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The nature of the lattice instability connected to the structural transitionand superconductivity of (Sr,Ca)$_3$Ir$_4$Sn$_{13}$ is not yet fullyunderstood. In this work density functional theory (DFT) calculations of thephonon instabilities as a function of chemical and hydrostatic pressure showthat the primary lattice instabilities in Sr$_3$Ir$_4$Sn$_{13}$ lie at phononmodes of wavevectors $\mathbf{q}=(0.5,0,0)$ and $\mathbf{q}=(0.5,0.5,0)$.Following these modes by calculating the energy of supercells incorporating themode distortion results in an energy advantage of -14.1 meV and -9.0 meV performula unit respectively. However, the application of chemical pressure toform Ca$_3$Ir$_4$Sn$_{13}$ reduces the energetic advantage of theseinstabilities, which is completely removed by the application of a hydrostaticpressure of 35 kbar to Ca$_3$Ir$_4$Sn$_{13}$. The evolution of these latticeinstabilities is consistent with experimental phase diagram. The structuraldistortion associated with the mode at $\mathbf{q}=(0.5,0.5,0)$ produces adistorted cell with the same space group symmetry as the experimentally refinedlow temperature structure. Furthermore, calculation of the deformationpotential due to these modes quantitatively demonstrates a strongelectron-phonon coupling. Therefore, these modes are likely to be implicated inthe structural transition and superconductivity of this system.

Electronic structure and phonon instabilities in the vicinity of the structural transition and superconductivity of (Sr,Ca)$_3$Ir$_4$Sn$_{13}$