First-principles investigation of electronic, structural, and vibrational properties of $a{\text{-Si}}_3{\text{N}}_4$

Using a density-functional scheme, we investigate the electronic, structural, and vibrational properties of amorphous silicon nitride. Through a Car-Parrinello molecular-dynamics simulation, we generate a model structure formed mainly by a network of ${\text{SiN}}_4$ tetrahedra a large fraction of which are edge sharing. Only a small fraction of atoms are overcoordinated and undercoordinated. First, the structural properties such as angular distributions, atomic arrangements in first-neighbor shells, the neutron total structure factor, the radial distribution function, and pair-correlation functions are examined. Next, the electronic properties are analyzed by considering the quasiparticle density of states which is calculated through the $GW$ method. Good agreement is found with experimental data when available. Successively, we focus on a range of vibrational spectra. First, the vibrational density of states is analyzed in terms of its decomposition into N and Si contributions. Then, we investigate the Born effective charge tensors, the high-frequency, and static dielectric constants and calculate the real and imaginary parts of the dielectric function in the infrared. Therefrom we obtain the infrared-absorption spectrum and the refractive index that are found to be in accord with experimental measurements. Moreover, we address the Raman spectrum which is compared with available experimental data. Electronic structure and vibrational properties of the point defects present in our model are also discussed. Density-functional and $GW$ schemes appear to be appropriate for modeling materials based on silicon nitride. In particular, our modeling of silicon nitride achieved a successful level of comparison with experiments. This allows us to infer that $a{\text{-Si}}_3{\text{N}}_4$ features a high content of edge-sharing tetrahedra, which are absent in the crystalline phases of silicon nitride at ambient conditions.

Figure 1

(Color online) Snapshots of the $a{\text{-Si}}_3{\text{N}}_4$ model. Si atoms and N atoms are colored with dark and light gray, respectively. Threefold and fivefold coordinated Si atoms are colored in purple and yellow, respectively. Twofold and fourfold coordinated N atoms are colored in red and green. Hydrogen are colored in pink. Defect atoms are also labeled according to the text.

Figure 2

(a) Distribution of Si-N bond lengths. A Gaussian broadening of $0.015\text{Å}$ is used. (b) Distributions of the Si-N-Si (solid) and N-Si-N angles (dot dashed). Contributions from angles Si-N-Si and N-Si-N in twofold rings are shown (dotted and dashed, respectively). Normalization to unity and a Gaussian broadening of $2.5°$ are used.

Figure 3

Neutron structure factor at room temperature (solid) compared to the experimental data (circles) from Ref. 12.

Figure 4

Radial distribution function of $a{\text{-Si}}_3{\text{N}}_4$ calculated at room temperature (solid) and experimental data (dotted) from Ref. 12.

Figure 5

Pair-correlation functions of $a{\text{-Si}}_3{\text{N}}_4$ calculated at room temperature (solid).

Figure 6

(Color online) (a) Electronic density of states (black) and partial DOS obtained by projecting electronic states onto $\text{N}2s$ (blue/dotted), $\text{N}2p$ (red/dot-dashed), $\text{Si}2s$ (purple/dashed) and $\text{Si}2p$ (green/double dot-dashed). The highest occupied state is aligned at 0 eV. Gaussian broadening of 0.25 eV is used. $GW$ energies are used. (b) Inverse participation ratio (IPR) of electronic states in silicon nitride.

Figure 7

Partial DOS obtained by projecting the electronic states onto (a) $1s$ orbitals of H atoms, (b) $2p$ and $2s$ orbitals of twofold (solid) and fourfold (dotted) coordinated N atoms, (c) $2p$ and $2s$ orbitals of threefold (solid) and fivefold (dotted) coordinated Si atoms. The highest occupied state is aligned at 0 eV. Gaussian broadening of 0.25 eV is used. $GW$ energies are used.

Figure 8

(a) Vibrational density of states (v-DOS) of $a{\text{-Si}}_3{\text{N}}_4$ and its decomposition into Si (dotted), N (dot-dashed), and H (dashed) contributions. (b) Decomposition of the N contribution (solid) to the v-DOS into rocking motions out of the plane of Si neighbors (dotted) and into stretching motions in the plane of Si neighbors (dashed). (Only threefold coordinated N atoms are taken into account.) A Gaussian broadening of $25{\text{cm}}^{-1}$ is used.

Figure 9

(Color online) (a) Contributions to the v-DOS of each H atom in our model. (b) Contributions to the v-DOS of each twofold coordinated (upper panel) and fourfold coordinated N atom (lower panel) (c) Contributions to the v-DOS of each threefold (upper panel) and fivefold coordinated Si atom (lower panel). A Gaussian broadening of $25{\text{cm}}^{-1}$ is used.

Figure 10

Calculated (solid) (a) real and (b) imaginary parts of the dielectric function of $a{\text{-Si}}_3{\text{N}}_4$ in the infrared region compared to experimental results of Ref. 83 (dotted). A Lorentzian and a Gaussian broadening of $25{\text{cm}}^{-1}$ are used.

Figure 11

(a) Calculated (solid) and experimental (dotted, Ref. 83) refraction index $n\left(\omega \right)$ of $a{\text{-Si}}_3{\text{N}}_4$. (b) Calculated (solid) and experimental FTIR (dotted, Ref. 85) absorption spectra of $a{\text{-Si}}_3{\text{N}}_4$. The experimental absorption spectrum (originally given in arbitrary units) was rescaled so that its maximum coincide with the maximum of the calculated spectrum.

Figure 12

(Color online) Calculated (solid) and experimental Raman spectra of $a{\text{-Si}}_3{\text{N}}_4$ (a) in parallel and (b) in cross polarization configurations. Experimental data for composition ratios $r=1.35$ (circles) and $r=1.4$ (squares) are taken from Ref. 86. The theoretical spectra are scaled to match the integrated intensity of the experimental spectra. A Gaussian broadening of $25{\text{cm}}^{-1}$ is used.

Figure 13

Calculated reduced Raman spectra of $a{\text{-Si}}_3{\text{N}}_4$ in parallel (solid) and in cross (dot-dashed) polarization configuration. Contribution from two-membered rings (dotted) is also shown for parallel polarization configuration. A Gaussian broadening of $25{\text{cm}}^{-1}$ is used.