diusion coecientD can be written as follows:[64]
D= A 6
X
N
d2·kET(d) (2.12)
where the factorAis accounting for disorder,dis the distance of a single hop, andkET is the energy transfer rate to a specied lattice point of set N. Furthermore, in the absence of second order processes such as annihiliation and ssion for instance, exciton diusion can be modeled with a second order dierential equation as follows:[38]
dn
dt =D· 52n(r)−n(r)
τ +G(r) (2.13)
where n is denoted as the exciton density, τ is the exciton lifetime, r is the position in space, and Gis the exciton generation rate. The rst term on the right represents exciton diusion which is driven by the gradient in exciton density, the second term accounts for exciton recombination, and the third term represents excition generation (equation 3.11) upon photon absorption as discussed in section 3.2. Finally, the charac-teristic length scale for exciton diusion, the exciton diusion lengthLD, can be written as follows:
LD =
√
D·τ (2.14)
As mentioned in the beginning of this section on exciton diusion,LD is a very critical material property for organic semiconductors employed in photovoltaics, since it re-ects the characteristic distance over which an exciton can migrate thereby limiting the absorption-diusion product (i.e., the exciton bottleneck) with respect to the employed donor-acceptor architecture (cf. Figure 2.1) for a specic material composition.
2.4. Exciton Dissociation
The primary process in electrical power generation of solar cells is the utilization of incident solar energy to generate free electronic charges in a semiconducting device. As already mentioned in the previous section 2.1, organic solar cells are often classied as excitonic solar cells due to their relatively high exciton binding energies (EexcB ) in comparison to the thermal energy (kBT) at room temperature (EBexckBT w25meV) of photogenerated excitons along with the consequences on exciton dissociation (Fig-ure 2.3). Therefore, excitons play a decisive (even limiting) role as previously discussed in the context of the exciton bottleneck (section 2.3).
The attractive electron-hole Coulomb interaction (i.e., EBexc) needs to be overcome for ecient exciton dissociation. Following a simple point charge picture, the Coulomb potential (VC) of an exciton can be described as follows:[65]
VC(r) = e2
4·π·r·0·r (2.15)
whereeis the elementary charge,ris the dielectric constant of the surrounding medium, 0 is the permittivity of free space, and r is the electron-hole separation distance. As discussed by Gregg, the fundamental dierent working principles of organic solar cells in comparison to their inorganic counterparts arises from the dierence in EBexc (i.e., VC).[19] The dierence inVC (equation 2.15) is caused by the intrinsic dielectric screen-ing porperties of the employed materials on the one hand, which is macroscopically expressed byr, and additionally due to the dierence in localization (i.e., the average r in equation 2.15) of the involved electronic states on the other hand.
Overcoming the Coulomb attraction in solar cells based on silicon pn junctions for
in-Donor Acceptor
Energy
Evacuum= 0
Donor Acceptor Donor Acceptor
Exciton Charge Transfer (CT) State
Charge Separated (CS) State
Figure 2.3.: Simplied schematic of an exciton, a charge transfer state, and a charge separated state at an organic donor-acceptor interface. Coulomb attraction is indicated as a dashed line and the exciton and CT binding energies are indicated due to the vertical position of the electron as well as the hole with respect to the molecular energy levels, respectively. The CT state can be split up if the energy gain due to the occupation of the nal CS state ("free" electron and hole polaron in the acceptor and donor, respectively) exceeds the binding energy of the charge transfer state. Based on Schmidt-Mende and Weickert.[66]
stance, is facile due to the high dielectric screening (r w12) and because the involved electronic states already being highly delocalized.[67] However, overcoming the Coulomb attraction in solar cells based on molecular materials is signicantly more demanding due to their typically smaller dielectric screening (rw2−4) and because of the more localized nature of the involved electronic states. In literature, there is a huge variety of theoretically calculated and experimentally determined values forEBexc ranging from less than 0.1 eV to over 1 eV for conjugated polymers such as poly(para-phenylene viny-lene) (PPVs) for instance.[68, 69] The binding energy for a CT state (EBCT) is usually estimated to be lower than EBexc, due to the increased electron-hole separtion distance at the donor acceptor interface (Figure 2.3). Literature values for EBCT ranging from 0.1 eV to 0.5 eV are typically estimated.[7073] However, source of diering results, at least partially originate from the disordered nature of polymers, dierent processing conditions, dierent donor-acceptor ratios, dierent raw material sources, and dierent materials that all together inuence the morphology and thereby the observed proper-ties. For instance, Loi et al. noted that the CT state energy can be tuned by altering the
2.4 Exciton Dissociation
blend composition in a F8DTBT:PCBM BHJ device.[74] The reduction of CT state en-ergy due to an enhanced PCBM concentration in polymer:fullerene BHJs was observed by dierent groups and is attributed to eectively increased r and higher delocaliza-tion (averager in equation 2.15) of electronic states in PCBM.[7577] The shown state
CT0
Figure 2.4.: State energy level diagram showing the kinetic competition of processes oc-curing during charge photogeneration in organic solar cells. After photoexcitation a charge transfer state can be populated, followed by separation of electron and hole into a charge separated state (cf. Figure 2.3). Based on intersystem crossing (ICS) it is also possible that a singlet-triplet transission occurs. Depending on the donor-acceptor combination there is a certain energy distri-bution of CT states and an activation energy is necessary in order to reach a charge separated (CS) state from the lowest charge transfer state CT0. Based on Schmidt-Mende and Weickert.[66]
energy level diagram (Figure 2.4) summarizes the contemporary picture of all compet-ing dynamic processes (for dierent ratesk) involved in charge separation at an organic donor-acceptor interface. For simplicity, formation of excitons in the acceptor phase are neglected. Therefore, the absorption of a photon in the donor (Figure 2.3) promotes the generation of the S1 singlet exciton (Figure 2.4) due to the photoexcitation of an electron from the donor's HOMO into the LUMO. This donor S1 state can be quenched due to an electron transfer from the donor (at a rate kCT) to the acceptor by forming an interfacial CTn state at dierent thermal excitation levels that are often referred to as cold (n=0) and hot (n=1,2,3,...). Based on the relatively weak electronic coupling, a reasonable rapid spin mixing between its singlet (1CT) and triplet states (3CT) is considered and thereby the transition from 3CTn states to the lowest triplet state T1 in the donor can occur at a reasonable rate kT.[78, 79] Beside the transition to the T1 state, CTn states can undergo geminate recombination to form the S0 ground state at a rate kGR. Note, the energy of the CT0 state can be experimentally determined from its radiative recombination (i.e., electroluminescence) to the S0 ground state.[80]
Alternatively, CTn states can undergo transitions to one of many possible CSn states and vice versa at rates kCS. Furthermore, charge separated states CSn can undergo non-geminate charge recombination at a rate kCR.
The complexity of the state energy level diagram (Figure 2.4) along with the kinetic competition of dierent pathways and most importantly the dierence of the investi-gated material systems limits a clear understanding of charge photogeneration in or-ganic solar cells, which is therefore still under debate. Object of discussion between the two school of thought is the question whether CSn is populated from the lowest (thermally relaxed and more localized) CT0 state or from a hot (and more delocalized) CTn state.[8183]
A fundamental theoretical description on electron-hole separation has been proposed by Onsager in 1938 and was extended by Braun in 1984.[84] The Onsager-Braun the-ory treats an electron-hole pair as a pair of point charges bounded through attractive Coulomb interaction (equation 2.15). The charge carriers are considered to be free if a critical radius (rc) is reached that corresponds to a Coulomb potential which is equal to the thermal energy kBT. A probability distribution for charge separation is obtained that depends on the initial thermalization lengthra (ra < rc). Based on the work of Braun and Tachiya, the initial probability distribution was modied by a possible de-formation of the attractive Coulomb potential in the presence of an electric eld.[85, 86]
Furthermore, nite lifetimes of the excited electron-hole pair along with geminate and non-geminate recombination of charge carriers were considered. However, a quantita-tive description of experimental values typically fails for organic systems, most likely due to energetic disorder, hence overestimatingrc.[8789]