  Ca(Fe$_{1-x}$Rh$_x$)$_2$As$_2$ undergoes successive phase transitions with
increasing Rh doping in the $T$ $=$ 0 limit. The antiferromagnetic-metal phase
with orthorhombic structure at 0.00 $\le$ $x$ $\le$ 0.020 is driven to a
superconducting phase with uncollapsed-tetragonal (ucT) structure at 0.020
$\le$ $x$ $\le$ 0.024; a non-superconducting collapsed-tetragonal (cT) phase
takes over at $x$ $\geq$ 0.024. The breakdown of Fermi-liquid transport is
observed in the ucT phase above $T_{\rm c}$. In the adjacent cT phase,
Fermi-liquid transport is restored along with a disappearance of
superconductivity. This interplay of superconductivity and Fermi-liquid
transport suggests the essential role of magnetic fluctuations in the emergence
of superconductivity in doped CaFe$_2$As$_2$.
