Quantum theory of elastic strings and the thermal conductivity of glasses

TL;DR

The paper develops a quantum continuum theory for glasses that combines phonons with extended line defects (Volterra dislocations) whose vibrational modes account for the boson peak. By mapping the BP density of states to a dislocation-length distribution $p(L)$ and computing phonon self-energies via the Peach-Koehler coupling, the authors derive the thermal conductivity using a Kubo-formalism framework and show that a long-$L$ tail $p(L) \propto L^{-5}$ yields a linear $\kappa(T)$ rise at low temperatures. They test the model in glycerol and silica, first in a ballistic/low-dissipation limit and then with full microscopic inputs; glycerol is well described with line defects alone, while silica requires both line defects and a temperature-dependent point-defect term to fit $\kappa(T)$ over $30$--$750$ K. The work provides a quantitative, testable bridge between the boson peak and thermal transport in glasses, and it outlines how measurements of the BP can constrain defect distributions and coupling strengths in a predictive framework.

Abstract

We study the thermal conductivity of amorphous solids by constructing a continuum model whose degrees of freedom are propagating vibrational modes (phonons) and extended Volterra dislocation line defects with their own vibrational degrees of freedom which do not propagate in space. Our working assumption is that these additional degrees of freedom account for the "boson peak" that is observed in glassy materials. This identification allows us to obtain the length distribution of dislocations from experimental data of the boson peak for each material, which we use as input to calculate the phonon self-energy in a quantum field theory framework using that the phonon-dislocation interaction is given by the Peach-Koehler force. The tail of the distribution for long dislocations is consistent with an $L^{-5}$ power law. Our results show that this power law yields a linear rise in the thermal conductivity, as observed in glasses at low temperatures. We then consider two approaches to describe thermal conductivity data quantitatively. In the simplest approach we only keep the low-frequency behavior of the phonon self-energy with one free parameter, plus an adjustable UV cutoff. In the more realistic approach we keep the full frequency dependence of the phonon self-energy as dictated by the phonon-dislocation interaction plus an additional contribution due to scattering with point defects, with a cutoff set by the typical interatomic spacing of the material. We obtain a satisfactory description of thermal conductivity data with both approaches. We conclude by discussing prospects to test the predictive power of this model.

Figures (12)

• Figure 1: Diagram depicting the various quantities involved in the description of the string that embodies the line defect in an otherwise homogeneous material. The ends of the string are pinned at positions ${\bf x}_0$ and ${\bf x}_0 + {\bf L}$. Its displacement from its equilibrium position is ${\bf X}(s,t)$, and $\sphericalangle$ is the 3D rotation angle that characterizes the orientation of the string relative to a fixed reference frame. All of these quantities (${\bf x}_0, {\bf L}, \sphericalangle$) will be averaged over in the following sections.
• Figure 2: String length distribution in Glycerol determined from the BP density of states determined by BGLBianchi2020 (see their Figure 1). For comparison, we display the length distribution BGL extracted from the same BP data, accounting for the factor of $2$ in the mode counting but maintaining the (mild) discrepancy between the dispersion relation of the string modes. For this figure, we choose $c_T$ to be equal to the physical speed of shear waves in Glycerol.
• Figure 3: Data from Fig. 2 in Wischnewski et al.Wischnewski1998, showing the boson peak data in amorphous SiO${}_2$ (open symbols), over which we display the parametrization we use to describe it in our model (solid colored lines). This parametrization is a linear (affine) function of the temperature. The BP density of states, which we use to obtain $p(L)$, is the difference between the data and the dashed horizontal lines that yield the Debye density of states at 51 K and 1673 K (drawn with their corresponding colors in the fit).
• Figure 4: String length distribution in Silica determined from the BP density of states in Fig. \ref{['fig:silica-DOS-fit']} (solid lines), for the same temperatures as shown in Fig. \ref{['fig:silica-DOS-fit']}. The temperature dependence of the distribution is determined from Eq. \ref{['eq:pL-interp']} using the distribution at $T_1 = 51$ K and $T_2 = 1673$ K as input. For comparison, we display the length distribution BGLBianchi2020 extracted from the $T = 1673$ K data using a different parametrization (dashed line).
• Figure 5: Comparison of data in Cahill et al. Cahill1987, reproduced from Figure 5 of that paper, with the model defined by Eq. \ref{['eq:cond-simple']} overlaid on top. The $y$-axis scale for SiO${}_2$ is on the right, whereas for all the others it is on the left. The values of $\Lambda$ and $A_{\rm eff}$ used for each model curve are presented in Table \ref{['tab:fit-params']}.