  Inspired by $\Xi_{cc}$ reported by LHCb Collaboration and $X(5568)$ reported
by $D0$ Collaboration, the $QQ\bar{q}\bar{q}$ ($Q=c,b,s$, $q=u,d$) tetraquark
states, are studied in the present work. With the help of gaussian expansion
method, two structures, diquark-antiquark and meson-meson, with all possible
color configurations are investigated systematically in a chiral quark model to
search for the possible stable states. The results show that there is no bound
state in the isovector $QQ\bar{q}\bar{q}$ system, while there are rather deep
bound states in the isoscalar $bb\bar{q}\bar{q}$, $cc\bar{q}\bar{q}$ and
$bc\bar{q}\bar{q}$ systems. There are also several shallow bound states in
$QQ^{\prime}$ system. Mixing two structures of diquark-antidiquark and
meson-meson can introduce more attractions and convert some unbound isoscalar
states into shallow bound states. The large mass of the heavy quark is
beneficial to the formation of the bound state. The separations between quarks
are calculated to unravel the spacial structure of the system.
