Anharmonic phonons and anomalous thermal expansion of graphite

We have investigated the anisotropic thermal expansion of graphite using ab-initio calculations of the implicit and explicit anharmonicity of phonons, which occur due to change in volume and increase in the thermal amplitudes, respectively. We find that the negative thermal expansion (NTE) in the a-b plane below 600 K and very large positive thermal expansion along the c-axis up to high temperatures arise due to various phonons polarized along the c-axis. While the NTE arises from the anharmonicity of transverse phonons over a broad energy range up to 60 meV, the large positive expansion along the c-axis occurs largely due to the longitudinal optic phonon modes around 16 meV. We find that these phonons have large implicit anharmonicity, but very little explicit anharmonicity. We also find very significant increase in the linear compressibility along the c-axis with increase in volume, and this has important role to quantitatively explain the thermal expansion behavior along the c-axis. The hugely anisotropic bonding in graphite is found to be responsible for wide difference in the energy range of the transverse and longitudinal phonon modes polarized along the c-axis, which are responsible for the anomalous thermal expansion behavior. This behaviour is in contrast to other nearly isotropic hexagonal structures like water-ice, which show anomalous thermal expansion in a small temperature range arising from a narrow energy range of phonons.