Electron Density Response to Phonon Dynamics in MgB₂: An Indicator of Superconducting Properties

Electron density differences resulting from atom displacement patterns aligned with phonon modes in MgB₂ have been calculated using density functional theory (DFT). The extent of phonon anomalies, identified as indicators of the superconducting transition temperature, T_c, under a range of conditions in AlB₂-type structures, reduce as boron atoms are displaced from their equilibrium positions along E_2g mode directions. The Fermi energy for displacements along the directions of the E_2g phonon mode accounts for changes in the covalent B-B bond electronic charge density. We applied differential atom displacements to show that the shifted σ band structure associated with the light effective mass became tangential to the Fermi level and that the Fermi surface undergoes a topological transition at a critical relative displacement of ~0.6% of the boron atoms from equilibrium. The difference in Fermi energies at this critical displacement and at the equilibrium position correspond to the superconducting energy gap. The net volume between tubular σ surfaces in reciprocal space correlated with the depth of the phonon anomaly and, by inference, it is a key to an understanding of superconductivity. This ab initioapproach offers a phenomenological understanding of the factors that determine