Superconducting phase transitions in rare-earth compounds

The superconducting transition temperature $T_c$ and the upper critical field $H_{c2\mathrm{}}$ in the rare-earth ($R$) compounds $R_x{\mathrm{Mo}}_6{\mathrm{S}}_8$, $R_x{\mathrm{Mo}}_6{\mathrm{Se}}_8$, and $R{\mathrm{Rh}}_4{\mathrm{B}}_4$, 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 $T_c$ and $H_{c2\mathrm{}}$ 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 $T_{c1\mathrm{}}$ and $T_{c2\mathrm{}}$ in Er${\mathrm{Rh}}_4$ ${\mathrm{B}}_4$ and ${\mathrm{Ho}}_{1.2}$ ${\mathrm{Mo}}_6$ ${\mathrm{S}}_8$, and the concentration dependence of $T_c$ in ${\mathrm{La}}_{1-x}{\mathrm{Gd}}_x{\mathrm{Mo}}_6{\mathrm{Se}}_8$ and ${\mathrm{Sn}}_{1.2(1-x)\mathrm{}}{\mathrm{Eu}}_x{\mathrm{Mo}}_{6.35}{\mathrm{S}}_8$. The suppression of the spin fluctuation by application of a magnetic field causes an increase of the BCS coupling. In calculating $H_{c2\mathrm{}}$, 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 $H_{c2\mathrm{}}$ observed in ${\mathrm{Sn}}_{1.2(1-x)\mathrm{}}{\mathrm{Eu}}_x{\mathrm{Mo}}_{6.35}{\mathrm{S}}_8$. It is proposed that in certain compounds a superconducting state may appear only in the presence of a magnetic field.