Structural properties of Ge-S amorphous networks in relationship with rigidity transitions: An ab initio molecular dynamics study

We investigate the amorphous ${\mathrm{Ge}}_x{\mathrm{S}}_{100-x}$ (with $10\le x\le 40)$ system from ab initio simulations. Results show a very good agreement with experimental findings from diffraction and the topology of the obtained structural models is further analyzed and compared with the selenide analog. Differences emerge, however, from a detailed molecular dynamics analysis showing that the ring statistics and the homopolar defects do not evolve similarly. The findings are also connected to rigidity theory, which provides a topological approach to decoding the physics of network glasses, and the effects of composition and temperature are analyzed.

Figure 1

A snapshot of an amorphous ${\mathrm{GeS}}_3$ showing Ge atoms (orange) cross linking the sulfur network. A fivefold ring has been highlighted.

Figure 2

Simulated interference function $k\left[S\right(k)-1]$ (black curves) in 300 K Ge-S glasses. The results are directly compared with measured structure factors (colored curves) from neutron diffraction: ${\mathrm{Ge}}_{10}{\mathrm{S}}_{90}$ [67], ${\mathrm{Ge}}_{20}{\mathrm{S}}_{80}$ [67], ${\mathrm{Ge}}_{33}{\mathrm{S}}_{67}$ (green [67, 68] and red [69]), and ${\mathrm{Ge}}_{40}{\mathrm{S}}_{60}$ [68]. The shoulder peak at $k_0\simeq 7.5$ Å is highlighted with an arrow.

Figure 3

(a) Calculated total pair-correlation function $g\left(r\right)$ in amorphous Ge-S (300 K) glasses (black lines) at 300 K, compared to experimental measurements from neutron diffraction: ${\mathrm{Ge}}_{10}{\mathrm{S}}_{90}$ [67], ${\mathrm{Ge}}_{20}{\mathrm{S}}_{80}$ (red curve [39]), ${\mathrm{Ge}}_{33}{\mathrm{S}}_{67}$ (circles [67, 68] and red curve [69]), and ${\mathrm{Ge}}_{40}{\mathrm{S}}_{60}$ [68]. (b) Comparison of the present simulated ${\mathrm{GeS}}_2$ (33%) and ${\mathrm{GeS}}_4$ (20%) with previous simulations from Blaineau et al. [37, 38], Celino et al. [35], and Bouzid et al. [39].

Figure 4

Computed pair-distribution functions $g_{ij}\left(r\right)$ $(i,j=$ Ge,S) as a function of composition $x$ in ${\mathrm{Ge}}_x{\mathrm{S}}_{1-x}$ glasses (300 K). The red curves are results for Ge-Se glasses [61] and serve for comparison. They have been rescaled in distance in order to match the sulphide-related functions (see also Fig. 6). The blue arrows indicate structural features discussed below.

Figure 5

Partial pair-correlation function $g_{\mathrm{GeGe}}\left(r\right)$ of ${\mathrm{Ge}}_{40}{\mathrm{S}}_{60}$ for the four independent quenches. The lowest-energy minimum corresponds to the green curve $(E=-26.415$ keV) and energy differences are provided.

Figure 6

Left: Calculated neighbor distributions in ${\mathrm{Ge}}_{25}{\mathrm{S}}_{75}$ (black lines) and ${\mathrm{Ge}}_{25}{\mathrm{Se}}_{75}$ (red curves, adapted from [61]) as a function of a rescaled distance $r/d_{\mathrm{XX}}$, where $d_{\mathrm{XX}}$ is the first peak of the partial chalcogen-chalcogen correlation function (2.19 Å and 2.36 Å for S-S and Se-Se, respectively). The thick lines correspond to the partial $g_{SS}\left(r\right)$ and $g_{SeSe}\left(r\right)$ also displayed in Fig. 4. Blue bars define the first- and second-nearest-neighbor distances $d_1$ and $d_2$ for the Ge-S glasses. Right: A typical fragment of S-rich Ge-S glasses showing the distances $d_1$ and $d_2$.

Figure 7

Calculated fraction of Ge-Ge (filled symbols, solid line) and chalcogen-chalcogen (open symbols, broken lines) homopolar bonds in Ge-S (red), Ge-Se [61] (black), and Ge-Te [77] (green) glasses as a function of Ge content. The cutoff has been chosen at the minimum of each pair-correlation function. The thin lines represent the RCN model of Eqs. (2) and (3).

Figure 8

Calculated number of rings $R^{\left(n\right)}$ in amorphous ${\mathrm{Ge}}_x{\mathrm{S}}_{1-x}$ (blue bars) for different Ge compositions using the rings method [81]. Results are compared to the calculated statistics of Ge-Se glasses [61, 76] (red bars).

Figure 9

Number of small rings $R^{\left(n\right)}$ $\left(3<n<8\right)$ in (a) amorphous ${\mathrm{Ge}}_x{\mathrm{Se}}_{1-x}$ [61] and (b) ${\mathrm{Ge}}_x{\mathrm{S}}_{1-x}$ (present work) as a function of Ge composition $x$. The gray zone indicates the experimentally determined intermediate phase [29, 32].

Figure 10

Evolution of the number of constraints $n_c$ for different isotherms as a function of Ge content in Ge-S glasses and liquids. The dotted line is the mean-field estimate $n_c=2+5x$ [1].

Figure 11

Temperature evolution of the number of constraints in ${\mathrm{Ge}}_{40}{\mathrm{S}}_{60}$ for Ge (black) and S (red). The broken lines are a least-squares fit using the Mauro-Gupta form [Eq. (4)].