Multiple superconducting transitions in Yb$_{3+x}$Co$_4$Sn$_{13-x}$

Abstract of the accompanying article: In the Yb-Co-Sn system a Remeika phase Yb$_{3+x}$Co$_{4}$Sn$_{13-x}$ is found with solid solution extending within $0\le\,x\,\le0.5$. All these compounds crystallize with primitive cubic [space group $Pm\bar{3}n$, $a\approx9.54$ \AA] strongly disordered Sc$_{3}$Ir$_4$Si$_{13+x}$ type. Stoichiometric Yb$_3$Co$_4$Sn$_{13}$ is a superconductor with critical temperature $T_c\,=\,3.1(2)$\,K, lower- [$B_\mathrm{c1}=1.90(2)$\,mT] and upper [$B_\mathrm{c2}=2.16(5)$\,T] critical fields. The specific heat jump $\Delta c_p/\gamma T_c\,=\,1.72(9)$ together with the fact that $B_\mathrm{c2}(T_c)$ dependence is described by the Werthamer-Helfand-Hohenberg (WHH) model hint towards a conventional mechanism with a weak electron-phonon coupling. Although a more precise analysis is hampered by the presence of multiple superconducting transitions observed in the specific heat of this Remeika phase, the non-stoichiometric Yb$_{3.2}$Co$_4$Sn$_{12.8}$ reveals $T_c\,=\,2.4(2)$\,K and $B_\mathrm{c2}=4.79(9)$\,T exceeding the classical Pauli limit. Strong electron-phonon coupling in this superconductor is confirmed by the high value of the $\lambda_\mathrm{AD}=0.92(2)$ parameter determined from the Allen-Dynes formula. However, the electronic specific heat of Yb$_{3.2}$Co$_4$Sn$_{12.8}$ follows the exponential law [$c_\mathrm{el}(T)\propto e^{-\Delta(0)/k_\mathrm{B}T}$] and can be described by the $\alpha$-model [$\alpha\equiv\Delta(0)/k_\mathrm{B}T_c=2$] indicating this stannide to be a conventional BCS-superconductor. Both Yb$_{3+x}$Co$_{4}$Sn$_{13-x}$ ($x=0, 0.2$) compounds reveal complex phonon spectra with possible `rattling' behavior. They are also found to be metallic systems, some aspects of which can be described by a free electron gas model.