† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11474218 and 11575116).
By using a multi-configurational time-dependent Hartree–Fock (MCTDHF) method for the time-dependent Schrödinger equation and a Newtonian equation of motion for lattice, we investigate the disorder effects on the dissociation process of excitons in conjugated polymer chains. The simulations are performed within the framework of an extended version of the Su–Schrieffer–Heeger model modified to include on-site disorder, off-diagonal, electron–electron interaction, and an external electric field. Our results show that Coulomb correlation effects play an important role in determining the exciton dissociation process. The electric field required to dissociate an exciton can practically impossibly occur in a pure polymer chain, especially in the case of triplet exciton. However, when the on-site disorder effects are taken into account, this leads to a reduction in mean dissociation electric fields. As the disorder strength increases, the dissociation field decreases effectively. On the contrary, the effects of off-diagonal disorder are negative in most cases. Moreover, the dependence of exciton dissociation on the conjugated length is also discussed.
In recent years, optoelectronic devices based upon conjugated polymers, for instance, organic light-emitting diodes (OLEDs),[1] field-effect transistors (FETs),[2] and photovoltaic cells,[3] have attracted sustained attention in global society.[4] Specifically, organic photovoltaic cells (OPVs) are particularly attractive because of their outstanding advantages such as low cost, light weight, easy fabrication, and flexible features, which are emerging as as a promising solution to the energy crisis and environmental pollution.
The photovoltaic effect involves the generation of electron and hole pairs and their subsequent collection at the opposite electrodes. In inorganic materials, free charges are produced by the photon absorption directly, while in organic materials, due to the strong electron–phonon interactions, photo-excitations will induce a generation of an electron-hole pair bound by lattice distortion, i.e., the exciton.[5] Therefore, the dissociation of excitons is considered to be one of the most important processes for photovoltaic conversion in polymer-based solar cells, and is a key influencing factor in determining the strength of the photocurrent.[6]
The generation of separated charges in photovoltaic materials generally results from the dissociation of these strongly bound excitons through the assistance of bulk trap sites, interface of materials with different electron affinities, or a high electric field. Based on these properties, a considerable amount of both experimental and theoretical research works has been done to effectively improve the charge generation efficiency of organic photovoltaic devices,[7–14] in which, the dissociation field of the exciton is a primary measure for photovoltaic conversion. However, studies have shown that it is practically impossibly to expect the theoretical predicted dissociation fields could occur in the electric fields usually encountered in organic solar devices.[15,16] Moreover, disorder commonly occurs in organic materials, and considerable research has indicated that the disorder effects are crucial for improving organic devices.[17–20] For example, theoretically, Yuan et al. simulated the polaron dynamics in a system of two randomly coupled polymer chains, and suggested that the inter-chain coupling disorder can achieve higher charge transport depending on the coupling strength and the electric field. Moreover, they also indicated that the diagonal disorder depresses polaron motion in most cases while off-diagonal disorder can open energetically easier pathways for charge transport.[21,22] Böhlin, et al. studied the effect of dynamic disorder on charge transport along a pentacene chain, and given the picture that the evolution of charge transport process may vary from an adiabatic polaron drift process to a combination of sequences of adiabatic drift and non-adiabatic hopping events depending on the degree of disorder.[23] Experimentally, Menon et al. studied the controlling polydispersity and demonstrated that energetic disorder has a negative effect on the device efficiency.[24] However, Konezny et al. indicated with experimental and numerical model results that increasing the energetic disorder in the active layer can lead to an increase in the device efficiency depending on the device parameters.[25] Clearly, the structural disorder is an important factor in determining the charge transport for polymer-based devices, and the disorder can be controlled but not eliminated. In this work, we focus on the effects of disordered structure on exciton dissociation under the influence of electric field in polymer chains. Two kinds of disorder, on-site disorder and off-diagonal disorder, are under consideration in this work, which are introduced into the simulations by randomly chosen on-site energies and hopping integrals between nearest-neighbor sites, respectively. In additional, the electron–electron correlation effects also play an important role in transport/recombination/dissociation processes,[26–32] which are taken into consideration in our simulations by employing the multi-configurational time-dependent Hartree–Fock (MCTDHF) method.[33,34]
This paper is arranged as follows. The model and method are present in Section
We consider a polymer chain with structural disorder, which can be well modelled by Hubbard extension of a Su–Schrieffer–Heeger (SSH)-type Hamiltonian.[35–38] The Hamiltonian is given as follows:
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The first part is to describe the lattice elastic potential energy and kinetic energy, respectively,
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The second part in Eq. (
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![](cpb_26_10_107103/cpb_26_10_107103_eqn005.gif)
![](cpb_26_10_107103/cpb_26_10_107103_eqn006.gif)
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The electric field E(t) is included in the Hamiltonian as a scalar potential. This gives the following contribution to the Hamiltonian:
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![](cpb_26_10_107103/cpb_26_10_107103_eqn010.gif)
Now, we briefly describe the numerical method in our simulations. The static structure is determined by the minimization of the system energy in the absence of an external electric field. In the non-adiabatic evolution processes, the atoms are considered as classical particles, governed by the Newtonian equation of motion, and move under the effects of lattice and electronic contributions:
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![](cpb_26_10_107103/cpb_26_10_107103_eqn016.gif)
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Our numerical simulations are mainly performed on a conjugated polymer chain of length N = 100. As is known, charge injection or photo-excitation generates an electron–hole pair, i.e., the exciton, which is trapped by lattice distortion because of the strong electron–lattice interactions in organic materials. For convenience, a pre-existing exciton is deliberately situated around the middle of the chain. The initial geometry is acquired by minimizing the total energy of the chain with static disorder before dynamic evolution. Because we simulate a chain including structural disorder, the exciton should adjust the position to ensure the system energy reaches its minimum. When the external electric field is applied, but in order to avoid abrupt changes and reduce the lattice vibration, the electric field is turned on smoothly in the form of Gaussian function, that is, the field strength changes as
The disorder is introduced according to a Gaussian distribution, and the larger the standard deviation σε (σt), the more disordered the system. When the σε (σt) is fixed, the random seeds produce a set of random sequence, {εi} ({ti}), which determines the on-site energies (hopping integrals). The dynamical process of dissociation should be different under diverse disordered configurations. Theoretically, more seeds lead to less random error. However, numerical simulations are finite and time-consuming, especially involving electron–electron correlation effects. As described in Ref. [22], most of the simulations are performed repeatedly with 16–20 random seeds. For some regions of strong disorder, more than 32 samples are analyzed to ensure the convergence of results.
At first, it should be stated that we find the field required to dissociate a singlet exciton reduces with the increasing of on-site Coulomb repulsion in present Su–Schrieffer–Heeger + Hubbard model, which is contrary to the results of Su–Schrieffer–Heeger + Pariser–Parr–Pople (PPP) model.[41] In the PPP model including long-range Coulomb attraction, the electron and hole are not easily separated under the presence of an electric field, such that eventually an over-large field is required to dissociate the exciton. Furthermore, the Hubbard model can be regarded as an infinite shielding factor limit of the PPP model. As illustrated in Fig. 6 of Ref. [41], the critical dissociation electric field decreases quickly with an increase in shielding factor β. For the case of singlet exciton, the net electron charge density (defined by
We are now investigating the dynamic evolution of excitons under the influence of an external electric field. We consider first the dynamic process of singlet excitons with on-site disorder. Figure
In the following, we analyze the dominant mechanism of exciton dissociation process under different Coulomb interactions and on-site disorder degree. Since different disorder configuration of {ɛi} may cause fluctuations in the potential gradient, it is clear that a saturated or constant critical electric field strength is no longer expected when a system with static disorder is considered. Generally, the stronger the disorder, the larger the standard deviation(variance) is. In Fig.
Since the conjugation length of polymers also plays a vital role in determining the dissociation of exciton states, it is necessary to study the relationship of static disorder to conjugated length. In the inset of Fig.
Finally, for the case of off-diagonal disorder, because the hopping integrals determine the charge hopping rate most directly, the mean critical dissociation electric fields should be quite sensitive to the off-diagonal disorder, as shown in Fig.
Within a Su–Schriffer–Heeger model modified to include disorder effect, electron–electron interaction, and an external electric field, we have investigated the dependence of exciton dissociation on the Coulomb interaction strength, degree of disorder, and the conjugated length. The simulations are performed using a multi-configurational time-dependent Hartree–Fock method for the time-dependent Schrödinger equation and a Newtonian equation of motion for lattice. Our results suggest that the on-site disorder effects contribute to the dissociation of exciton, thereby leading to a distinct lower mean critical dissociation field compared to an ordered system. However, the off-diagonal disorder does not favor the exciton dissociation process in most cases. We hope the theoretical results are able to provide productive guidance for organic solar cells.
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