Structure prediction of stable sodium germanides at 0 and 10 GPa

Abstract: In this work we used $\textit{ab-initio}$ random structure searching (AIRSS) to carry out a systematic search for crystalline Na-Ge materials at both 0 and 10 GPa. The high-throughput structural relaxations were accelerated using a machine-learned interatomic potential (MLIP) fit to density-functional theory (DFT) reference data, allowing $\sim$1.5 million structures to be relaxed. At ambient conditions we predict three new Zintl phases, Na$3$Ge$_2$, Na$_2$Ge and Na$_9$Ge$_4$, to be stable and a number of Ge-rich layered structures to lie in close proximity to the convex hull. The known Na$\delta$Ge${34}$ clathrate and Na$_4$Ge${13}$ host-guest structures are found to be relatively stabilized at higher temperature by vibrational contributions to the free energy. Overall, the low energy phases exhibit exceptional structural diversity, with the expected mixture of covalent and ionic bonding confirmed using the electron-localisation function (ELF). The local Ge structural motifs present at each composition were determined using Smooth Overlap of Atomic Positions (SOAP) descriptors and the Ge-K edge was simulated for representatives of each motif, providing a direct link to experimental x-ray absorption spectroscopy (XAS). Two Ge-rich phases are predicted to be stable at 10 GPa; NaGe$_3$ and NaGe$_2$ have simple kagome and simple hexagonal Ge lattices respectively with Na contained in the pores. NaGe$_3$ is isostructural with the MgB$_3$ and MgSi$_3$ family of kagome superconductors and remains dynamically stable at 0 GPa. Removing the Na from NaGe$_2$ results in the hexagonal lonsdalite Ge allotrope, which has a direct band gap.