  The spontaneous exchange bias (SEB) effect is a remarkable phenomenon
recently observed in some reentrant spin-glass materials. Here we investigate
the SEB in La$_{1.5}$(Sr$_{0.5-x}$Ba$_{x}$)CoMnO$_{6}$ double-perovskites, a
system with multifarious magnetic phases for which a notable increase in the
exchange bias field is observed for intermediate Sr/Ba concentrations. The Ba
to Sr substitution leads to the enhancement of the crystal lattice, which is
accompanied by the raise of both the effective magnetic moment ($\mu_{eff}$)
and the antiferromagnetic (AFM) transition temperature that is observed below
the ferromagnetic ordering. Such increases are likely related to the increased
fraction of Co$^{3+}$ in the high spin configuration, leading to the
enhancement of Co$^{3+}$--O--Mn$^{4+}$ AFM phase and to the reduction in the
uncompensation of the AFM coupling between Co and Mn. The combined effect of
the increased $\mu_{eff}$ and AFM phase plausible explains the changes in the
SEB effect.
