Vibrational spectra of vitreous germania from first-principles

We report on a first-principles investigation of the structural and vibrational properties of vitreous germania $\left(v\text{−}\mathrm{Ge}{\mathrm{O}}_2\right)$. Our work focuses on a periodic model structure of 168 atoms, but three smaller models are also studied for comparison. We first carry out a detailed structural analysis both in real and reciprocal spaces. Our study comprises the partial pair correlation functions, the angular distributions, the total neutron correlation function, the neutron and $x$-ray total structure factors, and the Faber-Ziman and Bhatia-Thornthon partial structure factors. We find overall good agreement with available experimental data. We then obtain the vibrational frequencies and eigenmodes. We analyze the vibrational density of states in terms of Ge and O motions, and further in terms of rocking, bending, and stretching contributions. The inelastic neutron spectrum is found to differ only marginally from the vibrational density of states. Using a methodology based on the application of finite electric fields, we derive dynamical Born charge tensors and Raman coupling tensors. For the infrared spectra, we calculate the real and imaginary parts of the dielectric function, including the high-frequency and static dielectric constants. The Raman spectra are shown to be sensitive to the medium-range structure and support an average Ge-O-Ge angle of 135°. We identify the shoulder $X_2$ as a signature of breathing O vibrations in three-membered rings. Four-membered rings are found to contribute to the main Raman peak. We advance an interpretation for the shoulder $X_1$ in terms of delocalized bond-bending modes. We derive bond polarizability parameters from the calculated Raman coupling tensors and demonstrate their level of reliability in reproducing the spectra. The calculated vibrational spectra all show good agreement with the respective experimental spectra.

Figure 1

Electronic density of states (DOS) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$. The highest occupied states are aligned at $0\mathrm{eV}$. A Gaussian broadening of $0.25\mathrm{eV}$ is used.

Figure 2

Distribution of the $\mathrm{O}\text{−}X\text{−}\mathrm{O}$ angle and of bond length $d_{X\mathrm{O}}$ $(X=\mathrm{Ge},\mathrm{Si})$ for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$ (solid) and for a model of $v\text{−}\mathrm{Si}{\mathrm{O}}_2$ generated according to a similar procedure (dashed) (Ref. 39). The bond lengths are rescaled with respect to their average value. Gaussian broadenings of 2.5° and $0.005\mathrm{\AA }$ are used.

Figure 3

The distribution of the distortion parameter ${\delta }_{\mathrm{V}}$ is given for the same models considered in Fig. 2. We found a spread of 0.011 and 0.020 for $v\text{−}\mathrm{Si}{\mathrm{O}}_2$ and $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, respectively.

Figure 4

Distribution of the Ge-O-Ge intertetrahedral angle for our four models of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$: model I (solid), model II (dot-dashed), model III (dotted), and model IV (dashed). A Gaussian broadening of 2.5° is used.

Figure 5

Ring statistics of our four models of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$ according to the minimal path analysis (Ref. 5).

Figure 6

(a) Pair-correlation functions (PCF) $g_{\mathrm{Ge}\mathrm{O}}\left(r\right)$ (solid), $g_{\mathrm{Ge}\mathrm{Ge}}\left(r\right)$ (dashed), and $g_{\mathrm{O}\mathrm{O}}\left(r\right)$ (dot-dashed), calculated in the harmonic approximation for $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$ (model I). (b) Theoretical neutron total correlation function $T_{\mathrm{N}}\left(r\right)$ (dot-dashed). Its counterpart $T_{\mathrm{N}}^{\prime }\left(r\right)$ (solid) accounts for the experimental resolution and permits comparison with experiment (dotted) (Ref. 45).

Figure 7

Theoretical total X-ray structure factor (solid) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, compared with its experimental counterpart (circles) (Ref. 49). The structure factor is given as a function of an adimensional scattering vector, scaled by the average Ge-O bond distance $d_{\mathrm{Ge}\mathrm{O}}$ ($d_{\mathrm{Ge}\mathrm{O}}^{\mathrm{expt}}=1.73\mathrm{\AA }$, $d_{\mathrm{Ge}\mathrm{O}}^{\mathrm{theo}}=1.78\mathrm{\AA }$).

Figure 8

Theoretical Faber-Ziman partial structure factors (solid) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, compared with experimental data (circles) of Ref. 53: (a) $S_{\mathrm{Ge}\mathrm{O}}\left(Q\right)$, (b) $S_{\mathrm{O}\mathrm{O}}\left(Q\right)$, and (c) $S_{\mathrm{Ge}\mathrm{Ge}}\left(Q\right)$. The scattering vector is scaled as described in Fig. 7.

Figure 9

Theoretical Bhatia-Thornton partial structure factors (solid) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, compared with experimental data (circles) of Ref. 53: (a) $S_{\mathrm{N}\mathrm{N}}$, (b) $S_{\mathrm{C}\mathrm{C}}$, and (c) $S_{\mathrm{N}\mathrm{C}}$. The scattering vector is scaled as described in Fig. 7.

Figure 10

Vibrational density of states $Z\left(\omega \right)$ (solid) and its decompositions. In (a), $Z\left(\omega \right)$ is first decomposed into O and Ge weights. In (b), the O weight is further decomposed according to the O motion along the rocking (R, dashed), bending (B, dot-dashed), and stretching (S, dotted) directions. Inset: Projection on symmetry-adapted modes of the $\mathrm{Ge}{\mathrm{O}}_4$ tetrahedra: $T_2$ (dashed) and $A_1$ (dotted). A Gaussian broadening of $19{\mathrm{cm}}^{-1}$ is used.

Figure 11

Calculated neutron vibrational density of states $\left(v\text{−}\mathrm{D}\mathrm{O}\mathrm{S}\right)$ of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$ (solid) at room temperature compared to the actual $v\text{−}\mathrm{D}\mathrm{O}\mathrm{S}$ (dashed). In the calculation of the neutron spectrum (Ref. 9) we used transferred momenta (Ref. 60) in the range $0.5–4.5\mathrm{\AA }$.

Figure 12

Isotropic oxygen Born charge vs Ge-O-Ge angle for our four models of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$: model I (open squares), model II (open disks), model III (closed disks), and model IV (closed squares).

Figure 13

(a) Real and (b) imaginary parts of the dielectric function for our four models of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$: model I (solid), model II (dotted), model III (dot-dashed), and model IV (dashed). A Lorentzian broadening of $25.7{\mathrm{cm}}^{-1}$ is used for the real part, a Gaussian broadening of $19{\mathrm{cm}}^{-1}$ is used for the imaginary part.

Figure 14

Theoretical reduced HH and HV Raman spectra (solid) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, compared to corresponding experimental spectra from Ref. 15 (dotted). The theoretical HH spectra is scaled to match the integrated intensity of the experimental spectrum. The same scaling factor is then applied to the HV spectrum. A Gaussian broadening of $19{\mathrm{cm}}^{-1}$ is used in the theoretical spectra.

Figure 15

Reduced HH and HV Raman spectra (solid) compared to spectra obtained for specific approximations of the coupling tensors, for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$. The dotted curve in the HH spectrum corresponds to retaining only the isotropic component of these tensors associated to oxygen bending motions. The dot-dashed curves correspond to the result obtained within the bond-polarizability model with optimal parameters.

Figure 16

Coupling factor $f_I$ vs Ge-O-Ge angle for all O atoms in model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$ (disks). The curve corresponds to the best fit with Eq. (25), obtained for $\alpha =101{\text{bohr}}^2$.

Figure 17

HH Raman spectra for our four models of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$: model I (solid), model II (dot-dashed), model III (dotted), and model IV (dashed). A Gaussian broadening of $19{\mathrm{cm}}^{-1}$ is used.

Figure 18

(a) HH reduced Raman spectrum (solid) for model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$, together with HH Raman spectra obtained for vibrational eigenmodes projected on breathing O vibrations in three- (dotted) and four-membered rings (dashed). (b) Projections as in (a) for three-membered rings belonging to model I (dotted), model II (solid), and model III (dashed). In model IV these rings are absent. (c) Projections as in (b) but only for the most planar ring in each model. A Gaussian broadening of $19{\mathrm{cm}}^{-1}$ is used.

Figure 19

Average Raman intensity (solid) obtained for vibrational eigenmodes projected on specific motions of the Ge-O-Ge unit. The average is carried out over all units in model I of $v\text{−}\mathrm{Ge}{\mathrm{O}}_2$. The illustrated motions correspond to the choices which maximize the intensity in either the (a) low- (below $\sim 350{\mathrm{cm}}^{-1}$) or (b) high-frequency band (between 350 and $700{\mathrm{cm}}^{-1}$), corresponding to the $B_1$ and $B_2$ motions discussed in the text. The relative weight of Ge and O motions in the $B_1$ and $B_2$ motions has been taken to correspond to the relative contribution of the two species in the respective frequency range of the $v\text{−}\mathrm{D}\mathrm{O}\mathrm{S}$ (Fig. 10). The dotted curve in both panels corresponds to a projection on O bending motions irrespective of the Ge motions.