Thermal Conductivity and Specific Heat of Noncrystalline Solids

TLDR

The origin of the low‑temperature anomaly in glassy solids remains unclear, and impurities or surface effects are unlikely causes. The authors model the glass as a crystal with all atoms displaced, treating each as an interstitial or vacancy scatterer whose cross‑section is set by the missing mass, explaining the observed mean‑free‑path behavior. The study finds that vitreous SiO₂, Se, and related glasses have comparable thermal conductivities (within a factor of five) with a plateau near 10 K and a T¹·⁸ dependence below 1 K, and that their specific heats follow AT + BT³ between 0.1 and 1 K, indicating a universal low‑temperature anomaly that cannot be explained by impurities or electronic states but is consistent with Rayleigh‑type phonon scattering.

Abstract

The thermal conductivity of vitreous Si${\mathrm{O}}_{2}$, Se, and silica- and germania-based glasses has been measured between 0.05 and 100 \ifmmode^\circ\else\textdegree\fi{}K. Comparison with earlier work on noncrystalline solids shows that they all have the same conductivity within a factor of 5 over the entire temperature range investigated, with the same characteristic plateau around 10 \ifmmode^\circ\else\textdegree\fi{}K, and that their conductivity varies as ${T}^{n}$, $n\ensuremath{\sim}1.8$, below $T=1$\ifmmode^\circ\else\textdegree\fi{}K. Furthermore, the average phonon mean free path is large by comparison with the phonon wavelength, about ${10}^{\ensuremath{-}4}$ cm at 2 \ifmmode^\circ\else\textdegree\fi{}K and decreasing as ${T}^{\ensuremath{-}4}$ at larger $T$, suggesting a Rayleigh-type scattering mechanism. Such a mean free path can be quantitatively explained by approximating the glassy structure with that of a crystal in which every atom is displaced from its lattice site. Then every atom scatters like an interstitial atom, or---even simpler---like one that is missing at its regular lattice site, with a scattering cross section determined by the missing mass (isotopic defect). The specific heat of amorphous Si${\mathrm{O}}_{2}$, Ge${\mathrm{O}}_{2}$, and Se has been found to vary as $AT+B{T}^{3}$ between 0.1 and 1 \ifmmode^\circ\else\textdegree\fi{}K, with $A=10$ erg/g \ifmmode^\circ\else\textdegree\fi{}${\mathrm{K}}^{2}$ within a factor of 2. This departure from the Debye specific heat may be characteristic of the glassy state, as all earlier measurements of other glasses [polystyrene, glycerol, Lucite (PMMA)] indicate a similar anomaly. Its origin is not clear. Impurities or surface effects through adsorbed gases are unlikely because of the many samples and experimental techniques used in different laboratories. We have tried to attribute the anomaly to low-lying electronic states, motional states of ions, trapped atoms or large groups of atoms, or one-dimensional vibrations within a three-dimensional solid, so far without success. At the present time, the only independent evidence for these excitations appears to be in the low-temperature thermal conductivity at $T<1$ \ifmmode^\circ\else\textdegree\fi{}K described above.