# Correct determination of charge transfer state energy from luminescence spectra in organic solar cells

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### Optical simulation of radiative out-coupling

Emitted electromagnetic waves are partially reflected at material interfaces in multilayer structures due to differences in their optical coefficients, i.e., refractive indices (n) and extinction coefficients (k). The direct and reflected waves superimpose leading to constructive and destructive interference. This can alter the wavelength dependency of the spectrum measured outside the device I(λ) in comparison to the intrinsic emission line shape Ihom(λ) of the emitting material, i.e., the emitted spectrum in absence of any reflecting interfaces. The radiative out-coupling factor γOC is defined as the quotient of these two quantities:

$$gamma _{{mathrm{OC}}}(lambda ) = frac{{I(lambda )}}{{I_{{mathrm{hom}}}(lambda )}}$$

(4)

As the emitted radiation in the organic solar cells investigated in this study originates from the entire PAL (bar gamma _{{mathrm{OC}}}) is defined as the average out-coupling factor over all emission positions within the PAL as outlined in detail below. For simulations of the radiance enhancement within the active layer of the OSCs, an implementation33 of the scattering matrix (S matrix) method34,35 based on custom code was applied. Within this method, the exact configuration of the investigated OSC layer stacks can be simulated. Experimentally determined n and k values are used as input parameters. All used parameters can be found in the Supplementary Fig. 1, 2. Super- and substrate are air (n = 1) and glass (n = 1.67), respectively, or vice versa depending on the layer stack. When the luminescence is observed through the 1.1 mm thick glass substrates, the latter was modeled as superstrate with infinite elongation. Thereby, no reflection at the glass–air interface is included in the simulation, i.e., the back reflection at the 1.1 mm distant glass–air boundary is assumed to be not coherent and does not contribute to interference effects within the stack.

The used layer stacks are one-dimensional photonic structures, therefore a one-dimensional simulation along the direction perpendicular to the electrodes (referred to as x-axis) fully describes these systems. Luminescence within the photoactive layer (PAL) was implemented by a dipole-like emitter and its position x was varied throughout the whole PAL. For this, the layer stack was split at position x and two S matrices, one for the lower and one for the upper layers, were calculated. The out-coupled radiance IPAL(x) from the device into the superstrate for each dipole position was obtained from the time-averaged Poynting vector ({bar{mathbf S}}) with a sampling of Δx = dPAL/N, with dPAL being the thickness of the photoactive layer. The error was proven to be <1% for N ≥ 100 (relative to N= 1000), thus N = 100 was used for the results shown in this work. To obtain a spherically symmetrical emission the dipole was oriented in x-, y-, and z-direction successively and the resulting radiances were averaged. Under the assumption of a relative permeability of μr = 1 for electromagnetic plane waves (left| {{bar{mathbf S}}} right|) can be calculated by the amplitude |E| of the electric field:36

$$left| {{bar{mathbf S}}} right| = frac{1}{2}sqrt {frac{{varepsilon _{mathrm{0}}varepsilon _{mathrm{r}}}}{{mu _0}}} left| {mathbf{E}} right|^2$$

(5)

with ε0, εr, and μ0 being the vacuum and relative permittivity and the permeability of free space, respectively. As reference, the free space radiance Ihom of a homogeneous system with infinite elongation was calculated. To obtain the radiative out-coupling factor γOC(x) the radiance IPAL(x) for an emission from the full stack was divided by the reference radiance Ihom37 according to Eq. (4). Assuming spatially homogeneous recombination and thus homogeneous luminescence within the PAL of an OSC, the radiative out-coupling factor of the system was calculated as spatial average of γOC(x), i.e., the spatial average over all emission positions x in the PAL:

$$bar gamma _{{mathrm{OC}}} = frac{1}{{d_{{mathrm{PAL}}}}}{int}_0^{d_{{mathrm{PAL}}}} {gamma _{{mathrm{OC}}}(x){kern 1pt} {kern 1pt} {mathrm{d}}x} = frac{1}{{d_{{mathrm{PAL}}}}}{int}_0^{d_{{mathrm{PAL}}}} {frac{{I_{{mathrm{PAL}}}left( x right)}}{{I_{{mathrm{hom}}}}}{kern 1pt} {kern 1pt} {kern 1pt} {mathrm{d}}x}$$

(6)

Since γOC and thus (bar gamma _{{mathrm{OC}}}) are relative quantities, the amplitude of the dipole emitter can be chosen arbitrarily and was set to unity.

### Luminescence spectroscopy

To investigate CT state emission, EL spectroscopy was performed for a variation of photoactive D–A systems and layer stacks. The EL spectrum IEL(λ) measured in power per unit area and per unit wavelength for a device with a PAL consisting of P3HT:PC61BM with PEDOT:PSS as transparent electrode (device type A in Fig. 1a is plotted as blue solid line in Fig. 2). The emission features a rather wide peak with a maximum at ~1360 nm.

After measuring the EL spectrum, the very same device (type A) was modified by depositing a 10 nm thick (and thus semitransparent) Ag layer on top of the transparent electrode and renamed to type B, depicted in Fig. 1b. When measuring EL again, drastic changes are observed in the spectrum which can be seen from the solid red line in Fig. 2. It shows a narrower peak with a central wavelength of about 1250 nm. It should be pointed out that the PAL and contact materials were not changed and morphologic changes are not expected by mere deposition of such a thin Ag layer on top of the PEDOT:PSS layer and thus the recombination properties are also not expected to change. Accordingly, the different spectrum has to originate from the change of the optical properties of the system. In addition, the calculated radiant out-coupling factor (bar gamma _{{mathrm{OC}}}) for the corresponding layer stacks are plotted in Fig. 2 (dashed lines). In case of device type B, the out-coupling factor (bar gamma _{{mathrm{OC}}}) resembles the spectral shape quite accurately proving further that the spectrum is very strongly governed by out-coupling effects. Therefore, the data hardly allows for any statement about the spectral fingerprint of the emitting photoactive material, i.e., the energetics of the CT complex.

For type A, the wavelength dependence of (bar gamma _{{mathrm{OC}}}) originates from the interference of the direct emission and the reflected portion at the back mirror, i.e., wide-angle interference. In this case, the path difference between direct and reflected light depends on the distance of the photon emission from the reflective electrode, thus, the averaging over all emission positions within the PAL leads to a quite homogeneous out-coupling. Still, aspects of the spectral shape like the asymmetry of the spectrum can be partially explained by optical out-coupling. In contrast, the additional highly reflective surface in the type B stack forms an optical microcavity leading to a pronounced peak in (bar gamma _{{mathrm{OC}}}) (multibeam interference). While the amplitude of the out-coupled radiance IPAL(x) and thus (bar gamma _{{mathrm{OC}}}(x)) still depend on the position of emission within the cavity the peak position is determined by the optical distance of the mirrors.

The previously discussed case has a more exemplary character, since a reflective surface at the illumination side reduces photon absorption and hence is avoided in the design of an OSC. The following part therefore focuses on regularly used architectures of OSC. The EL spectra of P3HT:PC61BM devices (type A) for varying PAL thicknesses dPAL are shown in Fig. 3a along with the corresponding (bar gamma _{{mathrm{OC}}}left( lambda right)) and the corrected spectra (I_{{mathrm{EL,corr}}} = I_{{mathrm{EL}}} cdot bar gamma _{{mathrm{OC}}}^{ , ,- 1}). The corrected intensity IEL,corr represents the expected free space radiance without the optical environment, i.e., the emission in a homogeneous medium. In Fig. 3b, the same quantities are plotted for ITO-based devices (type C in Fig. 1c).

The measured spectra depend sensitively on dPAL and show peak positions ranging from 1240 to 1370 nm for device type A and from 1050 to 1380 nm for type C devices. In addition, peak amplitudes range from 3.4 to 15.3 for type A and from 1.5 to 23.8 for type C.

In great contrast, all-corrected intensities IEL,corr are quite similar regarding the shape and amplitude for the whole range of different values of dPAL and for the two different device architectures as could be expected for the same photoactive material used in these devices. This is a clear indication that the measured spectra are strongly influenced by the optical properties of the layer stacks. The measured spectra using PEDOT:PSS as transparent contact (type A) are wider compared to the spectra of the ITO-based devices (type C), which show more narrow peaks. For both architectures, the overall intensities vary as a function of dPAL but more severe for the ITO-based ones. While the out-coupling (bar gamma _{{mathrm{OC}}}) in the PEDOT:PSS device is quite homogeneous as discussed previously, the out-coupling of the ITO-based devices feature clear peaks. These peaks originate from the high reflectance of ITO at the relevant wavelengths as shown in Fig. 4 (measured on glass), resulting in the formation of an optical microcavity. In contrast, PEDOT:PSS (measured on glass) does not show a comparable increase of its reflectance in the infrared part of the spectrum (green curve in Fig. 4).

It should be noted that varying dPAL can result in morphological and also energetic changes which may influence IEL and thereby IEL,corr38. The deviations present in the corrected spectra are due to slight offsets in peak position and peak width between measurement and simulation. The results shown in Fig. 3 clearly demonstrate the strong impact of interference effects on the observed phenomena.

Additionally, the polymer PV2000 as an example for a high-performing polymer in combination with PC61BM and the state of the art non-fullerene DA system PBDB-T:ITIC are investigated in ITO-based solar cells39 (full names listed under cell preparation further below). This is shown in Fig. 1d (type D) yielding PCE between 4.9% (dPAL = 80 nm) and 7.9% (dPAL = 240 nm) in the case of PV2000:PC61BM and 7.2% (dPAL = 50 nm), 8.2% (dPAL = 132 nm), and a maximum of 9.3% (dPAL = 88 nm) for PBDB-T:ITIC. Electroluminescence spectra for the PV2000:PC61BM devices are shown in Fig. 5a for layer thicknesses in the range of 80 nm ≤ dPAL ≤ 240 nm alongside with (bar gamma _{{mathrm{OC}}}) and IEL,corr. While IEL peaks between 1050 and 1500 nm with maximum peak intensities in the range of 40–150 (a.u.) all-corrected spectra feature a peak between 1120 and 1170 nm with maximal intensities between 26 and 34 (a.u.). In the uncorrected spectra, these devices show a shift in wavelength of the peak of almost 500 nm and a significant change in overall emission for different thicknesses of the absorber layer. This can clearly be ascribed as to originate from changes in the optical out-coupling properties upon variation of absorber thickness. The spectra IEL, IEL,corr, and (bar gamma _{{mathrm{OC}}}) for the PBDB-T:ITIC devices for absorber thicknesses 50 nm ≤ dPAL ≤ 132 nm are shown in Fig. 5b. Thicker photoactive layers could not be realized due to a poor resulting film quality. IEL peaks between 940 and 1070 nm with maximum peak intensities in the range of 100–600 (a.u.), whereas all-corrected spectra (I_{{mathrm{EL}}} cdot bar gamma _{{mathrm{OC}}}^{ , ,- 1}) feature a peak at ~1070 nm with maximal intensities between 85 and 12 (a.u.). Note that the strong s-shape in the corrected spectrum of the device with an average absorber thickness of 132 nm originates from strong fluctuations of dPAL of about 30 nm (as measured by profilometry), resulting in an inaccurate correction when only the average thickness is taken into account.

OSCs usually feature a reflective back contact. In case of no interference in the device, i.e., incoherent emission of light, γOC equals 2 since the radiation is reflected at the backside mirror and emitted only into one hemisphere. Hence, the total emitted power which is proportional to the square of the electric field of the wave |E|2 remains unchanged. In contrast, for coherent light, |E| is doubled when the criterion for constructive interference is fulfilled. For this reason, the radiance is enhanced by a factor of 4 compared to the free space emission thus resulting in a factor of 2 in total emitted power. As could be shown experimentally by Drexhage, this doubling of the total emitted power is accompanied by a decreased radiative lifetime for the emission40. This decreased radiative lifetime for emission does also apply for the devices investigated in this study. However, the total recombination and thus the lifetime of electrons and holes in typical organic solar cells are strongly dominated by non-radiative recombination, which can clearly be seen from the previously mentioned low-quantum yields for radiative CT recombination in the range of only 10−9–10−6 in typical D–A systems. For this reason, interference effects do not have any measurable impact on the overall charge carrier lifetimes in these materials.

For the determination of the CT state energy, the previously shown spectra IEL(λ) (in W/nm/m2) were converted to If(E) given in photons per unit spectral energy and per area by dividing by the photon energy E341. The normalized reduced spectra If/E are plotted in Fig. 6 alongside with the corrected spectra If,corr/E for all systems. The corrected spectra of PBDB-T:ITIC are smoothened by a floating average over 20 data points to reduce noise and to ensure proper normalization. Gaussian functions are fitted by means of Eq. (1) to all spectra and added to Fig. 6. The resulting CT state energy ECT, reorganization energy λRO and the energy of maximum CT emission (E_{{mathrm{CT}}}^{{mathrm{em}}} = max left( {I_{mathrm{f}}/E} right)) from the fits for all systems and the corresponding photoactive layer thicknesses are listed in Table 1.

(E_{{mathrm{CT}}}^{{mathrm{em}}}) is strongly influenced by interference as previously seen from the wavelength-dependent spectra in Figs. 3 and 5. From Fig. 6 it gets obvious that the high energetic tails, i.e., the apparent reorganization energy λRO, and thus the estimated CT energy (E_{{mathrm{CT}}} = E_{{mathrm{CT}}}^{{mathrm{em}}} + lambda _{{mathrm{RO}}}) are significantly affected by interference effects as well. The spectra of the ITO-based P3HT:PC61BM and PBDB-T:ITIC devices (Fig. 6b–d) show that the uncorrected data sets differ from the corrected spectra for all absorber thicknesses and in case of P3HT:PC61BM for dPAL > 135 nm microcavity effects can result in observed spectra significantly deviant from the expected Gaussian shape. This results in a strong dependence of the fit results on the chosen data range making a fit of a Gaussian function questionable in general. In case of PV2000:PC61BM (Fig. 6c), the uncorrected spectra look like a superposition of two peaks for dPAL ≥ 175 nm due to the resonant peak of the microcavity. These strongly altered spectral shapes make a meaningful analysis of the uncorrected data difficult and prone to misinterpretation. Interestingly, there are cases where two effects compensate each other when the CT state energy (E_{{mathrm{CT}}} = E_{{mathrm{CT}}}^{{mathrm{em}}} + lambda _{{mathrm{RO}}}) is determined by the applied fitting procedure. As an example, the uncorrected PBDB-T:ITIC spectra feature higher (E_{{mathrm{CT}}}^{{mathrm{em}}}) values with lower λRO (peak width) compared to the corrected spectra. This results in comparable CT energies although the values for (E_{{mathrm{CT}}}^{{mathrm{em}}}) and λRO differ significantly. Hence, even in this case the correction procedure is indispensable to achieve a deeper understanding of the underlying processes and material properties.

It can be seen in Table 1 that the corrected spectra resemble each other quite accurately for all absorber thicknesses and the fitted ECT,corr coincide within <0.1 eV. In contrast, the raw ECT energies deviate up to more than 0.35 eV for the investigated systems (note that it could be even higher for other systems). This deviation should by no means be misinterpreted as an upper limit for the error of the CT state energy as quantities derived from uncorrected spectra are not meaningful if interference effects are dominant. In general, cell architectures featuring ITO as transparent electrode show more pronounced microcavity effects due to the high reflectivity of ITO for wavelengths longer than 1000 nm as explained in detail before. The described effects are expected to be negligible if devices which do not differ significantly in terms of their optical properties are compared among each other as could be the case when different interlayers with comparable optical constants are compared or when mere morphological changes are induced by additives or annealing procedures. Nevertheless, consistent layer thicknesses should be ensured in such cases. In contrast, correction for optical effects is crucial if the actual spectral shape of the luminescence spectrum is of interest in organic thin-film devices.

It should be stated here, since out-coupling of light is ultimately the reverse process of light absorption, the results will be affected in a comparable manner if low energetic CT state absorption tails are analyzed.