# Optics of Inhomogeneous Thin Films with Defects: Application to Optical Characterization

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. Inhomogeneous thin Films without Defects

#### 2.1.1. Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) Method

#### 2.1.2. Approximate Method Based on Using Multilayer Systems

#### 2.1.3. Approximate Method Based on Modification of Recursive Formulae of Multilayer Systems

#### 2.1.4. Approximate Method Based on Multiple-Beam Interference Model

#### 2.2. Inhomogeneous Thin Films with Defects

#### 2.2.1. Transition Layers and Overlayers

#### 2.2.2. Thickness Nonuniformity

#### 2.2.3. Random Roughness of Film Boundaries

#### 2.2.4. Uniaxial Anisotropy

## 3. Examples of Optical Characterization

#### 3.1. Optical Characterization of the Transition Layer

#### 3.2. Optical Characterization of the Inhomogeneous SiO${}_{x}$C${}_{y}$H${}_{z}$ Thin Film

#### 3.2.1. Sample Preparation and Experimental Arrangement

#### 3.2.2. Dispersion Model

#### 3.2.3. Data Processing

#### 3.2.4. Results and Discussion

#### 3.3. Optical Characterization of the Inhomogeneous SiN${}_{x}$ Thin Films

#### 3.3.1. Sample Preparation and Experimental Arrangements

#### 3.3.2. Influence of Boundary Roughness

#### 3.3.3. Dispersion Model

#### 3.3.4. Data Processing

#### 3.3.5. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the inhomogeneous thin film nonuniform in thickness (

**left**). Schematic diagram of the positions of the samples during ellipsometric measurements (

**right**), the dashed line represents the intersection of the plane of incidence with the sample plane.

**Figure 3.**Schematic diagram of the system of thin homogeneous layers approximating the inhomogeneous layer.

**Figure 4.**Spectral dependencies of the optical constants of the transition layer (

**left**) and the associated ellipsometric parameters measured at angle of incidence ${70}^{\circ}$ on the sample pretreated in the argon discharge (

**right**). The points represent the experimental values while the lines represent the theoretical values.

**Figure 5.**The spectral dependencies (

**left**) and the depth profile at $E=4$ eV (

**right**) of the optical constants of the inhomogeneous layer SiO${}_{x}$C${}_{y}$H${}_{z}$ thin film.

**Figure 6.**The profiles of local thicknesses of the SiO${}_{x}$C${}_{y}$H${}_{z}$ film. The solid lines correspond to result obtained by the ellipsometric method, the dashed lines correspond to results obtained by imagining spectroscopic reflectometry (ISR).

**Figure 7.**Maps of local thicknesses determined by the ellipsometric method (

**left**) and by ISR (

**right**) for the SiO${}_{x}$C${}_{y}$H${}_{z}$ film.

**Figure 8.**Three-dimensional representations of the shape of the upper boundary determined by the ellipsometric method (

**left**) and by ISR (

**right**) for the SiO${}_{x}$C${}_{y}$H${}_{z}$ film.

**Figure 9.**Agreement between the experimental and theoretical values for the associated ellipsometric parameters of the SiO${}_{x}$C${}_{y}$H${}_{z}$ film at incidence angle ${70}^{\circ}$ (

**left**) and the degree of polarization at three selected angles of incidence (

**right**). The plots correspond to azimuth angle $\beta ={0}^{\circ}$. The points represent the experimental values while the lines represent the theoretical values.

**Figure 10.**Spectral dependencies ofthe optical constants of the SiN${}_{x}$ film at the upper (

**top**) and lower (

**bottom**) boundaries for the ordinary (

**left**) and extraordinary (

**right**) waves.

**Figure 11.**Profiles of the optical constants of the SiN${}_{x}$ film at $E=3.5$ eV for the ordinary (

**left**) and extraordinary (

**right**) waves.

**Figure 12.**The spectral dependencies of reflectance for all three samples (

**left**) and the ellipsometric quantities for sample 2 at incidence angle ${70}^{\circ}$ (

**right**). The points represent the experimental values while the lines represent the theoretical values.

**Table 1.**Values of the parameters describing the thickness nonuniformity of the characterized film of SiO${}_{x}$C${}_{y}$H${}_{z}$ (left). Values of quantity $\chi $ for the individual total corrections in the formulae for the reflection coefficients of the inhomogeneous SiO${}_{x}$C${}_{y}$H${}_{z}$ thin film (right).

Ellipsometry | ISR | $\mathit{\chi}$ | |||
---|---|---|---|---|---|

${d}_{0}$ | [nm] | $960.7\pm 1.1$ | $959.5\pm 0.3$ | WKBJ | 5.813 |

${d}_{\mathrm{x}}$ | [nm] | $9.7\pm 0.1$ | $14.78\pm 0.06$ | with term 1 | 5.419 |

${d}_{\mathrm{y}}$ | [nm] | $9.5\pm 0.1$ | $11.10\pm 0.06$ | with terms 1 + 2 | 5.418 |

${d}_{\mathrm{xx}}$ | [nm] | $2.01\pm 0.09$ | $2.73\pm 0.03$ | with terms 1 + 2 + 3 | 5.418 |

${d}_{\mathrm{xy}}$ | [nm] | $1.35\pm 0.09$ | $1.30\pm 0.03$ | wedge | 5.807 |

${d}_{\mathrm{yy}}$ | [nm] | $1.69\pm 0.09$ | $2.23\pm 0.03$ |

Sample 1 | Sample 2 | Sample 3 | |||
---|---|---|---|---|---|

deposition time | t | [min] | 30 | 45 | 90 |

thickness ellipsometry | ${d}_{\mathrm{e}}$ | [nm] | $111.21\pm 0.05$ | $167.10\pm 0.08$ | $323.9\phantom{0}\pm 0.1\phantom{0}$ |

thickness reflectance | ${d}_{\mathrm{r}}$ | [nm] | $112.12\pm 0.06$ | $167.91\pm 0.07$ | $320.80\pm 0.09$ |

roughness (rms) | $\sigma $ | [nm] | $\phantom{00}3.0\phantom{0}\pm 0.2\phantom{0}$ | $\phantom{00}4.6\phantom{0}\pm 0.2\phantom{0}$ | $\phantom{00}6.3\phantom{0}\pm 0.2\phantom{0}$ |

profile parameter | $\xi $ | [nm] | $\phantom{0}39.3\phantom{0}\pm 1.8\phantom{0}$ | $\phantom{0}65.6\phantom{0}\pm 2.8\phantom{0}$ | $\phantom{0}57.6\phantom{0}\pm 2.6\phantom{0}$ |

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**MDPI and ACS Style**

Ohlídal, I.; Vohánka, J.; Čermák, M. Optics of Inhomogeneous Thin Films with Defects: Application to Optical Characterization. *Coatings* **2021**, *11*, 22.
https://doi.org/10.3390/coatings11010022

**AMA Style**

Ohlídal I, Vohánka J, Čermák M. Optics of Inhomogeneous Thin Films with Defects: Application to Optical Characterization. *Coatings*. 2021; 11(1):22.
https://doi.org/10.3390/coatings11010022

**Chicago/Turabian Style**

Ohlídal, Ivan, Jiří Vohánka, and Martin Čermák. 2021. "Optics of Inhomogeneous Thin Films with Defects: Application to Optical Characterization" *Coatings* 11, no. 1: 22.
https://doi.org/10.3390/coatings11010022