Effect of glass stability on the low frequency vibrations of vapor deposited glasses

Abstract

Ultra-stable glasses prepared from the physical vapor deposition of organic molecules present a very low density of two-level states, the kind of glass defects that determine their peculiar low temperature thermal properties. Numerical simulations suggest that quasi-localized harmonic vibrational modes emerge in the soft regions associated with two-level states. However, the connection between the low frequency vibrational modes and the local structural instabilities of glasses remains unexplained. Here we exploit a recently developed spectrograph for nuclear resonant analysis of inelastic X-ray scattering to probe the density of vibrational states of amorphous thin films of ultra-stable and conventional glasses down to an exceptionally low frequency of $\sim 70$ GHz. We show that the glass stability does not affect the harmonic vibrational modes at the lowest frequencies, despite a reduction of almost an order of magnitude in the density of two-level states. At the same time, the vibrational modes at higher frequencies, around the boson peak maximum, are extremely sensitive to the glass stability. Although we cannot exclude the possible existence of quasi-localized modes in glasses, we show that their presence is not strictly necessary to describe the measured density of low frequency vibrations. The experimental developments here presented pave the way to the solution to the long-standing debate on the low frequency vibrations in glasses.

Figures (13)

• Figure 1: Inelastic X-ray scattering with nuclear resonance analysis. a): simplified sketch of the experimental setup at the sample stage (the drawing is not to scale) article99. b)-c): The raw data measured on the USG sample at 300 K (b)) and at 150 K (c)). The red and blue data represent the scattered intensity collected by the detector located above the sample, while the green data correspond to the instrumental function measured by the forward detector. The intensity scale refers to the sample signal.
• Figure 2: The reduced density of vibrational states. a): ultra-stable glass measured at 300 K (red diamonds) and at 150 K (blue squares); b): ordinary glass at 300 K (red triangles). The horizontal lines in panel b) are the Debye levels estimated from the elastic moduli at 300 K (red dotted), and at 30 K (black dashed). The inset of panel a) highlights the low frequency region, where a difference between the two temperatures is visible below $\sim 0.15$ THz. The parabolic black lines are an estimate of the low frequency part of the harmonic DOS, based on eq. \ref{['Eq: par']}. The grey rectangle (panel b)) represents the energy region accessible with standard X-ray monochromators.
• Figure 3: Specific heat calculation assuming the measured DOS to be harmonic. Panel a): reduced DOS of the OG (red triangles) and USG (blue diamonds) at room temperature. In this first attempt we treat the DOS as temperature independent and we consider a constant value at low frequencies (lines). Panel b): specific heat divided by the cube of temperature of the OG and USG. The lines are computed using eq. \ref{['eq:Cp']} with the measured room temperature DOS and the low frequency extrapolation shown in a). The curves do not accurately describe $C_p/T^3$, because the DOS is affected by a temperature dependence at low frequencies and the specific heat reflects the DOS at low temperatures, much lower than 300 K, as discussed in the text.
• Figure 4: The specific heat over $T^3$. The calculated $C_p$ (red solid line) compared to the experimental ones for the USG (blue squares) in a) and for the OG (red triangles) in b). The harmonic contribution (yellow dashed line) and the TLS contribution (violet dotted line) are shown as well. The Debye level found with the fitting procedure is plotted as a black horizontal line. A significant discrepancy with the $T \to 0$ K estimate from the elastic moduli (black dashed line) is found for the USG, while the two values coincide for the OG. The specific heat of the crystal is also shown for comparison (light-blue circles) in panel a) together with the fit to a parabola at low temperatures (green line), as discussed in the text. The specific heat data are from ref Moratalla2023.
• Figure 5: The reduced density of states in Debye units. As indicated in the legend, the red triangles represent the OG measured at 300 K, while the blue squares the USG measured at 150 K. The frequency is reported in units of the Debye frequency, $\nu_D$, and $g(\nu)$ is divided by the Debye DOS, $g_D(\nu) = A_D \nu^2$. The low frequency estimate of the harmonic DOS based on eq. \ref{['Eq: par']} is shown as a dotted red line for the OG and a dash-dotted blue line for the USG. The shaded violet region indicates the contribution of the excess modes, $A_{ex} \nu^4$ in eq. \ref{['Eq: par']}, to the reduced harmonic DOS, highlighting the fact that this contribution is insensitive to the glass stability. The dashed black line is the HET model with parameters optimized to describe the measured DOS, as detailed in appendix \ref{['App:HET']}.