling on exciton dissociation deserves to be investigated within GW+BSE in the future with the development of computational method. However, the accuracy of our calculated time scale for exciton dissociation should not influence our main conclusion in this article that 𝑇1PF−F is unable to dissociate at the pentacene/TiO2 interface. Dissociation of excitons as discussed above is dominated by the transfer of photoelectron which initially occupies PF LUMO. For comparison, Figure 8 shows time propagation of a single electron, i.e. in the absence of hole, which originally occupies PF LUMO at 𝑡 = 0 for four wave vectors: (0.0, 0.0), (0.0, 0.25 (0.0, 0.5

2

) and (0.4

2

, 0.375

2

2

),

), where a and b are the dimensions of the

supercell along the [001] and [1 ̅ 0] directions respectively. Decay of the single electron is on the order of ~10 fs, which is apparently much faster than the exciton. The presence of hole in the exciton might retard the transfer of photoelectron across the interface. After the single electron transfers into TiO2, part of it may return back into PF again as the result of reflection from the bottom surface of the substrate model. This therefore leads to oscillation of the electron population with time as shown in Figure 8. 20


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Figure 8. Time evolution of the population on PF in the single electron state at the pentacene/TiO2 interface with the PF LUMO at four wave vectors as the initial state: (a) (0.0, 0.0) , (b) (0.0, 0.25

0.375

2𝜋 𝑏

2𝜋 𝑏

) , (c) (0.0, 0.5

2𝜋 𝑏

) and (d) (0.4

2𝜋 𝑎

,

). The red straight line denotes a fit to the exponential decay as a guide to

eye. (e), (f) and (g): spatial distribution of the single electron at t=0, 15 fs and 25 fs for (0.0, 0.5

2𝜋 𝑏

).

Our result that 𝑇1PF cannot split into separated hole and electron at the pentacene/TiO2 interface seems to account for the experimental phenomena from Pace et al in which no photoelectron from 𝑇1PF dissociation is detected.43 Then what is the situation for the pentacene/C60 bilayer heterojunction in which 𝑇1PF dissociation has been reported? In this heterojunction, pentacene and C60 are arranged in the “head-on” molecular orientation, i.e. the long molecular axis of pentacene molecule is nearly perpendicular to the fullerene crystal surface.95 The lattice match between pentacene and C60 crystals at their interface is very complicated.95 It has been found that regular crystalline interface is important for the delocalization of excitons and therefore for the charge separation,34,

87

as emphasized by Tamura and Burghardt in their

calculations.7 However, to construct a supercell that can well represent the crystalline 21



periodicity of their interface, a large number of pentacene and C60 molecules are required, which is beyond the capability of current first-principles approaches for computing electronic excitations. From our above investigations on the pentacene/TiO2 interface, 𝑇1PF cannot dissociate at the interface since its energy is far below and therefore off-resonant with interfacial CT and CS states. Here, we use an approximate pentacene/C60 interface model (Figure 9 and Figure S13) to estimate the energy alignment between 𝑇1PF and the lowest interfacial CT state so as to predict the dissociation behavior of 𝑇1PF at this interface. This model (named as P/C60) is constituted by a pentacene molecule and a monolayer of the (111)-oriented face-centered cubic C60 crystal at the experimental lattice constant (10.02 Å),96 with pentacene and C60 arranged in the “head-on” orientation. The lowest singlet excited state of the interface is a CT one at 1.90 eV (𝑆1𝑖𝑛𝑡−CT in Figure 9, Figure S15). The lowest triplet state is a Frenkel-type exciton localized on the pentacene molecule at 1.28 eV (𝑇1𝑖𝑛𝑡−F in Figure 9, Figure S16). This triplet energy is very close to the lowest triplet in the bulk pentacene crystal calculated 𝑖𝑛𝑡−CT in our previous work (1.2 eV).97 The lowest interfacial triplet CT state (𝑇lowest in

Figure 9, Figure S16) emerges at 1.90 eV with Eb of 0.44 eV. To estimate the possible 𝑖𝑛𝑡−CT influence of donor density and acceptor thickness on the energy of 𝑇lowest , we

construct two larger pentacene/C60 interfaces (2P/C60 and P/2C60 shown in Figure S13). In 2P/C60 the density of pentacene is doubled compare to P/C60, with the intermolecular distance between pentacene molecules set to 4.34 Å. In P/2C60 the acceptor contains two fullerene layers. HOMO-LUMO band gaps of the three pentacene/C60 interface models, where HOMO and LUMO are contributed by the donor and acceptor respectively, differs little from each other (0.30 eV, 0.25 eV and 0.28 eV for P/C60, 2P/C60 and P/2C60 within DFT, as shown in Figure S13). It is also reasonable to anticipate that the exciton binding energies of interfacial CT states of 2P/C60 and P/2C60 should not deviate much from those of P/C60 (due to the huge computational cost of GW+BSE only the excited states of P/C60 are computed here). 𝑖𝑛𝑡−CT Since the energy of 𝑇lowest equals the difference between the HOMO-LUMO gap

22


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Figure 9. Structure (left) and excitonic levels (right) of the pentacene/C60 interface. 𝑖𝑛𝑡−CT 𝑆1𝑖𝑛𝑡−CT, 𝑇1𝑖𝑛𝑡−F and 𝑇lowest are the lowest singlet excited state which is of CT

character, the lowest triplet excited state and the lowest triplet CT state of the pentacene/C60 interface. 𝑇1 is a Frenkel-type exciton. 𝑖𝑛𝑡−CT and the CT exciton binding energy, we can estimate that 𝑇lowest at the real

pentacene/C60 interface in experiments is around 1.9 eV. Therefore, 𝑇1PF should be 𝑖𝑛𝑡−CT ~0.6 eV lower in energy than 𝑇lowest at the real pentacene/C60 interface. We might

deduce that 𝑇1PF cannot dissociate at the pentacene/C60 interface directly just like that at the pentacene/TiO2 interface. Then how does the 𝑇1PF dissociation proceed at the pentacene/C60 interface in experiments? We suspect that 𝑇1PF is promoted to higher triplets (𝑇𝑛PF ) of pentacene which thereafter dissociate via scattering as discussed above for pentacene/TiO2 (Figure 5). 𝑇1PF → 𝑇𝑛PF absorption has been detected in the visible region from 1.2 to 2.4 eV for pentacene and its derivatives.36, 38-39 Besides the external light source, fluorescence from the lowest singlet exciton of pentacene, which is at ~2 eV, might be another pump for the 𝑇1PF → 𝑇𝑛PF transition. Involvement of this additional 𝑇1PF promotion process should affect its dissociation rate significantly. This might account for the experimental phenomenon why the electron transfer time from 𝑇1PF to C60 is longer than that from the singlet exciton by more than one order of magnitude.

4 Conclusions Mechanism for the dissociation of excitons into free charge carriers at the donor/acceptor interface in solar cells has been long debated and still remains elusive. After theoretically studying the pentacene/TiO2 heterojunction by many-body Green’s 23



function theory and time-dependent Schrödinger equation, we propose that while other excitons of pentacene film can dissociate at the time scale of 100 fs through their resonance with long-range interfacial charge-separation (CS) states after their scattering with the interface, the lowest triplet exciton of pentacene film (𝑇1PF ) cannot dissociate since it fails to resonate with interfacial CS states. This may explain the recent experimental observations on singlet fission in the pentacene/TiO2 solar cell. We also suggest that 𝑇1PF may not dissociate itself as well in the pentacene/C60 heterojunction due to the large energy gap between 𝑇1PF and interfacial CT states. In experiments, it is most likely that 𝑇1PF is first promoted to higher triplet states which then dissociate through resonance with CS states. This two-step dissociation process might interpret why the time constant of 𝑇1PF dissociation is more than one order of magnitude longer than the singlet exciton at the pentacene/C60 interface as measured in experiments.

Supporting Information Computational methods, convergence tests, spatial distributions of electronic orbitals and excitons, and energies of some excited states. This material is available free of charge via the Internet at http://pubs.acs.org.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grants Nos 21433006, 21573131 and 21833004), the Natural Science Foundation of Shandong Province (Grant No. JQ201603).

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Different Dissociation Rates of Singlet and Triplet Excitons of

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