Structure of the Intermediate Phase Glasses GeSe3 and GeSe4: The Deployment of Neutron Diffraction With Isotope Substitution

The method of neutron diffraction with isotope substitution was used to measure the full set of partial pair-correlation functions for each of the network-forming glasses GeSe₃ and GeSe₄, which lie at the boundaries of the so-called intermediate phase in the Ge-Se system. The results show the formation of chemically ordered networks, where selenium chains are cross-linked by Ge(Se₄)_1/2 tetrahedra, in contrast to glassy GeSe₂ where the chemical order is broken. In all of these materials, the Ge-centered structural motifs are distributed unevenly on an intermediate length scale, as indicated by the appearance of a first sharp diffraction peak in the Bhatia-Thornton concentration-concentration partial structure factor. The new experimental work provides benchmark results for guiding in the development of realistic structural models, which can be used to explore the network rigidity and other structure-related properties of the glass. In this context, there are discrepancies between experiment and the predictions of first-principles molecular dynamics simulations that are particularly marked in respect of the Ge-Ge correlation functions, which are sensitive to the connectivity of the Ge-centered structural motifs.

1. Introduction

The Ge_xSe_1−x (0 ≤ x ≤ 1) system is a prototype for understanding the atomic-scale organization in covalently-bonded network glass-forming materials (Tronc et al., 1973; Bresser et al., 1986; Sugai, 1987; Penfold and Salmon, 1991; Zhou et al., 1991; Salmon and Liu, 1994; Feng et al., 1997; Wang et al., 1998; Petri et al., 1999a,b, 2000; Bureau et al., 2003; Salmon and Petri, 2003; Salmon, 2007a; Sartbaeva et al., 2007; Shatnawi et al., 2008; Inam et al., 2009; Lucas et al., 2009; Gjersing et al., 2010a,b; Bauchy et al., 2011; Hosokawa et al., 2011; Micoulaut et al., 2013; Salmon and Zeidler, 2015; Kaseman et al., 2016; Zeidler et al., 2017; Whittaker et al., 2018), where the network-forming motifs provide some of the essential building-blocks for technologically relevant chalcogenide glasses (Hilton, 1966; Seddon and Laine, 1997; Zakery and Elliott, 2003, 2007; Gopinath et al., 2004; Lezal et al., 2004; Kohoutek et al., 2008; Troles et al., 2009; Eggleton et al., 2011). According to mean-field constraint-counting theory, the glass network will undergo the transition with increasing x from an under-constrained elastically floppy phase to an over-constrained stressed-rigid phase at x = 0.2, where the mean number of Lagrangian bonding constraints per atom ${\bar{N}}_c$ = 3, the number of degrees of freedom per atom in three dimensions (Phillips, 1979; Thorpe, 1983). At the x = 0.2 composition the mean nearest-neighbor coordination number $\bar{n}$ = 2.4. In comparison, spectroscopic and calorimetric measurements on the Ge_xSe_1−x system suggest that the floppy and stressed-rigid phases are separated by an ‘intermediate phase’ that extends over the finite composition range 0.20 ≤ x ≤ 0.26 (Boolchand et al., 2001a,b; Bhosle et al., 2011, 2012). For this range, it is argued that the system lowers its free energy on glass formation by self-organizing to incorporate structural configurations that minimize the occurrence of either over or under constrained regions, to give a glass network that is isostatically rigid and stress-free (Thorpe et al., 2000). Here, self-organization is facilitated by structural variability, i.e., an ability of the system to accommodate different atomic conformations (Sartbaeva et al., 2007; Massobrio et al., 2009). The intermediate phase should therefore have a structural origin, but its signature has not been identified from diffraction or extended X-ray absorption fine structure (EXAFS) spectroscopy experiments (Shatnawi et al., 2008; Zeidler et al., 2017), or from molecular dynamics simulations (Micoulaut et al., 2013). From a dynamical perspective in the Ge_xSe_1−x system, intermediate phase compositions correspond to a maximum in the viscosity at the liquidus temperature, and to a minimum in the fragility index (Zeidler et al., 2017).

In this paper we take advantage of recent advances in neutron diffraction instrumentation to investigate the structure of two glasses that delimit the intermediate phase. Specifically, neutron diffraction with isotope substitution (NDIS) (Fischer et al., 2006; Salmon and Zeidler, 2013) is combined with singular valued decomposition (SVD) (Ludwig et al., 1987; Zeidler et al., 2010) to measure the full set of partial structure factors for glassy GeSe₃ (x = 0.25) and GeSe₄ (x = 0.20). These structure factors represent the maximal information that can be extracted from experiment at the pair-correlation function level. The results are compared to those obtained from high resolution ⁷⁷Se magic angle spinning (MAS) nuclear magnetic resonance (NMR) (Gjersing et al., 2010b) or two-dimensional ⁷⁷Se NMR (Kaseman et al., 2016) spectroscopy. The results are also compared to those obtained from recent density-functional-theory based first-principles molecular dynamics (FPMD) simulations (Kibalchenko et al., 2011; Bouzid and Massobrio, 2012; Micoulaut et al., 2013; Wezka et al., 2014; Bouzid et al., 2015; Chaker et al., 2018). This simulation approach is favored when modeling the structure of materials in the Ge_xSe_1−x system on account of the similarity between the electronegativity values for Ge and Se, although there are issues regarding the sensitivity of FPMD models to the size of the simulated system, and to the protocol used to prepare a glass from the liquid state (Bouzid et al., 2015; Le Roux et al., 2016).

The paper is organized as follows. The essential theory for the NDIS experiments is given in section 2. The experimental methods are then described in section 3, where the choice of neutron diffractometer enabled the diffraction patterns to be measured with excellent count-rate stability over a wide range of scattering vectors. The results are presented in section 4 at both the first-difference function and partial structure factor levels in order to check for self-consistency in the measured parameters. The results are discussed in section 5 by reference to those obtained from other experiments and FPMD simulations, where issues include the nature and connectivity of the structural motifs, and whether there is a structural signature of the intermediate phase (Shatnawi et al., 2008; Zeidler et al., 2017). Conclusions are drawn in section 6.

2. Theory

In a neutron diffraction experiment on glassy GeSe₃ or GeSe₄, the total structure factor (Fischer et al., 2006)

$\begin{array}{ll}F(k)={\displaystyle \sum _{\alpha }}{\displaystyle \sum _{\beta }}{c}_{\alpha }{c}_{\beta }{b}_{\alpha }{b}_{\beta }\left[S_{\alpha \beta }(k)-1\right],& (1)\end{array}$

is measured, where α and β denote the chemical species, c_α and b_α represent the atomic fraction and coherent neutron scattering length of chemical species α, respectively, S_αβ(k) is a Faber-Ziman (Faber and Ziman, 1965) partial structure factor, and k is the modulus of the scattering vector. Consider the case of four samples that are identical in every respect, except for their isotopic compositions. Let the Ge-Se isotope combinations be represented by nat-nat, nat-mix, 70-nat, and 73–76, where nat denotes the natural isotopic abundance and mix denotes a mixture of Se isotopes, with enrichments (see section 3.1) that give coherent scattering lengths of b_natGe = 8.185(20), b_70Ge = 9.93(10), b_73Ge = 5.16(4), b_natSe = 7.970(9), b_mixSe = 10.10(5), and b_76Se = 12.19(10) fm (Sears, 1992). Then, if the total structure factors are denoted by ${}_{\text{nat}}{}^{\text{nat}}F(k),{}_{\text{mix}}{}^{\text{nat}}F(k),{}_{\text{nat}}{}^{\text{70}}F(k),\text{ and}{}_{\text{76}}{}^{\text{73}}F(k)$, respectively, it follows that

$\begin{array}{ll}\left(\begin{array}{l}{}_{\text{nat}}{}^{\text{nat}}F(k)\hfill \\ {}_{\text{mix}}{}^{\text{nat}}F(k)\hfill \\ {}_{\text{nat}}{}^{\text{70}}F(k)\hfill \\ {}_{\text{76}}{}^{\text{73}}F(k)\hfill \end{array}\right)=W\left(\begin{array}{c}{S}_{\text{GeGe}}(k)-1\\ S_{\text{SeSe}}(k)-1\\ S_{\text{GeSe}}(k)-1\end{array}\right).& (2)\end{array}$

In the case of GeSe₃ the weighting factor matrix is given by

$\begin{array}{ll}\frac{\text{W}}{\text{barn}}=\left(\begin{array}{ccc}0.0419(2)& 0.3573(8)& 0.2446(8)\\ 0.0419(2)& 0.574(6)& 0.310(2)\\ 0.0616(12)& 0.3573(8)& 0.297(4)\\ 0.0166(2)& 0.836(14)& 0.236(4)\end{array}\right)& (3)\end{array}$

(1 barn = 10⁻²⁸ m²), and in the case of GeSe₄ the weighting factor matrix is given by

$\begin{array}{ll}\frac{\text{W}}{\text{barn}}=\left(\begin{array}{ccc}0.0268(1)& 0.4065(9)& 0.2088(7)\\ 0.0268(1)& 0.653(7)& 0.265(2)\\ 0.0394(8)& 0.4065(9)& 0.253(3)\\ 0.0106(2)& 0.951(16)& 0.201(3)\end{array}\right).& (4)\end{array}$

The partial structure factors S_αβ(k) are related to the partial pair-distribution functions g_αβ(r) via the Fourier transform relation

$\begin{array}{ll}{g}_{\alpha \beta }(r)=1+\frac{1}{2{\pi }^2\rho r}{\displaystyle {\int }_0^{\infty }}\text{d}kk\left[S_{\alpha \beta }(k)-1\right]\text{ sin}(kr)& (5)\end{array}$

where ρ is the atomic number density and r is a distance in real space. The mean coordination number of atoms of type β, contained in a volume defined by two concentric spheres of radii r₁ and r₂ centered on an atom of type α, is given by

$\begin{array}{ll}{\bar{n}}_{\alpha }^{\beta }=4\pi \rho c_{\beta }{\displaystyle {\int }_{r_1}^{r_2}}\text{d}r{r}^2{g}_{\alpha \beta }(r).& (6)\end{array}$

It is possible to eliminate one of the partial structure factors by combining two total structure factors. For example, S_SeSe(k) can be eliminated by forming a first-difference function such as

$\begin{array}{ll}\begin{array}{l}\Delta F_{\text{Ge}}\left(k\right)\equiv {}_{\text{nat}}^{\text{70}}F\left(k\right)-\left(b_{{\text{nat}}_{\text{Se}}}^{\text{2}}/b_{{\text{76}}_{\text{Se}}}^{\text{2}}\right){}_{\text{76}}^{\text{73}}F(k)\hfill \\ =c_{\text{Ge}}^{\text{2}}\left(b_{{\text{70}}_{\text{Ge}}}^{\text{2}}-b_{{\text{73}}_{\text{Ge}}}^{\text{2}}{b}_{{\text{nat}}_{\text{Se}}}^{\text{2}}/b_{{\text{76}}_{\text{Se}}}^{\text{2}}\right)\left[S_{\text{GeGe}}(k)-1\right]\hfill \\ +2c_{\text{Ge}}{c}_{\text{Se}}\left(b_{{70}_{\text{Ge}}}{b}_{{\text{nat}}_{\text{Se}}}-b_{{73}_{\text{Ge}}}{b}_{{\text{nat}}_{\text{Se}}}^{\text{2}}/b_{{76}_{\text{Se}}}\right)\hfill \\ \left[S_{\text{GeSe}}(k)-1\right].\hfill \end{array}& (7)\end{array}$

Similarly, S_GeSe(k) can be eliminated by forming a first-difference function such as

$\begin{array}{ll}\begin{array}{l}\Delta F\left(k\right)\equiv {}_{\text{76}}^{\text{73}}F\left(k\right)-\left(b_{{73}_{\text{Ge}}}{b}_{{76}_{\text{Se}}}/b_{{70}_{\text{Ge}}}{b}_{{\text{nat}}_{\text{Se}}}\right){}_{\text{nat}}^{\text{73}}F(k)\hfill \\ =c_{\text{Ge}}^{\text{2}}\left(b_{{\text{73}}_{\text{Ge}}}^{\text{2}}-b_{{70}_{\text{Ge}}}{b}_{{73}_{\text{Ge}}}{b}_{{76}_{\text{Se}}}/b_{{\text{nat}}_{\text{Se}}}\right)\left[S_{\text{GeGe}}(k)-1\right]\hfill \\ +c_{\text{Se}}^{\text{2}}\left(b_{{\text{76}}_{\text{Se}}}^{\text{2}}-b_{{73}_{\text{Ge}}}{b}_{{\text{nat}}_{\text{Se}}}{b}_{{76}_{\text{Se}}}/b_{{\text{70}}_{\text{Ge}}}\right)\left[S_{\text{SeSe}}(k)-1\right].\hfill \end{array}& (8)\end{array}$

The corresponding real-space functions, denoted by ΔG_Ge(r) and ΔG(r), are obtained from Equations (7) and (8), respectively, by replacing each S_αβ(k) by the corresponding g_αβ(r) function. The weighting factors for the Ge-Ge and Ge-Se partial structure factors in Equation (7) are 0.054(1) and 0.196(3) barn for GeSe₃ vs. 0.0349(8) and 0.167(3) barn for GeSe₄, respectively. The weighting factors for the Ge-Ge and Se-Se partial structure factors in Equation (8) are −0.0323(6) and 0.552(12) barn for GeSe₃ vs. −0.0207(4) and 0.628(14) barn for GeSe₄, respectively. The first-difference functions are obtained from pairs of measured F(k) functions, whereas the partial structure factors are obtained by combining all of the measured F(k) functions. Self-consistent results are expected if systematic errors are small (Salmon and Petri, 2003).

It is also informative to decompose a total structure factor into its contributions from the Bhatia-Thornton (Bhatia and Thornton, 1970) number-number, concentration-concentration, and number-concentration partial structure factors denoted by S_NN(k), S_CC(k) and S_NC(k), respectively, where

$\begin{array}{l}F(k)={\langle b\rangle }^2\left[S_{\text{NN}}(k)-1\right]+c_{\text{Ge}}{c}_{\text{Se}}{\left(b_{\text{Ge}}-b_{\text{Se}}\right)}^2\{\left[S_{\text{CC}}(k)/c_{\text{Ge}}{c}_{\text{Se}}\right]\\ -1\}+2\langle b\rangle (b_{\text{Ge}}-b_{\text{Se}})S_{\text{NC}}(k)& (9)\end{array}$

and 〈b〉 = c_Geb_Ge + c_Seb_Se is the mean coherent scattering length. Equation (2) can then be re-written as

$\begin{array}{ll}\left(\begin{array}{l}{}_{\text{nat}}{}^{\text{nat}}F(k)\hfill \\ {}_{\text{mix}}{}^{\text{nat}}F(k)\hfill \\ {}_{\text{nat}}{}^{\text{70}}F(k)\hfill \\ {}_{\text{76}}{}^{\text{73}}F(k)\hfill \end{array}\right)=W\prime \left(\begin{array}{c}{S}_{\text{NN}}(k)-1\\ [S_{\text{CC}}(k)/c_{\text{Ge}}{c}_{\text{Se}}]-1\\ S_{\text{NC}}(k)\end{array}\right),& (10)\end{array}$

where in the case of GeSe₃ the weighting factor matrix is given by

$\begin{array}{ll}\frac{W^{\prime }}{\text{barn}}=\left(\begin{array}{ccc}0.644(1)& 0.00009(2)& 0.035(1)\\ 0.926(7)& 0.0069(4)& -0.369(11)\\ 0.716(4)& 0.0072(7)& 0.331(2)\\ 1.088(16)& 0.093(3)& -1.468(31)\end{array}\right),& (11)\end{array}$

and in the case of GeSe₄ the weighting factor matrix is given by

$\begin{array}{ll}\frac{W^{\prime }}{\text{barn}}=\left(\begin{array}{ccc}0.642(1)& 0.00007(2)& 0.034(2)\\ 0.945(8)& 0.0059(3)& -0.373(11)\\ 0.699(3)& 0.0061(6)& 0.327(2)\\ 1.16(2)& 0.079(2)& -1.518(33)\end{array}\right).& (12)\end{array}$

The corresponding partial pair-distribution functions are denoted by g_IJ(r) (I, J = N, C), and are related to the g_αβ(r) functions via the relations

$\begin{array}{ll}{g}_{\text{NN}}(r)=c_{\text{Ge}}^2{g}_{\text{GeGe}}(r)+c_{\text{Se}}^2{g}_{\text{SeSe}}(r)+2c_{\text{Ge}}{c}_{\text{Se}}{g}_{\text{GeSe}}(r),& (13)\end{array}$

$\begin{array}{ll}{g}_{\text{CC}}(r)=c_{\text{Ge}}{c}_{\text{Se}}\left[g_{\text{GeGe}}(r)+g_{\text{SeSe}}(r)-2g_{\text{GeSe}}(r)\right],& (14)\end{array}$

$\begin{array}{ll}{g}_{\text{NC}}(r)=c_{\text{Ge}}\left[g_{\text{GeGe}}(r)-g_{\text{GeSe}}(r)\right]-c_{\text{Se}}\left[g_{\text{SeSe}}(r)-g_{\text{GeSe}}(r)\right].& (15)\end{array}$

Equation (9) shows that S_NN(k) would be measured directly in a diffraction experiment if b_Ge = b_Se, which is almost the case for the ^natGe^natSe₃ and ^natGe^natSe₄ samples because Ge and Se of natural isotopic abundance have very similar neutron scattering lengths. It follows that g_NN(r) will describe the sites of the scattering nuclei but cannot distinguish between the chemical species that occupy those sites, i.e., it gives information on the topological ordering. The mean coordination number $\bar{n}$ is given by

$\begin{array}{ll}\bar{n}=4\pi \rho {\displaystyle {\int }_{r_1}^{r_2}}\text{d}r{r}^2{g}_{\text{NN}}(r)=c_{\text{Ge}}{\bar{n}}_{\text{Ge}}+c_{\text{Se}}{\bar{n}}_{\text{Se}},& (16)\end{array}$

where the mean coordination number of Ge atoms ${\bar{n}}_{\text{Ge}}\equiv {\bar{n}}_{\text{Ge}}^{\text{Ge}}+{\bar{n}}_{\text{Ge}}^{\text{Se}}$ and the mean coordination number of Se atoms ${\bar{n}}_{\text{Se}}\equiv {\bar{n}}_{\text{Se}}^{\text{Se}}+{\bar{n}}_{\text{Se}}^{\text{Ge}}$. According to the “8-N" rule, in which ${\bar{n}}_{\text{Ge}}$ = 4 and ${\bar{n}}_{\text{Se}}$ = 2, it follows that $\bar{n}$ = 2.5 for GeSe₃ and $\bar{n}$ = 2.4 for GeSe₄. In comparison, S_CC(k) will give information on the chemical ordering and, according to Equation (14), there will be a positive peak or negative trough in g_CC(r) if there is a preference at a given distance for like or unlike chemical species, respectively. Finally, g_NC(r) will describe the cross-correlation between sites and their occupancy by a given chemical species. A full description of the Bhatia-Thornton formalism as applied to binary network glass-forming systems is given elsewhere (Salmon, 1992, 2007b).

Equations (2) and (10) can be solved to give the full set of S_αβ(k) or S_IJ(k) functions by using the SVD method, which is described by Ludwig et al. (1987) and Zeidler et al. (2010). In the case of Equation (2), the two-norm condition number for the normalized weighting factor matrix is ${\kappa }_2^{\prime }$ = 144 for GeSe₃ vs. ${\kappa }_2^{\prime }$ = 236 for GeSe₄, respectively, i.e., the weighting factor matrix for GeSe₃ is better conditioned (Zeidler et al., 2010). In the case of Equation (10), ${\kappa }_2^{\prime }$ = 63 for GeSe₃ vs. ${\kappa }_2^{\prime }$ = 76 for GeSe₄, respectively, i.e., the weighting factor matrices for the S_IJ(k) functions are better conditioned than those for the S_αβ(k) functions.

We note that if Equation (2) is re-written to include only the total structure factors for the nat-mix, 70-nat and 73-76 samples, then the two-norm condition number for the normalized weighting factor matrix becomes ${\kappa }_2^{\prime }$ = 133 for GeSe₃ vs. ${\kappa }_2^{\prime }$ = 221 for GeSe₄. The resultant S_αβ(k) functions are in agreement with those obtained by using all four of the total structure factors and the SVD method (Rowlands, 2015). Inclusion of the total structure factor for the nat-nat sample does not improve the conditioning of the weighting factor matrix because ${}_{\text{nat}}{}^{\text{nat}}F(k)$ does not contain information on the chemical ordering in the glass: As b_natGe ≃ b_natSe it follows from Equation (9) that ${}_{\text{nat}}{}^{\text{nat}}F(k)\simeq {\langle b\rangle }^2[S_{\text{NN}}(k)-1]$.

In practice, a diffractometer can measure over only a finite scattering vector range up to a maximum value k_max, which is equivalent to the multiplication of the ideal diffraction pattern by a window function M(k) where M(k ≤ k_max) = 1 and M(k > k_max) = 0. The Fourier transform of this window function is given by

$\begin{array}{ll}M(r)=\frac{1}{\pi }{\displaystyle {\int }_0^{k_{\text{max}}}}\text{d}kcos(kr)=\frac{k_{\text{max}}}{\pi }\text{sinc}(k_{\text{max}}r)& (17)\end{array}$

where sinc(y) ≡ sin(y) / y. To help in identifying those features in a measured g_αβ(r) function that are an artifact of M(r), it is convenient to consider the density function

$\begin{array}{ll}\begin{array}{c}{d}_{\alpha \beta }^{\prime }(r)=\frac{2}{\pi }{\displaystyle {\int }_0^{\infty }}\text{d}k\left[S_{\alpha \beta }(k)-1\right]kM(k)sin(kr)\\ =d_{\alpha \beta }(r)\otimes M(r),\hfill \end{array}& (18)\end{array}$

where d_αβ(r) ≡ 4π ρr [g_αβ(r) − 1] and ⊗ denotes the one-dimensional convolution operator. A least-squares fit to $d_{\alpha \beta }^{\prime }(r)$ can then be made by using a sum of Gaussian functions, representing the peaks in d_αβ(r), convolved with M(r) (Martin et al., 2003; Salmon and Petri, 2003). A similar procedure can also be adopted for the first-difference density functions $\Delta D_{\text{Ge}}^{\prime }(r)\equiv 4\pi \rho r\left[\Delta G_{\text{Ge}}(r)/|\Delta G_{\text{Ge}}(r\to 0)|\right]\otimes M(r)$ and ΔD′(r) ≡ 4π ρr [ΔG(r) / |ΔG(r → 0)|] ⊗ M(r). Other window functions can be used to reduce the severity of Fourier transform artifacts.

3. Experimental

3.1. Synthesis and Characterization of Glassy GeSe₃ and GeSe₄

The Ge isotopes used for the sample preparation were processed in order to remove oxide impurities. Finely powdered isotope was loaded into a glass crucible, with a fritted glass bottom of 3 μm pore size, and placed within a reduction furnace. The furnace was flushed with Ar gas to remove air, and the gas flow was switched to hydrogen. Next, the furnace temperature was increased to 873 K at a rate of 4 K min⁻¹, left for 48 h, and reduced to ambient at a rate of 1 K min⁻¹. The gas flow was then switched back to Ar. This procedure gave a yield of ~83% and delivered Ge powder that had changed in appearance from black to metallic grey. Infra-red transmission spectroscopy experiments on the powder showed no indication of the Ge-O absorption band at ~879 cm⁻¹ (Madon et al., 1991; Micoulaut et al., 2006), consistent with test experiments in which the reduction procedure was used to remove the oxygen from crystalline GeO₂, or the oxygen impurities from crystalline Ge.

Two diffraction experiments were performed, the first on glassy GeSe₄ and the second on glassy GeSe₃. Each GeSe₄ sample was prepared by loading elemental ^natGe (99.999%, Alfa Aesar), ⁷⁰Ge (95.30% ⁷⁰Ge, 4.10% ⁷²Ge, 0.51 % ⁷⁶Ge, 0.05% ⁷⁴Ge and 0.04% ⁷³Ge, Isoflex USA), or ⁷³Ge (95.60% ⁷³Ge, 2.34% ⁷²Ge, 2.03% ⁷⁴Ge, 0.025% ⁷⁰Ge and 0.005% ⁷⁶Ge, Isoflex USA) together with ^natSe (99.999+%, Sigma Aldrich), ⁷⁶Se (99.8 % ⁷⁶Se and 0.20 % ⁷⁷Se, Isoflex USA), or ^mixSe (a 49.5:50.5 mixture by mass of ^natSe and ⁷⁶Se) in the required mass ratio into a silica ampoule within a high-purity argon-filled glove box. The ampoule, of 5 mm inner diameter and 1 mm wall thickness, had previously been etched using a 48 wt% solution of hydrofluoric acid, rinsed using water then acetone, and baked dry under vacuum at 1073 K for 2 h. Once loaded, the ampoule was isolated using a Young's tap, and transferred to a vacuum line where it was sealed under a pressure of 10⁻⁵ Torr. The sealed ampoule was placed in a furnace that was rocked at a rate of 0.57 rpm with a maximum rocking angle of 30° to the horizontal. The furnace was heated at a rate of 1 K min⁻¹ from ambient to a temperature T = 1,248 K, dwelling for 4 h each at T = 494 K, T = 958 K, and T = 1,211 K, near to the melting and boiling points of Se and the melting point of Ge, respectively. The highest temperature was maintained for 48 h before the rocking motion was stopped, the furnace was placed vertically and left for 4 h, and the furnace was then cooled at a rate of 1 K min⁻¹ to T = 928 K (~100 K above the melting point of GeSe₄) where the sample was left to equilibrate for 4 h. The ampoule was then dropped into an ice/water mixture. The glassy samples separated cleanly from the their silica ampoules, which were broken open in a high-purity argon-filled glove box. The GeSe₃ samples were prepared from the GeSe₄ samples by adding elemental Ge of matched isotopic abundance, quenching from 1,027 K, and otherwise following the same method. The mass of the prepared samples was ~1.5–2.2 g, and the precise sample stoichiometry, as determined by mass, was 0.2496(3) Ge: 0.7504(3) Se for GeSe₃ vs. 0.1998(3) Ge: 0.8002(3) Se for GeSe₄.

The measured infrared transmission spectra for the GeSe₃ and GeSe₄ glasses showed no indication of Ge-O or Se-O impurity bands, e.g., in the region around 735–781 cm⁻¹ (Savage and Nielsen, 1965). A similarly prepared sample of GeSe₄ glass, but with only 12 h of rocking, was characterized using energy dispersive X-ray spectroscopy and Raman spectroscopy, and was found to be homogeneous on a sub-micron to centimeter length scale (Lucas et al., 2019). The mass density, as measured using a helium pycnometer, was 4.309(3) g cm⁻³ or 4.334(4) g cm⁻³ for the GeSe₃ and GeSe₄ glasses, respectively, corresponding to an atomic number density of ρ = 0.03353(2) Å⁻³ or ρ = 0.03359(3) Å⁻³, respectively. A single glass transition temperature of T_g = 523(3) K for GeSe₃ or T_g = 463(3) K for GeSe₄ was measured by using modulated differential scanning calorimetry with a scan rate of 3 K min⁻¹ and modulation of 1 K per 100 s.

3.2. Neutron Diffraction Experiments

The diffraction experiments were performed using the D4c instrument at the Institut Laue-Langevin (Fischer et al., 2002). The Cu(220) reflections from a monochromator gave an incident neutron wavelength λ = 0.4990(1) Å, and the λ/2 reflections were suppressed by using a Rh filter. The monochromator crystals were orientated to focus the neutron beam in both the horizontal and vertical directions, thus maximizing the flux of neutrons at the sample position. The same cylindrical vanadium container, of inner diameter 4.8 mm and wall thickness 0.1 mm, was used for each set of experiments to ensure a reproducible scattering geometry. The samples were loaded into the container within a high-purity argon-filled glove box. Diffraction patterns were measured at room temperature (≃298 K) for each of the samples in its container, the empty container, the empty instrument, and a cylindrical vanadium rod of diameter 6.08 mm for normalization purposes. The diffraction pattern was also measured for a slab of neutron absorbing ¹⁰B₄C of dimensions similar to the sample in order to estimate the effect of the sample's attenuation on the background count-rate at small scattering angles (Bertagnolli et al., 1976). The relative counting times for the sample-in-container and empty container measurements were optimized in order to minimize the statistical error on the container-corrected intensity (Salmon et al., 2016). The counting times for each of the sample-in-container measurements was ~18 h for the GeSe₃ experiment vs. ~22 h for the GeSe₄ experiment. The count-rate stability was checked by saving the measured intensity for a given setup at regular intervals, and a comparison between these intensities showed no deviation within the statistical error. The count-rate stability of the D4c instrument is measured to be ±0.012(8)% (Zeidler et al., 2012).

The total structure factors were obtained by following the data analysis procedure described by Salmon et al. (1998), and self-consistency checks were made to assess the reliability of the measured functions. For instance, it is necessary that (i) the measured intensities are greater than or equal to zero which leads to the condition $S_{\text{NN}}(k)S_{\text{CC}}(k)\ge S_{\text{NC}}{(k)}^2$; (ii) each of the Faber-Ziman partial structure factors satisfies the sum-rule relation ${\int }_0^{\infty }\text{d}k{k}^2\left[S_{\alpha \beta }(k)-1\right]=-2{\pi }^2\rho$ as found by taking the low-r limit of Equation (5); (iii) the measured g_αβ(r) functions oscillate about zero at r-values smaller than the distance of closest approach between two atoms; and (iv) when these oscillations in g_αβ(r) are set to zero, the back Fourier transform should be in good overall agreement with the original partial structure factor.

4. Results

There is significant contrast between the total structure factors F(k) measured for the different isotopically enriched samples of glassy GeSe₃ or GeSe₄ (Figure 1). A so-called first sharp diffraction peak is observable at k_FSDP = 1.06(1) Å⁻¹ for GeSe₃ vs. k_FSDP = 1.12(2) Å⁻¹ for GeSe₄, and is a signature of ordering on an intermediate length scale (Salmon, 1994). As shown in Figure 2, the FSDP appears as a peak in ΔF_Ge(k), for which the Ge-Ge and Ge-Se partial structure factors receive positive weighting factors [Equation (7)], and as a trough in ΔF(k), for which the Ge-Ge and Se-Se partial structure factors receive negative and positive weighting factors, respectively [Equation (8)]. These observations are consistent with the FSDP having contributions from both S_GeGe(k) and S_GeSe(k).

FIGURE 1

Figure 1. The total structure factors F(k) for glassy (A) GeSe₃ and (B) GeSe₄. The solid curves with vertical error bars represent the measured functions, where the size of an error bar is smaller than the curve thickness at most k values.

FIGURE 2

Figure 2. The first-difference functions ΔF_Ge(k) and ΔF(k) for glassy (A) GeSe₃ and (B) GeSe₄. The solid curves with vertical error bars represent the measured functions, where the size of an error bar is smaller than the curve thickness at most k values.

Figure 3 shows the $\Delta D_{\text{Ge}}^{\prime }(r)$ and ΔD′(r) functions for glassy GeSe₃ and GeSe₄, obtained by Fourier transforming the spline-fitted reciprocal-space functions shown in Figure 2 with k_max = 23.5 Å⁻¹. The first peaks in $\Delta D_{\text{Ge}}^{\prime }(r)$ and ΔD′(r) were assumed to have contributions solely from Ge-Se and Se-Se nearest-neighbors, respectively, i.e., homonuclear Ge-Ge bonds were assumed to be rare. The first peak in each function was then fitted to a single Gaussian function convoluted with the modification function M(r). In the case of GeSe₃, the fits gave a Ge-Se nearest-neighbor distance ${\bar{r}}_{\text{GeSe}}$ = 2.37(2) Å with ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 4.02(2) and a Se-Se nearest-neighbor distance ${\bar{r}}_{\text{SeSe}}$ = 2.35(2) Å with ${\bar{n}}_{\text{Se}}^{\text{Se}}$ = 0.72(2), corresponding to a goodness-of-fit parameter R_χ (Wright, 1993) of 0.055 or 0.048, respectively, for a fitted range of 1.5–3.0 Å. In the case of GeSe₄, the fits gave ${\bar{r}}_{\text{GeSe}}$ = 2.37(2) Å with ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 4.02(5) and ${\bar{r}}_{\text{SeSe}}$ = 2.35(2) Å with ${\bar{n}}_{\text{Se}}^{\text{Se}}$ = 1.01(2), corresponding to an R_χ value of 0.022 or 0.044, respectively, for a fitted range of 1.5–3.0 Å.

FIGURE 3

Figure 3. The first-difference density functions $\Delta D_{\text{Ge}}^{\prime }(r)$ and ΔD′(r) for glassy (A) GeSe₃ and (B) GeSe₄ where the solid (black) curves were obtained by spline-fitting and Fourier transforming the ΔF_Ge(k) and ΔF(k) functions shown in Figure 2. In each case, the chained (red) curve shows a fit of the first peak in the measured function to a single Gaussian function [light solid (green) curve] convoluted with M(r).

Figure 4 shows the full set of partial structure factors S_αβ(k) for glassy GeSe₃, as extracted from the total structure factors shown in Figure 1A by using the SVD method. The results show that S_GeGe(k) has the most prominent FSDP at k_FSDP = 1.01(1) Å⁻¹, and that S_GeSe(k) also has a small FSDP at k_FSDP = 1.11(1) Å⁻¹. The density functions $d_{\alpha \beta }^{\prime }(r)$ of Figure 5A were obtained by Fourier transforming the spline fitted S_αβ(k) functions of Figure 4 with k_max = 23.5 Å⁻¹, where a Lorch window function (Lorch, 1969; Salmon, 2006) was also applied to S_GeGe(k) in order to reduce Fourier transform artifacts. The first peak in $d_{\text{GeSe}}^{\prime }(r)$ and $d_{\text{SeSe}}^{\prime }(r)$ was fitted to a single Gaussian function convoluted with M(r). The fits gave ${\bar{r}}_{\text{GeSe}}$ = 2.37(2) Å with ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 4.00(2) and ${\bar{r}}_{\text{SeSe}}$ = 2.35(2) Å with ${\bar{n}}_{\text{Se}}^{\text{Se}}$ = 0.70(2), corresponding to an R_χ value of 0.048 or 0.047, respectively, for a fitted range of 1.5–3.0 Å. These values are consistent with those obtained from the first-difference functions, and point to a network structure that is built from Ge(Se₄)_1/2 tetrahedra and Se_n chains (n is an integer ≥2) where the homonuclear Se-Se bond distance is 2.35(2) Å. The peak in $d_{\text{GeGe}}^{\prime }(r)$ at 2.70(2) Å has a position that is larger than expected for homonuclear Ge-Ge bond distances of 2.33–2.42 Å (Salmon, 2007a), and it was therefore discounted as a real feature. In the structure of high-temperature crystalline GeSe₂, the Ge-Ge distances for edge-sharing (ES) and corner-sharing (CS) Ge(Se₄)_1/2 tetrahedra are 3.05 and 3.51–3.60 Å, respectively (Dittmar and Schäfer, 1976). The peaks in $d_{\text{GeGe}}^{\prime }(r)$ at 3.04(3) and 3.58(2) Å were therefore assigned to the Ge-Ge distances for ES and CS Ge(Se₄)_1/2 tetrahedra, respectively.

FIGURE 4

Figure 4. The Faber-Ziman partial structure factors S_αβ(k) for glassy GeSe₃, obtained from the measured total structure factors of Figure 1A by using the SVD method. The vertical error bars represent the measured data points with statistical errors. The solid (red) curves are the back Fourier transforms of the corresponding partial pair-distribution functions g_αβ(r) given by the solid curves in Figure 7.

FIGURE 5

Figure 5. The density functions $d_{\alpha \beta }^{\prime }(r)$ for glassy (A) GeSe₃ and (B) GeSe₄ where the solid (black) curves were obtained by Fourier transforming the spline-fitted S_αβ (k) functions shown in Figures 4, 6, respectively. For $d_{\text{GeGe}}^{\prime }(r)$, the broken (magenta) line gives the locus of points for which $d_{\text{GeGe}}^{\prime }(r)=-4\pi \rho r$. For $d_{\text{GeSe}}^{\prime }(r)$ and $d_{\text{SeSe}}^{\prime }(r)$, the chained (red) curve shows a fit of the first peak to a single Gaussian function [light solid (green) curve] convoluted with M(r). The Ge-Se and Ge-Ge data sets have been shifted vertically for clarity of presentation.

Figure 6 shows the full set of partial structure factors S_αβ(k) for glassy GeSe₄, as extracted from the total structure factors shown in Figure 1B by using the SVD method. The results show that S_GeGe(k) has the most prominent FSDP at k_FSDP = 0.97(2) Å⁻¹, and that S_GeSe(k) also has a small FSDP at k_FSDP = 1.08(2) Å⁻¹. The density functions $d_{\alpha \beta }^{\prime }(r)$ of Figure 5B were obtained by Fourier transforming the spline fitted S_αβ(k) functions of Figure 6 with k_max = 23.5 Å⁻¹, where a cosine window function was also applied to S_GeGe(k) over the range 21.45 ≤ k (Å⁻¹) ≤ 23.5 in order to reduce Fourier transform artifacts. The first peak in $d_{\text{GeSe}}^{\prime }(r)$ and $d_{\text{SeSe}}^{\prime }(r)$ was fitted to a single Gaussian function convoluted with M(r). The fits gave ${\bar{r}}_{\text{GeSe}}$ = 2.37(2) Å with ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 4.04(5) and ${\bar{r}}_{\text{SeSe}}$ = 2.35(2) Å with ${\bar{n}}_{\text{Se}}^{\text{Se}}$ = 1.00(2), corresponding to an R_χ value of 0.055 or 0.052, respectively, for a fitted range of 1.5–3.0 Å. These values are consistent with those obtained from the first-difference functions, and point to a network structure that is built from Ge(Se₄)_1/2 tetrahedra and Se_n chains. The $d_{\text{GeGe}}^{\prime }(r)$ function shows no evidence of homonuclear Ge-Ge bonds, and does not display a peak at ≃3.04 Å that can be assigned to ES Ge(Se₄)_1/2 tetrahedra. The first peak at 3.59(1) Å is, however, broader than the corresponding feature for GeSe₃, with a low-r cutoff of 3.07(3) Å that is typical of ES tetrahedral distances.

FIGURE 6

Figure 6. The Faber-Ziman partial structure factors S_αβ(k) for glassy GeSe₄, obtained from the measured total structure factors of Figure 1B by using the SVD method. The vertical error bars represent the measured data points with statistical errors. The solid (red) curves are the back Fourier transforms of the corresponding partial pair-distribution functions g_αβ(r) given by the solid curves in Figure 8.

The final Ge-Se and Se-Se d_αβ(r) functions were obtained by merging smoothly the Gaussian representation of d_GeSe(r) or d_SeSe(r) in the first-peak region, where the effect of M(r) on $d_{\text{GeSe}}^{\prime }(r)$ and $d_{\text{SeSe}}^{\prime }(r)$ is measurable, with the $d_{\alpha \beta }^{\prime }(r)$ functions at higher-r values. The $d_{\text{GeSe}}^{\prime }(r)$ function at these higher-r values was smoothed by the application of a Lorch window function. In comparison, d_GeGe(r) was set equal to the measured $d_{\text{GeGe}}^{\prime }(r)$ function at all r values. In each case, the unphysical oscillations at r values less than the distance of closest approach between the centers of two atoms were set to d_αβ(r) = −4π ρr. The final d_αβ(r) functions were then converted to the g_αβ(r) functions shown in Figures 7, 8 for glassy GeSe₃ and GeSe₄, respectively. The back Fourier transforms of these g_αβ(r) functions are compared to the measured partial structure factors for glassy GeSe₃ and GeSe₄ in Figures 4, 6, respectively.

FIGURE 7

Figure 7. The partial pair-distribution functions g_αβ(r) for glassy GeSe₃ after the effect of M(r) has been removed.

FIGURE 8

Figure 8. The partial pair-distribution functions g_αβ(r) for glassy GeSe₄ after the effect of M(r) has been removed.

The Bhatia-Thornton partial structure factors S_IJ(k) and corresponding partial pair-distribution functions g_IJ(r) are shown in Figures 9, 10, respectively. As discussed in section 2, the weighting factor matrix used to obtain the S_IJ(k) functions is better conditioned than the matrix used to obtain the S_αβ(k) functions, which accounts for the reduction in size of the error bars for S_CC(k) as compared to S_GeGe(k). For each material, the inequality relation $S_{\text{NN}}(k)S_{\text{CC}}(k)\ge S_{\text{NC}}{(k)}^2$ is fulfilled within the statistical error at all k values (section 3.2). Each set of partial structure factors shows an FSDP in the range 1.01(2)–1.05(2) Å⁻¹ for GeSe₃ vs. 0.91(2)–1.11(2) Å⁻¹ for GeSe₄, and a principal peak or trough in the range 2.01(2)–2.04(2) Å⁻¹ for GeSe₃ vs. 1.99(2)–2.03(2) Å⁻¹ for GeSe₄.

FIGURE 9

Figure 9. The Bhatia-Thornton partial structure factors S_IJ(k) for glassy (A) GeSe₃ and (B) GeSe₄, obtained from the measured total structure factors of Figure 1 by using the SVD method. The vertical error bars represent the measured data points with statistical errors. The solid (red) curves are spline fits used to obtain the partial pair-distribution functions g_IJ(r) shown in Figure 10.

FIGURE 10

Figure 10. The Bhatia-Thornton partial pair-distribution functions g_IJ(r) for glassy (A) GeSe₃ and (B) GeSe₄. The solid black curves were obtained by Fourier transforming the spline fitted partial structure factors shown in Figure 9 and setting the unphysical oscillations at r values smaller than the distance of closest approach between the centers of two atoms [broken (red) curves] to the g_IJ(r → 0) limit. The insets show the larger-r features in rg_CC(r), and the arrowed lines indicate the periodicity expected from the position of the principal peak in S_CC(k) at k_PP ≃2.03 Å⁻¹.

As in the case of glassy GeSe₂, the FSDP in S_CC(k) for glassy GeSe₃ and GeSe₄ indicates the presence of concentration fluctuations on an intermediate length scale (Penfold and Salmon, 1991; Salmon, 1992). This feature arises from the Ge-Ge and Ge-Se pair-correlation functions, in keeping with the relation

$\begin{array}{ll}{S}_{\text{CC}}(k)=c_{\text{Ge}}{c}_{\text{Se}}\left\{1+c_{\text{Ge}}{c}_{\text{Se}}\left[S_{\text{GeGe}}(k)+S_{\text{SeSe}}(k)-2S_{\text{GeSe}}(k)\right]\right\}& (19)\end{array}$

and the measured S_αβ(k) functions shown in Figures 4, 6. From the properties of Fourier transforms, a sharp peak in S_CC(k) at k_i is associated with real-space ordering of periodicity 2π/k_i that decays with a correlation length of 2π/Δk_i, where Δk_i is the peak width (Salmon, 1994). The most prominent feature in S_CC(k) is the principal peak at k_PP ≃2.03 Å⁻¹ with a height that decreases in the order from GeSe₂ to GeSe₃ to GeSe₄. The Fourier transform of S_CC(k) shows oscillations of periodicity 2π/k_PP that are readily discernable for GeSe₂ (Salmon, 2006) and GeSe₃ (inset to Figure 10A). Similarly, the small FSDP in S_CC(k) at k_FSDP will originate from a real-space length scale that is commensurate with 2π/k_FSDP, i.e., the Ge atoms in the network structures are not distributed uniformly on an intermediate length scale. In the context of FPMD simulations, the appearance of this FSDP is significant because its reproduction has necessitated the use of exchange and correlation energy functionals that enhance charge transfer between the Ge and Se atoms (Massobrio et al., 1999, 2000, 2010; Micoulaut et al., 2009, 2013).

5. Discussion

5.1. Network Structure

The measured Ge-Se coordination numbers for glassy GeSe₂ (Petri et al., 2000; Salmon and Petri, 2003), GeSe₃ and GeSe₄ indicate the formation of Ge(Se₄)_1/2 tetrahedra (Table 1). The Ge-Se:Se-Se distance ratio, obtained from the nearest-neighbor Ge-Se and next-nearest-neighbor Se-Se peak positions, is 0.607(6), 0.616(6), or 0.619(6) for GeSe₂, GeSe₃, and GeSe₄, respectively, and the intra-tetrahedral Se-Ge-Se bond angle obtained from these peak positions is 111(1)°, 109(1)°, or 108(1)°, respectively. In comparison, for a regular tetrahedron, the Ge-Se:Se-Se distance ratio is $\sqrt{3/8}$ = 0.612, and the intra-tetrahedral Se-Ge-Se bond angle is 109.47°. The Ge-Se-Se bond angle, estimated from the first and second peak positions in g_GeSe(r), is 98(2)° for both GeSe₃ and GeSe₄.

TABLE 1

Table 1. The positions ${\bar{r}}_{\alpha \beta }$ and coordination numbers ${\bar{n}}_{\alpha }^{\beta }$ found from the first few peaks in the partial pair-distribution functions g_αβ(r) measured by NDIS in the work of Petri et al. (2000) and Salmon and Petri (2003) for glassy GeSe₂, and in the present work for glassy GeSe₃ and GeSe₄.

For all of the glasses, the peak in g_SeSe(r) at ${\bar{r}}_{\text{SeSe}}\simeq$2.34 Å originates from homonuclear Se-Se bonds, and the corresponding coordination number increases with the Se content of the glass. A comparison of the ${\bar{n}}_{\text{Se}}^{\text{Se}}$ values with those found from ⁷⁷Se NMR experiments (Gjersing et al., 2010b; Kaseman et al., 2016) is given in Table 2. Here, the NMR experiments provide the fractions of Se atoms in Se-Se-Se vs. Ge-Se-Se connections, and the corresponding Se-Se coordination numbers follow from ${\overline{n}}_{\text{Se}}^{\text{Se}}(\text{Se-Se-Se})=\left(N_{\text{Se},\text{Se-Se-Se}}/N_{\text{Se}}\right)\times 2$ and ${\bar{n}}_{\text{Se}}^{\text{Se}}(Ge-Se-Se)=(N_{\text{Se},Ge-Se-Se}/N_{\text{Se}})\times 1$, where N_{Se, Se-Se-Se} is the number of Se atoms bound to two other Se atoms in a Se-Se-Se connection, N_{Se, Ge-Se-Se} is the number of Se atoms bound to one Ge atom and one Se atom in a Ge-Se-Se connection, and N_Se is the total number of Se atoms in the glass.

TABLE 2

Table 2. The composition dependence of the nearest-neighbor coordination numbers.

For GeSe₂, the peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}$=2.42(2) Å originates from homonuclear Ge-Ge bonds, but there is no evidence of these bonds in either GeSe₃ or GeSe₄. For GeSe₂ and GeSe₃, the peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}\simeq$3.02–3.04 Å originates from ES Ge(Se₄)_1/2 tetrahedra, but this feature is absent for GeSe₄. The associated inter-tetrahedral Ge-Se-Ge bond angle, estimated from the nearest-neighbor Ge-Se and ES Ge-Ge distances, is 80(1)°.

For all of the glasses, the peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}\simeq$3.57 Å will have a contribution from CS Ge(Se₄)_1/2 tetrahedra (Petri et al., 2000; Salmon and Petri, 2003), and the associated inter-tetrahedral Ge-Se-Ge bond angle, estimated from the nearest-neighbor Ge-Se and CS Ge-Ge distances, is 98(2)°. In comparison, for the high-temperature crystalline phase of GeSe₂ (Dittmar and Schäfer, 1976), the Ge-Se-Ge bond angle is in the range of 80.2–80.6° for ES tetrahedra vs. 96.2–100.1° for CS tetrahedra.

For GeSe₃ and GeSe₄, there will be contributions to g_GeGe(r) from the Ge-Ge distances within Ge-Se_n-Ge connections. For instance, in the case of GeSe₄ there are peaks at 4.34(2) Å and 5.06(2) Å that do not appear to be Fourier transform artifacts, with coordination numbers of ${\bar{n}}_{\text{Ge}}^{\text{Ge}}$ = 0.33(2) and ${\bar{n}}_{\text{Ge}}^{\text{Ge}}$ = 1.28(2), respectively. If these peaks originate from Ge-Se-Se-Ge connections, then the associated Ge-Ge distance will depend on the relative rotation of these Ge atoms about the Se-Se bond, i.e., on the dihedral angle ϕ between two planes that contain both of the Se atoms and either the first or the second Ge atom in a Ge-Se-Se-Ge connection. For a Ge-Se-Se bond angle of 98° and bond distances of ${\bar{r}}_{\text{GeSe}}$ = 2.37 Å and ${\bar{r}}_{\text{SeSe}}$ = 2.35 Å, a Ge-Ge distance of 4.34 Å is obtained for ϕ ≃84°, and a Ge-Ge distance of 5.06 Å is obtained for ϕ ≃120°.

For each glass, the NDIS experiments give ${\bar{n}}_{\text{Ge}}$ = 4 and ${\bar{n}}_{\text{Se}}$ = 2 within the experimental error (Table 2), and the overall mean coordination numbers $\bar{n}$ are in agreement with those obtained from the first peak in the g_NN(r) functions of Figure 10 [see Equation (16)]. The NDIS results are therefore consistent with the expectations of the “8-N" rule (Zeidler et al., 2017). In comparison, the fraction of Se atoms in Ge-Se-Ge connections has been measured for several Ge_xSe_1−x glasses with x ≤ 1/3 by using ⁷⁷Se NMR spectroscopy (Gjersing et al., 2010b; Kaseman et al., 2016). The Se-Ge coordination numbers for CS and ES connections are given by the expressions ${\bar{n}}_{\text{Se}}^{\text{Ge}}$(CS) = (N_{Se, CS} / N_Se) × 2 and ${\bar{n}}_{\text{Se}}^{\text{Ge}}$(ES) = (N_{Se, ES} / N_Se) × 2, respectively, where N_{Se, CS} and N_{Se, ES} are the numbers of Se atoms in CS and ES connections, respectively. The ${\bar{n}}_{\text{Se}}^{\text{Ge}}$ values extracted from the ⁷⁷Se NMR results for glassy GeSe₂ and GeSe₄ are compared to the NDIS results in Table 2. Both techniques give a mean coordination number ${\bar{n}}_{\text{Se}}={\bar{n}}_{\text{Se}}^{\text{Se}}+{\bar{n}}_{\text{Se}}^{\text{Ge}}$ = 2, although there is a discrepancy for GeSe₄ between the relative contributions to ${\bar{n}}_{\text{Se}}$ from ${\bar{n}}_{\text{Se}}^{\text{Se}}$ vs. ${\bar{n}}_{\text{Se}}^{\text{Ge}}$. A similar set of findings was obtained by re-interpreting the measured ⁷⁷Se MAS NMR spectra for glassy GeSe₂ and GeSe₄ (Gjersing et al., 2010b) by using the isotropic chemical shifts calculated for the ⁷⁷Se nucleus via density-functional-theory based NMR calculations for amorphous model structures generated by FPMD (Kibalchenko et al., 2011) (Table 2). The associated Ge-Se coordination numbers are given by ${\bar{n}}_{\text{Ge}}^{\text{Se}}=(c_{\text{Se}}/c_{\text{Ge}}){\bar{n}}_{\text{Se}}^{\text{Ge}}=\left[(1-x)/x\right]{\bar{n}}_{\text{Se}}^{\text{Ge}}$, and are also shown in Table 2. For the case of GeSe₄, the ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ values extracted from the ⁷⁷Se NMR results are larger than expected from the “8-N” rule.

Within the framework of the “8-N” rule, the chemically ordered network (CON) and random covalent network (RCN) models provide simple but contrasting viewpoints of the network structures (Salmon, 2007a). In the CON, Ge-Se bonds are favored. It follows that only Ge-Se and Se-Se bonds are allowed for compositions with x < 1/3 such that ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 4, ${\bar{n}}_{\text{Se}}^{\text{Se}}=2(1-3x)/(1-x)$, and ${\bar{n}}_{\text{Ge}}^{\text{Ge}}$ = 0, whereas only Ge-Se and Ge-Ge bonds are allowed for compositions with x > 1/3 such that ${\bar{n}}_{\text{Ge}}^{\text{Se}}$ = 2(1−x) / x, ${\bar{n}}_{\text{Se}}^{\text{Se}}$ = 0, and ${\bar{n}}_{\text{Ge}}^{\text{Ge}}$ = 2(3x−1) / x. By contrast, in the RCN there is a purely statistical distribution of bond types such that Se-Se bonds are allowed for x > 1/3 and Ge-Ge bonds are allowed for x < 1/3. In this case, the coordination numbers are given by ${\bar{n}}_{\text{Ge}}^{\text{Se}}=4(1-x)/(1+x)$, ${\bar{n}}_{\text{Se}}^{\text{Se}}=2(1-x)/(1+x)$ and ${\bar{n}}_{\text{Ge}}^{\text{Ge}}=8x/(1+x)$. Table 2 shows that the coordination numbers from the NDIS method are consistent with the CON for GeSe₃ and GeSe₄, and that there is a breakdown of the chemical ordering for GeSe₂.

The fraction of Ge atoms in ES tetrahedral motifs can be estimated by assuming that there are no extended chains of ES motifs. Then, each ES conformation will involve two ES Ge atoms, two ES Se atoms and four other Se atoms, i.e., each ES Ge atom will be associated with one other ES Ge atom. The corresponding Ge-Ge coordination number is given by ${\bar{n}}_{\text{Ge}}^{\text{Ge}}=(N_{\text{Ge},\text{ES}}/N_{\text{Ge}})\times 1$, where N_{Ge, ES} is the number of Ge atoms in ES motifs and N_Ge is the total number of Ge atoms in the glass. In order to make a quantitative comparison with the results from ⁷⁷Se MAS NMR experiments (Gjersing et al., 2010b), the fraction of Se atoms in ES Ge-Se-Ge connections needs to be estimated from the NDIS results. With the assumption that there are no extended chains of ES motifs, each pair of ES Ge atoms will be associated with two ES Se atoms, i.e., N_{Ge, ES} = N_{Se, ES} such that $N_{\text{Se},\text{ES}}/N_{\text{Se}}=(N_{\text{Ge}}/N_{\text{Se}}){\bar{n}}_{\text{Ge}}^{\text{Ge}}=x{\bar{n}}_{\text{Ge}}^{\text{Ge}}/(1-x)$. Hence, the measured NDIS ${\bar{n}}_{\text{Ge}}^{\text{Ge}}$ values give N_{Se, ES} / N_Se = 0.17(3) for GeSe₂ vs. N_{Se, ES} / N_Se = 0.05(2) for GeSe₃. In comparison, the ⁷⁷Se MAS NMR experiments give N_{Se, ES} / N_Se values of 0.23(3) and 0.10(3) for GeSe₂ and GeSe_3.78, respectively. For the case of GeSe₄, there is no evidence of ES conformations from the NDIS experiments, although the low-r tail on the first peak in g_GeGe(r) does cover part of the distance range expected for ES tetrahedra. In comparison, ⁷⁷Se MAS NMR experiments give N_{Se, ES} / N_Se = 0.08(3) for GeSe₄ (Gjersing et al., 2010b), and Raman spectroscopy experiments report the existence of ES tetrahedra over the entire composition range 0.1 ≤ x ≤ 0.33 (Sugai, 1987; Wang et al., 1998; Lucas et al., 2009; Gjersing et al., 2010b; Bhosle et al., 2012).

We note that the presence of ES conformations and the appearance of an FSDP in S_CC(k) (Figure 9) are in contradiction with the chain-crossing model for Ge_xSe_1−x glasses with x <1/3 in which a uniform distribution of Ge-centered structural motifs is hypothesized (Tronc et al., 1973; Bureau et al., 2003).

5.2. Comparison With FPMD Models

In Figures 11–13, the measured S_αβ(k) and g_αβ(r) functions are compared to those obtained from N = 120 atom FPMD simulations of GeSe₂ (Bouzid and Massobrio, 2012; Micoulaut et al., 2013; Wezka et al., 2014), GeSe₃ (Micoulaut et al., 2013), and GeSe₄ (Micoulaut et al., 2013; Bouzid et al., 2015) and from N = 480 atom simulations of GeSe₄ (Bouzid et al., 2015). In all of this FPMD work, the Becke-Lee-Yang-Parr (BLYP) exchange and correlation energy functional was employed (Becke, 1988; Lee et al., 1988), and steps were taken to ensure that the simulated systems were free from residual pressure (Bouzid and Massobrio, 2012). The simulated nearest-neighbor coordination numbers are compared to the measured values in Table 2.

FIGURE 11

Figure 11. The measured vs. simulated (A) S_GeGe(k) and (B) g_GeGe(r) functions for glassy GeSe₂, GeSe₃ and GeSe₄. The measured functions [dark (black) curves] are from the work of Petri et al. (2000) and Salmon and Petri (2003) in the case of GeSe₂, or from the present work in the cases of GeSe₃ and GeSe₄. In (A) the vertical bars represents the statistical uncertainty on the measured data points. The FPMD results are from (i) Micoulaut et al. (2013) with N = 120 [solid (red) curves], (ii) Bouzid and Massobrio (2012) and Wezka et al. (2014) for GeSe₂ with N = 120 [solid (green) curves], and (iii) Bouzid et al. (2015) for GeSe₄ with either N = 120 [solid (green) curves] or N = 480 [broken (blue) curves]. The data sets for GeSe₃ and GeSe₄ are offset vertically for clarity of presentation.

FIGURE 12

Figure 12. The measured vs. simulated (A) S_GeSe(k) and (B) g_GeSe(r) functions for glassy GeSe₂, GeSe₃, and GeSe₄. The measured functions [dark (black) curves] are from the work of Petri et al. (2000) and Salmon and Petri (2003) in the case of GeSe₂, or from the present work in the cases of GeSe₃ and GeSe₄. In (A) the vertical bars represent the statistical uncertainty on the measured data points. The FPMD results are from (i) Micoulaut et al. (2013) with N = 120 [solid (red) curves], (ii) Bouzid and Massobrio (2012) and Wezka et al. (2014) for GeSe₂ with N = 120 [solid (green) curves], and (iii) Bouzid et al. (2015) for GeSe₄ with either N = 120 [solid (green) curves] or N = 480 [broken (blue) curves]. The data sets for GeSe₃ and GeSe₄ are offset vertically for clarity of presentation.

FIGURE 13

Figure 13. The measured vs. simulated (A) S_SeSe(k) and (B) g_SeSe(r) functions for glassy GeSe₂, GeSe₃, and GeSe₄. The measured functions [dark (black) curves] are from the work of Petri et al. (2000) and Salmon and Petri (2003) in the case of GeSe₂, or from the present work in the cases of GeSe₃ and GeSe₄. In (A) the vertical bars represent the statistical uncertainty on the measured data points. The FPMD results are from (i) Micoulaut et al. (2013) with N = 120 [solid (red) curves], (ii) Bouzid and Massobrio (2012) and Wezka et al. (2014) for GeSe₂ with N = 120 [solid (green) curves], and (iii) Bouzid et al. (2015) for GeSe₄ with either N = 120 [solid (green) curves] or N = 480 [broken (blue) curves]. The data sets for GeSe₃ and GeSe₄ are offset vertically for clarity of presentation.

Each model gives a good account of the measured Ge-Se and Se-Se correlation functions, but agreement with the measured Ge-Ge correlation functions is less impressive. In general, it is precisely the Ge-Ge correlations that show the greatest sensitivity to the details of the modeling procedure (Salmon and Zeidler, 2015), e.g., to the exchange and correlation energy functional used in the simulations (Massobrio et al., 1999, 2000, 2010; Micoulaut et al., 2009, 2013), to the effect on the structure of any residual pressure (Bouzid and Massobrio, 2012), and to the effect on the structure of both the system size and quench protocol that is employed, especially for small Ge content glasses (Bouzid et al., 2015; Le Roux et al., 2016).

In the case of GeSe₂, the models reproduce all of the measured features in g_GeGe(r), although the CS tetrahedral peak at ${\bar{r}}_{\text{GeGe}}~$3.66 Å is shifted toward a larger-r value as compared to experiment. In the case of GeSe₃, the model gives homonuclear Ge-Ge bonds in addition to ES and CS tetrahedral motifs, but homonuclear bonds were not found in the NDIS results, and the height of the CS peak in the simulated g_GeGe(r) function is smaller than the measured value. In the case of GeSe₄, homonuclear Ge-Ge bonds were found in one FPMD model but not in the others, connections that were not found by experiment. Each model features a clearly defined ES tetrahedral peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}~$3.0 Å that is absent in the measured function, although there is some overlap of the modeled ES peak with the low-r tail of the first peak in the measured g_GeGe(r) function. The height of the CS peak in the simulated g_GeGe(r) function is smaller than the measured value. These general findings from the NDIS experiments on GeSe₃ and GeSe₄ support those obtained from reverse Monte Carlo (RMC) models that were used to analyze the results obtained from anomalous X-ray scattering (AXS) experiments (Hosokawa et al., 2011).

The inclusion of dispersion or van der Waals (vdW) forces in FPMD simulations has recently been considered by Chaker et al. (2018) for the case of glassy GeSe₄. Two different theoretical schemes were used for the description of these forces, one based on the work of Grimme (2006) (vdW_G) and the other based on a Wannier function analysis (vdW_W) (Silvestrelli, 2008). The results were compared to those obtained for the N = 480 atom FPMD model of Bouzid et al. (2015) in which dispersion forces were not included. The main effect of dispersion forces is to (i) increase the height of the CS tetrahedral peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}\simeq$3.6 Å, making it comparable to the height found in the FPMD simulations of Micoulaut et al. (2013), and (ii) either leave constant or increase the height of the second peak in g_SeSe(r) at ${\bar{r}}_{\text{SeSe}}\simeq$3.8 Å, depending on the theoretical scheme. The dispersion corrections lead to ${\bar{n}}_{\text{Ge}}$ values that are in better accord with the “8-N” rule (Table 2).

5.3. Intermediate Phase

High-energy X-ray diffraction, neutron diffraction and EXAFS spectroscopy experiments have been performed to look for a structural signature of the intermediate phase in Ge_xSe_1−x glass (Shatnawi et al., 2008; Zeidler et al., 2017). The composition dependence of several parameters was investigated across the stiffness transition. These parameters included descriptors of the local structure such as (i) the first peak position and width in the total pair-distribution function, (ii) the mean coordination number $\bar{n}$, and (iii) the Debye-Waller factor describing the root-mean-square displacement (static or thermal) of atoms about their equilibrium positions. They also included descriptors of the non-local structure on length scales that are associated with the FSDP and principal peak in the measured total structure factors, where ordering associated with the principal peak can persist to a nanometer length scale (Salmon et al., 2005; Salmon, 2006). All of these parameters were found to evolve smoothly with the glass composition, i.e., a structural signature of the intermediate phase was not identified.

The present work provides benchmark results for guiding in the development of accurate models for the structures of glassy GeSe₃ and GeSe₄, which can be used to explore the network rigidity and other structure-related properties of the glass. The investigated glasses lie at the boundaries of the intermediate phase in the Ge_xSe_1−x system, which is reported for the composition range 0.20 ≤ x ≤ 0.26 (section 1). Often, models for the glass structure are provided by FPMD simulations, and the onset of a stressed-rigid phase has been attributed to a substantial change in the distribution of bond angles associated with Ge-centered tetrahedra, such that bond-angle constraints are broken for x > 0.25 (Micoulaut et al., 2013). However, as emphasized by the discussion of section 5.2, there is mismatch between the measured and FPMD results that is particularly noticeable in respect of the Ge-Ge correlation functions, which are sensitive to the connectivity of the Ge-centered structural motifs. In addition, different FPMD models can lead to different conclusions about the network structure. Thus, it will be interesting to see the predictions of FPMD models for intermediate phase glasses that are consistent with all of the measured partial pair-correlation functions.

6. Conclusions

Full sets of partial structure factors for glassy GeSe₃ (x = 0.25) and GeSe₄ (x = 0.20) were measured by using the NDIS method. These glass compositions lie at the boundaries of the intermediate phase in the Ge_xSe_1−x system. The results are consistent with chemically ordered network structures, in contrast to GeSe₂ where the chemical ordering is broken by the formation of homonuclear Ge-Ge and Se-Se bonds (Petri et al., 2000; Salmon and Petri, 2003). For all of these compositions, the nearest-neighbor Ge and Se coordination numbers are given by ${\bar{n}}_{\text{Ge}}$ = 4 and ${\bar{n}}_{\text{Se}}$ = 2 within the experimental error (Table 2), in accordance with the expectations of the “8-N" rule. In the case of GeSe₄, larger ${\bar{n}}_{\text{Ge}}$ values are deduced from ⁷⁷Se MAS NMR experiments (Table 2), which suggests the need for a re-interpretation of the spectroscopy results. A distinct peak in g_GeGe(r) at ${\bar{r}}_{\text{GeGe}}~$3.03 Å that originates from ES tetrahedra was observed for both GeSe₂ and GeSe₃, but was not found for GeSe₄. This finding appears to contradict the results obtained from both ⁷⁷Se MAS NMR (Gjersing et al., 2010b) and Raman spectroscopy (Sugai, 1987; Wang et al., 1998; Lucas et al., 2009; Gjersing et al., 2010b; Bhosle et al., 2012) experiments, although the low-r tail on the first peak in the measured g_GeGe(r) function does have a contribution in the distance range associated with ES tetrahedra. The appearance of an FSDP in S_CC(k) indicates a non-uniform distribution of Ge-centered tetrahedra on an intermediate length scale, in contradiction to the chain-crossing model (Tronc et al., 1973; Bureau et al., 2003).

The new experimental work provides benchmark results for aiding in the development of realistic structural models for establishing the network rigidity and other structure-related properties of the glass. A comparison of the NDIS results for GeSe₃ and GeSe₄ with those from recent FPMD simulations shows mismatch that is particularly marked in respect of the Ge-Ge correlation functions, i.e., there is an ingredient missing from the FPMD simulations that may be related to the system size.

Data Availability

The data sets created during this research are openly available from the University of Bath Research Data Archive at https://doi.org/10.15125/BATH-00664 (Salmon and Zeidler, 2019). Raw neutron diffraction data sets for GeSe₃ and GeSe₄ are available at https://doi.ill.fr/10.5291/ILL-DATA.6-05-922 (Salmon et al., 2013).

Author Contributions

PSS and AZ designed the research. AZ prepared the samples. All authors contributed to the neutron diffraction experiments. RFR and AZ analyzed the data. PSS wrote the paper.

Funding

The Bath group received support from the EPSRC via Grant No. EP/J009741/1 and from the University of Bath for a PhD studentship (RFR). AZ and PSS are grateful to Corning Inc. for the award of Gordon S. Fulcher Distinguished Scholarships, during which this work was completed. AZ is supported by a Royal Society − EPSRC Dorothy Hodgkin Research Fellowship.

Conflict of Interest Statement

PSS and AZ were employed by Corning Inc. during the time this work was completed.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

We thank Annalisa Polidori (Bath) for help with the diffraction experiments, and Sabyasachi Sen (UC Davis) for pointing out (Kaseman et al., 2016).

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