First-principles investigation of the structural and vibrational properties of vitreous $\mathrm{Ge}{\mathrm{Se}}_2$

Using a density-functional scheme, we study the structural and vibrational properties of vitreous germanium diselenide $\left(v\text{−}\mathrm{Ge}{\mathrm{Se}}_2\right)$. Through the use of classical and first-principles molecular-dynamics methods, we generate a set of structural models showing a varying degree of chemical disorder. In particular, two types of structural concepts are represented: one in which the tetrahedral order is preserved to a very large extent, and one which reproduces the high degree of disorder in first-neighbor shells found in first-principles molecular-dynamics simulations of the liquid. The investigated structural properties include the angular distributions, the atomic arrangements in the first-neighbor shells, and the pair-correlation functions. In reciprocal space, we have calculated the x-ray and neutron total structure factors and the partial structure factors. Comparison with experiment gives overall good agreement for the models of either structural conception. We then investigate the vibrational properties via the vibrational density of states and the inelastic neutron spectrum. The considered models yield similar spectra and agree with experimental data. We also obtain infrared and Raman spectra through a density-functional scheme based on the application of finite electric fields. For these spectra, significant differences appear among the models. The comparison with experiment favors a model showing a high degree of chemical order. The Raman intensity is analyzed in terms of the underlying atomic vibrations. The assignment of the Raman companion line to Se motions in edge-sharing tetrahedra is supported.

Figure 1

(a) Electronic density of states (DOS) and (b) participation ratio $p$ of our models of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$: model I (solid), model II (dotted), and model III (dashed). The highest occupied states are aligned at $0\mathrm{eV}$. Gaussian broadenings of 0.25 and $0.1\mathrm{eV}$ are used in (a) and (b), respectively.

Figure 2

Distribution of Ge-Se bond lengths in model I (solid), model II (dotted), and model III (dashed). A Gaussian broadening of $0.01\mathrm{\AA }$ is used.

Figure 3

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

Figure 4

Partial pair-correlation functions of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ at room temperature: model I (solid) and experimental data from Ref. 4 (dotted).

Figure 5

Comparison between calculated partial pair-correlation functions for models I (solid), II (dotted), and III (dashed).

Figure 6

(a) X-ray and (b) neutron structure factors at room temperature for model I (solid) compared to the experimental data (circles) obtained in Refs. 20, 40, respectively. (c) Comparison between calculated neutron structure factors of models I (solid), II (dotted), and III (dashed). A Gaussian broadening of $0.1{\mathrm{\AA }}^{-1}$ is used. The bond lengths in Table are used to rescale the transferred momenta.

Figure 7

Faber-Ziman partial structure factors at room temperature: model I (solid) vs experiment (Ref. 4) (circles). Transferred momenta are rescaled as in Fig. 6.

Figure 8

(a) Bhatia-Thornton concentration-concentration partial structure factors at room temperature: model I (solid) vs experiment (Ref. 21) (circles). Transferred momenta are rescaled as in Fig. 6. (b) Comparison between calculated concentration-concentration partial structure factors of models I (solid), II (dotted), and III (dashed). A Gaussian broadening of $0.1{\mathrm{\AA }}^{-1}$ is used.

Figure 9

Normalized vibrational density of states $\left(v\text{−}\mathrm{DOS}\right)$ of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$: a comparison between models I (solid), II (dotted), and III (dashed). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 10

(a) Vibrational density of states $\left(v\text{−}\mathrm{DOS}\right)$ of model I of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ and its decomposition into Ge (dotted) and Se (dashed) weights. (b) Further decomposition of the Se weight into bending (dotted), rocking (dot-dashed), and stretching (dashed) motions. Only Se atoms twofold coordinated by Ge are considered. A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 11

Neutron vibrational density of states of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ at $13\mathrm{K}$ calculated for model I (solid), compared to corresponding experimental data of Ref. 46 (open squares). Experimental data obtained at room temperature (Ref. 47, closed circles) are shown for comparison. Transferred momenta are taken in the range $0.6–8.5{\mathrm{\AA }}^{-1}$, corresponding to the experimental interval (Ref. 46).

Figure 12

Isotropic Se Born charge $\left(Z_{\mathrm{Se}}^{*}\right)$ vs Ge-Se-Ge angle for models I (open circles), II (open squares), and III (closed circles). Only Se atoms twofold coordinated by Ge are considered.

Figure 13

Dielectric function of model I of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ (solid) compared to experimental data from Ref. 55 (dotted) and Ref. 54 (dashed): (a) real part, (b) imaginary part, and (c) longitudinal response $-\omega \mathrm{Im}(1∕ϵ)$. A Lorentzian broadening of $8{\mathrm{cm}}^{-1}$ is used in (a), while Gaussian broadenings of $4{\mathrm{cm}}^{-1}$ are used in (b) and (c). In (c), the experimental and theoretical spectra are rescaled to show the same integrated intensity.

Figure 14

Comparison between the calculated imaginary parts of the dielectric functions of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ for models I (solid), II (dotted), and III (dashed). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 15

Reduced HH and HV Raman spectra of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ for model I of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ (solid), compared to experimental data of Ref. 54 (dotted). The calculated HH spectrum is scaled to match the integrated intensity of the experimental spectrum. The same scaling factor is used for the theoretical HV spectrum. For clarity, the HV spectra are magnified by a factor of 2. A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 16

Comparison between the calculated reduced HH Raman spectra of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ for models I (solid), II (dotted), and III (dashed). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 17

(a) Decomposition of the reduced HH Raman spectrum of model I (solid) into Se (dotted) and Ge (dashed) weights following the procedure described in the text. (b) Further decomposition of the Se weight into rocking (dot-dashed), bending (dotted), and stretching (dashed) contributions (see text). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used.

Figure 18

(a) Raman intensity (shaded) associated with in-phase Se bending vibrations in four-atom rings (see inset), compared to the full reduced HH Raman spectrum of model I of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ (solid). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used. (b) Isotropic coupling factor for Se bending motions vs Ge-Se-Ge angle for Se atoms in model I. Only Se atoms twofold coordinated by Ge are considered, either inside (closed circles) or outside (open squares) of four-atom rings. The dotted line corresponds to a linear fit of the data.

Figure 19

(a) Raman intensities associated with homopolar bonded Se atoms (dotted) and in particular to their Se-Se stretching vibrations (shaded), compared to the full reduced HH Raman spectrum of model I of $v\text{−}\mathrm{Ge}{\mathrm{Se}}_2$ (solid). A Gaussian broadening of $4{\mathrm{cm}}^{-1}$ is used. (b) Se-Se stretching-mode frequency vs bond-length $d_{\mathrm{Se}\mathrm{Se}}$ for models I (closed circles), II (closed squares), and III (open circles). The dotted line corresponds to a linear fit of all the data.