  The deposition and intercalation of metal atoms can induce superconductivity
in monolayer and bilayer graphenes. For example, it has been experimentally
proved that Li-deposited graphene is a superconductor with critical temperature
$T_{c}$ of 5.9 K, Ca-intercalated bilayer graphene C$_{6}$CaC$_{6}$ and
K-intercalated epitaxial bilayer graphene C$_{8}$KC$_{8}$ are superconductors
with $T_{c}$ of 2-4 K and 3.6 K, respectively. However, the $T_{c}$ of them are
relatively low. To obtain higher $T_{c}$ in graphene-based superconductors,
here we predict a new Ca-intercalated bilayer graphene C$_{2}$CaC$_{2}$, which
shows higher Ca concentration than the C$_{6}$CaC$_{6}$. It is proved to be
thermodynamically and dynamically stable. The electronic structure,
electron-phonon coupling (EPC) and superconductivity of C$_{2}$CaC$_{2}$ are
investigated based on first-principles calculations. The EPC of
C$_{2}$CaC$_{2}$ mainly comes from the coupling between the electrons of
C-$p_{z}$ orbital and the high- and low-frequency vibration modes of C atoms.
The calculated EPC constant $\lambda$ of C$_{2}$CaC$_{2}$ is 0.75, and the
superconducting $T_{c}$ is 18.9 K, which is much higher than other
metal-intercalated bilayer graphenes. By further applying -4\% biaxial
compressive strain to C$_{2}$CaC$_{2}$, the $T_{c}$ can be boosted to 26.6 K.
Thus, the predicted C$_{2}$CaC$_{2}$ provides a new platform for realizing
superconductivity with the highest $T_{c}$ in bilayer graphenes.
