Accurate Determination of the Uniaxial Complex Refractive Index and the Optical Band Gap of Polymer Thin Films to Correlate Their Absorption Strength and Onset of Absorption

Abstract

The advanced development of optoelectronic devices requires a methodical knowledge of the fundamental material properties of the key active components. Systematic investigations and correlations of such basic optical properties can lead to new insights for the design of more potent materials. In this perspective, we provide a systematic overview of the uniaxial anisotropic complex refractive indices and the absorption coefficients obtained by ellipsometry as well as the optical band gap energies derived from Tauc plots of six selected solution‐processed polymer thin films. While the optical band gap energies are intentionally distributed over the visible spectral range, we found that the absorption strength of all polymer samples are grouped in a random distribution within a rather uniform range of values.

A rainbow of polymers: An overview is given of the basic optical and dielectric properties of six selected polymers that have been processed into thin films. Their optical band gap energies are distributed over the entire visible spectral range. Is there a correlation between absorption onset and strength?

Introduction

Organic semiconducting polymers have become indispensable for modern optoelectronic applications such as photovoltaic cells (OPV), light emitting diodes or transistors (OLED, OLEFET), field‐effect or electrochemical transistors (OFET, OECT), and bioelectronics.[ ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ ] The core components of such optoelectronic devices are thin π‐conjugated polymer layers that enable light to be harvested or emitted. Advanced device design requires knowledge of basic optical properties such as complex refractive index, optical band gap or fluorescence quantum yield. Systematic investigations have led, for example, to the semi‐emperical Energy Gap Law, which relates the nonradiative decay rate of photoluminescence to the electronic band gap of organic and some metal‐organic materials. It predicts an exponential increase in the nonradiative decay rate as the band gap narrows, which is an obstacle to the development of red and near‐infrared emitters with high quantum yields.[ ¹¹ , ¹² , ¹³ ] The development of materials to overcome this inherent limitation is still ongoing.[ ¹⁴ , ¹⁵ , ¹⁶ ]

For reasons of analogy and curiosity, it is of interest to investigate whether there is a similar correlation between absorption strength and absorption onset that may constrain the design of efficient near‐infrared absorbing photovoltaic devices. The absorption strength can be quantified by the absorption coefficient, while the onset of absorption is described by the optical band gap. For this initial systematic overview, six neat polymers processed into thin films were selected whose optical band gaps are distributed over the entire visible spectral range. These polymers are sketched in Figure 1(a), and are briefly introduced below, ordered according to their optical band gap energies, starting with the widest band gap.

Figure 1.

(a) Structural formulae of repeating units and acronyms of the selected polymer materials. (b) Normalized absorbance spectra (− log(T)) averaged over multiple recordings (minimum N=3) of polymer thin films with varying thicknesses doctor‐bladed on glass substrates.

The polyfluorene‐benzo‐thiadiazole copolymer F8BT (or PFBT)[ ¹⁷ , ¹⁸ ] and the polyphenylene‐vinylene type polymer MDMO‐PPV (or OC₁C₁₀–PPV)[ ¹⁹ , ²⁰ ] are both emissive polymers, and they are typically used for green‐yellow and yellow‐orange emitting OLEDs, respectively. In OLEFETs, they act as ambipolar materials that are capable of transporting both electrons and holes concurrently.[ ⁸ , ²¹ ] F8BT is also used as interfacial electron transport layer in perovskite solar cells reducing hysteresis effects. ^[22]

PDCBT belongs to the polythiophene class of p‐type donor polymers and has lead to more than 10 % device power conversion efficiency in fullerene‐free solar cells. ^[23] Furthermore, it is also inserted as interfacial hole transport layers in perovskite solar cells to relieve strain in the pervoskite layer ^[24] and to improve light stability of the devices. ^[25]

PBDB−T‐2F (or PBDB−T‐F, PBDB‐TF, PM6) belongs to the family of PBDB−T family p‐type copolymers for organic solar cells.[ ²⁶ , ²⁷ ] The polymer repeating unit is composed of two building blocks that are a benzo‐dithiophene‐dione (BDD) and a core‐substituted benzo‐dithiophene (BDT) unit. The latter carries two additional fluorine atoms conjugated to the side groups in case of PBDB−T‐2F. Combined with non‐fullerene acceptors, solar cells with record power conversion efficiencies up to 18 % could be obtained.[ ²⁸ , ²⁹ ] Such organic solar cells have also been sucessfully integrated into perovskite‐organic tandem solar cells, with certified power conversion efficiences beyond 23 %. ^[30]

PTB7 is considered a low‐bandgap copolymer containing alternating electron‐rich benzo‐dithiophene (BDT) and electron‐poor thieno‐thiophene (TT) blocks. It has been implemented in inverted organic solar cells as a p‐type donor material in combination with a fullerene acceptor with an efficiency over 9 %, which is one of the highest values reported for polymer:fullerene solar cells. ^[31] It was also successfully tested in so‐called ternary blends, in which two donor materials are blended with a fullerene acceptor. ^[32] Since then PTB7 has been widely used as prototypical compound to study various aspects of photovoltaic devices such as processing techniques, charge carrier mobility and recombination, as well as structural properties.[ ³³ , ³⁴ , ³⁵ , ³⁶ , ³⁷ ]

ZZ50 (or better known as c‐PCPDTBT) was one of the first low‐bandgap poylmers enabling the preparation of high‐performance polymer solar cell devices. ^[38] It was designed by Konarka Technologies Inc. and is one of the classical examples of a p‐type donor‐acceptor copolymer consisting of alternating electron‐rich cyclopenta‐dithiophene (CPDT) and electron‐poor benzo‐thiadiazole (BT) building blocks within the polymer repeating unit.[ ² , ³⁹ ] Since then the PCPDTBT polymer familiy has been of general interst to study structure‐property‐device relationship.[ ⁴⁰ , ⁴¹ ]

Results and Discussion

Thin films of the six selected polymers, whose structural formulae of the repeating units are sketched in Figure 1(a), have been processed from chlorobenzene solutions via blade coating onto glass substrates at elevated temperature. The blade velocity was varied to obtain at least three layer thicknesses for each polymer, roughly ranging from 5 nm to 120 nm to ensure sufficient transparency. Preheating of the polymer solutions and heating of the substrate holder during coating to 80 °C allowed homogeneous film formation and promoted self‐assembly or aggregation of the polymers during drying of the thin films. Normalized absorbance spectra averaged over multiple recordings for each compound are plotted in Figure 1(b). Absorbance (or optical density, OD) is calculated from normal incidence transmission intensity spectra T according to: ^[42]

$${\displaystyle OD=-\log \left(\frac{T}{T_{\mathrm{ref}}}\right)}$$ (1)

where the reference beam is typically left blank $T_{\mathrm{ref}}=1$ for thin film measurements. Absorbance should not be confused with absorption A, which is for thin films not a directly measurable quantity, but is calculated as follows assuming $T_{\mathrm{ref}}=1$ : ^[42]

$${\displaystyle A=1-T-R-S}$$ (2)

where R stands for reflection and S for scattering. The reflection is practically absent for solution samples, if a solvent‐filled cuvette is used as reference, and scattering is mostly relevant for rough surfaces and colloidal dispersions.

All absorbance spectra of the six polymer thin films show quite broad absorption bands, whose maxima are distributed over the visible spectral range from about 450 nm to 800 nm, and correspondingly the absorption onsets from about 500 nm to 850 nm. The absorption bands may consist of multiple convoluted absorption features, which can be attributed to inter‐ and intra‐chain excitonic coupling as well as vibronic progressions. ^[43] These aggregation related features depend strongly on processing conditions – such as solvent and additives, annealing temperature, choice of deposition method – as well as on polymer molecular mass distribution and purity. For blended layers, as typically required for device applications, the number of tunable parameters to control morphology and aggregation increases further. A smart design of this parameter space is a key strategy for optimized device performance.[ ⁴⁴ , ⁴⁵ ] What we want to say at this point is that a given polymer can have different absorbance spectra that correlate with synthesis and processing. Therefore, to avoid over‐interpretation of all the optical spectra discussed, we refrain from providing physical models of the nature of the spectral features.

In the next step, the complex refractive indices of all six polymer thin films have been determined by variable angle spectroscopic ellipsometry (VASE), and the results are shown in Figure 2. The analogeous representation as dielectric function versus photon energy is shown in the Supporting Information in Figure S1 as well as all measured and modeled VASE and T data in Figures S2 to S7. The complex refractive index $\tilde{N}$ is the square root of the complex dielectric function $\widetilde{ϵ}$ and quantifies how light interacts with a material: ^[46]

$${\displaystyle \tilde{N}=\sqrt{\widetilde{ϵ}}=n-\mathrm{i}k}$$ (3)

Figure 2.

Uniaxial anisotropic complex refractive index (real part n and extinction coe‐cient k) versus wavelength of (a) F8BT (MSE=4.2, N=28), (b) MDMO‐PPV (MSE=6.6, N=12), (c) PDCBT (MSE=5.9, N=16), (d) PBDB‐T‐2F (MSE=9.9, N=23), (e) PTB7 (MSE=5.3, N=22), and (f) ZZ50 (MSE=7.7, N=19). The ordinary (in‐plane) components no and k _o are plotted with solid lines, while the extra‐ordinary (out‐of‐plane) components n _eo and k _eo are graphed with dotted lines. The analogeous representation as dielectric function versus photon energy is shown in the Supporting Information in Figure S1 as well as all measured and modeled VASE and T data in Figures S2 to S7.

where n is the real part (or just refractive index) and k is the imaginary part or extinction coefficient, and the time dependence of the electrical field is supposed to be $\exp \left(+i\omega t\right)$ . This is the engineering or “Nebraska” convention, which is the sign convention used by J.A. Woollam ellipsometers and CompleteEASE software. ^{[47]   1} The real part n describes the ability to refract and reflect light at interfaces and determines the speed of light within a material. The extinction coefficient k represents the extent of absorption of light as it propagates through a material. All quantities are functions of wavelength λ or photon energy E.

For an isotropic material, the complex refractive index is a scalar, meaning it has the same value in all directions. For anisotropic materials, it becomes a tensor, which is a mathematical tool to describe how the optical response varies for the different directions in a three‐dimensional coordinate system. ^[42] Uniaxial anisotropy refers to a type of anisotropy in which a sample exhibits different physical properties along a single preferred direction, which can manifest in various material's properties. The complex refractive index splits into two orthogonal components, the ordinary (in‐plane) components within the plane of the thin film, n _o and k _o, and the extra‐ordinary (out‐of‐plane) components perpendicular to the plane of the thin film, n _eo and k _eo. This uniaxial type of anisotropy due to horizontal molecular alignment in non‐crystalline thin films is very common in polymers, conjugated molecules and low‐dimensional Van‐der‐Waals materials.[ ⁴⁸ , ⁴⁹ , ⁵⁰ ] A “spaghetti model” is a conceptual way to illustrate the idea of uniaxial anisotropy in polymer thin films, in which the polymer chains are imagined as cooked spaghetti distributed on a plate. The polymer chains preferably lie on the substrate and point randomly in all horizontal directions along the surface, which is the plane of the ordinary optical components. Consequently, the extra‐ordinary optical components are located perpendicular to the polymer chains and the substrate plane.

However, the accurate determination of an uniaxial anisotropic complex refractive index for absorbing materials by ellipsometry is a non‐trivial task.[ ⁵¹ , ⁵² , ⁵³ , ⁵⁴ ] Approaches based solely on unpolarized transmission and reflection spectra, in the weakest case considering samples with only one layer thickness, may fail to capture the uniaxial anisotropy. ^[55] A spectroscopic ellipsometry (SE) measurement at a fixed angle of incidence (AOI) records two independent pieces of information: the real and the imaginary part, Ψ and Δ, of the complex reflectance ratio given by the complex Fresnel coefficients for p‐ and s‐polarized light, ${\tilde{R}}_p$ and ${\tilde{R}}_s$ : ^[56]

$${\displaystyle \rho =\tan \left(\Psi \right)e^{\mathrm{i}\Delta }=\frac{{\tilde{R}}_p}{{\tilde{R}}_s}}$$ (4)

Even for the simpler case of an isotropic, absorbing thin film there are three unknown parameters: n, k and layer thickness d, so that the two measured values Ψ and Δ do not provide a unique solution. Organic thin films typically are tranparent in the near‐infrared spectral range, and in this transparent regime the problem is reduced to the determination of n and d. Once the thickness is determined and fixed by a Cauchy model, both n and k can also be determined in the absorbing spectral range. By including normal incidence transmission intensity spectra T in the fitting procedure, the parameter correlation between thickness and complex refractive index is greatly reduced, so that results with greatly improved significance can be achieved.[ ⁵⁷ , ⁵⁸ ] However, this requires a transparent substrate, which must be carefully characterized in advance to prevent small absorptions in the substrate (such as the iron content in float glass) from mixing into the optical properties of the thin film.

A uniaxial, absorbing thin film has at least 5 unknown parameters: n _o, k _o, n _eo, k _eo and d, assuming that the substrate is already fully characterized. Measurements with variable AOIs (VASE) increase the amount of measured information by providing multiple optical path lengths. More effective in this sense is a multi‐sample analysis (MSA), which performs a simultaneous data regression on the same material with a single set of uniaxial optical components, but with different layer thicknesses. ^[59] Again, the inclusion of transmission intensity spectra in the simultaneous data regression considerably reduces the parameter correlation between thickness and complex refractive index, but offeres a further advantage. The sensitivity to the out‐of‐plane components in z‐direction is inherently low: The s‐polarized light has no component in the z‐direction, and the z‐component of the p‐polarized light is small even for large AOIs, as the AOI becomes much steeper within the thin layer due to refraction. At normal incidence, the condition for the transmission intensity spectra T, only the in‐plane components are excited, so that an unambiguous solution is possible at least for the in‐plane optical components. ^[60]

For the presented ellipsometric investigation, the MSA plus T strategy was employed, combining multiple VASE measurements of multiple layer thickness samples with transmission spectra for each polymer compound, to obtain dependable optical response functions.[ ⁵⁷ , ⁶⁰ ] It is a prerequisite that the samples combined in the regression analysis vary in layer thickness only and otherwise are expected to have the same complex refractive index. Instead of an oscillator model, a B‐spline regression was implemented to accurately model the ellispometric data without enforcing a specific physical interpretation of the spectral features.[ ⁶¹ , ⁶² ] At this stage we are not trying to interpret the excitonic and vibronic nature of the optical spectra, but rather to provide an accurate analytical representation of the complex refractive index curves. The optical properties of the float glass substrate have been thoroughly characterized in advance, and a detailed presentation of the complex refractive index can be found in the Supporting Information in Figure S8. We paid particular attention to the fact that float glass has two sides, the “air” side and the “tin” side, with slightly different optical surface properties due to its processing conditions, ^[63] and applied all polymer layers preferably to the “air” side. For each of the six polymer thin films, at least three different film thicknesses in the range of approximately 5 nm to 120 nm were prepared. Layer thicknesses were determined by an isotropic Cauchy model in the transparent spectral range as a first step of the MSA regression procedure. The final thickness values were obtained after expansion to the full spectral range and conversion to a B‐spline function taking uniaxial anisotropy into account.

Since this polymer layer thickness is a critical parameter in the early phase of the fitting process, it deserves attention and validation. Thickness was therefore also determined using a complementary tactile method, namely profilometry. A detailed comparison of thickness values determined using profilometry $d_{\mathrm{P}}$ and ellipsometry $d_{\mathrm{E}}$ can be found in the Supporting Information in Table S1. Both methods find thicknesses in the same order of magnitude, which reflect the dependence on doctor‐blading velocities well. However, we found systematically lower thickness values when using profilometry, which also exhibit greater experimental spread. Profilometry has the undoubted disadvantage of being destructive and localized, as it uses a needle with a diameter of 2 μm to perform line scans, and the dependence on the experimenter appears to be greater. A single ellipsometric measurement gives an average value over a larger sample area, with the illuminated area varying from about 4.4 mm² to 12 mm² for AOIs from 45° to 75°. Ellipsometry is more abstract, but the routines are also more formalized and gain reliability through the MSA plus T strategy. However, the inherent correlation of n and $d_{\mathrm{E}}$ cannot be completely overcome, leaving room for residual systematic errors. Further explanations can be found in the Supporting Information in Section “Notes on the Correlation between Complex Refractive Index and Layer Thickness”.

For all six polymer samples, we have found complex refractive indices exhibiting a clear uniaxial anisotropy, see Figure 2. The difference between the ordinary components, n _o and k _o shown with solid lines, and the extra‐ordinary, n _eo and k _eo shown with dotted lines, is particularly large in the absorbing spectral regions. The ordinary components are always significantly larger, indicating a strongly favored horizontal orientation of the optical transition dipole moments and consequently a horizontal orientation of the conjugated polymer chains. In the transparent longer wavelength range, all polymer samples exhibit a negative uniaxial birefringence, i. e. n _o is larger than n _eo. This means that the fast ray travels along the extra‐ordinary direction, which is then the direction of the optical axis, i. e. the optical axis of all polymer samples is orientend perpendicular to the surface. Such significant uniaxial anisotropy has been previously documented for neat F8BT and MDMO‐PPV thin films. ^[64] In case of F8BT, the anisotropy was found to increase with increasing polymer molecular weight. ^[65] It was also found that the molecular weight and polydispersity index of PTB7 influence the anisotropy of the thin films. ^[66] Furthermore, the uniaxial anisotropy is significantly reduced in fullerene‐blended PTB7 thin films, as required for photovoltaic applications. The same trend was observed for neat and fullerene‐blended ZZ50 (PCPDTBT) thin films. ^[54] Overall, we note good general agreement with the complex refractive indices reported in the literature. However, we could not find any reference data for PDCBT and PBDB−T‐2F.

The maxima of the obtained ordinary extinction coefficients are between 0.5 (F8BT) and 1.1 (ZZ50), which corresponds to the typical range for conjugated polymer thin films. For a large variety of polymer thin films analyzed by spectroscopic ellipsometry, a remarkable uniformity of extinction coefficients within this range was observed. ^[67] This was attributed to their persistence length, i. e. the average distance over which the polymer chain maintains a relatively straight or persistent structure before it becomes significantly bent or coiled. For a given polymer, the extinction coefficient in this context can increase with increasing molecular weight. It has been found that diketopyrrolopyrrole‐containing (DPP) polymer types have an increased extinction coefficient of up to 1.5 due to their high persistence length. ^[67] However, the polymers selected in this study exhibit extinction coefficients within the representative “uniformity range”, which allows a certain generalizability of the correlation between the onset of absorption and the absorption strength.

To quantify the absoprtion strength, only the ordinary extinciton coefficient k _o is taken into account to calculate the ordinary, or for simplicity, just absorption coefficient α as follows:

$${\displaystyle \alpha =\frac{4\pi k_{\mathrm{o}}}{\lambda }}$$ (5)

The absorption coefficients α for all six polymer samples are shown in Figure 3(a). In addition, their maximum values ${\alpha }_{\max }$ spectrally closest to their absorption onsets, as well as their respective spectral positions in wavelength ${\lambda }_{\alpha }$ and photon energy $E_{\alpha }$ are listed in Table 1. The maxima of the absorption coefficients ${\alpha }_{\max }$ are in the same order of magnitude for all six polymer samples, approximately between 138000 cm⁻¹ (F8BT) and 202000 cm⁻¹ (ZZ50). This means that the difference between the smallest and the largest ${\alpha }_{\max }$ value is only about 38 %.

Figure 3.

(a) Ordinary absorption coefficients $\alpha =4\pi k_{\mathrm{o}}/\lambda$ versus photon energy. (b) Maximum ordinary absorption coefficient values ${\alpha }_{\max }$ versus optical band gap energies E _g determined from Tauc plots (see Supporting Information Figure S9) considering the ordinary absorption coefficient only. The numeric values are listed in Table 1. The error bars consider also experimental spread, and for ${\alpha }_{\max }$ are concealed by the dots. (c) “Maximum absorbable photon energy per pathlength” ${\alpha }_{\max }·E_{\alpha }$ versus spectral position of $E_{\alpha }$ in photon energy.

Table 1.

Maximum ordinary absorption coefficient values α_max (spectrally nearest to the optical band gap E _g) and their spectral positions in wavelength λ_α and photon energy E _α as well as optical band gap energies Eg determined from Tauc plots considering the ordinary absorption coefficient only, see Supporting Information Figure S9. Error bars consider experimental spread for α_max and E _g, but solely relate to the spectral resolution in case of λ_α and E _α. Data are graphed in Figure 3. Also some reference optical band gap values E _gR (preferably based on Tauc plot method) are listed.

Polymer	α max (103 ⋅ cm−1)	λ α (nm)	E α (eV)	band gap E g (eV)	references E gR (eV)
F8BT	138±1	459±2	2.701±0.013	2.45±0.14	2.4(17), 2.51(18)
MDMO‐PPV	166±1	500±2	2.479±0.008	2.18±0.18	2.17(19), 2.23(20)
PDCBT	190±1	551±2	2.248±0.007	1.97±0.18
PBDB−T‐2F (PM6)	147±1	574±2	2.161±0.006	1.90±0.19	1.82(68), 1.80(69)
PTB7	148±1	609±2	2.036±0.006	1.73±0.10	1.58(37), 1.64(36)
ZZ50 (PCPDTBT)	202±1	679±2	1.826±0.005	1.53±0.12	1.44(70), 1.45(71), 1.5 (72)

To quantify the onset of absorption, the optical band gap energies E _g were estimated from the absorption coefficients α of all polymer thin film samples using the commonly applied Tauc plot method. ^[73] Details on procedure as well as Tauc plots can be found in the Supporting Information Figure S9. As the Tauc method refers to the onset of absorption, this results in the energy of the optical band gap. This is generally smaller than the energy of the electronic band gap, which refers to the HOMO‐LUMO gap.[ ⁷⁴ , ⁷⁵ ] The values of the optical band gap energies E _g are listed in Table 1 together with reference values E _gR taken from literature, also applying the Tauc plot method. Note that we were unable to find a reference value for PDCBT. The uncertainties of the optical band gap energies E _g given in this study may appear quite large, especially since none of the literature references give uncertainties. However, the estimative nature of the Tauc plot method, as described in the Supporting Information, easily leads to a deviation of the determined optical band gap energies of about ±0.1 eV. Together with an estimation of the experimental spread based on the MSE of the ellipsometric fitting, we arrive at the given uncertainties. Nevertheless, we found good agreement between the determined E _g and the reference values, which are all within the stated uncertainties of our data. For PBDB−T‐2F, PTB7, and ZZ50 the determined E _g values have a tendency towards higher energies. The reasons for this may also be due to different processing conditions and molecular weights for the literature reference samples.

As intended, the optical band gap energies are distributed over the visible spectral range, approximately between 2.5 eV and 1.5 eV, in the descending order F8BT, MDMO‐PPV, PDCBT, PBDB−T‐2F, PTB7 and ZZ50. The spectral positions $E_{\alpha }$ of the maxima of the absorption coefficients ${\alpha }_{\max }$ follow the same order. However, the values of ${\alpha }_{\max }$ tend to scatter randomly within their rather narrow “uniformity range”. In Figure 3(b), ${\alpha }_{\max }$ is plotted versus the optical band gap energies E _g for each of the six polymers samples, which exhibit no noticeable correlation. Thus, we could not detect any “absorptive” analogy to the “emissive” Energy Gap Law, which relates the nonradiative decay rate of photoluminescence to electronic band gap energy.[ ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ ]

The maximization of energy conversion, i. e. the conversion of photon energy into photocurrent, is of central importance for photovoltaics. We introduce the quantity of the “maximum absorbable photon energy per pathlength”, which corresponds to the maximum absorption coefficient multiplied by its spectral position ${\alpha }_{\max }·E_{\alpha }$ . This quantity is plotted in Figure 3(c) against the spectral position $E_{\alpha }$ , which follows the same order as the spectral distribution of the optical band gap energies E _g. Here, too, there is no recognizable correlation between the spectral position and the “maximum absorbable photon energy per pathlength”. The latter values are in the same order of magnitude for all six polymer samples, approximately between 300 eV cm⁻¹ (PTB7) and 430 eV cm⁻¹ (PDCBT). The difference between the smallest and the largest value is only about 35 %, which is even a smaller range than for the maximum absorption coefficients ${\alpha }_{\max }$ . However, if the quantity of absorbable photons (${\alpha }_{\max }$ ) is weighted with their energies per photon ($E_{\alpha }$ ) changes the distribution pattern of the values, see Figure 3(b) and (c). For example, the compound with the largest absorption coefficient ZZ50, but the smallest optical band gap, drops from first to fourth place when the “maximum absorbable photon energy per pathlength” is taken into account. In contrast, the compound F8BT with the smallest absorption coefficient but with the largest optical band gap moves from last place to third place in this respect for the given selection of polymer samples.

Conclusions

We have given an overview of the basic optical and dielectric properties of six selected donor‐type polymers that have been processed from solution into thin films. These polymers are F8BT, MDMO‐PPV, PDCBT, PBDB−T‐2F (PM6), PTB7 and ZZ50 (PCPDTBT), which are established compounds for optoelectronic applications. Nevertheless, we could not find any valid data on complex refractive indices obtained by ellipsometry for PBDB−T‐2F and PDCBT in the literature, and for the latter also no values for the optical band gap. Through a facile multi‐sample approach using VASE ellipsometry, which we have described in great detail, we obtain representative complex refractive indices that exhibit strong uniaxial anisotropy. From the ordinary extinction coefficients, the absorption coefficients can be calculated, and a subsequent Tauc plot analysis leads to a reasonable estimate of the optical band gap energies. These optical band gap energies are intentionally distributed over the visible spectral range. We wanted to elucidate whether there is a correlation between the absorption onsets (optical band gap energies) and the absorption strength (maxima of the absorption coefficients). What we were able to confirm is an uncorrelated distribution of maximum absorption coefficients within a certain “uniformity range” of absorption strengths observed in a previous systematic study. This limitation is due to the similarities of the electronic structure of the repeating molecular units as well as the electronic conjugation and physical conformation of the polymer chains (persistence length) in all compounds. In order to obtain an absorption strength beyond this uniformity range, new structural design approaches for the polymer compounds must be considered.

Experimental

Materials

F8BT (or PFBT):

Poly(9,9‐dioctylfluorene‐alt‐benzothiadiazole).

(C₃₅H₄₂N₂S)n.

Luminescence Techology Lot‐Nr. S957‐130108.

MDMO‐PPV (or OC₁C₁₀–PPV):

Poly‐[2‐(3,7‐dimethyloctyloxy)‐5‐methyloxy]‐para‐phenylene‐vinylene.

(C₁₉H₂₈O₂)n. $M_{\mathrm{n}}\approx$ 170000 g/mol.

Covion GmbH Germany.

PDCBT:

Poly[2,2””‐bis[[(2‐butyloctyl)oxy]carbonyl][2,2’:5’,2”:5”,2”’‐quaterthiophene] −5,5”’‐diyl].

(C₄₂H₅₆O₄S₂)n. 1‐Material, LOT YY1146.

PBDB‐T‐2F (or PBDB−T‐F, PBDB‐TF, PM6):

Poly[(2,6‐(4,8‐bis(5‐(2‐ethylhexyl‐3‐fluoro)thiophen‐2‐yl)‐benzodithiophene))‐alt‐(5,5‐(1’,3’‐di‐2‐thienyl‐5’,7’‐bis(2‐ethylhexyl)benzo[1’,2’‐c:4’,5’‐c’]dithiophene‐4,8‐dione)].

(C₆₈H₇₆F₂O₂S₈)n. 1‐Material LOT YY15020CH.

PTB7:

Poly[[4,8‐bis[(2‐ethylhexyl)oxy]benzo[1,2‐b:4,5‐b’]dithiophene‐2,6‐diyl][3‐fluoro‐2‐[(2‐ethylhexyl)carbonyl]thieno[3,4‐b]thiophenediyl]].

(C₄₁H₅₃FO₄S₄)n. $M_{\mathrm{n}}>$ 22000 g/mol.

Solarmer.

ZZ50 (or c‐PCPDTBT):

Poly[2,6‐(4,4‐bis‐(2‐ethylhexyl)‐4H‐cyclopenta[2,1‐b;3,4‐b’]dithiophene)‐alt‐4,7(2,1,3‐benzothiadiazole)].

(C₃₁H₃₈N₂S₃)n. $M_{\mathrm{n}}\approx$ 35000 g/mol.

Konarka Technologies.

Sample Preparation

All polymers were dissolved in chlorobenzene with a concentration of 5 mg/mL (F8BT, MDMO‐PPV, PDCBT, PBDB−T‐2F, PTB7) or 6 mg/mL (ZZ50) and stirred on hotplate set to 80 °C for at least 1 day before processing.

Glass objective slides (Marienfeld, 7.5 cm×2.5 cm, 1 mm thick) were cleaned in an ultrasonic bath for 15 minutes using subsequently acetone, iso‐propanol and DI water, and then blow‐dried with nitrogen. Immediatly before coating the substrate surface (“air” side of the glass, see Supporting Information Figure S8) was treated with oxygen plasma for 5 minutes.

All polymer layers were produced by doctor‐blading onto such glass objective slides. The substrate holder was heated to 80 °C and the coating blade was positioned with less than 1 mm gap above the glass surface. After applying a small amount of the heated polymer solution into this gap, where it was held by capillary forces, the blade was moved across the surface at a constant speed. Doctor‐blading velocities varied in principle from 2.5 mm/s to 40 mm/s in steps of 5 mm/s, but not all speeds were used for each polymer. In general, the faster the blade moves, the thicker the polymer layer becomes, but the reproducibility of a certain target thickness is rather low. A detailed comparison of polymer layer thickness values is given in the Supporting Information in Table S1.

Ellipsometry

A J.A. Woollam M2000‐DI single rotating compensator ellipsometer (PCSA) with a horizontal sample stage was used for variable angle spectroscopic ellipsometry (VASE) scans in reflection to determine Ψ and Δ, and for normal incidence transmission intensity measurements. Measuring the transmission intensity requires a completely transparent substrate, which leads to reflections on the backside of the substrate. These are incoherent with the reflections from the top side due to the substrate thickness of 1 mm and therefore cause an unwanted depolarization of the recorded signal. ^[76] However, these incoherent backside reflections can be included in the model. Angles of incidence (AOIs) from 45° to 75° in steps of 5° were chosen to record Ψ and Δ. With an approximate beam diameter of 2 mm the illuminated area varies from about 4.4 mm² to 12 mm² for AOIs from 45° to 75° assuming an elliptical shape. The spectral range from 193 nm to 1690 nm is recorded with two CCD‐detector arrays offering 705 wavelengths simultaneously with a bandwidth of 5 nm within the UV‐VIS and of 10 nm within the NIR spectral range. The vendor provided software CompleteEASE was used for data recording (version 5) and for analysis (version 6). A multi‐sample analysis (MSA) was implemented to combine multiple measurements per sample (minimum of 2) and measurements from samples with varying layer thicknesses (minimum of 3) to obtain a representative complex refractive index for a polymer material. The transmission intensity spectra T were included (fit weighting 1000 %) for parameter decorrelation of complex refractive index and layer thickness. ^[57] Optimization criteria were a “visual best match” for the measured and modeled data, especially transmission data, as well as minimization of the Root Mean Squared Error (MSE). The number N of all simultaneously fitted data sets for each of the six polymers is given in the caption of Figure 2 together with the respective MSE. Note that the determined complex refractive index is valid only for particular processing conditions such as solvent and annealing temperature, and may also vary for different polymer batches. It is a prerequisite that the samples combined in the MSA+T regression analysis vary in layer thickness only and otherwise are expected to have the same complex refractive index. Just the selected parameters layer thickness and incoherent backside reflections (fixed at zero for transmission intensity measurements though) were allowed to vary for each MSA dataset, all other fit parameters are collective. The general fitting procedure was as follows: ^[60] (1) fit layer thicknesses for transparent regime with Cauchy layer; (2) convert to transparent B‐spline (Kramers Kronig consistent mode, E2 positive) plus wavelength expansion fit with 0.1 eV resolution; (3) fix IR‐amplitude at zero, keep E‐infinity offset as fit parameter; (4) convert to anisotropic (type uniaxial) difference mode off; (5) adjust band gap; (6) customize node spacing, i.e more nodes for spectral regime with absorption features in Ex‐component (ordinary component), and set Ez‐component (extra‐ordinary component) to 0.2 eV. The latter accounts for the low sensitivity to the detailed shape of the extra‐ordinary component. ^[57] Surface roughness could be added optionally, which improved the fit only in case of MDMO‐PPV. The “model‐free” B‐spline approach is advantageous because it does not demand detailed knowledge about the electronic (and vibronic) transitions, which is required for choosing physical meaningful oscillator types and parameters.[ ⁶¹ , ⁶² ]

The refractive index of the 1.0 mm thick float soda‐lime glass substrates (7.5 cm×2.5 cm and 1 mm thick objective slides, Marienfeld) has been determined in advance using a Kramers‐Kronig‐consistent B‐Spline with transmission intensity data included. Both sides of two uncoated slides were inspected to distinguish between the “tin” and the “air” side as discussed in the Supporting Information and shown in Figure S8. ^[63] The polymer layers were preferably coated on the “air” side of the glass slides for consistency of the MSA.

Tauc Plots

Following the commonly applied Tauc method the optical band gap energies E _g were estimated from the absorption coefficients of the polymer thin film samples. ^[73] The ordinary absorption coefficient $\alpha =4\pi k_{\mathrm{o}}/\lambda$ valid only for normal incidence transmission were considered. Details on procedure as well as Tauc plots can be found in the Supporting Information Figure S9. Note that the Tauc method relates to the onset of absorption and therefore gives the optical band gap energy. It is usually smaller than the electronic band gap energy, which refers to the HOMO‐LUMO gap of a compound.[ ⁷⁴ , ⁷⁵ ]

Profilometry

Thickness measurements employing profilometry were performed to validate the polymer layer thicknesses determined via ellipsometric fitting. For this a Dektak XT profilometer was used with a scanning needle of 2 m diameter and a scanning speed of 100 μm/s. The polymer layers needed to be manually scratched with a pin needle to obtain grooves where the scanning needle could sweep across. A detailed comparison of polymer layer thickness values determined by profilometry and ellipsometry is given in the Supporting Information in Table S1.

Supporting Information

Dielectric functions versus photon energy as an alternative representation of the complex refractive index. Advanced ellipsometric characterization of the float glass substrate's refractive index. Comparison of layer thicknesses determined by ellipsometry and profilometry. Detailed description of Tauc plot method and Tauc plots.

Tabulated spectroscopic data of the complex refractive indices and absorption coefficients are attached as data‐files and can be downloaded from Zenodo: https://doi.org/10.5281/zenodo.13732081.

Conflict of Interests

We declare no conflicts of interest.

Supporting information

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Kamptner A., Scharber M. C., Schiek M., ChemPhysChem 2024, 25, e202400233. 10.1002/cphc.202400233

Data Availability Statement

The data that support the findings of this study are available in the supplementary material of this article.