Medium-range structure of vitreous ${\text{SiO}}_2$ obtained through first-principles investigation of vibrational spectra

Using a density-functional framework, we investigate the vibrational spectra of vitreous ${\text{SiO}}_2$ to determine to what extent these spectra provide information about the medium-range structure of the oxide network. We carry out a comparative study involving three model structures, which all feature a nondefective network of corner-sharing tetrahedra but differ through their Si-O-Si bond-angle distributions and ring statistics. We first address the results of typical diffraction probes. Fair agreement with experiment is achieved for the total neutron and total x-ray structure factors of all models, indicating limited sensitivity of these structure factors to the medium-range structure. The same consideration also applies to the Si-O and O-O partial structure factors. At variance, the Si-Si partial structure factor is found to be highly sensitive to the Si-O-Si bond-angle distribution. We then address typical vibrational spectra, such as the inelastic neutron spectrum, the infrared spectra, and the Raman spectra. For the inelastic neutron spectrum and the infrared spectra, the comparison with experiment is fair for all models, indicating poor sensitivity to the structural arrangement of tetrahedra. The only noticeable exception is the feature at $\sim 800{\text{cm}}^{-1}$ which shifts to higher frequencies with decreasing Si-O-Si angles. At variance, the Raman spectra are shown to be very informative about the medium-range organization of the network through their sensitivity to the concentrations of three-membered and four-membered rings. Our study indicates that the considered experimental data are globally consistent with a medium-range structure characterized by an average Si-O-Si bond angle of $148°$ and with small-ring concentrations as derived from the intensities of the experimental Raman defect lines. To describe the infrared and Raman couplings, our work also introduces parametric models which reproduce well the spectra calculated from first principles.

Figure 1

Distributions of (a) Si-O bond lengths, (b) O-Si-O bond angles, and (c) Si-O-Si bond angles for models I (solid curve), II (dotted curve), and III (dashed curve) of $v{\text{-SiO}}_2$. Gaussian broadenings of $0.005\text{Å}$ and $2.5°$ were used.

Figure 2

Ring statistics of models I–III of $v{\text{-SiO}}_2$. The size specifies the number of Si-O segments.

Figure 3

Electronic DOSs of our models of $v{\text{-SiO}}_2$: model I (solid curve), model II (dotted curve), and model III (dashed curve). The energy scales are aligned through the $\text{O}2s$ states. The energy scale is referred to the top of the valence band of model I. The DOSs of models II and III have been rescaled to match the normalization of the DOS of model I. A Gaussian broadening of 0.25 eV was used.

Figure 4

(a) Calculated vibrational density of states of model I (solid curve) of $v{\text{-SiO}}_2$ compared to those of model II (dotted curve) and model III (dashed curve), taken from Refs. 14, 16. (b) Calculated effective neutron density of states for model I (solid curve), compared to the experimental result from Ref. 49 (closed symbols) and to the actual vibrational density of states (dotted curve). Experimental data from Ref. 50 are also shown (open symbols). In the calculation, we used a temperature of 33 K and the $Q$ range between 6 and $13{\text{Å}}^{-1}$, as in the experiment of Ref. 49. A Gaussian broadening of 2.5 meV was used in the calculation.

Figure 5

Pair-correlation functions $g_{\text{SiO}}\left(r\right)$, $g_{\text{SiSi}}\left(r\right)$, and $g_{\text{OO}}\left(r\right)$ of model I of $v{\text{-SiO}}_2$, calculated in the harmonic approximation at room temperature.

Figure 6

(Color online) Comparison between (a) the Si-Si pair-correlation functions $\left[g_{\text{SiSi}}\left(r\right)\right]$ and (b) the Si-Si partial structure factors $\left[S_{\text{SiSi}}\left(Q\right)\right]$ of model I (solid curve), model II (dotted curve), and model III (dashed curve). Calculated results are obtained in the harmonic approximation at room temperature. We used a Gaussian broadening of $0.15{\text{Å}}^{-1}$ in (b). The partial structure factors are compared with the experimental data of Ref. 7 (disks with error bars).

Figure 7

(a) Calculated total neutron structure factor (solid curve) for model I of $v{\text{-SiO}}_2$ at room temperature, compared with its experimental counterpart (circles) (Ref. 58). (b) Calculated x-ray structure factor at room temperature (solid curve) for model I of $v{\text{-SiO}}_2$, compared with its experimental counterpart (circles) (Ref. 59).

Figure 8

(Color online) Faber-Ziman partial structure factors calculated for model I of $v{\text{-SiO}}_2$ at room temperature (solid curve), compared to experimental results from Ref. 7 (disks with error bars): (a) $S_{\text{SiO}}\left(Q\right)$, (b) $S_{\text{OO}}\left(Q\right)$, and (c) $S_{\text{SiSi}}\left(Q\right)$.

Figure 9

(Color online) Bhatia-Thornton partial structure factors calculated for model I of $v{\text{-SiO}}_2$ at room temperature (solid curve), compared to experimental results from Ref. 7 (disks): (a) $S_{\text{NN}}\left(Q\right)$, (b) $S_{\text{CC}}\left(Q\right)$, and (c) $S_{\text{NC}}\left(Q\right)$.

Figure 10

(a) Real $\left({ϵ}_1\right)$ and (b) imaginary $\left({ϵ}_2\right)$ parts of the dielectric function, and (c) energy-loss function for model I of $v{\text{-SiO}}_2$ (solid curve) compared to experimental data of Ref. 67 (dotted curve). Lorentzian and Gaussian broadenings of $19{\text{cm}}^{-1}$ were used for the real and imaginary parts, respectively.

Figure 11

(a) Real and (b) imaginary parts of the dielectric function and (c) energy-loss function for our models of $v{\text{-SiO}}_2$: model I (solid curve), model II (dotted curve), and model III (dashed curve). The results for model II were taken from Ref. 16. The imaginary part of model III was taken from Ref. 15. Lorentzian and Gaussian broadenings of $19{\text{cm}}^{-1}$ were used.

Figure 12

Components of the oxygen Born charge tensors vs Si-O-Si angle for model I of $v{\text{-SiO}}_2$: (a) isotropic component $⟨Z_{\text{iso}}^{\ast }⟩$ ($\ell =0$ term), (b) $\lambda$, and (c) $\mu$ components of the $\ell =2$ term (see text). The solid lines correspond to linear regressions.

Figure 13

Imaginary part of the dielectric function of model I of $v{\text{-SiO}}_2$ obtained using the full Born charge tensors (solid curve), compared to those obtained using solely the average values of the isotropic charges $Z_{\text{iso}}^{\ast }$ of Si and O atoms (dot-dashed curve) and using the parametric model described in the text (dotted curve). A Gaussian broadening of $19{\text{cm}}^{-1}$ was used.

Figure 14

Reduced (a) HH and (b) HV Raman spectra calculated for model I (solid curve), compared with the experimental data of Ref. 76 (dotted curve). (c) Nonreduced HH and HV Raman spectra of model I (solid curve) compared with the experimental data of Ref. 52 (dotted curve). The theoretical HH spectrum was scaled to match the integrated intensity of the experimental spectrum. The same scaling factor was then applied to the HV spectrum. A Gaussian broadening of $19{\text{cm}}^{-1}$ was used.

Figure 15

(Color online) Calculated reduced HH Raman spectra of model I (solid curve), model II (dotted curve), and model III (dashed curve), compared with the experimental data of Ref. 76 (gray/red). The result for model II is taken from Ref. 16. A Gaussian broadening of $19{\text{cm}}^{-1}$ is used.

Figure 16

Raman coupling factors $f_I$ vs Si-O-Si angle, calculated for O atoms in model I (circles), model II (diamonds), and model III (squares). The dashed curve corresponds to the fit of the coupling factors of model I according to Eq. (46).

Figure 17

Reduced HH Raman spectrum of model I calculated with globally optimized bond-polarizability parameters (dotted curve), compared to the corresponding spectrum calculated fully from first principles (solid curve). A Gaussian broadening of $19{\text{cm}}^{-1}$ was used.

Figure 18

Schematic illustration of the number of independent finite electric field calculations that are required for evaluating the diagonal (disks) and off-diagonal (filled squares) components of the tensors $\partial {\chi }_{ij}/\partial R_{Ik}$ pertaining to a couple of Cartesian directions.