Adjusting the energy of interfacial states in organic photovoltaics for maximum efficiency

A critical bottleneck for improving the performance of organic solar cells (OSC) is minimising non-radiative losses in the interfacial charge-transfer (CT) state via the formation of hybrid energetic states. This requires small energetic offsets often detrimental for high external quantum efficiency (EQE). Here, we obtain OSC with both non-radiative voltage losses (0.24 V) and photocurrent losses (EQE > 80%) simultaneously minimised. The interfacial CT states separate into free carriers with ≈40-ps time constant. We combine device and spectroscopic data to model the thermodynamics of charge separation and extraction, revealing that the relatively high performance of the devices arises from an optimal adjustment of the CT state energy, which determines how the available overall driving force is efficiently used to maximize both exciton splitting and charge separation. The model proposed is universal for donor:acceptor (D:A) with low driving forces and predicts which D:A will benefit from a morphology optimization for highly efficient OSC.

Modeling of ground state absorption (GA) and photoluminescence (PL) spectra is an important prerequisite for a correct interpretation of photobleaching (PB) and stimulated emission (SE) features, respectively, in transient absorption (TA) data. Supplementary   Fig. 1 shows PL and GA spectra of as-deposited WF3 thin films (solid lines). Fits are given as dashed lines. The PL spectra can be modeled by a single effective vibronic progression of Gaussians with a (00) transition at 1.692 eV, an energy of the effective vibronic progression of 157 meV, and an effective Huang-Rhys (HR) factor of 0.83, which might be slightly exaggerated due to reabsorption effects.
For the modeling of GA in WF3, we found that we can only obtain a good fit by assuming the simultaneous presence of an amorphous and an ordered phase, both showing similar HR factors as PL. Assuming a single phase requires an HR factor of 1.6, in stark contrast with the values from PL. The ordered phase is redshifted against the amorphous phase by about 180 meV and is characterized by narrower Gaussian bands.
7 In Supplementary Fig. 3, we show a GA spectrum of the blend (grey solid line). It can be fitted (black dashed line) quite well by a superposition of contributions from the pure as-deposited WF3 film and the annealed O-IDTBR film (red and blue lines, respectively). As Supplementary Table 1 shows, the center energies for both pure materials and the band widths had to be slightly adapted for a good fit.
Supplementary Transient absorption (TA) spectroscopy shows optical probes from various photoexcited states in a single experiment, thus making it an ideal tool to study complex photoexcitation dynamics. However, organic materials typically have broad TA bands, which is due to disorder and vibronic coupling. This generally leads to spectral congestion, which must be treated by matrix decomposition techniques coupled with nonlinear optimization schemes in order to obtain time-resolved concentrations of all contributing photoexcitations individually. i Our approach has been detailed in a previous publication. ii In short, we typically measure time-resolved TA spectra (twodimensional matrices as function of probe photon energy ω and pump-probe delay time t) varying a third parameter such as the pump intensity J. Following Lambert-Beer's Law, the experimental transient absorption Aexp is given by: where s ( )is the time-resolved area density of state m, and is the absorption crosssection of that state, which depends on the probe energy and follows a spectral model, for example, a combination of Gaussian bands whose parameters (relative strength, width, center, etc) are given in terms of the vector of hyperparameters .
Goal is to find s ( ) for all interacting states. To this end, we perform a singular value decomposition (SVD) yielding characteristic spectra and dynamics. We retain only the signal-related characteristic spectra and corresponding dynamics, given by the US, and VS matrices, respectively, rejecting noise. The characteristic spectra US are related to the matrix σ of absorption cross-sections via a rotation matrix R by The rotation matrix R is obtained by Moore-Penrose pseudo inversion while varying the hyperparameters χ of the cross-section spectra in σ in a nonlinear optimization loop.
Once the rotation matrix is obtained, the desired photoexcitation dynamics is obtained by where SS is the set of signal related singular values, and the matrix s is composed of the individual sm(t) in eq. S1. A detailed derivation is given in the Appendix of this Supporting Information.
We use as much experimental evidence as possible to reduce the number of free hyperparameters in χ. For example, for each state we train the hyperparameters using pure films. Then we use S2 is applied to predict ( ) for the blends, keeping most of the hyperparameters to those found in the pure films. If we allow hyperparameters to vary, then it is because of a clearly described physical reason. In the Appendix of this Supporting Information, we give the full theory and the rules which hyperparameters are allowed to be optimized. weighted superposition of simulated spectra of emissive singlet states (S, red solid lines) and polaron pairs (PP, blue solid lines). c) Resulting time-resolved concentrations of S and PP states, according to eq. S14.
In Supplementary Fig. 4, we show a SVD of a time-resolved TA spectrum (10 fs broadband pulses, parallel polarization of pump and probe pulses) of an as-deposited WF3 film. Prior to performing the SVD analysis, the spectra have been logarithmically binned along the time axis into 12 bins associated with an effective delay time each (X axis in panel b). The singular values are shown in panel c in descending order, each singular value associated with a column vector (basis spectra) and a row vector (dynamics). As shown in panel c, there are two singular values clearly offset in strength against the other ones, which means that the associated basis spectra and dynamics (shown in panel a and b, respectively) will capture nearly all spectral and dynamic information. The rest of the singular values will be dominated by noise, however, they might still contain some spectro-kinetic information, such as spectral migration and line narrowing effects. In the present case, it is acceptable to ignore these effects, because the strongest noise-related singular value is a factor of 5 weaker than the weakest signalrelated singular value, see panel c.
In Supplementary Fig. 5, we show time-resolved TA spectra of a WF3 film (solid lines of different color according to delay times, as given in legend). The TA spectra exhibit transient photobleach (PB) features around 1.9 eV, in the region of GA, an a broad, a structurelessPA band peaking in the near infrared, outside of the spectral range of our probe pulse. For delay times below 1 ps, a second band of formally negative differential absorption is present around 1.6 eV. As this is outside of the range of GA, it can only be associated with SE. However, the spectral position of the SE band in Supplementary  Fig. 5a does not exactly agree with that of PL in Supplementary Fig. 1, which peaks at around 1.7 eV. This might be explained by emission from a vibrationally hot state.
In Supplementary Fig. 6   As we explain by discussing Supplementary Fig. 11, these charged states will in general have different excited state absorption bands than the neutral excitations. However, they share the same PB and show no SE. In the absence of further effects, the PB region in The resulting dynamics are given in Supplementary Fig. 9bc; it is obvious that the EArelated features decay more slowly than the population-related features CT+CS and LE.
A relative increase of the EA contribution with respect to the population contribution has been ascribed to charge separation. iii The dynamics of this relative increase is therefore a measure of the initial charge separation kinetics, i.e., from an interfacially bound CT state to state where at least one of the charged states is no longer at the interface. With these spectral models, we achieve a good reproduction of the original TA spectra at all delay times, considering that all spectral shapes are derived from other optical probes and no new bands have been introduced ( Supplementary Fig. 10 a,b).   Supplementary Fig.10a with the colored dynamics in Supplementary Fig-10b,c.
Pump-probe delay times are given in the inset.
Supplementary Figure  In Supplementary Fig. 11, we give a schematic representation of the states contributing to the TA spectra in Supplementary Figs. 9 and 10 and explain their specific optical probes. We start our discussion with the formula for the calculation of the differential absorption, Here, d is the film thickness, σ is the absorption cross-section, ω is the probe photon energy, D 0 and A 0 are donor and acceptor molecules, respectively, in the ground state and not exposed to a static electric field, D F and A F ground states exposed to a static electric field, D* is a symmetric charge transfer exciton in the donor phase (nonemissive and not carrying net charges) and A* are emissive excitons in the acceptor phase, D + are positive charges in the donor phase and Aare negative charges in the acceptor phase.
We further assume that at our repetition rate of 1 kHz, all pump-induced excited states have decayed back to the ground state (verified by the fact that there is vanishing TA signal for negative delay times). Furthermore, there is no external electric field. Then, only D 0 and A 0 will contribute to Apump off, the concentration of all other states being zero. Therefore, which means that any state produced in the donor phase will, apart from the specific photoinduced absorption bands PA, also produce a ground state photobleach PB caused by the reduction of the concentration of ground state molecules. The same is true for the acceptor phase. Hence, PA bands are specific for the kind of photoexcitation but not for the phase, while PB bands are specific for the phase in which the excited states dwell but not for the kind of photoexcitation.
Turning now to the right part of Supplementary Fig. 11, all the electronically excited states WF3*, O-IDTBR*, or charged states CT and CS, will produce an individual unique set of PA bands (shown at lower probe energy). It is however possible, that due to the intramolecular CT character of low energetic transitions in low-bandgap moieties, these bands are relatively close together so that they often cannot be safely distinguished.
all the electronically excited states in the donor phase must exhibit a PB band which is the inverted replica of the GA of that phase (ignoring hole burning and spectral migration) same for all excited states in the acceptor phase.
This means that, CS and CT (and hence, their time-resolved populations) cannot be distinguished by looking at the PB. Both will produce PB in both phases. However, being charged these states cause static electric fields that red shift the ground state absorption (GA) in neighboring non-excited molecules (Stark effect). Since this shift is much smaller than the bandwidth, the resulting TA spectra can be approximated by first derivatives of the GA. A differential absorption upon modulation of static electric fields is called electroabsorption (EA); in our experiment we modulate the field by modulating the number of charged pairs causing it.
The number of molecules situated between the charged pair, where the field lines superpose constructively, is larger in a CS than in a CT state. This means that although we can only measure the sum of the populations of CS+CT (by virtue of their common PB), we can get an indication of their relative weight (and hence the transition from CT to CS) by comparing the evolution of the intensity of the EA bands to that of the PB bands (see point E at the end of this ESI for a simple electrostatic simulation). In order to distinguish them, we make use of the fact that the first derivative of a band has zero integral. (see dashed vertical lines in Supplementary Figure 11). Hence, if in the TA spectrum, the spectral weight of the negative and the positive lobe of the first derivative contribution are not equal, we know that the spectral region is superposed with PB. This distinction is systematically exploited by the matrix decomposition technique that we deploy.

SUPPLEMENTARY NOTE 3: Determination of the effective CT energy
We use a multi-objective optimization to reproduce the crucial observations of the main text (an EQE exceeding 80%, a charge separation time of 40 ps, and an EL spectrum which is indistinguishable from the pure LE spectrum) by the simplest possible rate equation model. Using the known overall driving force (difference between the LE and CS energies) and the experimental or calculated LE and CT deactivation rates, this allows us to determine the effective CT energy, relative to that of the LE and CS energies, as well as the rate constants for the two equilibration reactions (LE  CT,

CT CS).
Given the notion of an equilibrium between LE and CT states iv and between CT and CS states, v the simplest ordinary differential equation ( Single arrows refer to first-order processes while the double arrow refers to bimolecular Langevin recombination of charge carriers. Black arrows refer to processes whose rate constants are known in the present system. For the multi-objective optimization, we solve the ODE system (S6-S8) iteratively varying the four degrees of freedom and setting the initial conditions in order to model correctly the following three objectives: (1) To simulate EQE, we calculate the stationary state of the ODE system (S6-S8) by letting gpe=const and setting the left sides zero, and calculate IQE as where eand h + is the area flux of extracted electrons and holes, respectively, CSst is the stationary concentration of charge separated states under an absorbed stationary illumination flux gpe, which is set to one sun under AM1.5 conditions and a bandgap of 1.65 eV, considering an effective optical density of 0.6 due to above-bandgap transmission, reflection and parasitic absorption losses, and dAL is the active layer thickness.
(2) The observed EL spectra are calculated as the sum of the individual contributions from CT and LE states, ELLE and ELCT, respectively. For this experiment, injection is modeled via setting ginj(t) = const in eq. S8 and solving for stationary condition: Herein, LEst and CTst are the stationary concentrations of LE and CT states, respectively, PL,LE ( )and PL,CT ( )are the spectral shapes for emission from LE and CT states, respectively, normalized to unit spectral weight, and r is a geometrical constant comprising outcoupling efficiency and detector geometry. The spectral shape PL,LE ( )is known from PL/EL spectra of the pure o-IDTBR film/device, while for the spectral shape of PL,CT ( ), we assume a Gaussian band with the center at ECT and a width of 0.08 eV; this is chosen to be only slightly larger than the width of the LE emission (fitted to 0.063 eV) reflecting the strong LE-CT hybridization which should yield similar spectral shapes albeit shifted in energy.
(3) Finally, the rise of CS states is modeled by letting = (0), a delta pulse with unity intensity at time zero, and solving the dynamics of LE, CT, and CS states by diagonalizing the transfer matrix. Since no stationary charge density is created by this experiment, we do not consider charge recombination (kL=kCS,CT=0).
Multi-objective optimization is performed using linear scalarization of the offsets of the simulated targets with respect to the experimental targets: (1) an IQE of 85%, (2) the root mean square error between the actual EL spectrum and the simulated one, and (3) the deviation of the CS rise time from 40 ps.
The forward rate constant for charge separation is given by AECT,CS being a formal activation energy in the Marcus picture, and , 0 = 10 13 −1 the prefactor ("attempt-to-escape frequency").
The backward rate constants are obtained from the forward rate constants by considering the two Boltzmann equilibria: where DLE, DCT and DCS are the degeneracy factors of LE, CT, and CS states, respectively, kB is Boltzmann's constant, and T is the absolute temperature. The constant kL,0 is given by standard Langevin-type recombination: where q is the elementary charge, μn and μp are the electron and hole mobilities, respectively, and ε0 is the dielectric constant in vacuum and εr is the relative dielectric constant. Assuming μn.= μp = 10 -4 cm 2 /Vs and εr=3, we obtain kL,0 = 1.2.10 -10 cm 3 /s. In the multi-objective optimization, we allowed kL0 to float around this value. We found that for AECT,CS < 0.1, kL,0 tended towards the upper limit, otherwise towards the lower  1.65 eV iv 1.44 eV iv 1.2 . 10 6 s -1 iv 1.9 . 10 9 s -1 iv 3 . 10 5 s -1 iv 2.8 . 10 10 iv 0.1 iv 30 To get more insight into the optimization procedure, we do not blindly fit all 4 free parameters at once. Instead, we vary two of them ( LE, and , ) in a grid, fitting the remaining two parameters for each grid point.
In Supplementary Figure 13, we present the result of the multi-objective optimization.
In the grid, we have varied LE, from -0.3 to 0.3 eV, and , from 0 to 0.2 eV. For each grid point, kextr and kL0 was optimized to minimize the Euklidean distance (RMSE) of the three objectives to the perfect match. As can be seen in Supplementary showing too much CT contribution. Vice versa, we are able to perfectly match the EL spectrum and IQE, but then the charge risetime becomes too slow.
The occurrence of a pareto front clearly shows that the system that we are using, is not over-defined. It is in principle possible to add another degree of freedom, in order to achieve a perfect match of all three objectives.
Supplementary Figure 13. Result of the multi-objective optimization of the three objectives using the rate equation scheme (S17 -S19

SUPPLEMENTARY NOTE 5: Electrostatic calculations
In order to interpret the EA/(CT+CS) curve in Figure 2F of the main text, we need to predict the strength of electroabsorption features as function of the electron-hole separation. Scarognella et al have used an electrostatic model to predict the evolution of the local electric field at certain prominent positions close to the charged state in a cocrystal phase. xii In our study, we simulated the electrostatic field in a 10 X 10 X 100 (x,y,z) grid of molecules (edge dimensions 1X1X1 nm each). We assume a sharp interface between the donor (z50) and the acceptor (z>50)  Our high time resolution broadband TA experiments excited both moieties simultaneously. In order to distinguish charge generation after exciting the WF3 and the O-IDTBR phase, we performed TA spectroscopy with 100 fs narrowband pulses tuned to 2.07 and 1.7 eV, respectively (blue and orange curves, respectively, in Supplementary Fig. 21). The curves are normalized to the value at 1.85 eV, corresponding to the PB of the WF3 phase. Immediately after pumping, the relative amount of PB(O-IDTBR) at 1.7 eV is strongly dependent on the pumping condition, see lower part of Supplementary Fig. 21. After pumping at 1.7 eV, in resonance with the O-IDTBR* exciton, the corresponding PB is much stronger than that of WF3. After pumping at 2.07 eV, we predominantly excite WF3; in this case, the PB of both WF3 and O-IDTBR phases is observed. This agrees with the broadband experiment ( Figure   2c) and can be explained by the ultrafast formation of an interfacial CT state. If we pump the O-IDTBR phase, we observe a strong band at 1.7 eV, indicative of PB/SE of the singlet exciton, from which we conclude that such ultrafast CT formation does not occur after pumping the O-IDTBR phase (orange curves in Supplementary Fig. 21).
This was also observed in other small-molecule acceptors and explained by the diffusion of smaller excitonic systems towards the interface. iv Once full charge separation has taken place, we expect the PB ratio to be that of the fully charge separated states, and thus independent of the pumping conditions. Surprisingly, we find a significant "excitation memory effect" in the PB ratios even for J-V measurements: The J-V characteristics were measured using a source measurement unit from BoTest. Illumination was provided by a solar simulator (Oriel Sol 1A, from Newport) with AM1.5G spectrum at 100 mW/cm 2 . UV-VIS absorption was performed on a Lambda 950 spectrophotometer, from Perkin Elmer. EQEs were measured using an integrated system from Enlitech, Taiwan. All the devices were tested in ambient air.

FTPS
being a stochastic noise contribution to each data point. SA3 can be restated as a matrix equation: In SA4, ∈ {1,2, . . } indexes the different pump intensities Jj used, σ is an (Nω × Ns) matrix of individual cross-sections of all states, arranged as column vectors, while sj is an (Ns × Nt) matrix of individual area densities, arranged as row vectors, and N is an (Nω × Nt) matrix of measurement noise.
The goal is to solve S4 for sj, given the experimentally measured set of matrices Aj. To this end, we must reject the noise N and introduce our knowledge about σ: where −1 is the pseudo-inverse of . In eqns. S3 through S5, we have assumed that does not depend on J (only linear absorptions considered), while sj does. This allows us to concatenate all matrices along the time axis, forming one single matrix with such that where the matrix elements smr and Nir are given by the same definition of r as in eq. S6.
Note that the shape of σ in SA7 is still (Nω × Ns), while the shape of s is different from that of sj, namely (Ns × [Nt × Nj]). A singular value decomposition (SVD) of A exp yields Here U is a matrix of unitary basis column vectors, V a matrix of unitary row vectors, and S a diagonal matrix of weight coefficients ("singular values"). S can be decomposed into two matrices SS and SN containing only those singular values related to signal and noise, respectively, so that = + . Then: In S9, we have exploited the property that a truncated SVD is always optimal in a least Inserting the identity • −1 into SA10: and applying the pseudoinverse of • : we find that if In SA13, R is the rotation-scaling matrix transforming the signal related basis column vectors of the SVD (unitless) into the physically relevant absorption cross-sections (in cm 2 ) of the available states, while in SA14, the inverse of R transforms the signal related basis row vectors of the SVD into the physically relevant area densities (in cm -2 ) and thus into the photoexcitation dynamics, our desired quantity. We thus must solve SA13 to find s in SA14.
We solve S13 by assuming physically reasonable test functions for the column vectors in σ, according to the following rules: -We generally use Voigt or Gaussian band shapes. Vibronic coupling is introduced by assuming one essential vibronic progression with one effective Huang-Rhys factor. The intensity of the (00) vibronic progression can be reduced (H aggregation) or enhanced (J aggregation). In any case, the integral of a full progression is one.
-Thermal population of low-frequency vibrational or torsional modes can be considered by introducing a skewness of the bands.
-All states show a ground state photobleach (PB), which is an inverted and scaled (to area 1) replica of the ground state absorption (GA).
-Emissive singlet excitons show stimulated emission (SE), which is an inverted and scaled (to area 1) replica of the steady state photoluminescence (PL) spectrum.
-Charged states can (but not always) introduce electroabsorption-like (EA) features in molecules in the electronic ground state.
Once we have defined the tests functions in σ, we perform a non-linear optimization of S13 varying the bandshape parameters. During each iteration, optimum values for R are obtained by calculating the pseudo inverse = −1 • , until the residuals − • −1 are minimized. Success of the operation can be shown by calculating s from S14 and then comparing the fitted = • to the experimental one, compare thin and bold curves, respectively, in Supplementary Figs. 7c and 10. The following rules are applied: 59 -All states in the same phase share the same PB, in both shape and intensity. This dramatically reduces the number of free parameters, because the absolute crosssections and the approximate shape of PB can be derived from GA, because it is the same transition and thus has the same total oscillator strength. Only a small redshift of the center energy and some narrowing (both due to spectral migration effects) is allowed going from GA to PB. A broadening or blue shift is not allowed. A change of total oscillator strength or HR factor is also not allowed.
-For SE, a reduction of the (00) transition with respect to PL is allowed, caused by reabsorption of the typically optically thick films. If strong geometric reorganization in the excited state is expected, also the absolute area of SE can be different from one and therefore different from PB.
-For EA, we assume no spatial correlation of state energies and therefore we allow no spectral migration effects; therefore, exact first and second derivatives of GA are assumed, and no narrowing/redshift is allowed.
-If states have an entirely different electron configuration (neutral => charged; singlet =>triplet), they cannot have exactly the same PA bands.
These physically reasonable restrictions guarantee that a good fit yields correct physical states and at the same time reduces the number of free parameters. After the nonlinear optimization, the thus obtained preliminary rotation matrix should be scaled to the final one such that application of SA14 yields time-resolved concentrations that agree with known boundary conditions, such as the total number of excited states at time zero. If this is not done, the concentrations are not to scale, but still the normalized dynamics are correct. This means that first order rate constants can still be obtained absolutely but not second order (bimolecular) ones. Also, a branching ratio of a parallel reaction of type A->B; A->0 cannot be given, but the total loss dynamics of A and the formation dynamics of B will still be correct.
For the purpose of the present paper, relative concentrations are sufficient, because we are interested in a change in the population/EA ratio, which we use as a probe for charge separation. Therefore, a scaling of the rotation matrix has not been done, and all concentrations are given in arbitrary units.