2.1 T_g and the role of chemical bond strengths

A correlation between compositional trends in T_g and average chemical bond strengths in chalcogenide glasses was noted by Tichy and Ticha 〚25〛. Chemical bond strengths play a scaling role in determining T_gs, when comparing networks having the same global connectivity (r̄), such as GeS₂ with GeSe₂, or As₂S₃ with As₂Se₃. However, such chemical considerations alone cannot account for the observed changes in T_g as a function of r̄, which are structure-related. Two examples would serve to make the case. In binary As–S glasses T_g(x) are found to increase 〚20〛 monotonically (Fig. 3) with x in the 0 < x < 0.40 range. Pauling single bond strength 〚26〛 of a S–S bond of 50.9 kcal mol^–1 exceeds that of As–S single bond of 47.25 kcal mol^–1. Indeed, if average bond strengths alone were to determine T_gs of As_xS_1–x glasses as has been suggested 〚25〛, one would expect T_g to decrease with the As content x. On the other hand, the observed increase of T_g with x is precisely the behavior one expects as the S-chain network is progressively cross-linked by As resulting in an increase of the global connectivity in the 0 < x < 0.40 range.

Fig. 1 compares compositional trends in T_g in the Ge–Se binary 〚27〛 with the Si–Se one 〚28〛. The glaring difference between these trends is the presence of a threshold in T_g near x = 1/3 or $\overline{r}=2.67$ in the Ge–Se binary, but the absence of it in the Si–Se binary. The Pauling single bond strength 〚26〛 of a Ge–Se bond exceeds that of a Ge–Ge bond by 11.5 kcal mol^–1, while that of a Si–Se bond exceeds that of a Si–Si bond by 9.2 kcal mol^–1. If average bond strengths alone were the factor controlling compositional trends in T_g, one would be hard pressed to understand why T_gs decline at x > 1/3 in the Ge–Se binary, but increase at x > 1/3 in the Si–Se binary, when weaker homopolar bonds (Si–Si, Ge–Ge) emerge in both systems.

The T_g trends above are precisely the ones expected if Ge–Ge bonds were to segregate 〚29〛 from the Ge–Se backbone to nucleate a separate Ge-rich nanophase, and thus lower the global connectivity, while Si–Si bonds were to form part 〚30〛 of the Si–Se backbone and increase its global connectivity, and thus T_g of the glasses. Si–Se melts at x > 1/3 possess astronomically high viscosities 〚30〛 understandably because of the fully polymerized nature of the underlying backbones. The case of the Si_xSe_1–x binary glass system at x > 1/3, is the exception that proves the rule that T_gs are a good representation of network connectivity with chemical bond strengths playing the role of scaling these numbers between networks of similar global connectivities.

2.2 The vibration iso-coordination rule

Inelastic neutron scattering studies on ternary As_xGe_ySe_1–x–y glasses have proved an elegant probe of the low frequency vibrational density of states (VDOS) 〚31, 32〛. The excitations in the 5 meV region are usually identified with floppy modes 〚32〛 in undercoordinated networks. Effey and Cappelletti have found 〚33〛 scattering strength of these excitations to be strictly controlled by the global connectivities of glasses as measured by their mean coordination number, $\overline{r}=3\mathrm{x}+4\mathrm{y}+2\left(1-\mathrm{x}-\mathrm{y}\right)$, and to be independent of chemical compositions so long as $\overline{r}<2.40$. A parallel result was noted for vibrational lifetime of a H₂O guest molecule in these glasses, which increased 〚34〛 monotonically with r̄, regardless of chemical composition. These results support the notion that at low r (< 2.4), the Se-rich glasses are fully polymerized and continue to be so as the group IV and V elements are added to increase global connectivity of the backbone.

A striking exception to this rule emerged in binary As_xSe_1–x glasses at x = 0.6, for which the low frequency VDOS are found to be characteristic of a floppy glass ($\overline{r}=2.35$) even though the chemical composition suggests a rigid glass ($\overline{r}=2.60$). In the neutron VDOS 〚32〛, sharp modes are observed between 10 and 20 meV. These modes have contributions from characteristic As-rich clusters (As₄Se₄ and As₄Se₃) and amorphous As, for which confirmation is given by FT-Raman measurements 〚22, 35, 36〛. The sharpness of the modes suggests that these As-rich clusters are decoupled from the backbone. The exception to the vibration isocoordination rule falls in line with the sharply reduced glass transition of this composition 〚32, 19〛. These results are consistent with nanoscale phase separation of this As-rich glass.

Vibrational temperatures of ¹¹⁹Sn sites in ternary As–Ge–Se glasses

We have examined Ge-centered local structures in the As-Ge–Se ternary in Raman scattering and ¹¹⁹Sn Mössbauer spectroscopy 〚44〛 measurements. In Raman scattering, one observes modes of corner-sharing and edge-sharing Ge(Se_1/2)₄ tetrahedra to emerge with increasing x. At higher x (x > 0.21) modes of ethanelike Ge–Ge units manifest in the lineshape. In Mössbauer spectroscopy 〚44〛, tetrahedral Sn sites (labeled A) appear in the Se-rich 0 < x < 0.18 glasses, but non-tetrahedral Sn-sites (labeled B) to first appear once x > 0.18 in Ge-rich glasses as shown in Fig. 7. These non-tetrahedral sites constitute signature of Ge–Ge bonds in the network 〚45〛. Simple counting arguments reveal that a chemical threshold occurs at x_c = 0.182 above which Ge–Ge bonds are expected.

Lamb–Mössbauer factors of the A and B sites have been measured 〚44〛 by recording spectra of glasses at x > 0.18 as a function of temperature. From these results we obtain the vibrational temperature (effective Debye Temperature) of the A and B sites. Results of these experiments show that their vibrational temperatures are ${\theta }_{\mathrm{D}}^{\mathrm{A}}=125\pm 2\mathrm{K}$ and ${\theta }_{\mathrm{D}}^{\mathrm{B}}=110\pm 2\mathrm{K}$ i.e., about 15 K apart. We had performed similar measurements in binary Ge_xSe_1–x glasses earlier 〚32, 46〛, and found vibrational temperatures of these two sites, ${\theta }_{\mathrm{D}}^{\mathrm{A}}=130\pm 2\mathrm{K}$, and ${\theta }_{\mathrm{D}}^{\mathrm{B}}=100\pm 2\mathrm{K}$, to be 30 K apart in these nanoscale phase separated materials. Thus, vibrational temperatures of chemically inequivalent sites in nanoscale-phase separated glasses differ significantly more than in homogeneous glasses reflecting the intrinsic heterogeneity of the former systems.

5.1 Glass-transition temperature variation in the Ge_xSb_ySe_1–x–y ternary

In an impressive set of measurements of T_g and molar volumes in the titled ternary, Mahadevan and Giridhar 〚47〛 found that the threshold behavior of $T_{\mathrm{g}}\left(\overline{r}\right)$ near $\overline{r}=2.67$ observed at y = 0, i.e., binary Ge_xSe_1–x glasses (Fig. 2), shifts rather systematically (Fig. 8a) to lower values of r̄ as the Sb content (y) of titled ternary glasses increases in the 0 < y < 0.25 range. The existence of this threshold shift suggests nanoscale phase separation must occur. For a trivalent pnictide on strictly chemical grounds, one can always write:

In the present case, however, the first and second terms on the right hand side of equation (4) describe, respectively, the Sb-rich nanophase that segregates from the base glass phase. Because of such separation, the base glass can nanoscale phase separate 〚29〛 when the Ge/Se stoichiometry ratio exceeds the threshold value of $\frac{1}{2}$, i.e.:

Thus, for a given Sb content (y), equation (5) serves to define a critical mean coordination number:

5.2 Glass-transition temperature variation in the Ge_xAs_ySe_1–x–y ternary

A very different physical picture of alloying As as opposed to Sb in Ge–Se glasses emerges from T_g trends of the Ge_xAs_ySe_1–x–y ternary. The threshold behavior of T_gs so characteristic 〚29〛 of the base glass (Ge–Se) near $\overline{r}=2.67$, is no longer observed when as little as 5 atomic percent of As is alloyed, as shown in Fig. 9. The results of Fig. 9 taken from ref. 〚47〛 are based on the work of Borisova 〚48〛 and Frischat 〚49〛. These $T_{\mathrm{g}}\left(\overline{r}\right)$ trends suggest that As(Se_1/2)₃ pyramids now mix on a molecular scale with corner- and edge-sharing Ge(Se_1/2)₄ tetrahedra, and suppress nanoscale phase separation of the backbone. Ge–Ge bonds must still form at high r̄, but they apparently do so as part of the main backbone, as noted earlier (section 4) for the case when x = y, in the Ge–As–Se ternary.

With increasing As concentration y in the Ge–As–Se ternary, one also notes that the T_gs increase (Fig. 9), but more slowly with r near 2.67, i.e. the slope $\mathrm{d}{T}_{\mathrm{g}}/\mathrm{d}\overline{r}$ decreases. Important insights into the slopes dT_g/dr emerge from the stochastic agglomeration theory 〚14〛, which relates these to the number of ways building blocks (tetrahedra and pyramids) can agglomerate. Specifically, if there are several types of building blocks that can couple in many ways, the underlying entropy of mixing increases, and the calculations reveal 〚50〛 the slope dT_g/dr to invariably decrease. The pattern is also observed in binary glasses (Fig. 1), when one contrasts results on the two families of group IV selenides. For example, the slope near $\overline{r}=2.30$ for Si_xSe_1–x glass is less than for Ge_xSe_1–x glasses, probably because of a substantially larger concentration of edge-sharing tetrahedra in the Si-bearing glasses than in the Ge-bearing ones.

6.1 Discovery of the intermediate phase

Although the existence of floppy and rigid phases was recognized starting in the mid eighties 〚32, 51, 52, 53〛, it was not until 1999 that experimental evidence for intermediate phases evolved 〚5, 28, 54〛. These phases evolve in between floppy (n_c < 3) and stressed rigid (n_c > 3) phases, and their widths provide a measure of the degree of self-organization of a glass network. The widths of Intermediate Phases are apparently controlled by local and medium range structures that are stress-free. These phases were first observed in Raman scattering experiments on IV–VI binary glasses 〚27, 28, 54〛, wherein mode frequency (squared) variations as a function of r̄ display distinct power-laws.

An important development in identification of intermediate phases has been the application of T-modulated Differential Scanning Calorimetry (MDSC) 〚55, 56〛. The method permits deconvoluting the heat flow endotherm near T_g into two parts, a reversing heat flow and a non-reversing heat flow. The reversing heat flow is the fraction of the heat flow that tracks the programmed T-modulations and shows a text book glass transition on a flat baseline, independent of the instrument baseline. It permits a reliable scan rate and thermal history independent measurement of T_g. All (MDSC) T_g results reported in Figs. 1–4 were obtained from the inflexion point of the reversing heat flow using a model 2920 unit from TA Instruments, Incorporated at 3 °C min^–1 scan rate and 1 °C/100 s modulation rate. In these scans, the non-reversing heat flow signal usually displays a peak as a precursor to the actual glass transition. In fully relaxed samples, the area under the peak provides the latent heat of melting of a glass and will henceforth be noted as the non-reversing heat.

The novelty of the intermediate phase is that mean-field theories cannot access it 〚4〛. It is a rigid but stress-free phase of a glass. Glasses in the intermediate phase possess some very unusual physical properties including the near absence of a latent heat of melting. In other words, glasses (T < T_g) in these phases possess a configurational entropy that is nearly the same as in corresponding melts (T > T_g). This is revealed by the near absence of the non-reversing heat flow in MDSC measurements. Experiments on several glass systems have now been documented, and the results show the optical elastic thresholds to coincide 〚28, 54〛 with the thermal ones. To keep our discussion brief, we will therefore restrict to thermal thresholds, which can thus serve as a convenient probe of the rigidity transitions in glasses.

6.2 Thermally reversing windows in network glasses and rigidity transitions

Compositional trends of the non-reversing heat flow in binary Si_xSe_1–x 〚28〛 and ternary Ge_0.25S_0.75–yI_y 〚57〛 glasses appear in Fig. 10a. Noteworthy in these trends is the squarewell-like behavior observed in the binary glass. The intermediate phase here extends from x_c(1) = 0.20, or ${\overline{r}}_{\mathrm{c}}\left(1\right)=2.40$ to x_c(2) = 0.26 or ${\overline{r}}_{\mathrm{c}}\left(2\right)=2.52$, with glass compositions at x < x_c(1) reckoned as floppy, while those at x > x_c(2) as stressed rigid. The situation is replicated in the companion Ge_xSe_1–x glasses as discussed 〚54〛 elsewhere.

The case of the Ge–S–I ternary 〚57〛 is profound, because in this case the upper and lower bounds of the intermediate phase coincide, giving rise to a narrow peak in the non-reversing heat centered near r = 2.34. This is a remarkable result because it corresponds exactly to the value of r̄ where mean-field theory predicts the floppy to rigid transition to occur. A parallel circumstance occurs in the Ge–Se–I ternary 〚58〛. In Fig. 1, we note that T_gs in the Ge–Se–I ternary show a precipitous drop near $\overline{r}=2.34$, signaling the sharp rigid to floppy transition. The agreement between theory and experiment here is unparalleled in glass science. It raises the broader issue why does mean-field theory work so well in these ternary glasses when it completely breaks down in binary Ge–Se, and Si–Se glasses. The answer rests in the recognition that Iodine alloying randomly cuts S or Se bridges of the backbone, and the replacement inhibits the medium-range structure of the backbone in the form of rings to form. Constraint counting algorithms do not distinguish between S and Se in these glasses. Taken together, these results serve to demonstrate that the physical origin of the observed effects must indeed stem from rigidity effects.

Compositional trends in the non-reversing heat flow in binary As_xSe_1–x glasses 〚19〛 appear in Fig. 10b. Here the intermediate phase is localized between $\underset{¯}{\mathrm{x}}$_c(1) = 0.27 or ${\overline{r}}_{\mathrm{c}}\left(1\right)=2.27$ and x_c(2) = 0.38 or ${\overline{r}}_{\mathrm{c}}\left(2\right)=2.38$. This is also a fascinating result because it opens new ground. The result is inconsistent 〚19, 59〛 with the conventional picture of only pyramidal As-centered local structures present in the molecular structure of these Se-rich glasses. The downshift of the window onset to ${\overline{r}}_{\mathrm{c}}\left(1\right)=2.27$ that we have suggested 〚19〛 is consistent with inclusion of quasi-tetrahedral As-centered local Se=As(Se_1/2)₃ units in the structure of these glasses. In this case, MDSC measurements through the non-reversing heat flow have yielded a new feature of local structure that was apparently not recognized in earlier structure studies.

In all the three cases discussed above, it is noteworthy that rigidity transitions occur at ranges of r̄, below the regime where nanoscale phase separation effects manifest. These rigidity transitions are thus quite distinct from nanoscale phase separation effects.

6.3 The very special case of ternary As_xGe_xSe_1–2x glasses

The glass-forming region 〚48〛 in the As–Ge–Se ternary appears in Fig. 11. The hashed marked region shows the opening of the intermediate phase in between the stressed rigid and the floppy phases. The region straddles compositions near $\overline{r}=2.40$ shown as a broken tie-line connecting the binary compositions, GeSe₄ with As₂Se₃. Also shown in Fig. 11 are the regions (shaded) of glass formation where nanoscale phase separation effects manifest. Such effects are largely localized along the binary rich compositions, and for that reason even ternary compositions that are either As-rich or Ge-rich display features characteristic of nanoscale-separated glasses.

Compositional trends in the non-reversing heat flow in ternary As_xGe_xSe_1–2x glasses 〚39〛 appear in Fig. 12. The intermediate phase in this case extends from x_c(1) = 0.10 or r_c(1) = 2.30 to x_c(2) = 0.155 or r_c(2) = 2.46. This is a rather broad range, and it encompasses the intermediate phases of both the Ge–Se and the As–Se binary glasses. The width of the intermediate phase suggested from these thermal measurements, has recently been refined from the more precise Raman scattering measurements 〚44〛 and will be discussed in a forthcoming publication.

6.4 Electrical and phonon transport in ternary Ge_xAs_0.40–xSe_0.60 glasses

Several groups have examined thermal and electrical transport in ternary Ge_xAs_0.40–xSe_0.60 glasses and amorphous thin films, and rather complete results are now available for discussion. To orient the reader, it might be well to mention that this ternary joins the two binary end member compositions, As_0.40Se_0.60 and Ge_0.40Se_0.60, and furthermore it intersects the Ge_xAs_xSe_1–2x ternary (Fig. 11) of interest at right angles at x = 0.20. The line composed of points spans compositions in this ternary, as shown in Fig. 11. Thermal diffusivities, (α(x)), of bulk Ge_xAs_0.40–xSe_0.60 glasses and their S-counterparts were examined over the 0 < x < 0.40 range by Velinov et al. 〚60〛. They showed α(x) to display a global minimum near x = 0.15 for the selenide glasses, and near x = 0.18 for the sulfide glasses. Dark electrical conductivities, σ_d(x), in carefully annealed selenide thin-films over the full range (0 < x < 0.40) of compositions were studied by Nesheva and Skordeva 〚61〛. Their results show a rather spectacular narrowly defined deep global minimum in σ_d(x) near x = 0.22.

The end member compositions of the Ge_xAs_0.40–xSe_0.60 ternary are intrinsically nanoscale phase separated (Fig. 11). The molecular structure of Ge_0.40Se_0.60 glass inferred from Raman scattering and Mössbauer spectroscopy measurements has shown 〚46, 62〛 that it consists of two nanophases, an ethanelike phase and a distorted rocksalt phase. In section 2, we have discussed the case of nanoscale phase separation in As₄₀Se₆₀ glass. On the other hand, a glass composition at the midpoint, i.e., x = 0.20, which also forms part of the Ge_xAs_xSe_1–2x ternary, is rather special because it is spatially homogeneous. The most natural interpretation of the global minima in both α(x) and σ_d(x) near x = 0.20, or $\overline{r}=2.60$ is that it represents phonon and charge–transport localization in these fully polymerized networks. Carrier transport in glass compositions on either side of the global minimum is aided by the presence of internal surfaces in heterogeneous (nanoscale phase separated) structures. This is akin to increased diffusion of carriers at grain boundaries in a polycrystalline material as opposed to a single crystal. These results highlight the important role of glass structure morphology (homogeneous versus heterogeneous), on transport measurements, also noted earlier in work 〚63, 64〛 on chalcogenides.

The experimental evidence of a rigidity transition due to a dimensional regression from 2d to 3d, as suggested by Keiji Tanaka 〚12〛, would be best observed in a glass system, such as the Ge_xAs_xSe_1–2x ternary where nanoscale phase separation effects are absent. Neither Raman scattering nor MDSC results provide evidence of anomalies near $\overline{r}=2.67$ in the present ternary 〚39〛. The present results do not provide evidence of a rigidity transition due to a dimensionality change. Many of the glass systems where such an effect has been claimed 〚12, 47, 60, 61〛, upon close examination, are all found to be nanoscale phase separated 〚65〛. Finally, independent evidence for onset of nanophase separation in Ge–Se glasses at $\overline{r}>2.67$ has been obtained by T-dependent Raman measurements recently 〚66〛. Neutron structure factors of GeSe₂ glass 〚67〛 that have been analyzed by molecular dynamic simulations 〚68, 69〛 have to date not considered these nanoscale phase separation effects. Alternatively, structure studies on present ternary glasses that apparently do not phase separate, would appear to be rather attractive systems to investigate in future.