Density-driven defect-mediated network collapse of ${\mathrm{GeSe}}_2$ glass

The evolution in structure of the prototypical network-forming glass ${\mathrm{GeSe}}_2$ is investigated at pressures up to $\sim 16$ GPa by using a combination of neutron diffraction and first-principles molecular dynamics. The neutron diffraction work at pressures $\le 8.2$ GPa employed the method of isotope substitution, and the molecular dynamics simulations were performed with two different exchange-correlation functionals, the Becke-Lee-Yang-Parr (BLYP) and the hybrid Heyd-Scuseria-Ernzerhof HSE06. The results show density-driven structural transformations that differ substantially from those observed in common oxide glasses such as ${\mathrm{SiO}}_2$ and ${\mathrm{GeO}}_2$. Edge-sharing tetrahedra persist as important structural motifs until a threshold pressure of $\sim 8.5$ GPa is attained, whereupon a mediating role is found for homopolar bonds in the appearance of higher coordinated Ge-centered polyhedra. These mechanisms of network transformation are likely to be generic for the class of glass-forming materials where homopolar bonds and fragility-promoting edge-sharing motifs are prevalent in the ambient-pressure network.

Figure 1

The pressure-volume equation of state for ${\mathrm{GeSe}}_2$ glass under compression. The measured data from Mei et al. [17] [(black) $•$ with vertical error bars] are compared to the results obtained from FPMD in (i) this work using the BLYP functional [(red) $\blacksquare$] and (ii) the work by Durandurdu and Drabold [16] [(blue) $\diamond$]. The measured data were fitted using a second-order Birch-Murnaghan equation of state [solid (black) curve].

Figure 2

The difference functions (a) $\Delta F_{\mathrm{Ge}}\left(k\right)$/barn $=0.236\left(4\right)[S_{\mathrm{GeSe}}\left(k\right)-1]+0.099\left(2\right)[S_{\mathrm{GeGe}}\left(k\right)-1]$ and (b) $\Delta F_{\mathrm{Se}}\left(k\right)$/barn $=0.184\left(3\right)[S_{\mathrm{GeSe}}\left(k\right)-1]+0.588\left(11\right)[S_{\mathrm{SeSe}}\left(k\right)-1]$ as measured using the diffractometer D4 at ambient pressure (Ref. [8]) and at pressures of 3.0(5), 4.7(5), 6.3(5), 7.1(5), and 8.2(5) GPa (points with vertical error bars). The data sets are compared to FPMD results at ambient pressure and at 3.4, 4.88, 7.25, 8.73, and 9.87 GPa [light (green) curves]. The dark (black) curves are the back Fourier transforms of the measured $r$-space difference functions shown in Fig. 4.

Figure 3

The pressure dependence of (a) the total structure factor $F\left(k\right)$ and (b) the total pair-distribution function $G\left(r\right)$ for ${}^{\mathrm{N}}{\mathrm{Ge}}^{\mathrm{N}}{\mathrm{Se}}_2$ glass. (a) The data sets measured using the PEARL diffractometer are given by the points with vertical error bars. The solid (red) curves show spline fits to these measured data sets, except when $k<$ 1.55 Å${}^{-1}$ where they represent fitted Lorentzian functions because this region was not accessible in the diffraction experiments (see Ref. [31]). The solid light (green) curves show the FPMD results for pressures of 9.87, 11.56, 13.82, and 15.27 GPa. (b) The $G\left(r\right)$ functions shown by the solid (black) curves are the Fourier transforms of the measured $F\left(k\right)$ functions given in (a), and the broken (black) curves show the extent of the unphysical oscillations at $r$ values smaller than the distance of closest approach between the centers of two atoms. The solid light (green) curves show the Fourier transforms of the simulated functions shown in (a) after applying the same maximum cutoff value $k_{\max }$ = 19.55 Å${}^{-1}$ as used for the neutron diffraction data.

Figure 4

The pressure dependence of (a) $\Delta G_{\mathrm{Ge}}\left(r\right)$/barn $=0.236\left(4\right)[g_{\mathrm{GeSe}}\left(r\right)-1]+0.099\left(2\right)[g_{\mathrm{GeGe}}\left(r\right)-1]$ and (b) $\Delta G_{\mathrm{Se}}\left(r\right)$/barn $=0.184\left(3\right)[g_{\mathrm{GeSe}}\left(r\right)-1]+0.588\left(11\right)[g_{\mathrm{SeSe}}\left(r\right)-1]$. The solid (black) curves were obtained by spline fitting and Fourier transforming the measured $k$-space functions shown in Fig. 2, and the broken curves show the extent of the unphysical oscillations at $r$ values smaller than the distance of closest approach between the centers of two atoms. The light (green) curves give the Fourier transforms of the FPMD results shown in Fig. 2 after applying the same cutoff $k_{\max }$ = 20.5 Å${}^{-1}$ as used for the high-$P$ diffraction data.

Figure 5

The $\rho /{\rho }_0$ dependence of the nearest-neighbor (a) bond distance $\overline{r}$ and (b) coordination number $\overline{n}$ as obtained from (i) neutron diffraction using a ${}^{\mathrm{N}}{\mathrm{Ge}}^{\mathrm{N}}{\mathrm{Se}}_2$ sample on either the D4 [(red) $⧫$] or PEARL [(blue) $⧫$] diffractometer, or using ${}^{70}{\mathrm{Ge}}^{\mathrm{N}}{\mathrm{Se}}_2$ and ${}^{73}{\mathrm{Ge}}^{76}{\mathrm{Se}}_2$ samples and averaging the results [(black) $\blacksquare$]; (ii) x-ray diffraction [17] [(blue) $\circ$]; (iii) FPMD [solid (green) curves] where the results obtained before the high-temperature anneal at each pressure point are also given [broken (red) curves]; and (iv) FPMD ($\times$) performed by using the HSE06 hybrid functional. $\overline{n}$ was found from the simulated ${\overline{n}}_{\alpha }^{\beta }$ values by using a cutoff distance specified by the first minimum in $G\left(r\right)$. In (b), the horizontal chained line gives the “8-N” rule expectation of $\overline{n}$ = 2.67 [12]. Also given are the FPMD results for the fractions of $n$-fold coordinated (c) Ge and (d) Se atoms, along with the fractions of these $n$-fold coordinated (e) Ge and (f) Se atoms that have homopolar bonds. In (c)–(f), the symbols denote twofold [(red) $\square$], threefold [(green) $\diamond$], fourfold [(blue) $▵$], fivefold [(magenta)$⊲$], or sixfold [(black) $*$] coordinated species, and the error bars (usually smaller than the symbol size) were calculated according to Ref. [13]. The vertical broken lines correspond to pressures of $\sim 4$, 8, and 12 GPa.

Figure 6

FPMD results showing the $\rho /{\rho }_0$ dependence of (a) the intrapolyhedral Se-Ge-Se and (b) the interpolyhedral Ge-Se-Ge bond angle distributions where, from bottom to top in each panel, $\rho /{\rho }_0$ = 1, 1.204, 1.365, 1.463, 1.528, 1.575, or 1.639. In (c), the density dependence of the fractions of CS Ge atoms (${\mathrm{Ge}}_0$, $\square$) and ES Ge atoms ($\circ$) is given, together with a breakdown of the latter into its contributions from ${\mathrm{Ge}}_1$ ($\diamond$), ${\mathrm{Ge}}_2$ ($▵$), and ${\mathrm{Ge}}_{3/4}$ ($▿$) units. In (c), the error bars (usually smaller than the symbol size) were calculated according to Ref. [13], and the vertical broken lines correspond to pressures of $\sim 4$, 8, and 12 GPa.

Figure 7

The $\rho /{\rho }_0$ dependence of the mean intra-polyhedral Se-Ge-Se [(red) $•$ with vertical error bars] and inter-polyhedral Ge-Se-Ge [(black) $\blacksquare$ with vertical error bars] bond angles as estimated from the measured Ge-Se, Ge-Ge and Se-Se distances (see the text). The experimental results are compared to the mean values $\langle {\theta }_{\mathrm{SeGeSe}}\rangle$ [broken (red) curve with $\circ$ symbols] and $\langle {\theta }_{\mathrm{GeSeGe}}\rangle$ [broken (black) curve with $\square$ symbols] taken from the FPMD simulations.

Figure 8

Atomistic configurations from FPMD for ${\mathrm{GeSe}}_2$ glass at different reduced densities. Ge atoms are dark (purple) and Se atoms are light (yellow). Bonds are drawn when two atoms are separated by a distance $\le r_{\min }$ given by the position of the first minimum in $g_{\mathrm{GeSe}}\left(r\right)$.

Figure 9

The $\rho /{\rho }_0$ dependence of the $q$-parameter distribution as obtained from the FPMD simulations of ${\mathrm{GeSe}}_2$ glass. At each density, the distribution for all Ge atoms is broken down into its contributions from $n$-fold coordinated Ge atoms ($n$ = 3, 4, 5, or 6). The vertical (red) arrows mark the $q$ value expected for trigonal bipyramidal units.