Fig. 7: Velocity operator of vitreous silica and conductivity saturation with temperature.
From: Thermal conductivity of glasses: first-principles theory and applications
a Average square modulus of the velocity-operator elements \(\langle | {{\mathsf{v}}}_{{\omega }_{{{{\rm{a}}}}}{\omega }_{{{{\rm{d}}}}}}^{{{{\rm{avg}}}}}{| }^{2}\rangle\) for the 192(D) model of v-SiO2, computed from first principles and represented as a function of the energy differences (\(\hslash {\omega }_{{{{\rm{d}}}}}=\hslash (\omega {({{{\bf{q}}}})}_{s}-\omega {({{{\bf{q}}}})}_{{s}^{{\prime} }})\)) and averages (\(\hslash {\omega }_{{{{\rm{a}}}}}=\hslash \frac{\omega {({{{\bf{q}}}})}_{s}+\omega {({{{\bf{q}}}})}_{{s}^{{\prime} }}}{2}\)) of the two coupled eigenstates (having wavevector q and modes \(s,{s}^{{\prime} }\); see text for details). The one-dimensional projections in (b) show that the elements \(\langle | {{\mathsf{v}}}_{{\omega }_{{{{\rm{a}}}}}{\omega }_{{{{\rm{d}}}}}}^{{{{\rm{avg}}}}}{| }^{2}\rangle\) are almost unchanged at a given average frequency for increasingly large energy differences. For increasingly larger temperatures, these almost-constant elements drive the saturation the rWTE conductivity (Eq. (1) with the Voigt distribution), yielding results very close to the Allen-Feldman conductivity curve (Fig. 5), which is determined exclusively by velocity-operator elements with ℏωd → 0.
