  In this work we calculate the Sunyaev-Zel'dovich (SZ) effect due to the
$e^+e^-$ from dark matter (DM) annihilation in galaxy clusters. Two candidates
of DM particle, (1) the weakly-interacting massive particle (WIMP) and (2) the
light dark matter (LDM) are investigated. For each case, we also consider
several DM profiles with and without central cusp. We generally find smaller
signals than previously reported. Moreover, the diffusion of electrons and
positrons in the galaxy clusters, which was generally thought to be negligible,
is considered and found to have significant effect on the central
electron/positron distribution for DM profile with large spatial gradient. We
find that the SZ effect from WIMP is almost always non-observable, even for the
highly cuspy DM profile, and using the next generation SZ interferometer such
as ALMA. Although the signal of the LDM is much larger than that of the WIMP,
the final SZ effect is still very small due to the smoothing effect of
diffusion. Only for the configuration with large central cusp and extremely
small diffusion effect, the LDM induced SZ effect might have a bit chance of
being detected.
