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1.IntroductionOrganic solar cells (OSCs) combine the advantages of low-cost and large-area fabrication with the use of nonhazardous and environmentally friendly materials. Over the last few years, power conversion efficiencies improved continuously, now exceeding 10% in single-junction solar cells.1 Improvements often originated from the synthesis of new absorber materials that allowed for an enhanced spectral coverage of the solar spectrum or enabled higher internal quantum efficiencies.2–7 In the lab, screening and optimization of new materials and cell architectures often follow a trial and error approach, although only very little amounts of material are available. An alternative and material saving route to optimized OSCs is a comprehensive optoelectronic device simulation that reduces the experimental parameter space. This becomes particularly important for sophisticated device architectures, such as tandem solar cells.8 However, in order to carry out meaningful optical simulations, a profound knowledge of the refractive indices is mandatory.5,9–12 Unfortunately, optical constants from the literature often cannot be used to describe certain experiments since the optical properties strongly depend on the substrate, the underlying interface, the intrinsic polymer properties (molecular weight, polydispersity), the polymer:fullerene ratio, and the process conditions, such as the solvent system, the drying kinetics, and the postprocessing treatments.13–24 Therefore, an urgent need for flexible optical descriptions of bulk-heterojunctions emerged, which use parametric descriptions and allow for individual adjustments. If carried out purposefully, the optical model further reveals information about anisotropy within the bulk-heterojunction.25 Such models were recently utilized to determine the vertical gradient in the active layer of OSCs13,25 and to investigate the shape of [6,6]-phenyl -butyric acid methyl ester () inclusions in poly(3-hexylthiophene-2,5-diyl) (P3HT) solar cells.26 In this work, we present a modeling approach for spectroscopic ellipsometry (SE) data analysis, which facilitates the determination of the complex refractive indices of various polymer:fullerene blends. Therefore, in a first step, the relative dielectric functions of the fullerene [6,6]-phenyl -butyric acid methyl ester () and the polymers poly{[4,4’-bis(2-ethylhexyl)dithieno(3,2-b;2’,3’-d)silole]-2,6-diyl-alt-(2,1,3-benzothidiazole)-4,7-diyl} (PSBTBT) and poly[2,6-(4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b’’]dithiophene)-alt-4,7-(2,1,3-benzothiadiazole)] (PCPDTBT) are described by the superposition of Gaussian oscillators. In a second step, the relative dielectric functions of the polymer:fullerene blends are determined by superimposing the optical models of both constituents. PSBTBT and PCPDTBT are well-known donor-acceptor copolymers with good power conversion efficiencies when blended with .27–31 Their similar structures allow a direct comparison. For the active layer deposition and blend ratios, we follow literature-known process protocols, which are optimized for high-performance solar cells.27–29 Optical data are presented in the spectral region from 250 to 1,000 nm. We further compare isotropic and anisotropic optical models of neat polymers. The results of this comparison serve as input parameters for anisotropic optical descriptions of the blends, which are then analyzed with respect to the predominant polymer orientation and the degree of aggregation. 2.Spectroscopic EllipsometrySE is a contactless and nondestructive measurement technique that allows for deriving the complex refractive indices of thin-films and their thicknesses simultaneously. The complex refractive index , given by comprises the refractive index (real part) and the extinction coefficient (imaginary part) and is directly related to the complex relative dielectric function , assuming a relative permeability .The basic ellipsometry working principle relies on measuring the change of the light beam polarization state after interacting with the sample. This change is described by the two ellipsometric values and , which represent the amplitude ratio and phase difference between - and -polarization.32 In this study, various wavelengths and angles of incidence were investigated. In addition, transmission spectra were recorded at normal incidence on the very same spot and incorporated into the analysis. We use vendor-supplied software (WVASE32®, J. A. Woollam) to analyze the datasets. First, a multilayer model is constructed, where each individual layer describes a certain subset of physical properties of the sample, e.g., the level of interface intermixing, surface roughness, and the relative dielectric function of the material. In the following, “layer” will refer to a layer in the optical model and not necessarily to a physical layer. The relative dielectric functions are defined by oscillator models, i.e., the description of each optical transition by an appropriate oscillator. Then the relative dielectric function of a material can be described by the sum of these oscillators. To describe the transitions within the measurement range, we choose an oscillator type generating a Gaussian line shape in , Eq. (2), with a Kramers-Kronig consistent counterpart in , Eq. (3). where is the amplitude, is the center energy, is the width, and is the principal value.33 Higher-energy transitions outside the measurement range are summarized in a high-energy pole function, i.e., an oscillator with zero-width. Further, a real constant is added to .Based on this parameterized model, the optical response of the sample is calculated and compared to the measured and data and the transmission. The parameters for the individual layers (thickness, oscillator parameters, etc.) are fitted in order to achieve a good match between model-generated and measured values. The difference between these values is quantified by the mean squared error (MSE).34 MSE is used by the analysis software as weighted test function during the fitting procedure to further reduce the deviation between the three values by making use of the Marquardt-Levenberg algorithm.35,36 We verify the results of the fit and ensure physical significance of the calculated optical constants by, first, measuring the thicknesses of thin-films deposited on glass substrates in a transparent energy band by applying the Cauchy dispersion relation and by comparing them to profilometry results. (We note that the Cauchy model does not fulfill the Kramers-Kronig relation but is a valid approximation for transparent spectral regions.) Second, we perform a multisample analysis: several samples with varying thin-film thicknesses are characterized. All respective datasets are described with the same optical model, hence enabling a general and robust optical description and reducing parameter correlation.37,38 In the end, the oscillator model will guarantee full Kramers-Kronig consistency. Further, we include a wide range of angles of incidence in the analysis, typically around the Brewster angle (step width: 5 deg). To perform the multisample analysis, the datasets are measured on several samples with different layer thicknesses. All SE investigations cover the spectral region from 250 to 1,700 nm (1,000 nm for ) and layer thicknesses are determined in the transparent () wavelength regime from 1,000 to 1,700 nm. For reasons of clarity, we will show a reduced dataset in the following figures. In particular, we skip data in the infrared region beyond 1,000 nm, restrict to some representative angles of incidence, and only show typical data for one sample of the multisample analysis. 3.Experimental3.1.Spectroscopic Ellipsometryand were measured with a variable angle spectroscopic ellipsometer (VASE®, J.A. Woollam Co., Inc.), equipped with a rotating analyzer and two detectors. A rotating retarder plate ( Berek waveplate) in the beam path behind the fixed polarizer allows for a precise determination of .39 Datasets were recorded between 250 and 1,700 nm (1,000 nm for ). Transmission data of the samples were collected at normal incidence and by moving the VASE detector arm into the optical path behind the sample. Thus, the transmission spectra were determined on the very same spot as the ellipsometric data, allowing for their incorporation into the analysis. The datasets were analyzed with the software WVASE32® (J. A. Woollam, version 3.774). 3.2.Transmission SpectroscopyTransmission spectra of diluted solutions and of solid films spin cast onto quartz-glass substrates were recorded with a spectrophotometer (Lambda 1050, Perkin Elmer), enabling a lower noise level than the measurements carried out with the ellipsometer. 3.3.Materials and Sample PreparationSoda-lime glass substrates were cleaned with acetone and isopropyl alcohol and then dried in a nitrogen stream. All materials were used as received and were deposited under inert conditions in a nitrogen glovebox ( and ). (Solenne, ) was dissolved in DCB (anhydrous, 99%, Sigma-Aldrich®) at a concentration of , stirred overnight at 80°C, and spin cast on substrates at room temperature with different spin coater settings to achieve thicknesses between 85 and 120 nm. PSBTBT and PCPDTBT were both dissolved in DCB with a concentration of , stirred overnight at 70 or 90°C, respectively, and were deposited from hot solution on hot soda-lime glass substrates. Different spin coater settings allowed for tuning the layer thicknesses between 40 and 60 nm for PSBTBT or 15 and 30 nm for PCPDTBT. The blends (, layer thicknesses between 90 and 115 nm) and (, layer thicknesses between 90 and 130 nm) with a polymer concentration of in DCB were spin coated following the same conditions used for neat films. The blend ratios were chosen according to optimized process protocols.27,29 SE data of the neat polymer films and the blend films were recorded between 250 and 1,700 nm at different angles of incidence between 30 and 80 deg in 5 deg steps. Datasets were analyzed over the entire spectral range. For reasons of clarity, we only show a reduced dataset in the figures, i.e., we skip data in the infrared beyond 1,000 nm, restrict to some representative angles of incidence, and only show data for one sample of the multisample analysis. Thin-film thicknesses were cross-checked with a stylus profiler (DektakXT, Bruker AXS GmbH) and compared with ellipsometry results. 4.Results and Discussion4.1.Glass SubstrateSoda-lime glass is used as substrate for all thin-films.37 Therefore, we first set up an optical model for the glass substrate. As the substrate is thick, unwanted backside reflections are expected. They can either be suppressed by roughening the substrate’s backside or be accounted for in the analysis software. Here we use the latter approach as this allows measuring the transmission spectra on the very same spot after the ellipsometry characterization. The glass substrate is described with a Cauchy dispersion relation40 with an Urbach-tail41 and a surface roughness layer of . The derived complex refractive indices of the soda-lime glass substrates are depicted in Fig. 1 versus wavelength. In the following, we keep the parameters of the model constant and treat the surface roughness layer as an intermixed layer of the subsequently applied thin-film and the glass substrate. 4.2.PC71BMSolar cells based on the polymers PCPDTBT and PSBTBT in combination with the electron acceptor exhibit enhanced power conversion efficiencies. We first analyze . Therefore, the fullerene is dissolved in 1,2-dichlorobenzene (DCB) and then spin cast on the glass substrates as described in Sec. 3.3. A multisample SE analysis is performed on six samples with varying thicknesses. Figures 2(a) and 2(b) show the experimental data of a typical sample. Transmission spectra of in DCB solution and of a solid thin-film are recorded with a spectrophotometer, Fig. 2(c). From the absorption features present in all spectra, the starting values for the center energies of the Gaussian oscillators are derived, Table 1. Based on this estimation of , an optical model is set up comprising nine Gaussian oscillators , a pole taking into account higher-energy transitions, and a real constant . A stepwise fit (), starting from the infrared and progressing toward higher energies, results in the oscillator parameters given in Table 1. The fitted center energies differ slightly from their respective starting values . In this multisample analysis, all measured datasets are taken into account. If the fit is restricted to a single sample, lower MSEs can be achieved. As the multisample analysis only differs slightly from the results of the single-sample analysis and as we aim, here and in the following, at finding a widely applicable optical model, we utilize the multisample approach. The model-generated data , , and the optical constants are depicted in Figs. 2(a), 2(b), and 2(d), respectively. The good agreement with the optical constants in Ref. 42, supporting information, verifies the chosen approach. The good match between simulated and experimental transmission, which was measured on the same spot as the ellipsometry data, proves the fit quality, Fig. 3(a). Table 1From the transmission spectra in Fig. 1(c), the starting values En,input for the fit of the Gaussian oscillators’ center energies are derived. The optical model of [6,6]-phenyl C71-butyric acid methyl ester (PC71BM) comprises nine Gaussian oscillators (Gaun with center energy En, amplitude An, and width Brn,33 a pole, and the real constant εr,∞=2.00 eV.
All thin-film thicknesses as determined by profilometry are well within the 90% confidence limit of the thickness as determined by SE. 4.3.PolymersBoth polymers, PCPDTBT and PSBTBT, exhibit strong structural similarities. They differ only in the 5-position, where PCPDTBT comprises a carbon atom, whereas PSBTBT features a silicon atom, Fig. 4(a). It was shown that the silicon substitution in PSBTBT reduces the steric hindrance from the bulky alkyl groups and improves molecular packing.43 The main absorption features of both polymers are still comparable though. The measured datasets and for a PSBTBT thin-film spin cast on a glass substrate are shown in Figs. 5(a) and 5(b), respectively. The isotropic optical model is constructed in analogy to the procedure described for . The transmission spectra of PSBTBT in DCB solution and of a solid PSBTBT thin-film are recorded, Fig. 5(c). The analysis of the absorption features leads to the starting values for the center energies of the Gaussian oscillators. Then, all oscillator parameters, a pole for the high-energy transitions and are fitted. The respective values are summarized in Table 2. Based on this model, the isotropic optical constants are derived, Fig. 6(a). Table 2Isotropic, parameterized description of poly{[4,4’-bis(2-ethylhexyl)dithieno(3,2-b;2’,3’-d)silole]-2,6-diyl-alt-(2,1,3-benzothidiazole)-4,7-diyl} (PSBTBT). The starting values En,input are derived from the transmission spectra in Fig. 5(c). The center energies En change only slightly upon performing the fit. The parameters An and Brn are determined defining the Gaussian oscillators Gaun. The offset was found to be εr,∞=2.46 eV.
PCPDTBT is investigated accordingly. The respective isotropic datasets are shown in Fig. 7, Table 3, and Fig. 6(b). The spectral shape of the complex indices of refraction is comparable with the isotropic data presented by Guerrero et al.42 As they used PCPDTBT from a different supplier, the amplitudes are different though. Table 3Poly[2,6-(4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b′′]dithiophene)-alt-4,7-(2,1,3-benzothiadiazole] (PCPDTBT). Starting values En,input as derived from Fig. 4(c). The parameters En, An, and Brn determine the optical model. εr,∞=2.28 eV. Gau7 is outside the ellipsometer’s measurement range; the pole function covers its contribution.
Both polymer models are derived independently. However, the derived optical models utilize very similar sets of Gaussian oscillators, which reflect their strong structural similarity. A comparison of the isotropic optical constants further shows that the overall optical density of PSBTBT is somewhat higher than the optical density of PCPDTBT. This can be either attributed to a higher intrinsic absorption or to a denser packing of the polymer chains.43 For both polymers, we assign the transitions and to the 0–0 and 0–1 absorption peaks, while comprises higher vibronic progressions and an amorphous background. The nature of remains unclear, but may be attributed to transitions involving more localized states.44 A more detailed model has to take into account the anisotropy of the films. OSC polymers often exhibit elongated structures and, hence, form mostly flat two-dimensional molecules.45–48 Therefore, we probe the thin-films for anisotropy, assuming no preferential order in the in-plane direction as indicated by the invariance of under in-plane sample rotation.32,49,50 This assumption allows us to utilize a uniaxial optical description, only distinguishing between the - and the -contribution. [The xy-component is sometimes referred to as (in-plane) and z as (out-of-plane).] For this orientation, the off-diagonal elements of the Jones matrix vanish.32 In the uniaxial configuration, the optical constants and are determined by the -layer, and the -component by the -layer. The -optical constants describe light-matter interaction for light under normal incidence with its electrical field vector parallel to the sample surface. For light impinging at grazing incidence, is almost normal to the sample surface and the interaction with the polymer is described with and . For the anisotropic model, the isotropic oscillator parameters serve as input parameters for both the - and -layer, consequently doubling the number of fit parameters. Since doubling of the fit parameters does hamper finding a unique solution, we simplify our model by assuming that the same optical transitions will be observed in both directions ( and ), changing their relative amplitude only. Accordingly, the oscillator amplitudes and are defined as fit parameters. The fitted datasets and for PSBTBT are shown in Figs. 5(a) and 5(b) and the derived anisotropic optical constants in Fig. 5(d). The MSE improves from 7.23 to 6.37 upon utilizing an anisotropic model. The respective datasets for PCPDTBT are shown in Fig. 7. For PCPDTBT, the MSE improves from 7.77 to 7.51. Evidence of the fit quality is provided in Figs. 3(b) and 3(c), where we compare the experimental transmission data that were measured on the same spot as the ellipsometry data, with the simulated transmission based on the ellipsometric model. Both neat polymers exhibit pronounced uniaxial anisotropy. Since the polymer transitions, here modeled with —, are excited by an electrical field that is polarized parallel to the polymer backbone, i.e., the polymer axis as defined in Fig. 4(a), a detailed analysis of the anisotropic optical constants can further reveal morphological details.51,52,53 The optical constants and describe light-polymer interaction for polymers with their axes aligned parallel to the substrate surface, while the z-component describes upright polymers with their axes oriented in -direction, Fig. 4(b). The extinction coefficient of PSBTBT splits up in an - and a -component between and 800 nm, Fig. 5(d). Toward lower wavelengths, no splitting is observed, indicating a different nature of the involved transitions. According to similar observations in the literature, we attribute the shoulder around 761 nm () to polymer aggregation.43 As the contribution of the -component is more pronounced than the contribution of the -component, we conclude that aggregates of polymers with a axis aligned in-plane prevail, Fig. 4(c). For PCPDTBT, the - and -extinction coefficients also split up and show anisotropic behavior in the low-wavelength regime, Fig. 7(d). The low-energy shoulder, which is visible in the data, does not appear in the data. In analogy to PSBTBT, the shoulder in can be attributed to aggregates of polymers with axes aligned parallel to the -plane, whereas the interpretation of leads to ambiguous results. This can either be attributed to nonaggregated standing polymers or an isotropic amorphous background, Fig. 4(d). 4.4.BlendsThe measured spectroscopic ellipsometry data and of a and a thin-film are depicted in Figs. 8(a) and 8(c) and in Figs. 8(b) and 8(d), respectively. The optical properties of the polymer:fullerene blends are modeled within an effective medium approximation (EMA). Therefore, the optical constants of polymer and fullerene constituents are superimposed. As the analysis of neat polymers in Sec. 4.3 revealed a preferred orientation of the PSBTBT and PCPDTBT axes parallel to the substrate, we choose uniaxial models for the blends as well. The oscillator parameters in Tables 1, 2, and 3 are used as starting parameters. After determining the layer thicknesses in the transparent spectral regime, the volume ratio of polymer to is fitted. The resulting ratios are listed in Table 4 showing excellent agreement with volume ratios being calculated on the basis of material density as determined by helium pycnometry in the literature.54,55 This result shows that SE can be an alternative and contactless metrology approach to determine material density ratios in blend films. If the material density of one component is already known, the absolute material density of the other component can be determined. Table 4The normalized volume ratios in the blends as derived from ellipsometry fittings show excellent agreement with the volume ratio calculated from the material densities. ρPC71BM=1.52 g/cm3, ρPSBTBT=1.17 g/cm3, and ρPCPDTBT=1.10 g/cm3 as determined by helium pycnometry in the group of Dadmun et al.54,55
In the next step, we investigate whether the polymer’s anisotropy is preserved upon blending with . Therefore, the polymer oscillator amplitudes are fitted leading to the anisotropic optical constants shown in Fig. 8(e) for and in Fig. 8(f) for . As indicated in Figs. 8(a) and 8(c) and in Figs. 8(b) and 8(d), both multisample fits describe the experimental data very well ( and 4.64). Again, the comparison of the simulated and the experimental transmission in Figs. 3(d) and 3(e) confirms the good fit quality. Further, the anisotropic optical constants of match the shape of the isotropic optical constants known in the literature.56 When performing the fits, only the amplitudes of the oscillators to change considerably. In analogy to the analysis of the neat polymers, those oscillators are an indicator for the degree of polymer aggregation. For , a pronounced uniaxial anisotropy becomes visible in the optical constants, Fig. 8(e). Similar to neat PSBTBT, both the in-plane and the out-of-plane extinction coefficients exhibit shoulders around 760 nm, where the -contribution dominates the -contribution. The dominance of the -contribution can be attributed to polymer aggregates with their axes aligned in-plane as illustrated in Fig. 4(e). This interpretation of the polymer orientation from SE experiments fits well to the time-resolved x-ray analysis that we published earlier, where we found that PSBTBT nucleation takes place in defined orientation at an interface, while randomly oriented aggregates form in the bulk during solvent evaporation.29 As the degree of anisotropy is similar to the anisotropy of neat PSBTBT, the polymer orientation remains mostly unaffected upon blending with . For , the optical constants in - and in -directions differ slightly, Fig. 8(f). The PCPDTBT low-energy shoulder appears in both directions. This can be attributed to randomly oriented PCPDTBT aggregates as depicted in Fig. 4(f). In the blend model, the low-energy shoulder is more pronounced than in the model for neat PCPDTBT, indicating a higher degree of aggregation in the blend. The higher degree of the axis in-plane alignment within the blend as compared to the blend, probably induced by the substrate surface, may be explained by the lower volume fraction, Table 4. 4.5.LimitationsThe EMA approach requires negligible cross-coupling of optical transitions between the blend’s components. For example, charge transfer (CT) absorption could cause such a new excitation channel. Since CT absorption in state-of-the-art OSCs is some magnitudes lower than the transition,57 we disregard CT absorption. Both polymers that are investigated in this study exhibit a lower degree of crystallinity than other semicrystalline materials, such as P3HT.58 Upon blending more crystalline materials, the degree of crystallinity may be reduced significantly. Therefore, in order to generate a suitable model for highly crystalline materials, care should be taken that the degree of crystallinity between the neat material and the blend is comparable. For example, for neat P3HT, we suggest deposition from a low-boiling point solvent, such as chloroform, thus accelerating the film drying time and, thereby, effectively suppressing crystallization of the neat polymer. The determination of the anisotropic blend models benefits from the weak spectral overlap of the polymer and fullerene extinction coefficients, which allows separating both components and performing a precise fit. Fortunately, this applies to the vast majority of polymer:fullerene blends. Even more stable fit results can be achieved by taking into account advanced photophysical descriptions.59 So far, such a detailed description has been elaborated for P3HT only.60,61 A description of the complex absorption spectra of the low-band gap polymers is still lacking. Therefore, we suggest utilizing the phenomenological derived values of the oscillator center energies and robust multisample fits as discussed in this work. 5.ConclusionIn conclusion, we present a modeling approach for the analysis of ellipsometry data, which facilitates the precise determination of the optical constants of various polymer:fullerene blends. The optical constants of the neat constituents , PSBTBT, and PCPDTBT are determined, first, by parameterizing the constituents’ relative dielectric functions, where each optical excitation is described by a Gaussian oscillator. The center energies of the oscillators are derived phenomenologically from their respective transmission spectra. Based on this set of optical constants, the blends and are described within an EMA. An initial fit on the polymer-to-fullerene ratio allows to derive the volume density ratio of both constituents. Our approach further allows for considering a uniaxial anisotropic optical description of the neat polymers and their blends. The detailed analysis of the anisotropy provides deep insight into the PSBTBT, PCPDTBT, and blend morphology. Since many of the latest high-performance materials exhibit a lower degree of crystallinity than the vastly studied P3HT,62 we propose SE as a nondestructive bulk-heterojunction probing technique for future polymer:fullerene bulk-heterojunction investigations. AcknowledgmentsThe authors acknowledge funding by the Federal Ministry of Education and Research (BMBF) within the project Taurus, contract 03EK3504, the Deutsche Forschungsgemeinschaft (DFG) priority program 1355 “Elementary Processes of Organic Photovoltaics,” and the State of Baden-Württemberg through the DFG Center for Functional Nanostructures (CFN). Poly[2,6-(4,4-bis(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b′′]dithiophene)-alt-4,7-(2,1,3-benzothiadiazole)] and poly{[4,4’-bis(2-ethylhexyl)dithieno(3,2-b;2’,3’-d)silole]-2,6-diyl-alt-(2,1,3-benzothidiazole)-4,7-diyl} were supplied from Konarka. The authors thank Professor M. Dadmun, Professor G. Bazan, Dr. H. Chen, Dr. J. Peet, Dr. S. Hu, and Dr. J. Azoulay for the communication of their results54,55 prior to publication, Dr. B. Schmidt-Hansberg (Karlsruhe Institute of Technology) for fruitful discussions, and Dr. T. Wagner (LOT-QuantumDesign) for technical support. M.F.G.K. was supported by the Karlsruhe School of Optics and Photonics (KSOP). ReferencesM. A. Greenet al.,
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