#
Optical Properties of Silicon-Rich Silicon Nitride (Si_{x}N_{y}H_{z}) from First Principles

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## Abstract

**:**

_{x}N

_{y}H

_{z}have been calculated from first principles. Optical spectra for reflectivity, absorption coefficient, energy-loss function (ELF), and refractive index were obtained. The results for Si

_{3}N

_{4}are in agreement with the available theoretical and experimental results. To understand the electron energy loss mechanism in Si-rich silicon nitride, the influence of the Si/N ratio, the positions of the access Si atoms, and H in and on the surface of the ELF have been investigated. It has been found that all defects, such as dangling bonds in the bulk and surfaces, increase the intensity of the ELF in the low energy range (below 10 eV). H in the bulk and on the surface has a healing effect, which can reduce the intensity of the loss peaks by saturating the dangling bonds. Electronic structure analysis has confirmed the origin of the changes in the ELF. It has demonstrated that the changes in ELF are not only affected by the composition but also by the microstructures of the materials. The results can be used to tailor the optical properties, in this case the ELF of Si-rich Si

_{3}N

_{4}, which is essential for secondary electron emission applications.

## 1. Introduction

_{3}N

_{4,}the data have generally been limited to the evaluation of the index of refraction in the visible region and the infrared measurements. The results of these experiments have been used mainly for structural and chemical evaluation purposes [14]. Other studies also indicate the presence of chemically bound hydrogen in the film. Thus, the stoichiometry of an arbitrary film can be more accurately indicated by Si

_{x}N

_{y}H

_{z}. Thus the refractive index and other optical properties will be a function of x, y and z. To study secondary electron emission property of silicon rich silicon nitride, a complete set of optical data (ELF) in a wide energy range is necessary.

_{3}N

_{4}are validated against available theoretical and experimental results. The materials have been studied in the form of Si

_{x}N

_{y}H

_{z}(modified based on the β-Si

_{3}N

_{4}), so that the effect of the Si/N ratio, of the position of the access Si, and of the H in the bulk and on the surfaces can be taken into account. Furthermore, density of states (DOS) analysis has revealed that the change in the ELF has a direct link to the electronic structure of the materials. This work opens a window into the application in the fields of secondary electron emission by providing insight into the dielectric and optical properties of Si-rich silicon nitride.

## 2. Computational Methods and Structural Models

_{x}N

_{y}H

_{x}, modifications of the used unit cell have been made based on that of β-Si

_{3}N

_{4}. For example, Si1

_{3}N

_{15}has been modelled by replacing one N atom with one Si atom in a unit cell with 28 atoms; Si

_{7}N

_{7}has been modelled by replacing one N with one Si atom in the unit cell with 14 atoms. For Si

_{7}N

_{7}, two different distributions of the access Si have been included: Si distributed homogenously or forming a cluster (four N atoms next to each other are replaced by four Si atoms) in the unit cell. To study the influence of the H in the bulk Si-rich silicon nitride on the ELF, three types of H in the bulk in Si

_{7}N

_{7}, namel, H forming bonding with only Si, only N, and both Si and H, respectively. Geometric and electronic structures of clean and H terminated (10$\stackrel{\u2014}{1}$0) and (11$\stackrel{\u2014}{2}$0) β-Si

_{3}N

_{4}surfaces have been documented in our previous study [10] and details will not be shown here again.

_{3}N

_{4}has been calculated using density functional perturbation theory including local field effects. They are 4.21 (parallel to a and b) and 4.29 (parallel to c). This is in good agreement with previous theoretical values of 4.19 and 4.26 from Cai et al. [25] and of 4.24 and 4.32 from Watts [26].

_{3}N

_{4}, the components of the dielectric tensor parallel to a, b and c-axes are inspected.

_{ck}is the cell periodic part of the wave functions at the k-point k. The real part of the dielectric tensor is obtained by the usual Kramers-Kronig transformation,

## 3. Results AND Discussion

#### 3.1. Validation of β-Si_{3}N_{4}

_{3}N

_{4}is transparent in the energy range from 0.0 to about 5.0 eV and is featured by main absorption peaks between 8.5 eV and 10.5 eV, which continuously slopes down with increasing photon energies. It has slightly anisotropic characteristics parallel to the c-axis due to the hexagonal unit cells. The main optical spectra, i.e., the reflectivity, adsorption coefficient, energy-loss function (ELF), and refractive index can be calculated from the frequency dependent dielectric tensor shown in Figure 1. Due to the lack of experimental data for other spectra, only the refractive index and ELF have been plotted and compared with experimental references in Figure 2 and Figure 3, respectively.

**Figure 1.**The frequency dependent dielectric function ε(ω) = ε

_{1}(ω) + iε

_{2}(ω) of β-Si

_{3}N

_{4}as a function of the photon energy. (

**a**) and (

**b**) represent our calculated imaginary and real parts of dielectric function ε(ω), respectively. The black and red lines are polarization parallel to (a, b) and c-axes, respectively.

**Figure 2.**A comparison of the refractive index as a function of photon energy. The black curve represents the DFT results for β-Si

_{3}N

_{4}, and the red and blue lines are optical measurements for amorphous Si

_{3}N

_{4}. The blue line is reproduced from Philipp [29]. The red line was obtained by spectroscopic ellipsometry by Prodanović [30].

_{3}N

_{4}is barely affected by the inclusion of this part of the ELF [31].

#### 3.2. Si_{x}N_{y}—The Influence of the Si/N Ratio and Si Positions

_{3}N

_{4}and Si. The studied forms of Si-rich silicon nitride are Si

_{13}N

_{15}(one N atom is replaced by one Si in a unit cell with 28 atoms), Si

_{7}N

_{7}(one N atom is replaced by one Si atom in the unit cell with 14 atoms), and Si

_{7}N

_{7}cluster (the four N atoms are replaced by four Si atoms forming a Si cluster inside the unit cell with 56 atoms). Compared to that of β-Si

_{3}N

_{4}, the ELF of the Si-rich silicon nitride exhibits three main features:

- (1)
- The general shape and the position of the main peaks are similar, where the ELF increases from low energy and peaks at about 23 to 27 eV and slopes down at higher energy.
- (2)
- The main peaks of the Si-rich silicon nitride have shifted slightly to lower energy (at about 1–2 eV). Therefore, the magnitude of the ELF at low energies side of the slope is larger than that of β-Si
_{3}N_{4}. - (3)
- Extra energy loss peaks appear below 5 eV. The increase in the magnitude of the ELF and number of peaks are larger for Si
_{7}N_{7}than for Si_{13}N_{15}.

_{7}N

_{7}and Si

_{7}N

_{7}cluster. Compared with the ELF of Si

_{7}N

_{7}, that of Si

_{7}N

_{7}cluster exhibits more Si character which can be summarized as follows:

- (1)
- The position of the main peak of Si
_{7}N_{7}cluster is significantly sharper and is in a position at about 5 eV lower, and which is closer to that of Si. - (2)
- The positions of the adsorption peaks below 10 eV can be seen as a combination of those of Si, Si
_{3}N_{4}, and Si_{7}N_{7}, but the peaks are in general smaller and broader (below 1 eV and at about 7 eV).

_{x}N

_{y}has introduced (i) extra electronic states in the band gap just above the Fermi level which can be assigned to the unsaturated Si bonds; (ii) broader Si nonbonding feature in the range of 5–10 eV at conduction band. This observation is similar to a recent finding of Hintzsche [32] for amorphous Si

_{x}N

_{y}H

_{z}. For the Si

_{7}N

_{7}cluster, the intensity of the unsaturated Si feature is much smaller and the overall DOS plot is shifted to lower energy.

_{7}N

_{7}below 5 eV are due to the introduction of the electronic states at the band gap originating from the unsaturated Si atom. For the Si

_{7}N

_{7}cluster case, the ELF can be seen as a combination of Si and Si

_{3}N

_{4}and Si-rich Si

_{3}N

_{4}. Based on this calculated result, one can conclude that the optical properties of the Si-rich silicon nitride not only are determined by the composition of the materials, but are also determined by the microstructure at an atomic level.

**Figure 5.**Density of states (DOS) of Si-rich silicon nitride (Si

_{x}N

_{y}), compared to β-Si

_{3}N

_{4}.

#### 3.3. Si_{x}N_{y}H_{z}—H in Bulk Si-Rich Silicon Nitride

_{x}N

_{y}H

_{z}in the form of Si

_{7}N

_{7}H

_{1-2}in comparison with Si

_{7}N

_{7}. Three possibilities have been studied. They are one H forming bond with the dangling bond of the access Si, one H with N, and two H with both Si and N, respectively. The main peaks of the ELF of Si

_{7}N

_{7}H

_{1-2}are located at almost the same energies as that of Si

_{7}N

_{7}. The main differences can be found in the energy range below 5 eV. The sharp peak located at about 1 eV for Si

_{7}N

_{7}is now absent for all three, where the H has directly or indirectly saturated the defects in the Si-rich silicon nitride. This function of H in curing defects is the largest when bonded with Si, the smallest when bonded with N, and logically intermediate for a combination of the two.

**Figure 6.**Energy loss function of H in Si-rich silicon nitride Si

_{7}N

_{7}H

_{1-2}in comparison with Si

_{7}N

_{7}. Si

_{7}N

_{7}H

_{Si}, Si

_{7}N

_{7}H

_{N}, and Si

_{7}N

_{7}H

_{N}H

_{Si}represent H forming bonds with unsaturated Si atoms, N atoms, and the combination of the two, respectively.

**Figure 7.**Density of States (DOS) of Si-rich silicon nitride (Si

_{7}N

_{7}H

_{1-2}) in comparison with Si

_{7}N

_{7}.

#### 3.4. Si_{x}N_{y}H_{z}—H Termination on the Surfaces of Si_{3}N_{4}

#### 3.4.1. Surface vs. Bulk

_{3}N

_{4}with two surfaces: (10$\stackrel{\u2014}{1}$0) and (11$\stackrel{\u2014}{2}$0). Compared with bulk β-Si

_{3}N

_{4}, the main peaks of the ELFs (see Figure 8) of both surfaces have shifted energies which are about 3 to 5 eV lower. Similarly with defects in the bulk, the defects introduced by surfaces also lead to adsorption peaks at about 3 eV. However the peaks in the system with surface defects are much lower and broader that the ones observed in the system with bulk defects. This is because in both β-Si

_{3}N

_{4}with surfaces and Si-rich silicon nitride, defects (unsaturated bonds) occur and these introduce extra electronic states as seen in the DOS (see Figure 9). However, the defects on the surface are more often self-cured in the sense that the atoms on the surfaces tend to relocate to such positions to saturate the dangling bonds. In the bulk there is limited volume to allow the atoms to do so.

**Figure 8.**Energy loss function of surfaces with and without H termination in comparison with bulk β-Si

_{3}N

_{4}.

**Figure 9.**Density of states (DOS) of surfaces with and without H termination in comparison with bulk β-Si

_{3}N

_{4}.

#### 3.4.2. Clean Surface vs. H Termination

_{3}N

_{4}. This is mainly due to the large band gap, where the secondary electrons are more likely to escape to the surface without losing energy from interacting with impurities. As expected, the access Si in silicon rich silicon nitride and surfaces both cause decrease in the SEY. Whereas H adsorption in the bulk and H termination on the surface both have reduced the effect caused by the defects. Details of the electron transport mechanism and SEY results will be published in a separate paper [31]. In combination with Monte Carlo simulations, the significance of this work is to provide a guideline for tuning the material parameters to achieve favorable secondary electron emission properties.

## 4. Summary

_{x}N

_{y}H

_{z}have been studied using DFT, with a focus on the energy loss spectrum. Defects in the bulk (due to the excess Si) and on the surface, and the influence of hydrogen in the bulk, hydrogen termination on the surfaces have been investigated. Extra energy loss peaks have been found in the energy range lower than 10 eV in all the studied systems compared with β-Si

_{3}N

_{4}. The origins of these energy loss peaks have been identified in their corresponding electronic structures. It is concluded that the changes in ELF are not only affected by the composition but also by the microstructures of the materials. The excess Si in the bulk in Si-rich silicon nitride leads to a major increase both in the magnitude and the number of adsorption peaks in ELF, whereas those caused by surface defects are much milder. H in the bulk of Si-rich silicon nitride and H termination on surfaces has a healing effect by saturating the dangling bonds and therefore reducing the number and flattening the sharpness of the adsorption peaks. The results can be used as reference data for tuning the optical properties via controlling the composition and microstructures for secondary electron emission applications.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Tao, S.X.; Theulings, A.M.M.G.; Prodanović, V.; Smedley, J.; Van der Graaf, H.
Optical Properties of Silicon-Rich Silicon Nitride (Si_{x}N_{y}H_{z}) from First Principles. *Computation* **2015**, *3*, 657-669.
https://doi.org/10.3390/computation3040657

**AMA Style**

Tao SX, Theulings AMMG, Prodanović V, Smedley J, Van der Graaf H.
Optical Properties of Silicon-Rich Silicon Nitride (Si_{x}N_{y}H_{z}) from First Principles. *Computation*. 2015; 3(4):657-669.
https://doi.org/10.3390/computation3040657

**Chicago/Turabian Style**

Tao, Shu Xia, Anne M. M. G. Theulings, Violeta Prodanović, John Smedley, and Harry Van der Graaf.
2015. "Optical Properties of Silicon-Rich Silicon Nitride (Si_{x}N_{y}H_{z}) from First Principles" *Computation* 3, no. 4: 657-669.
https://doi.org/10.3390/computation3040657