interfaces. Alessandro Troisi*, Tao Liu, Domenico Caruso, David L. Cheung, David P. McMahon. Department of Chemistry, University of Warwick, U.K.. Abstract.
A predictive theory of charge separation in organic photovoltaics interfaces Alessandro Troisi*, Tao Liu, Domenico Caruso, David L. Cheung, David P. McMahon Department of Chemistry, University of Warwick, U.K.
Abstract The key process in organic photovoltaics cells is the separation of an exciton, close to the donor/acceptor interface into a free hole (in the donor) and a free electron (in the acceptor). In an efficient solar cell, the majority of absorbed photons generate such hole-electron pairs but it is not clear why such a charge separation process is so efficient in some blends (for example in the blend formed by poly(3hexylthiophene) (P3HT) and a C60 derivative (PCBM)) and how can one design better OPV materials. The electronic and geometric structure of the prototypical polymer:fullerene interface (P3HT:PCBM) is investigated theoretically using a combination of classical and quantum simulation methods. It is shown that the electronic structure of P3HT in contact with PCBM is significantly altered compared to bulk P3HT. Due to the additional free volume of the interface, P3HT chains close to PCBM are more disordered and, consequently, they are characterized by an increased band gap. Excitons and holes are therefore repelled by the interface. This provides a possible explanation of the low recombination efficiency and supports the direct formation of “quasi-free” charge separated species at the interface. This idea is further explored here by using a more general system-independent model Hamiltonian. The long range exciton dissociation rate is computed as a function of the exciton distance from the interface and the average dissociation distance is evaluated by comparing this rate with the exciton migration rate with a kinetic model. The phenomenological model shows that also in a generic interface the direct formation if quasi-free charges is extremely likely.
The mechanism for the generation of free charges in organic solar cell formed by a blend of electron donor and electron acceptor materials is not well understood. It is generally assumed that the photon absorption generates an exciton in the donor that, after reaching the donor-acceptor interface by diffusion, dissociates in a Coulombically bound hole-electron pair. However, simple theories 1, 2 indicate that the hole-electron attraction is too strong for the hole-electron pair to separate before de-excitation. Understanding why the generation of free charges can be so efficient is a key prerequisite to understand the difference between good and bad solar cells and to design better ones. According to some authors 2, the dissociation of the exciton at the interface generates a hole-electron pair with an excess of vibrational energy that can be used to overcome the Coulombic attraction. An alternative hypothesis is that the exciton leads directly to relatively delocalized charges (still not free charges), a hypothesis that allows a satisfactory modeling of the device 3. This latter idea was explored with microscopic models that assume the delocalization of the hole along a polymer chain 4 and include the possible effect of an interface potential between donor and acceptor materials that further reduces the attraction between the two 5. As in many material science problems the theoretical modeling can be approached with phenomenological models (focused on the description of the device physics with generic Hamiltonians) and computational chemistry/physics groups (trying to establish the differences between materials starting from an atomistic description of the morphology and the structure of the device). However, we have repeatedly noted that the solution of organic electronics problem often requires a combination of the two approaches 6. For example, computational studies of crystalline molecular semiconductors7 have revealed the role of dynamic disorder in determining the charge mobility in these materials and they have suggested the appropriate phenomenological model.8 A similar set of methods has
1
been used transport to describe the charge transport in liquid crystalline materials9 or to determine the best transport model in semicrystalline polymers.10 In the same spirit of our previous contributions we use detailed computational studies of specific systems to derive a generic model to describe the generation of free charges from an exciton in the donor. The atomistic system was studied in detail in ref.11 and only the most relevant results are summarized here. We have considered the prototypical polymer:fullerene interface used in photovoltaic cells where the polymer (electron donor) is P3HT and the fullelene derivative PCBM is the electron acceptor. As illustrated in Figure 1 we have observed that the electronic structure of P3HT in contact with PCBM is significantly altered compared to bulk P3HT. Due to the additional free volume of the interface, P3HT chains close to PCBM are more disordered and, consequently, they are characterized by an increased band gap. Excitons and holes are therefore repelled by the interface. The exciton must therefore dissociate tens of angstroms away from the interface leading to the generation of “quasi-free” charge whose Coulomb attraction is much weaker than in a charge transfer complex formed at the donor-acceptor interface.
Figure 1. (right panel) A snapshot from the simulation showing that, for the two PCBM/P3HT interfaces per snapshot, the P3HT chains are more disordered near the interface. (central panel) A schematics of the increased disorder of the polymer chains near the interface. (left panel) DOS (states monomer−1 eV−1) of P3HT for layers at different distances from the interface with PCBM. The plots are off-set for clarity. Because of the increased disorder near the interface the band gap is larger (mostly because of a reduction of the valence band edge energy). The black curve is the (rescaled) DOS for an idealized isolated chain with no disorder.
In the remaining part of this contribution we would like to explore the idea suggested by the computational modeling in a generic (i.e. non system specific) model. More specifically, we want to study the possibility that an exciton, located at a certain distance from the interface, can dissociate into a hole and electron that are already partially separated to start with. The electron transfer rate between a weakly coupled molecular donor and acceptor can be expressed as 12:
2π k= h
VDA
2
2π K BT λext
∑ e− S v
⎛ λ + vhω + ΔE ⎞ Sv exp ⎜ − ext ⎟ v! 4λext K BT ⎝ ⎠
(1)
where VDA is the electronic coupling between the donor and acceptor (initial and final) states, λext is the external reorganization energy, S is the effective Huang–Rhys factor, ω is frequency of one effective mode that incorporates in an average way the effect of all quantum modes, ∆E is the energy difference between initial and final states, KBT is the thermal energy, and ħ is the reduced Planck constant. This formula provides quantitatively correct rates for the process of generation of the charge separated state in a model containing a short polymer chain (where the exciton is initially localized) and a PCBM molecule in contact with the polymer chain 13. Eq. 1 should be generalized to the case where the exciton is far from the donor-acceptor interface and many states in the acceptor material may be occupied by the electron. As the hole and electron are generated at larger distances, the energy of the exciton dissociation process ∆E becomes less negative, the electron can be more delocalized in the acceptor material and the coupling between donor and acceptor states decreases.
2
Figure 2. (top) A simple 1D lattice model where the exciton can hop to neighboring sites with rate kHO, dissociate with rate kED(R) and relax to ground state with rate kR. (bottom) Exciton dissociation rate for an exciton at distance R from a cluster of acceptor molecules computed using eq. 3 for different values of the attenuation factor β.
We describe the acceptor as a cluster of N molecules, with one orbital the acceptor is described by a wavefunction
Ψi =
∑c
in
φn
φn
per molecule such that an electron in . A tight binding Hamiltonian describes the acceptor
n =1, N
φm H acc φn = γ
so that
if m and n are neighbors and
φm H acc φn = 0 otherwise.
The diagonal matrix elements
are influenced by the presence of a hole generated in the donor material and take the form of
φn H acc φn = α −
e2 4πε 0 ε r RDn 1
(2)
where ε0 and εr are the vacuum and relative dielectric constants, e is the electric charge, RDn is the distance between the donor and acceptor molecule n. α was set to zero, i.e. the origin of our energy scale, and γ was set to 0.0125 eV based on computations of bulk PCBM 14. The interface between donor and acceptor is assumed to be planar and the acceptor molecules are arranged in a semispherical cluster with FCC packing (the PCBM molecule is approximately spherical). Diagonalization of H acc provides the orbital energies Ei and the coefficient cin needed for the evaluation of kED(R). The total exciton dissociation rate N
kED
N
2π = ∑ ki =∑ i =1 i =1 h
kED VDi
is the sum of the electron transfer rate
2
2π K BT λext
∑ e− S v
ki to each of the acceptor orbitals:
⎛ λ + vhω + ΔEi ⎞ Siv exp ⎜ − ext ,i ⎟⎟ ⎜ v! 4λext ,i K BT ⎝ ⎠
(3)
Each rate ki is different because the energy difference, the electronic coupling, the intramolecular and polarization reorganization energy are different. ΔEi=Ei−ED, where ED is the energy of the donor state. In this work ED was set to
3
0.399 eV so that for an exciton-acceptor distance of R=3.5 Å and a “cluster” formed of a single acceptor ΔE1 matches the energy difference (−0.969 eV) computed from electronic structure calculation of a model containing a single donor and acceptor.13 In a system with many acceptors, an electron in state i of the acceptor is shared approximately by
⎛ N 4 ⎞ ⎜ ∑ ciA ⎟ ⎝ A=1 ⎠
−1
molecules (the inverse participation ratio) so that both the Huang-Rhys factor (Si) and the external
reorganization energy
λext ,i
are to be evaluated for each acceptor state. The electronic coupling between donor and
acceptor orbitals can be decomposed in terms of the coupling between donor orbital and each individual acceptor
VDi = ∑ cinVDn .
Since the donor orbital is assumed to be exponentially localized, VDn depends on the distance RDn
between donor and acceptor as
VDn = V0 exp ( − 12 β ( RDn − R0 ) )
where V0 and R0 were set to 0.0136 eV and 3.5 Å
13
based on the computation in and β is the so called attenuation factor. The decrease of the rate with the distance is illustrated in the bottom panel of Figure 2. A simple kinetic model that can be used to describe the possible fate of the exciton is illustrated in the top panel of Figure 2. In this one dimensional lattice model, the exciton occupies one of the lattice sites j (j>0) and it is assumed that site 1 is in contact with the acceptor material. Each exciton can hop to one of the neighbor sites with rate kHO, relax to ground state with de-excitation rate kR, or dissociate into a hole and electron with rate kED(R) as determined from Eq. 3. The rest of the parameters are set to be appropriate for the P3HT:PCBM cell. kR was set to 2.5*109 s-1 as determined from the exciton lifetime in bulk P3HT 15. The closest distance between donor and acceptor was taken to be R0=3.5 Å. The reported values of the exciton diffusion coefficient DE for P3HT range between 10-8 16 and 10-7 m2s-1 15 , meaning that if we set the distance between lattice sites to L=4 Å, the hopping rate can be set in the range kHO=[1011:1012] s-1 from the relation DE= kHOL2. The model shows that the average distance of exciton dissociation events is 10.6, 15.2, 34.4 Å for β values of 0.6, 0.4, 0.2 Å 1 respectively. We have to conclude that for β ~< 0.6 Å 1 long range charge separation is possible and indeed very likely. −
−
The most accurate evaluations of the factor β have been carried out in the study of the electron transfer kinetics in donor-bridge-acceptor system. β varies from ~1 Å when donor and acceptor are separated by an alkane bridge (high HOMO-LUMO gap) to values as low as ~0.2 Å for a connection made by a pi-conjugated fragment (small HOMOLUMO gap).17 The physics is very elementary: the electron can tunnel from the donor to the acceptor and the probability of this happening depends (exponentially) on the distance between donor and acceptor and on the barrier for tunneling which is high for large HOMO-LUMO gaps and small for small HOMO-LUMO gap bridges. For low β, charge separation over distances of many tens of Angstoms has been observed.17 We can already see that, in the case of the exciton, the barrier is expected to be very low and closer to the lower value of the range observed for donorbridge-acceptor systems. If we assume that the exciton is localized in a region where the LUMO is at ~0.1 eV lower energy than the average LUMO energy in the surrounding fragment (this is a typical value of disorder in polymeric systems) we can use Gamow model to estimate β~0.35 Å . More complicated estimates based on model Hamiltonian (not presented here) yield a similar range of β values. −1
−1
−1
In summary, it seems very likely that the exciton dissociates tens of angstrom away from the interface in a generic bulk heterojunction solar cell. We developed a phenomenological model of this process inspired by a computational chemistry study of a specific interface. This process explains the efficient formation of free holes and electrons observed in the best available organic solar cells. Acknowledgments. This work was supported by ERC and the Leverhulme Trust. References [1]
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