$d$-wave superconductivity as a model for diborides apart MgB$_2$

Abstract

Recently, Pei et al. (arXiv2105.13250) reported that ambient pressure ${\beta}$-MoB$_2$ exhibits a phase transition to ${\alpha}$-MoB$_2$ (space group: $P6/mmm$) at pressure P~70 GPa and this high-pressure phase is a high-temperature superconductor exhibited $T_c=32 K$ at P~110 GPa. Despite ${\alpha}$-MoB$_2$ has the same crystalline structure as ambient pressure MgB$_2$2 and the $T_c$'s of ${\alpha}$-MoB$_2$ and MgB$_2$ are very close, the first principles calculations showed that in ${\alpha}$-MoB$_2$ the states near the Fermi level, ${\epsilon}_F$, are dominated by the $d$-electrons of Mo atoms, while in MgB$_2$ the $p$-orbitals of boron atomic sheets dominantly contribute to the states near the ${\epsilon}_F$. More recently, Hire et al. (arXiv2212.14869) reported that the $P6/mmm$-phase can be stabilized at ambient pressure in $Nb_{1-x}Mo_{x}B_{2}$ solid solutions, and these ternary alloys exhibit $T_c=8 K$. In addition, Pei et al. (Sci. China-Phys. Mech. Astron. 65, 287412 (2022)) showed that compressed WB$_2$ exhibits $T_c=15 K$ at P~121 GPa. Here, we analyzed experimental data reported for $P6/mmm$-phases of $Nb_{1-x}Mo_{x}B_{2}$ (x = 0.25; 1.0) and highly-compressed WB$_2$, and showed that these three phases exhibit $d$-wave superconductivity. We also deduced the gap-to-transition temperature ratio for these three phases. We found that $Nb_{0.75}Mo_{0.25}B_{2}$ exhibits high strength of nonadiabaticity, which is quantified by the ratio of $T_{\theta}/T_F=3.5$, which is by one order of magnitude exceeds the ratio in MgB$_2$, ${\alpha}$-MoB$_2$, WB$_2$, pnictides, cuprates, and highly-compressed hydrides.