[Paper Review] Direct Calculation of Modal Contributions to Thermal Conductivity via Green-Kubo Modal Analysis: Crystalline and Amorphous Silicon

Wei Lv, Asegun Henry|arXiv (Cornell University)|Mar 27, 2015

Thermal properties of materials53 references155 citations

TL;DR

This paper introduces Green-Kubo Modal Analysis (GKMA), a first-principles method that directly computes modal contributions to thermal conductivity without relying on phonon velocity or the phonon gas model. By combining lattice dynamics with the Green-Kubo formalism, GKMA captures full anharmonicity and mode-mode correlations, enabling accurate prediction of temperature-dependent thermal conductivity in both crystalline and amorphous silicon, with the best agreement to experimental data to date for amorphous silicon when quantum corrections are applied.

ABSTRACT

In this letter we derive a new method for direct calculation of the modal contributions to thermal conductivity, which is termed Green-Kubo modal analysis (GKMA). The GKMA method combines the lattice dynamics formalism with the Green-Kubo formula for thermal conductivity, such that the thermal conductivity becomes a direct summation of modal contributions, where one need not define the phonon velocity. As a result the GKMA method can be applied to any material/group of atoms where the atoms vibrate around stable equilibrium positions, which includes not only crystalline line compounds, but also random alloys, amorphous materials and even molecules. The GKMA method provides new insight into the nature of phonon transport, as it casts the problem in terms of mode-mode correlation instead of scattering, and provides a general unified formalism that can be used to understand phonon-phonon interactions in essentially any class of materials or structures where the atoms vibrate around stable equilibrium sites.

Motivation & Objective

• To develop a general formalism for calculating modal contributions to thermal conductivity that does not rely on phonon velocity or the phonon gas model.
• To address the limitations of existing models in describing heat transport in non-periodic materials such as amorphous silicon, where phonon modes are not well-defined plane waves.
• To incorporate full anharmonicity and mode-mode correlations in thermal conductivity calculations, especially in systems where traditional approaches fail.
• To enable accurate prediction of temperature-dependent thermal conductivity by combining classical molecular dynamics with quantum corrections to mode-specific heat capacities.

Proposed method

• The method uses lattice dynamics to express atomic displacements and velocities as a superposition of normal mode contributions via transformation matrices.
• It derives the time-dependent contribution of each mode to the heat flux using the Hardy heat flux operator, based on modal velocity components.
• The Green-Kubo formula is applied to the mode-resolved heat flux, enabling direct calculation of each mode's contribution to thermal conductivity.
• Mode-mode cross-correlations are computed via time correlation functions, revealing interactions between modes of different frequencies.
• The method allows for quantum correction of classical MD results by scaling each mode’s contribution using the ratio of quantum to classical specific heat.
• The approach is validated by applying it to crystalline and amorphous silicon, with results compared to experimental data and prior theoretical models.

Experimental results

Research questions

• RQ1 Can modal contributions to thermal conductivity be computed directly without assuming well-defined phonon velocities or group velocities?
• RQ2 How do mode-mode correlations, particularly off-diagonal terms, contribute to thermal conductivity in amorphous materials like silicon?
• RQ3 To what extent does full anharmonicity, captured via molecular dynamics, improve predictions of thermal conductivity compared to harmonic approximations?
• RQ4 Can quantum corrections applied to classical MD results in the GKMA framework yield accurate temperature-dependent thermal conductivity in amorphous silicon?
• RQ5 Does the GKMA formalism reveal new physical insights into the role of localized (locon) versus delocalized (propagon/diffuson) modes in thermal transport?

Key findings

• The GKMA method successfully computes modal contributions to thermal conductivity without requiring phonon velocity, enabling application to non-periodic systems such as amorphous materials.
• For amorphous silicon, the GKMA method with quantum corrections achieves the best agreement with experimental data to date, outperforming previous models.
• At room temperature, mode-mode cross-correlations contribute approximately 30% of the total thermal conductivity, demonstrating the significant role of anharmonicity.
• The cross-correlation matrix reveals a distinct transition in mode interactions at ~16 THz, coinciding with the onset of localized modes (locons), without prior knowledge of mode character.
• The auto-correlations of locon modes are negligible, consistent with their minimal contribution to thermal conductivity.
• The method correctly captures the shift in modal contributions at low temperatures, showing that quantum corrections must be applied to frequency-resolved contributions obtained from GKMA to achieve accurate temperature dependence.