Reconciling models of interfacial state kinetics and device performance in organic solar cells: impact of the energy offsets on the power conversion efficiency

Achieving the simultaneous increases in the open circuit voltage (Voc), short circuit current (Jsc) and fill factor (FF) necessary to further increase the power conversion efficiency (PCE) of organic photovoltaics (OPV) requires a unified understanding of how molecular and device parameters affect all three characteristics. In this contribution, we introduce a framework that for the first time combines different models that have been used separately to describe the different steps of the charge generation and collection processes in OPV devices: a semi-classical rate model for charge recombination processes in OPV devices, zero-dimensional kinetic models for the photogeneration process and exciton dissociation and one-dimensional semiconductor device models. Using this unified multi-scale model in conjunction with experimental techniques (time-resolved absorption spectroscopy, steady-state and transient optoelectronic measurements) that probe the various steps involved in charge generation we can shed light on how the energy offsets in a series of polymer: non-fullerene devices affect the charge carrier generation, collection, and recombination properties of the devices. We find that changing the energy levels of the donor significantly affects not only the transition rates between local-exciton (LE) and charge-transfer (CT) states, but also significantly changes the transition rates between CT and charge-separated (CS) states, challenging the commonly accepted picture of charge generation and recombination. These results show that in order to obtain an accurate picture of charge generation in OPV devices, a variety of different experimental techniques under different conditions in conjunction with a comprehensive model of processes occurring at different time-scales are required.


Device Preparation and Characterization
All devices were fabricated in the inverse architecture on pre-patterned ITO covered glass substrates. After cleaning, ZnO was deposited from a zinc acetate anhydrous solution (110mg/ml in 2-methoxyethanol with 30 L ethanolamine per 1mL) by spin coating at 4000rpm, followed by annealing at 200 degrees. All blends were dissolved in chlorobenzene at a ratio of 1:1.25 D:A, and a concentration of 20mg/ml, and were stirred overnight at 50 degrees.. Blends were spin cast at spin speeds of 1500-2000rpm in a nitrogen atmosphere, and were soft-annealed at 80 degrees for 10 minutes after spin coating. The thickness of the active layers are around 80-100nm. To complete the devices, MoO 3 (10nm) and silver (100nm) were evaporated through shadow masks to yield devices with an area of 5mm 2 .

External Quantum Efficiency Measurements
EQE measurements were carried out using a grating spectrometer (CS260-RG-4-MT-D) to create monochromatic light from a tungsten halogen light source. The light was chopped at 300Hz, and a Stanford Research System SR380 lock-in amplifier was used to detect the photocurrent. Long pass filters at 610, 780, 850 and 1000nm were used to filter out scattered light from the monochromator. The spectra were calibrated using a Silicon photodiode.

Electroluminescence and Photoluminescence Measurements
EL and PL spectra were recorded using a Shamrock 303 spectrograph combined with a iDUS InGaAs array detector which was cooled to -90°C. The obtained EL intensity spectra were calibrated with the spectrum from a calibrated Halogen lamp. A 473nm diode laser was used as the excitation source for PL spectra, which were measured using the same spectrograph and detector system as for EL measurements at the open-circuit voltage of the devices.

Drift-diffusion model.
A one-dimensional drift diffusion model was implemented to simulate the results using MATLAB's built-in partial differential equation solver for parabolic and elliptic equations (pdepe). The full details can be found in the reference . 3 . This model was used to simulate TPV, J-V characteristics and the delayed charge extraction measurements. For all the simulations we fixed the carrier densities at the boundaries to be the same as that of the transport layers in equilibrium. TPQ and TPV was simulated using a symmetric model to allow the cell to be at open circuit conditions as described in Calado et al. 3 The laser length and light intensity were varied to keep the cell in the small perturbation regime, where the transient open circuit voltage did not exceed 30 mV. We used a uniform generation profile throughout the active layer of the device to simulate the bias light.

Optoelectronic characterization. (setup)
The TPV and TPQ experiment were carried in nitrogen atmosphere. Variable, continuous intensity illumination was provided using a ring of 1W white umiled LEDs incident on the ITO side of the device. For the TPV experiment, once the device output reached a steady state, the device was perturbed using a single diffuse pulse from aPhoxX 638-150 -laser diode whose intensity was adjusted for every background light bias to assure that the system is on the small perturbation regime. The resulting voltage transient ∆V(t) (∆V<<V OC ) was measured at the contacts using a DPO 5104B Tektronix digital phosphor oscilloscope. The 1 MΩ input impedance of the oscilloscope was used to hold the device at open-circuit throughout the measurement. Output transients were fitted to a single exponential function to obtain the time constant of the voltage decay. For the TPQ experiment, the cell was switch from open circuit to short circuit condition using a MOS-FET (ZVN4306A) that was controlled using a AFG3102C Tektronic pulse generator, the same signal from the pulse generator was used to control a MOS-FET (ZVN4306A) that switched off the LED ring. The current from the device was measured using the DPO 5104B Tektronix digital phosphor oscilloscope with a 46 ohm resistance. The excess current was measured by subtracting the Ω signal without a laser pulse, and the delay time between the laser pulse and the switch to short circuit was assured using the two output of the pulse generator.

J-V characterization:
Current-Voltage characteristics were measured using a Keithley 236 source-measure unit under AM1.5 solar irradiation (Oriel 300W solar simulator) at an intensity of 100mWcm-2. All electrical measurements were carried out in a nitrogen atmosphere. A diagram of the ultrafast transient absorption setup is provided in Fig. S4. The primary light source employed in this setup is a Ti:sapphire regenerative amplifier (Solstice, SpectraPhysics Inc.) operating at a repetition rate of 1 kHz, with a nominal temporal FWHM of c.a. 100 fs. A combination OPA/Frequency mixer (TOPAS, Light Conversion Inc.) is employed to create pump pulses in the 300-800 nm region. Ultra-broadband NIR (850-1450 nm) and visible (450-800 nm) probe pulses were created via bulk supercontinuum generation, respectively employing either a yttrium aluminium garnet (YAG) or sapphire crystal, respectively. The pump pulse was modulated at a frequency of 500 Hz using a mechanical chopper synchronised to the output of the primary light source. To introduce time resolution, the probe pulse was delayed by means of a mechanical delay stages. The pump and probe pulses were independently focused onto the sample using a glass lens and spherical focusing mirror, respectively. The spot diameter of both the pump and probe pulses was approximately 500 µm at the surface of the sample. Prior to arrival of the probe at the sample, a 'reference' replica pulse was obtained using a variable reflective neutral density filter. Both probe pulse replicas were thereafter collimated and passed to a spectrograph, where their spectra were taken and stored. The resulting 2D maps were then processed using the SurfaceXplorer package (Ultrafast Systems Inc.) as well as home-built analysis software.

MODEL PARAMETERS:
Based on the principle of detailed balance we can calculate the radiative dark saturation current using 4,5 where is the absorptance of the film, the energy of the photon ( is the reduced  8 . Radiative voltage Limit ( )

Figure S 3 A) Time correlated single photon counting (TCSPC) signal of the photoluminescence decay of pristine 8C-ITIC on a film measured at 950 nm and excited with a laser at 780 nm. The total model fit is a convolution of the IRF and a biexponential decay. B) PL of 8C-ITIC, excited with a laser at 410nm and a fit for the exciton PL using the parameters in table S6.
We measured the photoluminescence and transient photoluminescence of a film of pristine 8C-ITIC. The time correlated photon counting (TCSPC) of the luminescence of the film following an excitation with a laser pulse at 780 nm is shown in figure S2 a. The signal was fitted using a convolution of the IRF (measured at the same wavelength and under similar conditions) with a bi exponential decay function : The first exponential decay can be assigned to any process that occurs faster than the time resolution of our setup, ( it mainly controls the first fast drop in the signal). The second decay can be assigned to the recombination of the exciton. The lifetime of the exciton is ( ) which gives a recombination rate constant ( ). Using the model for the

SIMULATION OF PL AND EL IN THE SIMPLIFIED MODEL.
To simulate the luminescence of the devices under different conditions, we need to estimate the population of the LE and CT states (considering they are the only excited states that can recombine radiatively); and calculate The emission flux per energy of the device (units ).
is the sum of the Where the generation of free charge replace . This approximation is only valid when 1 ∂ ∂ we consider the device under low injection conditions, and that the impact of the field and the charge distribution is negligible. In this case, the EL would be simulated by considering ( =10 22 and =0), and the PL in the case where ( =0 and =10 22 ).

CT STATE PARAMETERS
Disclaimer: In the introduction of the model, we have introduced a set of free parameters (more than 15 just for the first part of the model (figure 1 c)), and these free parameters can impact the experimental results in similar ways 9 . Considering that these parameters are hard to accurately calculate for the systems studied 10 , we rely on the experimental results as well as a set of constraint for the choices of the free parameters. The parameters chosen in this study are therefore not unique and are only valid considering our set of approximations and assumptions.

Device as a whole
Thickness of the device (d) 10 -5 cm

EFFECTIVE DENSITY OF STATES
In this section we review the effective density of the CT and LE states resultant from the assumptions of the model. The value of initially considered in the model was to ensure that the absorption coefficient of the device is high enough (absorption coefficient around 2 10 5 cm -1 close to the band gap of the material).. In the model we calculate the population density of the CT and LE states based on the reciprocity principle and the radiative recombination rate constate for the two states (equation S2). We also consider that both the CT and LE state follow a Boltzmann distribution and are in thermal equilibrium. The population density of CT and LE states under a common chemical potential ( ) can be written as:

9.2.Global analysis details.
To The GSB can also be simplified considering a reduced impact of the transport as discussed in the EL and PL case section 5 of this document. We can just solve the kinetic model in equations (S3-5) considering a small generation laser pulse. Figure S10 shows a comparison between the two model and how they both agree well in the early times and only start differing after 1 ns. We use the simple model for the initial fitting procedure.

9.4.Perfect reproduction of the GSB; case of PFBDB-T:C8-ITIC
Considering the difference observed between the model considered in the paper and the GSB data inferred from the global analysis ( figure 5 in the main text). We discuss in this section the impact of a perfect reproduction of the GSB data on the device performances.
First, we noticed that the dynamics of the GSB after the first rise in the case where the reformation of the exciton is considerably slow compared to the other processes, is mainly related to the dissociation ,recombination and reformation of the CT state. In order to best reproduce the GSB data, we can adopt 2 different approaches, 1) reduce the CT recombination rate constant or 2) increase the CT reformation rate constant.
For the first case we need to then modify the properties of the CT state (i.e. its energy and reorganisation energies); the CT state properties are modified in a way to ensure that the recombination rate constant is in the order of 20 ns -1 which is an order of magnitude faster than the one we used in the paper. The impact of changing the CT energy and its reorganisation energy on the EL, PL and EQE is presented in figure S11. From Figure S11, the luminescence and absorption properties of the device can be reproduced significantly well with the new parameters. The fit to the GSB data is presented in figure S12, and as expected the new model better reproduces the hole GSB signal. However, when we use the model parameters and simulate the JVs, we find that the J sc is significantly lower, and this is due to the slow dissociation rate constant that is needed to reproduce the GSB data. The open circuit and FF for this case is higher than the one used in the model, this is related to the slower recombination rate constant of the CT state and the slower reformation rate constant.
For the second case, we consider that the decay in the GSB is related to a fast reformation of the CT state from the free charge carriers. For this case, the properties of the CT state are similar to the one considered in the main text, but the reformation rate constant is considered to be 6 orders of magnitude higher to better reproduce the GSB decay (red line in figure S10).
The impact of this change is rather detrimental to the device performances, as the J sc and FF drops significantly due to the high recombination rate of the free charge carries.

Device fabrication and J-V measurements
The

Estimated band and effective mobilities using trap-limited transport model
Space charge limited current method has often been used to extract the mobilities of organic solar cells. This method allows us to obtain mobility values via simply fitting the slope 2 region of the experimental data following Mott-Gurney law, i.e. . However, in Here we adapt the method of numerical fitting using drift-diffusion model coupled with exponential tail states, which explicitly includes the effect of trap states and has been proved to be useful for organic solar cells 1,13,14 . Shown in Figure S13 and S14 are measured and simulated J-V and slope characteristics for electron-and hole-only devices, log ( ) log ( ) respectively. As we can see that a slope of 2 is rarely seen in the devices measured, indicating that Mott-Gurney law cannot be used in the analysis here.
In the fitting process, we aim to fit the current density under both negative and positive bias for each device, which are indicated by the current density difference in the J-V curves.
We consider five fitting parameters as shown in Table S12, within which the trap-free mobility  Table S13. The results show that D2F:C8-ITIC blend presents the highest electron and hole (trap free) mobility compared to D0F:C8-ITIC and D4F:C8-ITIC blends. In general, trap free mobility is less useful as in the operation condition the effective mobility is the one controlling charge transport. And effective mobility is determined by the charge carrier density inside the devices.
Since we have measured the free charge carrier density at 1 sun condition using charge extraction as presented in Figure 6 in the main text, we can estimate the effective mobility 15,16 ( ) at 1 sun condition using Where is the trap free mobility, and is the trapped charge density in the exponential 0 tail, given by 16 The estimated values of effective electron and hole mobilities are shown in Table S13, showing that D2F:C8-ITIC blend has higher effective mobilities than the other two blends, especially in the case of hole transport, and the transport mobilities of D0F:C8-ITIC and D4F:C8-ITIC blends are like each other. We note here that the FF and Jsc of D0F:C8-ITIC and D4F:C8-ITIC blends are very similar (see Table S13), such differences on the effective hole transport mobilities (close to one order of magnitude difference) are inconsistent with the small device performance differences. Also, similar charge transport mobilities in D0F:C8-ITIC and D4F:C8-ITIC blends again don't agree with the difference on the FF and Jsc results in OPV devices. This indicates that early time scale process plays a more important role than charge transport processes. In addition, we have shown in the main text that the energetics between CT and CS states seem to be more important than mobilities in determining the reformation rate of CT excitons from free charges. Therefore, in the drift-diffusion modelling we consider that electron and hole mobilities are identical in each blend to rule out the effect of mobilities paying more attention on the early time scale processes, such as charge generation, exciton dissociation, and CT state exciton dissociation, CT state reformation from free charge carriers.    The short circuit current depends strongly on the ratio as the generation of free charge carriers is related to how fast the formed CT states can dissociate before recombining. Figure   S15 shows the dependence of the simulated J sc on the ratio . For a large amount of < 0.5 the photo-generated CT states recombines before generating free charges. This loss can is what is referred to as geminate recombination 18 . The loss in J sc for is strongly mobility   In figure S15, we plot the results for the FF, V oc , and against , which can Since in the simulation we kept N CB , N VB and G av constant, according to equation 10 V oc should be linear with . First, the change of both the FF and V oc with in figure S16

Rate of LE recombination
( ) is analogous to the one expected from the steady state equations: the FF drops with increased , and V oc is linear with the . higher , becomes mobility dependent and shows a reversed correlation for low mobility cases, whereas decay with the same slope as the case for low . ,1 The change of with light intensity is similar for the three devices, and show two different regime: 1) a first regime where the time constant is slightly dependent on light intensity and 2) a second regime where the time constant is strongly affected by the light intensity. This suggests that the dominant recombination mechanisms in these devices' changes from a first order process at low light intensity and a second order process for higher light intensity.

MARCUS RATE FOR THE DISSOCIATION RATES OF EXCITON AND CT STATES.
We can use Marcus rate constant formula to describe the dissociation of the excitons or the dissociation of the CT state 20 . The rate constant can be written as Where is the electronic coupling between the initial and final state, the reorganisation energy of the transition, the Planck constant, T the temperature and k B the Boltzmann's ℏ constant. is the difference in free energy between the initial state and the final state. We use Δ this formula to fit the change in the dissociation rates of the LE and the CT state along the series with the electronic coupling and reorganisation energy as fitting parameters. The results of the fit and the model results are presented in table S17.

REFORMATION RATE CONSTANT AND LANGEVIN ENCOUNTER RATE CONSTANT.
The reformation rate constant of CT state from free charge carrier ( ) could be , approached by the encounter probability in the Langevin recombination framework as has been done by Burke et al. 21 . In this framework can be approached by . We use this formula to back calculate the average charge carrier mobility needed to ensure for each of the devices studied (Table S11)  is also found to reduce with reduced . Δ , For the P4FBDB-T blend, is almost 3 times higher than that estimated using , . For an average charge carrier mobility of 3 10 -4 , This means that for this blend with the parameters considered the = 2.57 10 -10 Langevin reduction factor is >1. Now we discuss the case where the encounter probability is a limiting process ( i.e.

IMPROVING THE EFFICIENCY OF LOW OFFSET BLENDS.
We have found that reducing the offset between the HOMO of the donor and the HOMO of the acceptor affects different processes in the devices (dissociation of the CT and LE state and the reformation of the CT state from free charges). To improve the efficiency of the lowest offset system we propose to increase the CT state dissociation rate constant (  In this first case, when increasing we improve the charge generation efficiency. Since is increased simultaneously the device performance does not increase significantly due , to a reduced FF. Table S15 summarises these results. due to both an increase in J sc and the FF.

CASE WHERE THE CT TO CS STATE TRANSITION IS INDEPENDENT OF THE ENERGY OFFSET.
In this study we have found that changing the HOMO or the ionisation potential of the donor not only affects the LE to CT electron transfer process ( ) but also the transition , from the CT to CS (through both and ). In this section, we explore the case where , , we consider that changing the energetics of the donor impact both the CT and CS states energies in a similar way, so that is constant. This assumption means Δ , ≈ Δ 0that the rate of CT dissociation and reformation are the same for all the blends.
For this case we consider that the , and are the same for all the  The comparison of the GSB dynamics simulated with the two sets of parameters ( figure S24). show that the model with fixed has a slower rise for both PFBDB-T Δ , and P4FBDB-T blends. Moreover, due to the low recombination rate constant of the CT state ( ), the GSB does shows a small drop as compared to the model with , the 4 free parameters. This shows that for we need to consider a change in the CT to CS transition to accurately reproduce the experimental results.
When comparing the simulated luminescence and EQE spectra of the two sets of parameters (figure S25), we find that the EL and EQE spectra for case 2 is blue shifted as compared to both the experimental results and the spectra from case 1.  Finally for the device characteristics of the device, we find that if the CT to CS energy offset was kept the same (case 2) the lowest offset device would have a considerably higher PCE, mainly due to the strong increase in the V oc , without affecting the FF.