Enhanced magnetoelectric effects via strain engineering and structural softness

After describing a general theory of the magnetoelectric response, I will argue that inducing {\em structural softness} -- i.e., tuning a material so that it takes a small amount of energy to distort its structure -- constitutes a general and robust strategy to obtain very large effects. Further, I will argue that this design strategy will be effective at room temperature, and will not affect other desirable properties of the materials (i.e., their insulating character). I will illustrate this possibility with first-principles results for thin films of room-temperature multiferroic BiFeO$_3$, where the structural softness is induced by epitaxial strain. I will also present results for BiFeO$_3$-based solid solutions, discussing several alternative mechanisms by which their electromechanical and magnetoelectric responses can be enhanced. Finally, I will discuss the prospect of inducing in BiFeO$_3$ a so-called {\em morphotropic phase boundary}, where the material is expected to display very large functional responses in analogy to what occurs in strong piezoelectric PbZr$_{1-x}$Ti$_{x}$O$_{3}$.