# First-principles Theoretical Study of Lattice Thermal Conductivity of Crystals and Earth Minerals at High Temperatures and Pressures

## Abstract

The focus of this dissertation is the first-principles calculation of lattice thermal conductivity, a non-equilibrium thermal transport property, for a wide range of solid materials including both ideal crystals and iron (Fe)-bearing mineral solid solutions at high temperatures and pressures. Our computational technique combines first-principles density functional theory (DFT), quantum scattering theory, and the Peirls-Boltzmann kinetic transport theory (PBKTT) within the single mode excitation approximation (SMEA). Lifetimes of individual phonon modes have been directly evaluated over a wide range of temperature-pressure conditions, including those relevant to the Earth's deep interior, without any empirical extrapolation. Our first-principles calculated lattice thermal conductivity is directly derived from the lifetimes, group velocities, and heat capacities of individual phonon modes. An important input of our computational technique is the DFT-predicted microscopic inter-atomic potentials. In this study, inter-atomic potentials of solid materials were calculated up to third order using an efficient real-space super cell finite displacement (RSSFD) algorithm that we developed recently. This computational technique has predictive capability over the range of validity of the DFT calculated atomic energy and forces. The robustness of the predicted density dependence of harmonic force constants and third order lattice anharmonicity tensors governs the reliability of the derived pressure dependence of phonon lifetimes, group velocities and heat capacities. To study iron-bearing mineral solid solutions, the vibrational virtual crystal approximation (vVCA) was implemented to calculate the configurationally averaged harmonic force constant matrices. The most computationally intensive step of our calculation involves the evaluation of phonon scattering rates due to third order lattice anharmonicity. We have optimized our algorithm for speed, efficiency and massive parallelization on supercomputers. Our algorithm has been successfully tested on our PRISM computer cluster, the dense memory cluster (DMC) of the Alabama super computer authority (ASA) and the Ranger super computer at the Texas super computer center. Based on group symmetry theory, we only directly evaluate the phonon scattering rates at the irreducible q-points in the first Brillouin zone and rest are reconstructed from symmetry. For now our computational technique can handle systems as large as 20-atoms per unit cell. Our newly developed computational techniques have been successfully adopted to study the two most abundant lower mantle minerals: silicate perovskite Mg$_{(1-x)}$Fe$_{x}$SiO$_{3}$ (20-atoms per orthorhombic unit cell) with $x=0$ and $x=12.50$\%, and ferropericlase Mg$_{(1-x)}$Fe$_{x}$O (2-atoms per face centered cubic unit cell) with $x=0$ and $x=12.50$\%, as well as corundum structured Al$_{2}$O$_{3}$ (10-atoms per rhomohedral unit cell). Our calculation shows that the low frequency acoustic modes are more effective carriers of heat compared to the optical phonon modes. In MgO and Mg$_{(1-x)}$Fe$_{x}$O the acoustic phonon modes are seen to account for nearly 95\% of the overall lattice thermal conductivity. The effect of Fe-substitution on the thermal conductivity of Fe-free periclase has been discussed. Fe is observed to lower the thermal conductivity of ferropericlase by significantly reducing the phonon lifetimes of of the effective heat carrying phonon modes. The behavior of $\kappa_{\text{MgO}}$ and $\kappa_{\text{MgSiO}_3}$ have been studied. Both follow a T$^{-1}$ dependence typical of insulating materials however, the pressure increase of $\kappa_{\text{MgSiO}_3}$ is substantially weaker than $\kappa_{\text{MgO}}$. For example, the normalized thermal conductivity $\frac{\kappa}{\kappa_0}$ at 300 K from 0 to 135 GPa is seen to increase from 1 to 3 for MgSiO$_3$, while the same value goes from 1 to 8 for MgO. Implications for heat flow in the Earth's lower mantle have been discussed.