Mathematical Details in the application of Non-equilibrium Green's Functions (NEGF) and Quantum Kinetic Equations (QKE) to Thermal Transport

TL;DR

This thesis demonstrates that NEGF provides a comprehensive framework for thermal transport, deriving phonon interactions, anharmonic corrections, and kinetic equations, advancing the theoretical understanding of phonon-mediated heat transfer.

Contribution

It develops NEGF-based methods to generalize Landauer and kinetic theories, including anharmonic corrections, disorder effects, and phonon correlations, offering new analytical tools for thermal transport.

Findings

Derived conserving self-energy approximations for phonon interactions.

Obtained anharmonic corrections to energy current and noise.

Incorporated high mass disorder into ballistic energy current calculations.

Abstract

In this thesis we showed that Non-equilibrium Green's Function Perturbation Theory (NEGF) is really the overarching perturbative transport theory. This is shown in great detail by using NEGF as a starting point and developing in 3 directions to obtain the usual transport-related expressions. The 3 directions are: Landauer-like theory, kinetic theory and Green-Kubo linear response theory. This thesis is concerned with using NEGF to generalize the 2 directions of Landauer-like theory and the kinetic theory. Firstly, NEGF is used to derive phonon-phonon Hedin-like functional derivative equations which generates conserving self energy approximations for phonon-phonon interaction. Secondly, for the Landauer-like theory, using the perturbation expansion, we easily obtain anharmonic (or phonon-phonon interaction) corrections to the ballistic energy current and to the noise associated to the energy current. The lowest 3-phonon interaction, the lowest and the second lowest 4-phonon interaction corrections to the ballistic energy current are obtained. The lowest 3-phonon interaction correction to the noise is obtained. Along a seperate line, we found that we can incooperate high mass disorder into the ballistic energy current formula. The coherent potential approximation (CPA) for treating high mass disorder is found to be compatible with the ballistic energy current expression. Lastly, for the kinetic theory, Wigner coordinates + gradient expansion easily allow the reproduction of the usual phonon Boltzmann kinetic equation. It is also straightforward to derive phonon-phonon correlation corrections to kinetic equations. Kinetic equations lead to hydrodynamic (balance) equations and we derived phonon-phonon correlation corrections to the entropy, energy and momentum balance equations.