  In framework of heavy-diquark$-$light-quark endowed with heavy-pair binding,
we explore excited baryons $QQ^{\prime}q$ containing two heavy quarks
($QQ^{\prime}=cc,bb,bc$) by combining the method of Regge trajectory with the
perturbative correction due to the heavy-pair interaction in baryons. Two Regge
relations, one linear and the other nonlinear, are constructed semi-classically
in the QCD string picture, and a mass scaling relation based on the heavy
diquark-heavy antiquark symmetry are employed between the doubly heavy baryons
and heavy mesons. We employ the ground-state mass estimates compatible with the
observed doubly charmed baryon $\Xi_{cc}$ and spectra of the heavy quarkonia to
determine the trajectory parameters and the binding energies of the heavy pair,
and thereby compute the low-lying masses of the excited baryons
$\Xi_{QQ^{\prime}}$ and $\Omega_{QQ^{\prime}}$ up to 2S and 1P excitations of
the light quark and heavy diquark. The level spacings of heavy diquark
excitations are found to be smaller generally than that of the light-quark
excitations, in according with the nature of adiabatic expansion between the
heavy and light-quark dynamics.
