Charged particles can't have Majorana masses of any type because they would violate the charge conservation law. The Majorana mass is really a term that is converting a particle into its antiparticle. It implies that the particle must be considered "physically indistinguishable" from its antiparticle.

The Majorana mass term violates the lepton number or its generalization – the number of "particles minus antiparticles" – by $\Delta L = \pm 2$. It has the form 
$$ m \eta_A \eta_B \epsilon^{AB}  + \text{complex conjugate terms} $$
where the first term contains no complex conjugation of any factor, so it creates two equal particles (or annihilates two equal particles).

That's clearly impossible for particles that carry a nonzero conserved additive charge such as the electric charge. Only neutral fermions – in the Standard Model, only the neutrinos – may have a Majorana mass term. 

And because in the Standard Model, the visible left-handed neutrino is a component of a doublet with the charged particle that prohibits the Majorana term, as I just argued, the neutrino Majorana mass can't be there at a tree level, either. It has to be generated as an effective interaction.