The shandite structure hosts transition metal ions arranged in kagome layers. These layers are stacked rhombohedrally and are interspersed with post-transition metal ions and chalcogens. The electronic states near the Fermi level are dominated by the transition metal $d$ orbitals and feature saddle points near several of the high-symmetry positions of the Brillouin zone, most notably the F and L points. Combining symmetry considerations with ab initio methods, we study the electronic and structural properties of these materials with an emphasis on the connection between electronic saddle points at specific momenta and structural instabilities at these momenta. While the parent compounds studied are all found to be structurally stable under ambient conditions, we show that, in specific compounds, moving the saddle point closer to the Fermi level using either hydrostatic pressure or doping can induce a structural instability. The importance of the electronic degrees of freedom in driving this instability is supported by the dependence of the frequency of the soft phonon mode on the electronic smearing temperature. Our first-principles calculations show that as the smearing temperature is increased, the compound becomes structurally stable again, indicating that the electron-phonon coupling is playing an important role. Our findings survey the structural properties of a large family of shandite materials and shed light on the role played by saddle points in the electronic structure in driving structural instabilities in rhombohedrally stacked kagome layered materials.

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• Charge density waves
• Electronic structure
• Structural properties
• Van Hove singularity
• Kagome lattice
• Kagome metal
• First-principles calculations