  Recent experimental and theoretical work has shown that the double perovskite
NaLaMnWO$_6$ exhibits antiferromagnetic ordering owing to the Mn $d$ states,
and computational studies further predict it to exhibit a spontaneous electric
polarization due to an improper mechanism for ferroelectricity [King \textit{et
al., Phys.\ Rev.\ B}, 2009, \textbf{79}, 224428; Fukushima \textit{et al.,
Phys.\ Chem.\ Chem.\ Phys.}, 2011, \textbf{13}, 12186], which make it a
candidate multiferroic material. Using first-principles density functional
calculations, we investigate nine isostructural and isovalent
$AA^{\prime}$MnWO$_6$ double perovskites ($A$=Na, K, and Rb; $A^{\prime}$=La,
Nd, and Y) with the aim of articulating crystal-chemistry guidelines describing
how to enhance the magnitude of the electric polarization through chemical
substitution of the $A$-site while retaining long-range magnetic order. We find
that the electric polarization can be enhanced by up to 150\% in compounds
which maximize the difference in the ionic size of the $A$ and $A^{\prime}$
cations. By examining the tolerance factors, bond valences, and structural
distortions (described by symmetry-adapted modes) of the nine compounds, we
identify the atomic scale features that are strongly correlated with the ionic
and electronic contributions to the electric polarization. We also find that
each compound exhibits a purely electronic remnant polarization, even in the
absence of a displacive polar mode. The analysis and design strategies
presented here can be further extended to additional members of this family
($B$=Fe, Co, etc.), and the improper ferroelectric nature of the mechanism
allows for the decoupling of magnetic and ferroelectric properties and the
targeted design of novel multiferroics.
