  The longitudinal and transversal spin decoherence times, $T_1$ and $T_2$, in
semiconductor quantum dots are investigated from equation-of-motion approach
for different magnetic fields, quantum dot sizes, and temperatures. Various
mechanisms, such as the hyperfine interaction with the surrounding nuclei, the
Dresselhaus spin-orbit coupling together with the electron--bulk-phonon
interaction, the $g$-factor fluctuations, the direct spin-phonon coupling due
to the phonon-induced strain, and the coaction of the
electron--bulk/surface-phonon interaction together with the hyperfine
interaction are included. The relative contributions from these spin
decoherence mechanisms are compared in detail. In our calculation, the
spin-orbit coupling is included in each mechanism and is shown to have marked
effect in most cases. The equation-of-motion approach is applied in studying
both the spin relaxation time $T_1$ and the spin dephasing time $T_2$, either
in Markovian or in non-Markovian limit. When many levels are involved at finite
temperature, we demonstrate how to obtain the spin relaxation time from the
Fermi Golden rule in the limit of weak spin-orbit coupling. However, at high
temperature and/or for large spin-orbit coupling, one has to use the
equation-of-motion approach when many levels are involved. Moreover, spin
dephasing can be much more efficient than spin relaxation at high temperature,
though the two only differs by a factor of two at low temperature.
