Structure, Composition, and High-Field Superconductivity in Metal-Rich $\eta$-Carbide-Type Compounds

$\eta$-Carbide-type compounds have recently emerged as a diverse class of materials in the study of superconductivity. These phases contribute to a growing family of metal-rich quantum materials that exhibit unusual superconducting properties emerging from complex metallic bonding. Several members of the $\eta$-carbide-type phases have been found to be bulk superconductors – such as Nb${}_4$Rh${}_2$C${}_{1-\delta }$, Ta${}_4$Rh${}_2$C${}_{1-\delta }$, Ti${}_4$Ir${}_2$O${}_{1-\delta }$, and Ti${}_4$Co${}_2$O${}_{1-\delta }$ – with transition temperatures up to $T_{\mathrm{c}}\approx$ 10 K and upper critical fields as high as ${\mu }_0{H}_{\mathrm{c}2}(0)\approx$ 30 T. Whereas the transition temperatures may fall within the range typical for intermetallic superconductors, the pronounced violation of the weak-coupling Pauli limit in many of these crystallographically high-symmetry materials is noteworthy. Here, we review recent progress on superconducting $\eta$-carbide-type phases, emphasizing how crystal symmetry, synthetic challenges, transition-metal composition, and electronic structure govern their superconducting properties. Furthermore, we outline open questions and future directions, including the possible discovery of new $\eta$-carbide-type materials.