# Application of the Holomorphic Tauc-Lorentz-Urbach Function to Extract the Optical Constants of Amorphous Semiconductor Thin Films

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

^{®}and 3-mm-thick Borofloat

^{®}33) substrates using a commercial single-chamber magnetron-sputtering MV System, with a vertically adjustable cathode operated by RF power. The target–substrate distance was set to an appropriate value of $6.1\phantom{\rule{4pt}{0ex}}\mathrm{cm}$. The 3-inch diameter p-Si target (Si) was manufactured by Lesker Company (St Leonards-on-Sea, East Sussex, UK), with a purity of $99.999\%$. It had an electrical resistivity of $0.005$–$0.020\phantom{\rule{4pt}{0ex}}\mathsf{\Omega}\mathrm{cm}$ and a mass density of $2.32\phantom{\rule{4pt}{0ex}}\mathrm{g}/{\mathrm{cm}}^{3}$.

## 3. Transforming the NTLU Model to the ATLU Model

## 4. Fitting the Universal Transmission Formula to the As-Measured Spectrum

## 5. OCISPY (Optical Characterization by Inverse Synthesis): Python-Coded Computer Program for Determining the Optical Properties of Amorphous Semiconductor Films by Employing Multiple Dispersion Models

## 6. Extracting the Optical Constants n and k by Employing the Original NTLU and the Novel ATLU Dispersion Model

## 7. The Alternate Swanepoel Technique: Model-Free Determination of $n$ and $\alpha $, for Uniform Thin Films on Transparent Substrates

## 8. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) A schematic illustration of the wedge-shaped geometry of a thin layer. (

**b**) Raman spectra of four a-Si specimens deposited at $25\phantom{\rule{4pt}{0ex}}\xb0\mathrm{C}$ [12,13]. (

**c**) Grazing-incidence XRD diagram and (

**d**) micro-XRD pattern of an a-Si thin-film sample, grown at a temperature of $25\phantom{\rule{4pt}{0ex}}\xb0\mathrm{C}$. In the micro-XRD diagram of this particular sample, several pronounced diffraction peaks can be noticed, which can be undoubtedly attributed to ${\mathrm{SiO}}_{2}$ peaks (surface native oxide). The two diffraction peaks, $\left(400\right)$ and $\left(331\right)$, observed exclusively in this particular specimen, can correspond to crystalline planes of silicon indicative of a possible ordered nano-structure (nano-crystalline nature), embedded in the whole and absolutely dominant amorphous matrix (it has been considered negligible from the viewpoint of the assumed optical model) [14,15].

**Figure 2.**A very simplified flowchart corresponding to the optimization algorithm for the comprehensive optical characterization of amorphous-semiconductor thin layers. This specific algorithm is implemented in the Python-based computer program, OCISPY, presented and employed in this work. $\theta $ stands for the parameter vector introduced in the program. It very accurately solves the problem of extracting the optical properties, together with the average layer thickness, $\overline{d}$, and the wedging parameter, $\Delta d$, of non-crystalline-semiconductor thin films.

**Figure 3.**Two examples of the typical normal-incidence transmission spectrum of an approximately $1.1$-$\mathsf{\mu}\mathrm{m}$-thick amorphous Si thin film that have been sputtered on a transparent glass substrate, held at a temperature of (

**a**) $25\phantom{\rule{4pt}{0ex}}\xb0\mathrm{C}$ and (

**b**) $325\phantom{\rule{4pt}{0ex}}\xb0\mathrm{C}$, during the sputtering deposition process. Furthermore, the NATLU- and ATLU-generated transmission spectra of the previous two a-Si specimens, are also displayed. The relative difference between the experimental and simulated transmittance curves is also displayed in this figure for the two representative a-Si samples.

**Figure 4.**Refractive index, n, and extinction coefficient, k, versus wavelength, for a-Si thin layers (specimens (

**a**) $\mathrm{Si}\#02544$ and (

**b**) $\mathrm{Si}\#32540$), obtained by inverse synthesis, with the help of the holomorphic ATLU function adopted in this work. In addition, the values of n and k found by employing the model-free Swanepoel envelope technique are also shown for comparison in this figure.

**Figure 5.**Photon-energy dependence of the real and imaginary parts of the complex dielectric constant of two a-Si thin-film samples: (

**a**) $\mathrm{Si}\#02544$ and (

**b**) $\mathrm{Si}\#32540$. Shown also in this figure, the first and second derivative of ${\u03f5}_{1}$ and ${\u03f5}_{2}$ (

**c**–

**f**), demonstrating that they are certainly twice differentiable, as should be the case considering that the ATLU dielectric function is complex analytic.

**Figure 6.**Application of the Swanepoel technique as elaborated in the text: Using the as-measured normal-incidence transmission spectra of (

**a**) $\mathrm{Si}\#02544$ and (

**b**) $\mathrm{Si}\#32540$ a-Si thin-film samples, and that of the finite glass substrate alone. The two constructed, top- and bottom-transmission envelopes, are also displayed for the two a-Si specimens. Moreover, some exact values of the interference order, m, are indicated in this figure in order to illustrate the model-free Swanepoel technique.

**Table 1.**Values of all the ATLU model parameters for RFMS-a-Si thin layers, each prepared with a different Ar-gas pressure. In the sample identification (ID) code, the first adopted three numbers indicate the deposition temperature in degrees Celsius, whereas the next two numbers show the Ar-working pressure during the deposition in the particularly chosen unit decipascal. In addition, the values of the adopted cost or merit function, FoM, for those a-Si samples under study are presented. The values of the Urbach energy and amplitude, ${E}_{\mathrm{u}}$ and ${A}_{\mathrm{u}}$, respectively, obtained from the previous ATLU-model parameters, are also listed in the table. On the other hand, the value of the geometrical parameter $\overline{d}\phantom{\rule{4pt}{0ex}}(\equiv d$ for these a-Si samples), is also indicated. The best-fit values for $\Delta d$ are not shown in the table, since they are practically zero in all the roughly 1.1-$\mathsf{\mu}$m-thick-specimens investigated; only layers with thickness about or less than $800\phantom{\rule{4pt}{0ex}}\mathrm{nm}$ exhibited a fitted value of $\Delta d>0$. Moreover, the experimentally determined values of the surface roughness parameter, ${R}_{\mathrm{q},\mathrm{AFM}}$, for the seven a-Si specimens are also presented. In addition, the film thickness, ${d}_{\mathrm{SEM}}$, measured from the cross-sectional SEM micrographs corresponding to the present a-Si specimens under study, are also listed.

**Table 2.**The obtained values for three dielectric-constant parameters ${\u03f5}_{1}\left(0\right)$, ${\u03f5}_{2,\mathrm{max}}$, and $E\left({\u03f5}_{2,\mathrm{max}}\right)$ (see Figure 5), for the seven a-Si specimens under analysis, are all listed in this table. The fitted values have been found by adopting the holomorphic ATLU function.

**Table 3.**Optical and geometrical characterizations method for the two representative samples, Si$\#02544$ and Si$\#32540$, grown by using the highest Ar working-gas pressures of $4.4$ and $4.0$ Pa, respectively. Values of the optical and geometrical parameters, ${\lambda}_{\mathrm{tan}}$,${T}_{\mathrm{s}}$, s, ${T}_{+}$, ${T}_{-}$, ${n}_{\mathrm{crude}}$, ${d}_{\mathrm{crude}}$, ${m}_{0}$, m, ${d}_{\mathrm{acc}}$, $\alpha $, and k, calculated from their respective normal-incidence transmittance spectra, by employing to that end the Swanepoel transmission-envelope approach. The meaning of all the symbols are fully explained in the text. The values of ${T}_{+}$ and ${T}_{-}$ in bold character refer to the actual ‘measured’ value of transmission, while the other one belongs to the calculated envelope.

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**MDPI and ACS Style**

Ballester, M.; García, M.; Márquez, A.P.; Blanco, E.; Fernández, S.M.; Minkov, D.; Katsaggelos, A.K.; Cossairt, O.; Willomitzer, F.; Márquez, E. Application of the Holomorphic Tauc-Lorentz-Urbach Function to Extract the Optical Constants of Amorphous Semiconductor Thin Films. *Coatings* **2022**, *12*, 1549.
https://doi.org/10.3390/coatings12101549

**AMA Style**

Ballester M, García M, Márquez AP, Blanco E, Fernández SM, Minkov D, Katsaggelos AK, Cossairt O, Willomitzer F, Márquez E. Application of the Holomorphic Tauc-Lorentz-Urbach Function to Extract the Optical Constants of Amorphous Semiconductor Thin Films. *Coatings*. 2022; 12(10):1549.
https://doi.org/10.3390/coatings12101549

**Chicago/Turabian Style**

Ballester, Manuel, Marcos García, Almudena P. Márquez, Eduardo Blanco, Susana M. Fernández, Dorian Minkov, Aggelos K. Katsaggelos, Oliver Cossairt, Florian Willomitzer, and Emilio Márquez. 2022. "Application of the Holomorphic Tauc-Lorentz-Urbach Function to Extract the Optical Constants of Amorphous Semiconductor Thin Films" *Coatings* 12, no. 10: 1549.
https://doi.org/10.3390/coatings12101549