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Temperature Coefficient of Electronic Polarizability in Thin Polymer Films Deposited on Si and SiO_{2} Substrates Determined via Spectroscopic Ellipsometry

^{1}

^{2}

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## Abstract

**:**

_{2}substrates. The influence of thin-film thickness on the accuracy of TOC and LTEC determination is discussed.

## 1. Introduction

_{2}) and light-absorbing (Si (100)) substrates. The influence of the thin-film thickness on the accuracy of TOC and LTEC determination is discussed. Moreover, the analysis of temperature-dependent ellipsometry results developed here can be considered a convenient method for determining thermo-optical parameters such as the LTEC, TOC and TCEP.

## 2. Materials and Methods

_{2}substrates in order to determine the thermal properties of these layers. Importantly, the proposed method provides a convenient platform for inferring the microscopic properties of matter, such as its electronic polarizability and the volume of its single repeating unit, based on its macroscopic properties. In this section, we briefly present the basic relationships, following our earlier work [15]. The polarizability of a molecule, $\gamma ,$ is a measure of its ability to respond to the acting electric field, ${E}_{act},$ and can be used to ultimately obtain the electric dipole moment, p, which can be written as follows:

## 3. Results

#### 3.1. Unique Solution for the LTEC and TOC

#### 3.2. Analysis of the Accuracy of TCEP, LTEC and TOC Determination

## 4. Discussion

^{−1}and ${\beta}_{0}=-105\times {10}^{-6}$ °C

^{−1}which, according to Equation (9), gives ${\Xi}_{0}=23\times {10}^{-6}$ °C

^{−1}. On the other hand, from ref. [24], we have n

_{0}(632.8 nm)= 1.4916, ${\alpha}_{0}=248\times {10}^{-6}$ °C

^{−1}and ${\beta}_{0}=-141\times {10}^{-6}$ °C

^{−1}, which gives ${\Xi}_{0}=4\times {10}^{-6}$ °C

^{−1}. Interestingly, Berini et al. [25] reported the same value of the refractive index n(633 nm) = 1.4916 for PMMA as in [24]. However, in the case of the thermo-optical coefficient measured by a very sensitive interferometric technique, ${\beta}_{0}=-120\times {10}^{-6}$ °C

^{−1}was given in the tested temperature range from 20 to 70 °C

^{−1}[25]. As can be seen, values of ${\Xi}_{0}$ for PMMA as inferred from the cited data differ six fold. All of this clearly shows how important the accuracy of VTEC and TOC determination is.

#### 4.1. Effect of Substrate and Thin Layer Thickness on Precise Determination of LTEC and TOC in Temperature-Dependent Spectroscopic Ellipsometry—Optimal Spectral Windows

_{2}, respectively, for the reference temperature ${T}_{o}=25$ °C. Since our approach assumes that ${\rho}_{0}\left(\lambda \right)$ of the thin polymer film and the thermo-optical properties of the substrate are known, we will take the necessary inputs from the available literature. Namely, in the case of the PMMA film, we use the data of Soave et al. [24] and for the Si (100) substrate, we use data from the works of Jellison and Modine [26,27], while for SiO

_{2}, data are from Ghosh [28,29].

#### 4.1.1. Thin PMMA Films on a Si (100) Substrate

#### 4.1.2. Thin PMMA Films on a SiO_{2} Substrate

_{2}substrate [24]. To facilitate the comparison, we will present the results in a similar way to above for a thin PMMA film on a Si substrate. Thus, in Figure 3, we show the 3D graph in which the three coordinates ${h}_{p}$, ${\lambda}_{p}$ and ${\theta}_{p}$ of the drawn points define the maxima of the determinant $\left|D({h}_{p},{\lambda}_{p},{\theta}_{p})\right|.$

_{2}. Finally, in Figure 4, we show the spectral relationships for the coefficients ${c}_{11},$ ${c}_{12}$, ${c}_{21}$ and ${c}_{22}$ at the indicated $\theta $ values. These plots are dominated by the steep and linear behavior of ${c}_{12}$ and the nearly zero values of ${c}_{22}$. However, much more important is the reduction in the achievable accuracy of LTEC and TOC determination, which forced us to quadruple the accuracy thresholds to the value of $\pm 0.02$.

#### 4.1.3. Discussion of the Results

#### 4.1.4. Validity of the Linear Approximations

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

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**Figure 1.**Coordinates ${h}_{p}$, ${\lambda}_{p}$ and ${\theta}_{p}$, for which the determinant of the Cramer’s equation coefficients, see Equations (22) and (23), reaches the maximum.

**Figure 2.**The spectral dependence of coefficients ${c}_{11},$ ${c}_{12}$, ${c}_{21}$ and ${c}_{22}$ for six films with thicknesses of h = 120, 90, 80, 70, 60 and 55 nm, for figures (

**a**–

**f**), respectively, of PMMA deposited on a Si (100) substrate at indicated incidence angles $\theta $ detuned from the corresponding ${\theta}_{p}$ by 1°. A pair of adjacent arrows at different levels indicates the optimal spectral window.

**Figure 4.**The spectral dependence of coefficients ${c}_{11},$ ${c}_{12}$, ${c}_{21}$ and ${c}_{22}$ for six films with thicknesses of $h=120,90,80,70,60$ and 55 nm, for figures (

**a**–

**f**), respectively, of PMMA deposited on a SiO

_{2}substrate at indicated incidence angles $\theta $ detuned from the corresponding ${\theta}_{p}$ by 1°. A pair of adjacent arrows at different levels indicates the optimal spectral window.

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**MDPI and ACS Style**

Bednarski, H.; Hajduk, B.; Jarka, P.; Kumari, P.
Temperature Coefficient of Electronic Polarizability in Thin Polymer Films Deposited on Si and SiO_{2} Substrates Determined via Spectroscopic Ellipsometry. *Coatings* **2024**, *14*, 166.
https://doi.org/10.3390/coatings14020166

**AMA Style**

Bednarski H, Hajduk B, Jarka P, Kumari P.
Temperature Coefficient of Electronic Polarizability in Thin Polymer Films Deposited on Si and SiO_{2} Substrates Determined via Spectroscopic Ellipsometry. *Coatings*. 2024; 14(2):166.
https://doi.org/10.3390/coatings14020166

**Chicago/Turabian Style**

Bednarski, Henryk, Barbara Hajduk, Paweł Jarka, and Pallavi Kumari.
2024. "Temperature Coefficient of Electronic Polarizability in Thin Polymer Films Deposited on Si and SiO_{2} Substrates Determined via Spectroscopic Ellipsometry" *Coatings* 14, no. 2: 166.
https://doi.org/10.3390/coatings14020166