Specific heat, magnetic susceptibility, resistivity and thermal expansion of the superconductorZrB12

Abstract

In an attempt to clarify conflicting published data, we report new measurements of specific heat, resistivity, magnetic susceptibility, and thermal expansivity up to room temperature for the $6\mathrm{K}$ superconductor $\mathrm{Zr}{\mathrm{B}}_{12}$, using well-characterized single crystals with a residual resistivity ratio $>9$. The specific heat gives the bulk result $2\Delta \left(0\right)∕k_B{T}_c=3.7$ for the superconducting gap ratio, and excludes multiple gaps and $d$-wave symmetry for the Cooper pairs. The Sommerfeld constant ${\gamma }_n=0.34\mathrm{mJ}{\mathrm{K}}^{-2}{\mathrm{gat}}^{-1}$ and the magnetic susceptibility $\chi =-2.1\times {10}^{-5}$ indicate a low density of states at the Fermi level. The Debye temperature ${\theta }_D$ is in the range $1000–1200\mathrm{K}$ near zero and room temperature, but decreases by a factor of $\sim 2$ at $\sim 35\mathrm{K}$. The specific heat and resistivity curves are inverted to yield approximations of the phonon density of states $F\left(\omega \right)$ and the spectral electron-phonon scattering function ${\alpha }_{\mathit{tr}}^{2}F\left(\omega \right)$, respectively. Both unveil a $15\mathrm{meV}$ mode, attributed to Zr vibrations in oversized B cages, which gives rise to electron-phonon coupling. The thermal expansivity further shows that this mode is anharmonic, while the vanishingly small discontinuity at $T_c$ establishes that the cell volume is nearly optimal with respect to $T_c$.