2.1 Polymer solar cells
2.1.3 Charge generation in polymer solar cells
The performance of the PSC is controlled by the efficiencies of following main CT processes (Figure 2.4a): (i) light absorption by the eD material (Channel I in Fig- ures 2.4a and 2.4b, the light absorption of the eA material would occur via Channel II) and formation of bound electron-hole pairs, i.e. excitons; (ii) exciton diffusion to the eD–eA interfaces; (iii) exciton dissociation (ED) into the free charge carries (holes and electrons) and formation of a CT state; (iv) charge-separation and migration to- ward the electrodes, and (v) charge collection at the electrodes. The optimization of all of these steps are crucial for efficient charge-generation in the PSC, especially those related to ED and formation of free charge carriers.
In contrast to the inorganic materials (e.g. crystalline silicon), where the electron and the hole separate immediately upon photoexcitation, generation of free charge carri- ers in organic solar cells is not spontaneous at room temperature. This is due to low dielectric constants (εs≈2–4)[83]and the presence of strong electron-electron and electron-vibration interactions in organicπ-conjugated materials[84], which leads
- -
+ + +
-
eD
eA Channel I
Channel II
- -
+ +
(b) (a)
eD
eA Anode Cathode
-
+
-
+
-
+ +
+
-
+
- -
hv (i) hv
(ii) (iii) (iv)
Channel I Channel II
S0
S1
CS*
CT1
CT*
CR ED
CS CS ("hot")
CS ("cold") T1
E hν
(c)
Figure 2.4 Schematic illustrations of (a) the basic CT processes taking place at the interfaces of the eD and eA compounds in the photoactive layer together with the (b) one-electron and (c) many-electron state descriptions of the charge generation in a PSC. The "cold" charge separation (CS) process refer to the CS via the CT1state, wheres the "hot" CS occur via higher-energy CT states.
to large exciton binding energies at room temperature (ca. 0.35–0.50 eV)[84, 85].
At the interfacial CT state, the hole and electron, which are localized on the eD and eA molecules, respectively, are still electrostatically bound to each other and have to overcome their Coulomb attraction to be able to separate to free charge carriers [84]. Otherwise, charges will recombine to the ground state (GS)[86]. Recombina- tion via a triplet state is also one possible loss channel for PSCs (e.g. via the lowest triplet excited state, T1, see Figure 2.4c)[83]. Thus, maximizing the ED and charge- separation, while preventing charge recombination (CR) are of the great importance
In the PSCs, the ED can occur via two main channels referred to as Channels I and II[87]. In the Channel I, the eD compound is photoexcited followed by the elec- tron transfer (ET) from the eD to eA compound, while in the Channel II, the eA compound is photoexcited followed by the hole transfer (HT) from the eA to eD (Figures 2.4a and 2.4b). In the conventional fullerene-based PSCs, the role of the Channel II has been usually neglected due to overlap of the absorption spectra of the individual eD and eA materials, i.e. polymers and fullerenes, respectively. However, in emerging NF PSCs, where the spectrum of eA is more easily separated from that of eD, the Channel II has been more focus of interest lately. Suitable energy offsets are required for the ED. In Channel I, the offsets between the excited electron on the eD compound and the EA of the eA compound in its GS (also approximated by the EAs of the eD and eA) provide a driving force for the PET[7, 87]. In Channel II, the excited eA compound oxidizes the eD compound, which is in the GS, and thus the efficiency of the photoinduced hole transfer (PHT) is governed by the offsets be- tween the oxidation potential of the eD in its GS and the reduction potential of the excited eA (also approximated by the IEs of the compounds). Thus, for achieving efficient charge generation in PSCs, the eD and eA materials with the matching IEs and EAs should be selected.
In general, the mechanism of charge transport can be described by two different models depending on the material and temperature: a band transport model and localized hopping model[88, 89]. At very low temperature, the transport in in- organic crystals (or ultrapure organic crystals) occurs via coherent band transport, where charge carriers are fully delocalized at the valence or conduction band edges.
At higher temperatures, e.g. room temperature, the charge transport of weakly cou- pled organic crystals and disordered solids, e.g. polymer melts is described with the hopping model: initially, the charge carrier is localized on one site in the solid, which, depending on the degree of the localization, can be a single molecular unit, larger part of molecule, or several molecules. From there, the charge carrier moves to another site by discrete jumps.
In a hopping regime, several theoretical approaches have been developed to describe charge transport, including CT rate models[88]. As the transfer of electron induces changes in the nuclear geometry of the system, the initial (i) and final (f) states of the CT processes can be represented by potential energy surface (PES) curves, i.e.
eD*–eA
eD –eA+ -
eD–eA Evert
2Hif
E00
λi2, CR
λi2, ED
λi1, ED
QGSQR QP Q
E
λi1, CR
ΔG°ED
ΔGCR°
Figure 2.5 Schematic illustrations of changes in the potential energy of the system as a function of nuclear reaction coordinate Q of the states relevant to the ED and CR processes taking place at the interfacial eD–eA system. The black solid lines illustrate the diabatic potential energy curves, while the adiabatic ones are represented with gray, dashed lines. The equilibrium nuclear coordinates of the GS (i.e. eD–eA), reactants (i.e. eD*–eA), and products (i.e. eD+–eA-) are denoted by QGS, QR, and QP, respectively. The contributions to the CT parameters, i.e. inner reorganization energy (λi), Gibbs free energy of the reaction (∆G◦), and electronic coupling (Hif) are also indicated together with the lowest vertical excitation energy (Evert) and energy difference between the GS and lowest excited state (E00). Adapted from [90].
parabolas of the reactants and products as a function of nuclear coordinates Q (Fig- ure 2.5)[83]. In the high-coupling limit, the Born–Oppenheimer approximation is valid and the electronic wave function changes slowly, i.e. adiabatically, when the system crosses the activation barrier between the initial and final charge localized states[88]. The degeneracy at the crossing of the states is removed (i.e. avoided crossing) leading to two new separate (adiabatic) PESs, where the CT proceeds along lower surface[47].
In the weak-coupling limit (ħhωi ≪kBT; also denoted to as the high-temperature limit [91]or low-frequency regime), the CT is governed by the Franck–Condon
during the transfer. Thus, ET must take place at the intersection point of two cross- ing diabatic PES curves (Figure 2.5)[83]. On the basis of Fermi’s golden rule, the hopping rate for the diabatic CT can be expressed by the semiclassical Marcus theory [92–94]:
kED/CR= |Hif|2 ħh
s π λkBTexp
−(∆G◦+λ)2 4λkBT
(2.1)
whereHifis the electronic coupling between the initial (i) and final (f) states of the particular CT process;kBandħhare the Boltzmann and reduced Planck’s constants, respectively;λis the reorganization energy (consisting of the inner,λi, and outer, λs, contributions); and∆G◦is the Gibbs free energy of the CT reaction.
The Marcus theory has been applied in several theoretical studies for calculating the CT rates of the PSC systems[23, 54, 55, 95]. While a variety of the computational methods have been developed for deriving the parameters present in Equation 2.1, i.e. Hif,λ, and∆G◦, their calculation is not so straightforward. The methods that have been applied for calculating the CT parameters in this work are presented in more detail under the methods, see Section 3.2.6. However, as one of the main fo- cuses of this work has been calculating the electronic couplings, different theoretical coupling schemes will be shortly discussed here.
The electronic coupling (also referred to as a transfer integral) is the off-diagonal matrix element of the electronic Hamiltonian of the system (HÒ). The electronic coupling between the initial and final diabatic states|Ψi>and|Ψf>can be defined
Hif=<Ψi|HÒ|Ψf> (2.2) There exists different theoretical approaches for calculating electronic couplings, which differ in how they define the diabatic states [88]. In some of the schemes, e.g. constrained DFT[96], the diabatic states are constructed directly. Alternative, some schemes make use of the adiabatic eigenstates retrieved from the quantum me- chanical calculations in determining the diabatic ones. In the case of the symmetric systems, the simplest approach for the electronic coupling is to calculate it as half
the energy gap between the GS and excited state (see Figure 2.5). In the approaches based on Koopmans’ theorem, the coupling is taken as half the splitting between the two HOMO (or two LUMO) levels of the interacting compounds[91]. As these schemes are not applicable for asymmetric systems, other approaches should be em- ployed for them instead, such as a fragment orbital approach[97], which has been an efficient scheme for calculating the electronic couplings of the GS CT processes.
Furthermore, it has also been employed to approximate the couplings for the excited state CT processes of the PSC systems[95]. There exist also several diabatization schemes, where the adiabatic states are transformed to the diabatic ones by using ei- ther the wave-function for example in Boys-localization[98], Edmiston–Ruedenberg localization[99], and block diagonalization[100, 101]or an additional operator, e.g.
dipole moment (µ) in the GMH scheme[48, 49]or charge difference (∆q) in the FCD scheme[50].
As mentioned in Section 1.1, the eigenstate-based GMH and FCD have been em- ployed in this work, as they are useful approaches for describing the electronic cou- plings of the CT processes involving excited states[48, 50, 51]. Importantly, they can be applied for large molecules like those employed in the PSC systems, where they have been popular schemes for calculating the electronic couplings of different CT processes. Furthermore, multiple states can be included simultaneously in both the GMH and FCD schemes[48–51].