  We consider dark matter as singlet fermionic particles which carrying
magnetic dipole moment to explore its contribution on the polarization of
cosmic microwave background (CMB) photons. We show that Dirac fermionic dark
matter has no contribution on the CMB polarization. However, in the case of
Majorana dark matter this type of interaction leads to the B-mode polarization
in presence of primordial scalar perturbations which is in contrast with
standard scenario for the CMB polarization. We numerically calculate the B-mode
power spectra and plot $C_l^{BB}$ for different dark matter masses and the
$r$-parameter. We show that the dark matter with masses less than 100MeV have
valuable contribution on $C_l^{BB}$. Meanwhile, the dark matters with mass
$m_d\leq50MeV$ for $r=0.07$ ( $m_d\leq80MeV$ for $r=0.09$) can be excluded
experimentally. Furthermore, our results put a bound on the magnetic dipole
moment about $M\leq 10^{-16} e\,\,cm$ in agreement with the other reported
constraints.
