  Discovering materials that display a linear magnetoelectric effect at room
temperature is challenge. Such materials could facilitate novel devices based
on the electric-field control of magnetism. Here we present simple, chemically
intuitive design rules to identify a new class of bulk magnetoelectric
materials based on the 'bicolor' layering of $Pnma$ ferrite perovskites, e.g.,
LaFeO$_3$/ LnFeO$_3$ superlattices for which Ln = lanthanide cation. We use
first-principles density-functional theory calculations to confirm these ideas.
Additionally, we elucidate the origin of this effect and show it is a general
consequence of the layering of any bicolor, $Pnma$ perovskite superlattice in
which the number of constituent layers are odd (leading to a form of hybrid
improper ferroelectricity) and Goodenough- Kanamori rules. Here, the polar
distortions induce both weak ferromagnetism and a linear magnetoelectric
effect. Our calculations suggest that the effect is 2-3 times greater in
magnitude than that observed for the prototypical magnetoelectric material,
Cr$_2$O$_3$. We use a simple mean field model to show that the considered
materials order magnetically above room temperature.
