Electronic structure of Chevrel-phase high-critical-field superconductors

Using muffin-tin orbitals and the atomic-sphere approximation, we have studied the band structures of Chevrel-phase molybdenum chalcogenides, $M_m{\mathrm{Mo}}_6{X}_{8-x}$. Generally, these compounds exist for a broad variety of elements, $M=\mathrm{Pb},\mathrm{Sn},\mathrm{Ag},\mathrm{Cu}$ and $X=\mathrm{S},\mathrm{Se},\mathrm{Te}$. $m$ may be between 0 and 2, depending on the element $M$. We present level schemes, computed for a range of Mo and $X$ potentials, for three ${\mathrm{Mo}}_6{X}_{14}$ clusters appropriate for the crystal structures of ${\mathrm{Mo}}_6$${\mathrm{S}}_8$, ${\mathrm{Mo}}_6$${\mathrm{Se}}_8$, and Pb${\mathrm{Mo}}_6$${\mathrm{S}}_{7.5}$, respectively. Self-consistent Mo and $X$ potentials have been estimated. The cluster levels give the positions of the $\mathrm{Mo} 4d$-like bands, while the widths and dispersions are estimated analytically in the tight-binding approximation taking the covalent mixing with the $X p$ states into account. The 30 $\mathrm{Mo} d$ bands are grouped into narrow subbands derived from the levels for an isolated ${\mathrm{Mo}}_6$ octahedron. The Fermi level falls in a doubly degenerate $E_g$ band with Mo wave functions of $x^2-y^2$ character and the $E_g$ bandwidths vary between 65 and 35 mRy in the compounds considered. The $E_g$ band is probably crossed by a five times wider, singly degenerate $A_{1g}$ band of predominantly $3z^2-r^2$ character. The $E_g$ and $A_{1g}$ bands are the only ones crossing the Fermi level in the ternaries but, in the binaries, the octahedra are elongated and a 50-35 mRy wide $A_u$ band, split off from a triply degenerate $T_{2u}$ band, furthermore overlaps the $E_g$ band. The susceptibilities measured for Sn${\mathrm{Mo}}_5$${\mathrm{S}}_6$ and Pb${\mathrm{Mo}}_5$${\mathrm{S}}_6$ are in good agreement with our estimates, $N\left(0\right)=11\mathrm{}$ states/(spin Mo-atom Ry) and $I_{\mathrm{Mo}}^{\mathrm{Stoner}}=40\mathrm{}$ mRy, of the band density of states and the effective exchange-interaction parameter. From the measured electronic-specific-heat coefficients we deduce the value $\lambda =2.5\mathrm{}$ for the electron-phonon enhancement. In accord with experimental phonon spectra we estimate frequencies of 10 and 15 meV for a rocking mode of ${\mathrm{Mo}}_6$${\mathrm{Se}}_8$ and ${\mathrm{Mo}}_6$${\mathrm{S}}_8$ units, respectively. For the average electron-phonon matrix element in the Gaspari-Gyorffy and atomic-sphere approximations we find $〈I^2〉=3\mathrm{}\times {10}^{-3\mathrm{}}$ (Ry/bohr ${\mathrm{}(\mathrm{R}\mathrm{y}/\mathrm{b}\mathrm{o}\mathrm{h}\mathrm{r}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{u}\mathrm{s})\mathrm{}}^2$. The magnitude and extreme sensitivity to local environment effects of the spin-orbit coupling in the $E_g$ band offer an explanation for the high critical magnetic fields measured in the ternaries.