Superconductivity and structural variation of the electron-correlated layer systems Sr(Pd$_{1-x}T_x$)$_2$Ge$_2$ ($T$ = Co, Ni, or Rh; $0 \leq x \leq 1$)*

Superconductivity variations in the pseudoternary Sr(Pd$_{1-x}T_x$)$_2$Ge$_2$ layer system (Pd($4d^8$), $T$ = Co($3d^7$), Ni($3d^8$), or Rh($4d^7$); $0 \leq x \leq 1$) are reported. For the BaFe$_2$As$_2$-type tetragonal structure, the degenerate $nd^7$ or $nd^8$ orbital of transition metal $T$ is splitted by $c$-axis squeezed $T$Ge$_4$ tetrahedral crystal field in the $T$-Ge layer. For the isoelectronic Sr(Pd$_{1-x}$Ni$_x$)$_2$Ge$_2$ system, superconducting transition temperature $T_c$ decreases monotonically from 3.12 K for $4d$-band SrPd$_2$Ge$_2$ to 0.92 K for $3d$-band SrNi$_2$Ge$_2$, where major contribution of conduction electrons was from the half-filled dispersive 3D-like upper-lying $nd_{xz,yz}$ bands. For the Sr(Pd$_{1-x}$Rh$_x$)$_2$Ge$_2$ system, $T_c$ decreases to 2.40 K with 25\% of $4d^7$ Rh substitution. For the Sr(Pd$_{1-x}$Co$_x$)$_2$Ge$_2$ system, $T_c$ decreases sharply to 2.58 K with only 3\% of $3d^7$ Co substitution. The lower $T_c$ of the present electron-overdoped ($nd^7$ or $nd^8$) compound is due to dispersive 3D-like $nd_{xz,yz}$ conduction bands with weak electron correlation.