Superconducting phase transitions in rare-earth compounds

The superconducting transition temperature Tc and the upper critical field Hc2 in the rare-earth (R) compounds RₓMo₆S₈, RₓMo₆Se₈, and RRh₄B₄, are theoretically studied within a model in which the superelectrons mainly originate from the $4d$ electrons of Mo and Rh atoms. The fluctuation of local spins inside a Cooper pair weakens the BCS coupling. Thus, the quantitites Tc and Hc2 are strongly influenced by the fluctuation. These quantities are obtained as functions of the intra- and interatomic exchange interactions and the concentration of the spins. The theoretical results explain the existence of the upper and lower superconducting transition temperatures Tc1 and Tc2 in ErRh₄{B}₄$ and ${Ho}1.2Mo₆{S}₈$, and the concentration dependence of ${T}c$ in ${La}_{1{-}x}{Gd}ₓ{Mo}₆{Se}₈$ and ${Sn}_{1.2(1{-}x)}{Eu}ₓ{Mo}6.35{S}₈$. The suppression of the spin fluctuation by application of a magnetic field causes an increase of the BCS coupling. In calculating ${H}c2$, this fact is taken into account as well as the spin splitting of conduction bands and spin-orbit scattering. Our theory explains the anomalous behavior of the temperature dependence of ${H}c2$ observed in ${Sn}_{1.2(1{-}x)}{Eu}ₓ{Mo}6.35{S}₈$. It is proposed that in certain compounds a superconducting state may appear only in the presence of a magnetic field.