[Paper Review] Unearthing the foundational role of anharmonicity in heat transport in glasses

Alfredo Fiorentino, Enrico Drigo+2 more|arXiv (Cornell University)|Jul 14, 2023

Thermal properties of materialsMaterials Science3 citations

TL;DR

This paper demonstrates that the harmonic approximation in the Allen-Feldman theory of heat transport in glasses inherently predicts divergent thermal conductivity due to an infrared singularity, which is only regularized by anharmonic effects or finite-size cutoffs. Using quasi-harmonic Green-Kubo theory and Haydock's recursive method, the authors show that anharmonicity is essential for finite thermal conductivity in both crystalline and glassy materials, resolving the long-standing paradox of the theory's empirical success despite its theoretical flaw.

ABSTRACT

The time-honored Allen-Feldman theory of heat transport in glasses is generally assumed to predict a finite value for the thermal conductivity, even if it neglects the anharmonic broadening of vibrational normal modes. We demonstrate that the harmonic approximation predicts that the bulk lattice thermal conductivity of harmonic solids inevitably diverges at any temperature, irrespective of configurational disorder, and that its ability to represent the heat-transport properties observed experimentally in most glasses is implicitly due to finite-size effects. Our theoretical analysis is thoroughly benchmarked against careful numerical simulations. Our findings thus reveal that a proper account of anharmonic effects is indispensable to predict a finite value for the bulk thermal conductivity in any solid material, be it crystalline or glassy.

Motivation & Objective

• To resolve the long-standing paradox of the Allen-Feldman (AF) theory's empirical success despite its harmonic approximation predicting divergent thermal conductivity.
• To investigate the role of anharmonicity in regularizing the infrared singularity that plagues harmonic theories of heat transport in disordered solids.
• To clarify the extent to which finite-size effects and boundary scattering in simulations mimic physical regularization mechanisms.
• To benchmark the harmonic and quasi-harmonic approaches against numerical simulations using Haydock’s recursive method for large-scale glass models.

Proposed method

• The authors employ the quasi-harmonic Green-Kubo (QHGK) theory to incorporate anharmonic effects perturbatively into the harmonic AF framework, enabling regularization of the infrared divergence without relying on quantum tunneling or artificial cutoffs.
• They use Haydock’s recursive algorithm to compute the vibrational dynamical structure factor (VDSF) for large glass models (up to 13,824 atoms), enabling efficient evaluation of harmonic linewidths and sound damping with O(kN) scaling.
• Finite-size scaling analysis is performed on amorphous Si, SiO₂, and 4H-SiC to isolate the divergence of thermal conductivity in the harmonic limit and assess convergence toward the bulk limit.
• The VDSF is fitted to a Debye model to extract low-frequency contributions, and the resulting conductivity is evaluated using the Allen-Feldman formula (Eq. 1) in both harmonic and quasi-harmonic regimes.
• The study compares results from direct diagonalization and Haydock’s method to validate the accuracy of the recursive approach for large systems.
• A systematic convergence test is performed by increasing the number of Lanczos recursion steps (up to 600) to ensure numerical stability and reliability of the VDSF and linewidth calculations.

Experimental results

Research questions

• RQ1 Why does the Allen-Feldman harmonic theory of heat transport in glasses yield finite thermal conductivity values in practice, despite predicting a divergent result due to an infrared singularity?
• RQ2 To what extent do finite-size effects and boundary scattering in simulations artificially regularize the harmonic divergence, and how do they compare to physical anharmonic effects?
• RQ3 Can the quasi-harmonic Green-Kubo approach effectively regularize the infrared singularity in the bulk limit without relying on quantum tunneling or ad hoc cutoffs?
• RQ4 How significant is the contribution of low-frequency modes (ω < ω_min) to the thermal conductivity when anharmonicity is properly accounted for?
• RQ5 To what extent do standard smearing procedures in simulations mimic real boundary-scattering effects observed in thin-film experiments?

Key findings

• The harmonic Allen-Feldman theory predicts an infinite thermal conductivity in the bulk limit due to an infrared singularity arising from Rayleigh-like ω⁴ damping, regardless of configurational disorder.
• Finite-size effects in simulations introduce a natural low-frequency cutoff (ω_min ~ 2πc/L), which artificially regularizes the divergence and explains the apparent success of the harmonic theory in finite models.
• Anharmonic effects, properly accounted for via the quasi-harmonic Green-Kubo method, regularize the infrared singularity in the bulk limit without requiring quantum tunneling or arbitrary cutoffs.
• The contribution of modes below ω_min is found to be relatively small when anharmonicity is included, indicating that the harmonic divergence is not physically relevant in real materials.
• The commonly used numerical smearing procedure in simulations effectively mimics boundary-scattering effects observed in thin-film experiments, highlighting a hidden physical correspondence.
• Haydock’s recursive method enables accurate computation of VDSF and linewidths for large glass models (up to 13,824 atoms), with results validated against direct diagonalization, confirming numerical robustness and convergence.