  The two-orbital double-exchange model is employed for the study of the
magnetic and orbital orders in ($R$MnO$_3$)$_n$/($A$MnO$_3$)$_{2n}$ ($R$: rare
earths; $A$: alkaline earths) superlattices. The A-type antiferromagnetic order
is observed in a broad region of parameter space for the case of SrTiO$_3$ as
substrate, in agreement with recent experiments and first-principles
calculations using these superlattices. In addition, also a C-type
antiferromagnetic state is predicted to be stabilized when using substrates
like LaAlO$_3$ with smaller lattice constants than SrTiO$_3$, again in
agreement with first principles results. The physical mechanism for the
stabilization of the A- and C- magnetic transitions is driven by the orbital
splitting of the $x^2-y^2$ and $3z^2-r^2$ orbitals. This splitting is induced
by the $Q_3$ mode of Jahn-Teller distortions created by the strain induced by
the substrates. In addition to the special example of
(LaMnO$_3$)$_n$/(SrMnO$_3$)$_{2n}$, our phase diagrams can be valuable for the
case where the superlattices are prepared employing narrow bandwidth
manganites. In particular, several non-homogenous magnetic profiles are
predicted to occur in narrow bandwidth superlattices, highlighting the
importance of carrying out investigations in this mostly unexplored area of
research.
