  We investigate the microscopic structure of the low-lying isovector-dipole
excitation mode in neutron-rich $^{36,38,40}$Mg close to the drip line by means
of the deformed quasiparticle-random-phase-approximation employing the Skyrme
and the local pairing energy-density functionals. It is found that the
low-lying bump structure above the neutron emission-threshold energy develops
when the drip line is approached, and that the isovector dipole strength at
$E_{x} < 10$ MeV exhausts about $6.0 %$ of the classical Thomas-Reiche-Kuhn
dipole sum rule in $^{40}$Mg. We obtained the collective dipole modes at around
8-10 MeV in Mg isotopes, which consist of many two-quasiparticle excitations of
neutron. The transition density clearly shows an oscillation of the neutron
skin against the isoscalar core. We found significant coupling effects between
the dipole and octupole excitation modes due to the nuclear deformation. It is
also found that the responses for the compressional dipole and isoscalar
octupole excitations are much enhanced in the lower energy region.
