  We investigated the coexistence of superconductivity and antiferromagnetic
order in the compound Er$_{2}$O$_{2}$Bi with anti-ThCr$_{2}$Si$_{2}$-type
structure through resistivity, magnetization, specific heat measurements and
first-principle calculations. The superconducting transition temperature
$T_{\rm c}$ of 1.23 K and antiferromagnetic transition temperature $T_{\rm N}$
of 3 K are observed in the sample with the best nominal composition. The
superconducting upper critical field $H_{\rm c2}$(0) and electron-phonon
coupling constant $\lambda$$_{e-ph}$ in Er$_{2}$O$_{2}$Bi are similar to those
in the previously reported non-magnetic superconductor Y$_{2}$O$_{2}$Bi with
the same structure, indicating that the superconductivity in Er$_{2}$O$_{2}$Bi
may have the same origin as in Y$_{2}$O$_{2}$Bi. The first-principle
calculations of Er$_{2}$O$_{2}$Bi show that the Fermi surface is mainly
composed of the Bi 6$p$ orbitals both in the paramagnetic and antiferromagnetic
state, implying minor effect of the 4$f$ electrons on the Fermi surface.
Besides, upon increasing the oxygen incorporation in Er$_{2}$O$_{x}$Bi, $T_{\rm
c}$ increases from 1 to 1.23 K and $T_{\rm N}$ decreases slightly from 3 to
2.96 K, revealing that superconductivity and antiferromagnetic order may
compete with each other. The Hall effect measurements indicate that hole-type
carrier density indeed increases with increasing oxygen content, which may
account for the variations of $T_{\rm c}$ and $T_{\rm N}$ with different oxygen
content.
