Ambient-pressure hardness and superconductivity in sp2 and sp3 bonded Ce−

Stimulated by recent experimental synthesis efforts of ${\mathrm{LaB}}_{8}$, ${\mathrm{WB}}_{2}$, and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{MoB}}_{2}$, all characterized by intrinsic superconductivity (SC), we investigated the introduction of cerium (Ce) into various boron (B) concentrations by leveraging first-principles calculations in tandem with crystal-structure prediction techniques. Under ambient and external pressure, Ce atoms transfer electrons to B atoms, enabling the formation of distinct B configurations, namely honeycomb-like B layers in CeB, honeycomb-like B layers in ${\mathrm{CeB}}_{2}$, ${\mathrm{B}}_{20}$ cages in ${\mathrm{CeB}}_{5}$, ${\mathrm{B}}_{24}$ cages in ${\mathrm{CeB}}_{6}$, ${\mathrm{B}}_{23}$ cages in ${\mathrm{CeB}}_{7}$, ${\mathrm{B}}_{26}$ cages in ${\mathrm{CeB}}_{8}$, and ${\mathrm{B}}_{24}$ cages in ${\mathrm{CeB}}_{12}$. Except for ${\mathrm{CeB}}_{5}$ and ${\mathrm{CeB}}_{7}$, the other Ce-B compounds can be quenched to ambient pressure. Clathrate-like ${\mathrm{CeB}}_{\mathrm{n}} (\mathrm{n}=6, \mathrm{8}, \text{and} 12)$ compounds hold promising prospects as potential incompressible materials, with Vickers hardness $({H}_{v})$ of $20\ensuremath{\sim}39$ GPa. Based on the Migdal-Eliashberg (ME) theory, we discovered that both ${\mathrm{CeB}}_{2}$ and ${\mathrm{CeB}}_{8}$ are typical anisotropic single-gap superconductors with superconducting critical temperature $({T}_{c})$ values of 29 and 27 K, respectively. Because of the 29-K SC originating from the coupling of both $\mathrm{B}\text{\ensuremath{-}}2p$ and $\mathrm{Ce}\text{\ensuremath{-}}5d$ orbital states with several soft phonon modes induced by Fermi surface nesting and Kohn anomalies---fundamentally different from that in isostructural ${\mathrm{MgB}}_{2}$, ${\mathrm{WB}}_{2}$, and $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{MoB}}_{2}$---${\mathrm{CeB}}_{2}$ thus can be classified as another high-${T}_{c}$ superconductor.