  The recent discovery of pressure-induced superconductivity in the bilayer
La$_3$Ni$_2$O$_7$ (LNO) has opened a new platform for the study of
unconventional superconductors. In this publication, we investigate
theoretically the whole family of bilayer 327-type nickelates $R_3$Ni$_2$O$_7$
($R$ = Rare-earth elements) under pressure. From La to Lu, the lattice
constants and volume decrease, leading to enhanced in-plane and out-of-plane
hoppings, resulting in an effectively reduced electronic correlation $U/W$.
Furthermore, the Ni's $t_{2g}$ states shift away from the $e_g$ states, while
the crystal-field splitting between $d_{3z^2-r^2}$ and $d_{x^2-y^2}$ is almost
unchanged. In addition, six candidates were found to become stable in the Fmmm
phase, with increasing values of critical pressure as the atomic number
increased. Similar to the case of LNO, the $s_{\pm}$-wave pairing tendency
dominates in all candidates, due to the nesting between the {\bf M}=$(\pi,\pi)$
and the {\bf X}=$(\pi,0)$ and {\bf Y}=$(0,\pi)$ points in the Brillouin zone.
Then, $T_c$ is expected to decrease as the radius of rare-earth (RE) ions
decreases. Our results suggest that LNO is already the "optimal" candidate,
with Ce a close competitor, among the whole RE bilayer nickelates, and to
increase $T_c$ we suggest to grow on special substrates with larger in-plane
lattice spacings.
