  We present full atomistic calculations of the spin-flip time (T$_{1}$) of
electrons and holes mediated by acoustic phonons in self-assembled
In$_{1-x}$Ga$_x$As/GaAs quantum dots at zero magnetic field. At low magnetic
field, the first-order process is suppressed, and the second-order process
becomes dominant. We find that the spin-phonon-interaction induced spin
relaxation time is 40 - 80 s for electrons, and 1 - 20 ms for holes at 4.2 K.
The calculated hole-spin relaxation times are in good agreement with recent
experiments, which suggests that the two-phonon process is the main relaxation
mechanism for hole-spin relaxation in the self-assembled quantum dots at zero
field. We further clarify the structural and alloy composition effects on the
spin relaxation in the quantum dots.
