  In the framework of nonrelativistic QCD, the excited doubly heavy baryons are
thoroughly studied via the channel $e^{+} e^{-}\rightarrow \langle
QQ^{\prime}\rangle[n] \rightarrow \Xi_{QQ^{\prime}} +\bar{Q^{\prime}}
+\bar{Q}$, which takes place at the collision energy $Z$-pole. $Q^{(\prime)}$
represents $b$ or $c$ quark for the production of $\Xi_{cc}$, $\Xi_{bc}$, and
$\Xi_{bb}$, respectively. All of the intermediate diquark states $\langle
QQ'\rangle[n]$ in $P$-wave, $\langle cc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$,
$\langle cc\rangle[^{3}P_{J}]_{\mathbf{6}}$, $\langle
bc\rangle[^{1}P_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle
bc\rangle[^{3}P_{J}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb
\rangle[^{1}P_{1}]_{\mathbf{\bar 3}}$, and $\langle
bb\rangle[^{3}P_{J}]_{\mathbf{6}}$ with $J=0$, 1, or 2, are taken into account.
The cross sections and differential distributions, including the transverse
momentum, rapidity, angular, and invariant mass, are discussed for the excited
baryons production. We find that the contributions of $\langle cc \rangle$,
$\langle bc \rangle$, and $\langle bb \rangle$ in $P$-wave are found to be
3.97$\%$, 5.08$\%$, and 5.89$\%$, respectively, compared to $S$-wave. Supposing
that all excited states can decay into the ground state 100\%, the total events
$N_{\Xi_{cc}}=8.48 \times10^{4-6}$, $N_{\Xi_{bc}}=2.26\times10^{5-7}$, and
$N_{\Xi_{bb}}=4.12 \times10^{3-5}$ would be produced at the Super-$Z$ Factory
with a high luminosity up to ${\cal L} \simeq 10^{34-36}{\rm cm}^{-2} {\rm
s}^{-1}$.
