  We present a detailed discussion on the doubly charmed baryon $\Xi_{cc}$
production at the RHIC and LHC via the proton-nucleus ($p$-N) and
nucleus-nucleus (N-N) collision modes. The extrinsic charm mechanism via the
subprocesses $g+c\to (cc)[n]+\bar{c}$ and $c+c\to (cc)[n]+g$ together with the
gluon-gluon fusion mechanism via the subprocess $g+g\to(cc)[n]+\bar{c}+\bar{c}$
have been taken into consideration, where the intermediate diquark is in
$[n]=[^1S_0]_{\bf 6}$-state or $[^3S_1]_{\bar{\bf 3}}$-state, respectively.
Total and differential cross sections have been discussed under various
collision energies. To compare with the $\Xi_{cc}$ production via proton-proton
collision mode at the LHC, we observe that sizable $\Xi_{cc}$ events can also
be generated via $p$-N and N-N collision modes at the RHIC and LHC. For
examples, about $8.1\times10^7$ and $6.7\times10^7$ $\Xi_{cc}$ events can be
accumulated in $p$-Pb and Pb-Pb collision modes at the LHC within one operation
year.
