Comment on “Dynamic aspects of the liquid-liquid phase transformation in silicon” [J. Chem. Phys. 129, 104503 (2008)]

In a recent paper, Jakse and Pasturel (JP) (Ref. 1) presented a simulation study of supercooled liquid Si using first-principles molecular dynamics. The main objective was to provide supportive evidence that the first order liquid-liquid phase transition (LLPT) in the supercooled liquid Si, which was indicated in previous classical molecular dynamics studies,² is a fragile-to-strong transition. JP calculated the reduced vibrational density of states (VDoS) $g(ω)/ω2$ and judging from the presence/absence of a peak in $g(ω)/ω2$ at $∼2.9THz$ concluded that the LLPT in supercooled liquid Si is a fragile-to-strong transition.

Sastry and Angell (SA) (Ref. 2) had concluded on a fragile-to-strong transition in supercooled Si in an indirect way. Briefly, they called the low-density liquid (LDL) phase as strong liquid and the high-density liquid (HDL) phase as fragile liquid based on the existence of a dip in the intermediate scattering function between the short-time dynamics and the $α$-relaxation. Although the dip in the intermediate scattering function has circumstantially been associated with the Boson peak, there is still no physical basis to support this. Moreover, the dip transforms to an oscillatory behavior at the plateau of the intermediate scattering function between short-time dynamics and the $α$-relaxation when finite size effects of the simulated system come into play, a fact that makes its interpretation even more ambiguous. Further, SA suggested that the strength of the Boson peak compared to the quasielastic contribution could be a better benchmark for classifying the HDL and LDL as fragile and strong liquids, respectively.

JP used this idea to perform molecular dynamics simulations calculating the intermediate scattering function for the melt and the supercooled liquid state. Contrary to the previous work of classical molecular dynamics,² JP found an oscillatory behavior in the intermediate scattering functions for both HDL and LDL. The VDoS of supercooled liquid Si was found to contain two main peaks at $∼3.8$ and $∼13.0THz$ for HDL and at $∼3.2$ and $∼13.0THz$ for the LDL. Transforming the VDoS to the reduced form, i.e., $g(ω)/ω2$⁠, it was found that a peak near 2.9 THz is visible for LDL while a similar peak is not seen in the spectrum of HDL. By assuming that strong liquids show strong Boson peaks which persist to temperatures far above the glass transition temperature $Tg$⁠, JP concluded that since the HDL phase of Si does not show the Boson peak, it is a fragile liquid.

In this comment we show that (i) the assignment of the Boson peak by JP is erroneous, and (ii) the Boson peak strength is not generally useful for inferring the strong or fragile character of a supercooled liquid.

Raman experiments on amorphous Si $(a-Si)$ have shown the existence of two peaks at frequencies of $∼5.2THz$ $(170cm−1)$ and $∼14.0THz$ $(470cm−1)$⁠.³ These peaks have been unambiguously assigned to the transverse acoustic (TA) and transverse optic (TO) modes, respectively. The isotropic and depolarized reduced Raman spectra of $a-Si$ taken from Ref. 3 are shown in Fig. 1 together with the VDoS obtained by JP. There is a correspondence between the peaks in the VDoS emerging from simulations and the peaks in the Raman spectra. Since there are no experimental Raman spectra of supercooled Si, we made use of the Raman spectra of the ambient temperature amorphous state. Obviously, this is a very good approximation because it is generally valid that the Raman spectra between a glass and its corresponding supercooled liquid are almost identical apart from small band broadening and redshift in the latter. Indeed, the small redshift $(∼1.0THz)$ of the supercooled Si VDoS as compared to the Raman spectrum (see Fig. 1) is a reasonable effect of temperature. The TA peak in the reduced VDoS $g(ω)/ω2$ appears at $∼2.9THz$ due to the $ω2$ factor in the denominator. Therefore, it is clear that contrary to the premise by JP in Ref. 1, the peak at $∼2.9THz$ is not associated with the Boson peak in $a-Si$⁠.

That experiments do not clearly reveal a Boson peak in $a-Si$ is well known in the recent literature and this has been overlooked in both simulation studies.^1,2 It is obvious that a Boson peak in $a-Si$ is not observed in its Raman spectra.³ Moreover, amorphous Si is seen as an example of an amorphous solid that does not show significant low energy excitations in the VDoS obtained by inelastic neutron scattering.⁴ Such evidence was also provided in specific heat measurements,⁵ where it has been pointed out that “the $a-Si$ deviates from the Debye model significantly less than does crystalline Si, leaving little evidence for a large density of excess modes.” On inspecting the reduced spectra in the inset of Fig. 2 (Ref. 1) it becomes clear that the HDL phase exhibits also a TA peak at the same energy with the corresponding peak in LDL, which is, however, hidden under the quasielastic line. The finite (nonzero) value of $g(ω)$ at zero frequency of the HDL VDoS (see the inset of Fig. 2 in Ref. 1) is responsible for the steep increase in the quasielastic intensity in the $g(ω)/ω2$ representation, which hides the peak at $∼2.8THz$⁠. Concluding, in both the HDL and LDL phases there is a peak located at $∼2.9THz$⁠, which is the transformation of the TA peak seen in $g(ω)$⁠. Therefore none of these peaks are related to the Boson peak of supercooled liquid Si.

Even in the case where simulations in Ref. 1 were able to reveal the existence of the Boson peak in supercooled liquid Si, it would be impossible to infer the strong/fragile character of HDL and LDL only by the appearance of the Boson peak in $g(ω)/ω2$⁠. There seems to be no correlation between the Boson peak and fragility. Although a selected few strong liquids show strong Boson peaks that persist to temperatures above the glass transition temperature, it has been shown that when more data of a broader class of glasses are examined there is no correlation between the strong/fragile character of the liquid and the strength of the Boson peak as compared with the quasielastic contribution.⁶ A similar correlation between the strength (amplitude) of the Boson peak compared with the Debye contribution and the fragility of the liquid is also shown to be invalid.⁷ We conclude that there is no merit in determining a liquid’s fragility on the basis of the Boson peak intensity.

In summary, the inference that LLPT in supercooled liquid Si is a fragile-to-strong transition is unjustifiable.