Influence of static disorder of charge transfer state on voltage loss in organic photovoltaics

Spectroscopic measurements of charge transfer (CT) states provide valuable insight into the voltage losses in organic photovoltaics (OPVs). Correct interpretation of CT-state spectra depends on knowledge of the underlying broadening mechanisms, and the relative importance of molecular vibrational broadening and variations in the CT-state energy (static disorder). Here, we present a physical model, that obeys the principle of detailed balance between photon absorption and emission, of the impact of CT-state static disorder on voltage losses in OPVs. We demonstrate that neglect of CT-state disorder in the analysis of spectra may lead to incorrect estimation of voltage losses in OPV devices. We show, using measurements of polymer:non-fullerene blends of different composition, how our model can be used to infer variations in CT-state energy distribution that result from variations in film microstructure. This work highlights the potential impact of static disorder on the characteristics of disordered organic blend devices.

where EQE EL is related firstly to the rate constant of radiative ( ) and nonradiative ( nr ) decay of CT excitons, secondly to the outcoupling and optical properties of the cell through 1,6

Supplementary Method 3. Theoretical description of absorption and recombination from CT state with static disorder
To express the radiative recombination rate constant, we consider the rate of emission from the CT state to the ground state. We express the radiative rate constant using the operator ( = ⃗ ⃗ • ⃗ ⃗ ) , ⃗ ⃗ is the vector potential of the electromagnetic field and ⃗ ⃗ the momentum operator 6 . With static disorder involved, using the Fermi's Golden (FG) rule and the dipole approximation, the emission rate constant ( ) per unit photon energy (ℏω) can be expressed using the transition dipole moment. 8 The absorption rate constant per photon energy ( abs ) can be expressed in a similar way as except that we need to consider the photon density ( ) in the volume. We consider the states in each CT state manifold (denoted as in the subscript) share the same set of parameters. We assume quasi-thermal equilibrium (QTE) conditions, meaning that the occupation function of each electronic CT state should be considered in the expression for recombination, and that state occupation should follow Boltzmann statistics. Ultimately, when summing over all contributions from different CT manifolds (denoted as ), we have abs (ℏω) = 1 abs ∑ ∫ 3 0 ℏ 4 ( ℏω ) 3 ( CT ) 2 FCWD abs, (ℏω, CT ) ( CT ) ( CT ) , (5) (ℏω) = 1 rec ∑ ∫ 1 3 0 ℏ 4 ( ℏω ) 3 ( CT ) 2 FCWD rec, (ℏω, CT ) ( CT ) exp (− CT ) ( CT ) , (6) Where is the photon density and accounts for the strength of electro-magnetic field around the molecule; 9 0 is the permittivity of the free space; is the order of CT state manifold; is the transition dipole moment for CT manifold and is related to the oscillator strength of the CT manifold ( osc, ) under the dipole approximation 10 where , and , are the low frequency and high frequency reorganization energy for CT manifold , respectively, = , /ℏ is the Huang Rhys factor 12 , ℏ is the averaged harmonic energy spacing, typically 0.15-0.20 eV for molecules made of many carbon-carbon bonds 6,13 and and are the quantum numbers of the vibrational mode of the initial and final state.
respectively. − ( ) is then the generalized Laguerre polynomial of degree 6 . The factor abs and rec in Supplementary Equation (5) and Supplementary Equation (6) are the partition functions, and are defined as the sum of occupation of vibrational CT states for absorption, and as the sum of occupation of both vibrational and electronic CT states for recombination, and are given by  (12) where ( CT ) is the electronic coupling (EC) between CT and ground state described by generalized Mulliken-Hush method 16,17 . FCWD rec, (0, CT ) follows Supplementary Equation (8) with ℏω = 0.
We have now the rate constants and nr to model ∆ nr using Supplementary Equation (3). To predict oc , what we also need is oc,rad , which is calculated using EQE abs (ℏω) in Supplementary Equation (2).

Supplementary Method 4. Absorptance and emission probability
Knowing the absorption rate constant ( abs ) as a function of photon energy (ℏω), the absorption coefficient ( abs (ℏω)) can be calculated using Einstein coefficient as 6,14,17 abs (ℏω) = ℏ 3 2 2 1 (ℏ ) 2 abs (ℏω), (13) Where is the refractive index, and is the volume, CT  . (17) Supplementary Equation (17) correlates the emission ( (ℏω)) to absorptance ( (ℏω)). Another way of examining detailed balance is to utilize the absorption coefficient of CT states ( abs CT (ℏω)), which is more convenient using our model. Here, we rewrite Supplementary Equation In the case of absorptance from CT states, is much smaller than 1. Therefore, in order to obey detailed balance, the proportionality shown below should apply: Therefore, both Supplementary Equation (17) 18 We also note here that other models in Ref. [19][20][21] are essentially based on model by Kahle et al. 18   The equations used to perform the simulation in Supplementary Fig. 1 are the high temperature limit of Equation (8a-b) in Ref. 18 (i.e. Kahle model), which is also quoted as Equation (9) and Equation (10) in Ref. 19 , as shown here: Where (ℏω) and (ℏω) are the reduced rate constant of emission and reduced absorptance.
Note here we use the same notation for emission and absorptance as in our model to avoid confusion. In the simulation, the parameters were chosen as: CT  Therefore, the principle of detailed balance cannot be satisfied using Supplementary Equation (20) and Supplementary Equation (21) when is non-zero.        have been calculated following the method introduced previously 6 and shown in Supplementary   Fig. 10. The parameters chosen here for simulating those limits are listed in Supplementary Table   5. We consider that the driving force is zero for the simulations, i.e. CT = , based on the fact that efficient OPVs with zero driving force have been reported 24,25 . Supplementary Fig. 10a shows the calculated oc,rad and oc,nr as a function of / . Without static disorder, i.e. CT = 0 , oc,rad scales linearly with / with a slope smaller than 1, while oc,nr increases faster than oc,rad with / (slope > 1), consistent with previous findings 6,13 . With 0.1 eV of CT , we see a notable drop of both oc,rad and oc,nr , however energy gap law on ∆ nr is still obeyed 13 . As a result, both radiative (PCE rad ) and nonradiative (PCE nr ) efficiency limit are reduced significantly, as shown in Supplementary Fig. 10b.
To have an idea of how oc and PCE limits would scale with CT , we carry out further calculations for the devices with the optimum band gap ( = 1.4 eV) as derived from Shockley-Queisser (SQ) limit 5 , which is also close to of the best OPV 26 . Without static disorder, oc,rad and oc,nr are 1.05 and 0.9 V giving an ∆ nr = 0.15 eV, and PCE rad and PCE nr are 33% and 28%, respectively. Upon introducing a finite CT , we observe a notable decline of both limits for oc , as shown in Supplementary Fig. 10c. As a result, both PCE rad and PCE nr have been reduced notably due to extra radiative and nonradiative losses induced by static disorder in Supplementary Fig.   10d. With 0.1 eV of CT , we observe a 33% reduction of both oc,nr and PCE nr . These modelling results suggest that minimizing CT is crucial for maintaining high voltage and efficiency.  reproduced EQE using EL, via EL/ BB ) when temperature is varied in a large range, supporting the conclusion drawn using injection-dependent EL and temperature dependent EQE that static disorder is significant in the devices we studied here. We note here that for O-IDTBR devices, we didn't manage to acquire reliable T-dependent EL data.

Supplementary Note 3. Simulations of O-IDTBR devices
For the O-IDTBR devices, we propose that the position and relative density of both CT1 and CT2 remains constant based on the analysis in the main text. However, we suggest that osc of both CT states reduces as O-IDTBR crystal size grows upon increasing O-IDTBR wt%, based on the reduced emission intensity of CT states in Fig. 4b and previous observation 6,29 . The simulation results with changed osc are shown Supplementary Fig. 21.
Starting with injection dependent EL simulations, for all compositions, EL from CT states shows a clear two-peak transition with increased injection, which cannot be reproduced by single state model. We note here we only take account of the emission from CT states in the simulations, therefore the LE peak in experimental O-IDTBR device with 40% or 70% wt% is not reproduced.
For EQE, low injection EL, and voltage loss simulations, as shown in Fig. 6 e-g, the intensity of the EQE tail and EL emission from CT states reduce, and Δ nr increases as O-IDTBR wt% increases. This can be rationalized by the reciprocity relation, which relates emission to absorption, and the effect of high CT state emission in reducing Δ nr .

Supplementary Note 4. Injection dependent EL simulations
In the injection dependent electroluminescence simulations, the emission spectra are changing with increased injection (in both experiments in Fig. 5 and simulations in Fig. 6), which is a signature of a departure from quasi-thermal equilibrium (QTE) condition under high injection condition. That means the emission spectra are no longer describable by the QTE theory we introduced in the theory section (Eq. (11) in the main text), but in fact reflect a kinetically limited state filling effect 31 . In previous work, Gong et al. 31 addressed the problem by considering the bias dependent distributions of the electrons and holes that contribute to the EL, using Fermi-Dirac (FD) statistics. In a different approach, Burke et al. 22 argued that the limited capacity of interface states for excitons meant that the CT states should obey FD statistics. Here, we adopt an approach similar to Burke and use the formalism below to model the EL data:   Supplementary Fig. 20, and for simplicity in the modelling we assume it's unchanged while changing the composition as it's not the determining factor. The range of osc was chosen based on the change of the relative CT state emission intensity from Figure 4b, where roughly two orders of magnitude change was seen. The range of C1 was chosen based on the estimated interfacial density fraction changed based on Ref. 28 , where ~70% of crystalline interface was observed in the case of 20% O-IDFBR devices. This value was significantly reduced when more O-IDFBR was added in the blend. We here chose to have roughly two orders of magnitude changes to see the effect clearly.