Efficient energy transfer mitigates parasitic light absorption in molecular charge-extraction layers for perovskite solar cells

Organic semiconductors are commonly used as charge-extraction layers in metal-halide perovskite solar cells. However, parasitic light absorption in the sun-facing front molecular layer, through which sun light must propagate before reaching the perovskite layer, may lower the power conversion efficiency of such devices. Here, we show that such losses may be eliminated through efficient excitation energy transfer from a photoexcited polymer layer to the underlying perovskite. Experimentally observed energy transfer between a range of different polymer films and a methylammonium lead iodide perovskite layer was used as basis for modelling the efficacy of the mechanism as a function of layer thickness, photoluminescence quantum efficiency and absorption coefficient of the organic polymer film. Our findings reveal that efficient energy transfer can be achieved for thin (≤10 nm) organic charge-extraction layers exhibiting high photoluminescence quantum efficiency. We further explore how the morphology of such thin polymer layers may be affected by interface formation with the perovskite.


Supplementary Methods 1 Details of Absorption Measurements
The absorption spectra of the different polymers are depicted as Supplementary Figures below. Spectra were recorded using a range of different source/detector combinations (sources: tungsten halogen lamp (NIR source) and deuterium lamp, detectors: GaP and Si diode detectors) in order to cover the full range from the low-wavelength absorption of the polymers to the higher-wavelength absorption edge of MAPbI 3 . For most polymer:MAPbI 3 :quartz samples, the MAPbI 3 absorption is the dominant feature in the spectra and the specific polymer absorption cannot be extracted. However, for the P3HT samples on MAPbI 3 a much thinner (65 nm) MAPbI 3 layer was used to enable extraction of the polymer absorption by subtracting the absorption spectrum of a MAPbI 3 :quartz film from the absorption spectra of the P3HT:MAPbI 3 :quartz films (see Fig. S4). The resulting P3HT absorption spectra for films of different thicknesses, and the MAPbI 3 :quartz absorption spectrum used for the subtraction, are shown in Fig. S5. This approach enables us to investigate polymer aggregation effects in P3HT films deposited on a MAPbI 3 layer by comparing the P3HT-component of the absorption spectrum for P3HT films of different thickness and spin speed on the quartz:MAPbI 3 substrates. Table S2 lists the thicknesses of all polymer films investigated in this study.

Details of PL Measurements
In addition to the description of PL measurement details provided in the Methods section of the main text, we list here polymer-specific excitation wavelengths and fluences for the steady-state PL spectra, and the detection wavelengths for the PL transient measurements. Table S4 provides for the different polymers on quartz an overview of the excitation and detection wavelengths, and decay lifetimes obtained from mono-exponential fits to the data in the time ranges 0.1-3 ns (F8BT and Super Yellow) and 0.1-1.5 ns (P3HT and PTAA). S1

F8BT
The PL excitation decays of F8BT films of different thickness on quartz and on MAPbI 3 :quartz substrates are shown in Fig. 1 in the main text. The samples were excited at a wavelength of 400 nm. The PL excitation decay was observed at 540 nm. No fluence dependence was observed and the excitation fluence for the decays shown was 3 nJ cm −2 .

Super Yellow
The PL excitation decays of Super Yellow films of different thickness on quartz and on MAPbI 3 :quartz substrates are shown in Fig. 1 in the main text. The samples were excited at a wavelength of 400 nm. The PL excitation decay was observed at 600 nm. No fluence dependence was observed and the excitation fluence for the decays shown was 3 nJ cm −2 .

Details of Photoluminescence Quantum Efficiency Measurements
Photoluminescence Quantum Efficiency (PLQE) was measured for all polymer films on quartz. PLQE, φ, can be obtained by comparison of the integrated PL intensity from the sample with a known PLQE standard with φ ref , taking into account the difference in absorption at the excitation wavelength λ exc [1]: Supplementary Figure 1: Parasitic absorption of the solar spectrum from thin layers of PTAA and P3HT. The solar spectrum (orange) is taken from NREL [3] and the polymer transmission (blue dashed line for a 10 nm layer of P3HT and greeen dash-dotted line for a 10 nm layer of PTAA) is calculated from absorption data of each polymer. The relative losses incurred from parasitic absorption in the sun-facing polymer layer is calculated by integration over the polymer-layer filtered sun-light and division by the integral over the unfiltered light, within the spectral region up to 760 nm (the band edge of MAPbI 3 ). For a 10 nm polymer layer, 1% of the solar spectrum is lost for PTAA and 6% are lost for P3HT, which increases to 11% for a 20 nm-thick layer of P3HT.     FRET to model ET between the polymer and the MAPbI 3 . In a first step, the application of FRET to a two-layer geometry is outlined. The theory leads to an expression for the PL decay as a function of time, which is fitted to the PL decay traces in a second step. Using the results from the first two steps, the efficiency of ET from polymer to MAPbI 3 is calculated based on parameters of the polymer layer in a third step.

FRET in a Two-Layer Geometry
A theory of resonant ET from an energy donor to an energy acceptor based on their dipoledipole interactions was developed by Theodor Förster [9]. The rate of the ET, k(r) as a function of donor-acceptor distance r, donor lifetime τ d , and the Förster radius R 0 is expressed as: where κ 2 accounts for the geometric orientation of the donor and acceptor dipoles, φ d is the PLQE of a purely donor sample, N A denotes the Avogadro constant, n is the refractive index and J is the overlap integral between donor emission and acceptor absorption [9,10].
R 0 has units of nm if J is entered in units of dm 3 nm 4 mol cm [11]. R 0 defines the distance at which any other pathway of de-excitation of the donor is as likely as de-excitation via FRET. The orientation factor will be assumed to be κ = 0.845 2/3 for further discussions, which assumes random but fixed relative orientations of donor and acceptor dipoles within a solid film [12].
Förster's original expression for the rate of ET between donor and acceptor as stated above was derived for ET within a solution (3-dimensional space) with uniform donor and acceptor concentrations [13,14]. As FRET has become an important tool for measuring nano-scale distances, the theory has been extended to different applications and sample geometries [15,16,17,18,19,20,21].
A more general expression, which can also take into account the effect of restricted geometries, was formulated by Klafter and Blumen [19] ("KB-equation"), who introduced a dimensionality parameter β, which is linked to spatial dimensions as β = d/6 (in case that no further spatial restrictions but dimensionality exist) to describe the donor decay I d (t) as: where p is a parameter linked to the local acceptor concentration and also dependent on β. This formalism has been further extended to also incorporate arbitrary distributions of donors and acceptors of which a derivation and overview is given in Refs. 22, 21. As main result, the donor decay I d (t) as a function of time for ET between donors and acceptors, integrated over their positions r d and r a within a restricted volume, is given by: with: where C d (r d ) and C a (r a ) denote the heterogeneous donor and acceptor concentrations, respectively. Yekta et al. [23] applied the formalism to systems with planar geometry and used it to derive analytic expressions for fluorescence decay inter alia for the situation of "parallel slabs of donor and acceptor touching at a sharp interface". Their approach will S22 be outlined in the following and in the final step we will introduce a non-uniform donor concentration C d (z) to account for the fact that the number of initially excited donors as a function of film depth z is proportional to a Beer-Lambert profile.
For systems with planar symmetry, the donor and acceptor concentrations, C d (z) and C a (z) respectively, vary along a single spatial direction, here denoted as z-axis. For any thin vertical slice of the film, the donor and acceptor concentrations are assumed to be constant and Eqns. S.5 and S.6 can be rewritten as: Geometrical considerations lead to the general expression for I d (t) for systems with planar symmetry: (S.11) The planar system of interest for us consists of two extended layers touching at an interface. In our case, one layer consists of a luminescent polymer donating energy to the perovskite acceptors in the second layer. In order to arrive at a numerical expression for this situation, the interface is positioned at z = 0, the donor layer extends from z = 0 to L and the extended acceptor layer spans from z = 0 to −∞. The assumption of an infinite acceptor layer is well justified as most of the ET happens within a few R 0 , which is much smaller than the thickness of the MAPbI 3 layers we are considering. The concentration of donors and acceptors within each respective extended layer can be assumed to be uniform: C d and C a .
However, the number of initially excited donors will be assumed to follow a Beer-Lambert profile and is therefore not uniform. From these assumptions and geometrical definitions, the S23 concentrations can be written as: where α is the absorption coefficient of the polymer at the excitation wavelength used for the donor decay measurements and H(x) denotes the Heaviside function (0 for x < 0 and 1 for x ≥ 0). Using these definitions, the donor decay can be written as: where the substitution

PL Decay Modelling
Equations 1 to 4 in the main text enable modelling of the PL decay data from the polymers on MAPbI 3 :quartz. The modelling was performed in two steps: First, we used the expression for I d (t) in Eq. S.14 and R 0 as a fitting parameter to fit the data from SuperYellow:MAPbI 3 :quartz.
In a second step, we assume that the difference in R 0 between the polymers is a function of PLQE only (discussion below) and use the measured PLQE values to obtain I d (t) through Eq. S.14 for F8BT, P3HT and PTAA.
The parameters for calculating I d (t) are: the donor lifetime τ d , donor (polymer) and acceptor (MAPbI 3 ) concentration C d and C a , polymer film thickness L, Förster radius R 0 and absorption coefficient of the polymer α. We will discuss and analyze these parameters and how they were determined in the following.

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• The donor lifetime τ d in absence of any acceptors was obtained from polymer films on a quartz substrate. A mono-exponential decay was fitted to the TCSPC data over the following time ranges: 0.1-3 ns (F8BT and Super Yellow), 0.1-1.5 ns (P3HT) t 0 was set to the peak of the rise of the TCSPC data. The fits are indicated as dashed lines in Fig. 2 and the obtained lifetimes for the different polymers are listed in Table S4.
In addition, we fitted a mono-exponential decay at early times (<1.5 ns) and at later times (1.5-4 ns) of the PL transient for F8BT and SuperYellow, in order to test our assumption of a mono-exponential decay. Deviations from the donor lifetimes obtained are small (7% for F8BT and 13% for SuperYellow) and do not influence the modelled ET efficiency. This ensures that our assumption of a mono-exponential PL decay does not noticeably influence the ET efficiency we obtain through our modelling.
• The film thickness L of the polymer film on MAPbI 3 was obtained from DEKTAK measurements and is used as a specific input parameter. The values for the polymer films used in this study are given in Table S2.
• The donor concentration C d (in units of a number density) is assumed to be uniform within the polymer film and is used as a scaling factor only. C d scales the overall PL intensity in Eq. S.14. The fitting is performed with normalized PL decay traces, which are not determined in absolute terms. Therefore, we do not assign a specific value to C d . However, the concentration of excited donors as a function of film penetration depth, C d (z) was assumed to follow Beer's law (see Eq. S.12), which was incorporated into the expression for I d (t) in Eq. S.14.
• The absorption coefficient α was obtained from absorption data (see above) at the excitation wavelength for the PL decay measurements (see Table S4) and from the polymer film thickness (see Table S2). It is a fixed input parameter to the model, specific to each polymer. The values are listed in Table S4 in units of nm −1 as entered in I d (t). The unit nm −1 is used for consistency as the film thickness L and the film penetration depth z are entered in nm in Eq. S.14.
• The acceptor (MAPbI 3 ) concentration C a can be expressed a function of the refractive index n, PLQE φ d , the spectral overlap J and the Förster Radius R 0 . These physical quantities therefore become input parameters for modelling the PL decay data and will be additionally discussed. First, however, an equation to express C a in terms of these quantities will be derived. C a has units of nm −3 as we enter R 0 in units of nm in Eq. S.15. The Förster radius is defined in Eq. S.3 and can be rewritten in terms of units as [10,11,24]: with the overlap integral J as a function of wavelength λ in nm. J is given as: where I d (λ) is the normalized donor emission and a (λ) is the acceptor extinction coefficient. In addition to the overlap integral J, the Förster radius R 0 in Eq S.18 is determined by the refractive index n, the orientation factor κ and PLQE φ d . The orientation factor κ is fixed for all samples (see above) and we introduce the constant k = κ 2 /n 4 . Using these definitions, we rewrite Eq. S.18 as: Using the above, we can conclude with an expression for C a as a function of R 0 , φ d ,Ĵ and k: These physical quantities become additional input parameters for modelling the PL decay data and will be discussed in the following: • The refractive index n refers in original FRET theory to the solution in which donors and acceptors were dissolved [10,9], which no longer applies for the case of a twosolid-layers system. For films of π-conjugated polymers, an effective refractive index of n poly = 1.6 can be assumed [25,26,27] and MAPbI 3 has a refractive index of n MAPbI 3 = 2.5 at the wavelengths at which the polymers emit [28]. We use the average value from the polymer and the perovskite of n = 2.05 in our calculations assuming that the refractive indexes of both media play an averaged role for the ET process across their interface.
• The modified spectral overlapĴ between donor emission and acceptor absorption is defined through Eq. S.24.Ĵ was calculated from MAPbI 3 absorption (Supplementary Figures S14 and S15) and polymer emission data (Supplementary Figures S7-S9).Ĵ = 3.4 × 10 14 nm 3 was calculated for Super Yellow and entered in the calculation of C a (as in Eq. S.25) when fitting the PL decay data from SuperYellow:MAPbI 3 :quartz.

S27
• The Förster radius R 0 is defined in Equations S.3 and S.18. As mentioned above, modelling the PL decay of the polymers was done in two steps: In a first step, R 0 was used as a fitting parameter in Eqs. S.14 to S.17 to fit the PL decay data from SuperYellow:MAPbI 3 :quartz to obtain R SY 0 and through Eq. S.25 also C a (result from fitting: R SY 0 = 8.5 ± 0.6 nm, C a = 0.0016 nm −3 ). In a second step, I d (t) of the other polymers is modelled using Eqs. S.14 to S.17 with C a = 0.0016 nm −3 as a global input parameter and as a polymer specific input parameter. In this step, we assume the polymer PLQE to be the most sensitive parameter in determining R 0 between the different polymers to describe the ET from the polymer to the MAPbI 3 . R 0 (see Eq. S.18) is determined by the orientation factor κ, PLQE φ d , the refractive index n and the spectral overlap J.
κ is a fixed parameter and n is assumed to be the same for the investigated polymers.
The other parameter next to φ d which might change between the polymers, is the overlap integral J (see Eq. S. 19) or, alternatively,Ĵ, as defined in Eq. S.24. The spectral overlap is determined from normalized donor emission and acceptor absorption.
All polymers were deposited on MAPbI 3 , which acts as the acceptor. Therefore for these polymers range from 1% to 71%, which is clearly a much greater difference.

S28
The error on the Förster radius and on the modelled PL decay data was determined from uncertainties in the measured input parameters φ d (see Table S4) and L (10%). The error on R SY 0 was estimated from fitting the PL decay data with the upper and lower boundaries for φ SY and L SY . The error in the modelled PL decay data from the other polymers was then estimated from the resulting R SY 0 , polymer PLQE and polymer thickness. The error estimations are depicted as shaded area around the data in Fig. 1 in the main text.
To ensure that our approach of fitting the decay transients for SuperYellow: ). The results for the ET efficiency calculated with this fitting approach show only very small deviation (+/-1%) from the results obtained by fitting the data from SuperYellow:MAPbI 3 :quartz only, which strongly supports our approach.

Calculation of ET Efficiency
The ET efficiency η ET is defined in the main text in Eq. 6, here we briefly describe the numerical calculation of η ET . F sc is a scaling factor introduced to scale the mono-exponential PL decay to the scaling of I d (t) in Eq. S.14: F sc = I d (t 0 ). For the modelling of the transfer efficiency, t 0 is set to 1 × 10 −6 ns as zero cannot be used for practical calculation in the integral of Eq. S.14. Using a lower value for t 0 does not change the value of the efficiency in any case. The transfer efficiency takes values between 0 and 1. In terms of practical calculation, the integration is performed as a definitive integral using the python package scipy.integrate [29] and the boundaries were set to t = 0.000 001-100 ns. Convergence of the integral was thoroughly tested with a wide range of input parameters covering the whole range of parameter values that were used for the figures.

Supplementary Note 2: Aggregate Formation in P3HT
In this note, we provide a more in-depth discussion of the investigation of chain conformation in thin films of the hole-transport material poly(3-hexylthiophene)(P3HT) [30,31,32]. Film morphology and aggregate formation within polymer films are commonly studied through analysis of absorption and PL spectra [33,34,35,36,37,38].
The PL spectra of the P3HT films of different thicknesses on quartz and on MAPbI 3 are shown in Fig. 4 of the main text. The variation in thickness was achieved by changing spin speed in the fabrication routine from 2000 rpm to 6000 rpm in steps of 1000 rpm. The faster the spin speed, the thinner the polymer film was, which was confirmed with Dektak and absorption measurements (see Table S3). The films were excited at 460 nm with a fluence of 3-4 nJ cm −2 . A description of the emission spectra is given in the main text.
Normalized (at the absorption peak at 560 nm) and zoomed-in P3HT absorption components from P3HT:MAPbI 3 :quartz samples are shown in Fig. 6. The absorption peak around 615 nm changes relative to the absorption peak at 560 nm for the films on MAPbI 3 substrates suggesting a change in aggregation with spin speed/film thickness for these films, which is however not the case for the P3HT films on quartz substrates. The change is least prominent for the thickest film spun at the lowest spin speed, which agrees with changes in the PL data, on which the discussion will be mainly focused. The absorption spectra presented here are obtained by subtracting the MAPbI 3 absorption spectrum and might therefore be more prone to error than the PL, which stems only from P3HT. Furthermore, a recent study highlighted that PL is more suited for the study of aggregation in P3HT thin films of different thickness than absorption [37].
P3HT is a widely researched polymer. Centered around a report by Brown et al. [39] on the relative weakness and temperature behaviour of the lowest energy feature in P3HT absorption spectra, the origin of the spectral features of P3HT have been subject of much debate and investigation [40,41,39,42,43,44,45,46]. A widely used interpretation for the data is the model of weakly coupled H-aggregates, forming because of interchain interaction between the chromophores in thin films of P3HT, which was introduced and applied by S30 Spano and co-workers [44,45]. For ideal H-aggregates, the transition to the ground state is dipole-forbidden and there is no 0-0 peak in the emission, but disorder can lead to an increase of relative intensity of the 0-0 peak to the 0-1 peak in emission and absorption [44,34]. The model was extended to so called "HJ-aggregates" taking into account competition between interchain and intrachain interactions and therefore between the formation of Hand J-aggregates, respectively [46,35]. Here, intrachain interaction characterizes electronic excitations that are delocalized along the polymer chains (through bond) whereas interchain interaction relates to the electronic excitations delocalized between chains (through space) [36]. Sample morphology is thereby assumed to play a decisive role and for thin films in which interchain coupling is dominant, the weakly coupled H-aggregate model offers a prevalent framework for data interpretation [46,36].
Using the model of weakly interacting H-aggregates outlined above, we first compare the steady-state PL from P3HT films on a quartz substrate (dashed lines in Fig. 4 of the main text) with the PL from the films on MAPbI 3 :quartz (solid lines in Fig. 4 of the main text). For the films on MAPbI 3 :quartz we observe a blue-shift and an increase in 0-0 to 0-1 PL peak ratio corresponding to a weaker H-aggregate signature of suppressed 0-0 emission. We attribute the detected relative increase of the 0-0 peak of the PL from the films on MAPbI 3 to a decrease in interchain (through space) coupling, stemming from microstructural differences between those P3HT films [46,35,37]. A change in film morphology due to increased substrate surface roughness leading to less H-aggregate formation in P3HT films has been observed previously [47,37]. Our observations agree more specifically with the work of Ehrenreich et al. [37] who compared PL lineshapes from a thin and a thick P3HT film on MAPbI 3 and attributed the difference to a P3HT film structure composed of an amorphous region with less interchain interaction formed at the interface to the rough metal halide MAPbI 3 and a more ordered region further away from this interface. A thin P3HT film is more dominated by PL from the amorphous region than a thick P3HT film. In agreement with this, we note that the PL features from our thickest (2000 rpm) P3HT film on MAPbI 3 :quartz are most alike to the PL lineshape of the films on a smooth substrate while the thinner (3000 rpm to S31 6000 rpm) films show a more pronounced relative increase of the 0-0 PL peak and are more blue-shifted, which is a stronger signature of the amorphous phase at the MAPbI 3 interface.
So far, the focus was on the difference between P3HT films on a quartz substrate and on MAPbI 3 :quartz. However, we observe additional differences in the PL spectra between films with varying thickness on a MAPbI 3 :quartz substrate. Such a thickness dependence of the spectral lineshape of P3HT emission is observed only for P3HT films on the rough MAPbI 3 :quartz and not for P3HT films on smooth quartz implying that further changes in aggregation are connected to the roughness of the MAPbI 3 surface.
One such difference in PL spectra is a decrease in relative 0-0 PL peak intensity with decreasing film thickness for all but the thickest P3HT:MAPbI 3 :quartz film is. This is some- what opposite to what would be expected from the explanation of film-thickness-dependent emission stemming either mainly from the amorphous layer close to MAPbI 3 (thinner films) or from a more ordered and H-aggregated layer on top (thicker films) and needs further discussion.
Before discussing this observation, we further analyse PL spectral changes of the P3HT:MAPbI 3 :quartz films occurring with different film thickness: The relative intensity of the 0-0 peak decreases with decreasing film thickness and thus increasing spin speed. Within the framework of H-aggregates, this indicates that the interchain interaction is stronger in these thinner, faster spun-cast films [44,34]. Additionally, we observe a change in relative intensity of the 0-2 to 0-1 PL peak in Fig. 4 of the main text with thickness and therefore with spin speed for the P3HT film on MAPbI 3 : The 0-2 to 0-1 ratio is highest for the two thinnest films (5000 rpm and 6000 rpm), which also have a lower 0-0 to 0-1 peak ratio compared to the two intermediate films (3000 rpm and 4000 rpm).
For the discussion of these observations, we extended the framework to the HJ-aggregate model [46], which accounts for the interplay between inter-(through space) and intrachain (through the polymer bond) coupling. The strength of the two interactions can be assessed from the ratios of the 0-0 to 0-1 peak and of the 0-2 to 0-1 peak. It has been shown that S32 the 0-2 to 0-1 PL peak ratio is predominantly determined by intrachain (through bond) interactions and a decrease in relative peak intensity signifies stronger intrachain coupling [35]. The ratio of the 0-0 to 0-1 PL peak is an indication of the strength of interchain (through space) interaction as discussed above and still a main factor in the HJ-aggregate model [48].
In addition to that, when intrachain interaction is considered as well, the interplay between inter-and intrachain interaction influences the 0-0 to 0-1 peak ratio; stronger intrachain interaction leads to an increase in 0-0 to 0-1 peak ratio [35]. Applying this knowledge to our P3HT films on MAPbI 3 , interchain coupling strengthens (0-0 to 0-1 PL ratio decreases) and intrachain coupling weakens (0-2 to 0-1 PL ratio increases) for the two thinnest films spun at highest spin speeds. At this point, it is however important to notice that the changes PL intensity ratios observed here are far weaker than the changes for example observed for P3HT nanofibers displaying mainly J-aggregate behaviour [49]. We therefore still consider our P3HT thin films spun-cast on MAPbI 3 :quartz as predominantly exhibiting H-aggregation behaviour but in order to fully understand their PL spectral lineshape, intrachain interactions need to be taken into account through the HJ-aggregate model.
The impact of this discussion on film morphology and consequences for perovskite-based solar cells are illustrated in the main text.