Irregular orthorhombic twins and non-switchable polarization in multiferroic Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$

Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$ has been proposed as a prototypical magnet with hybrid improper ferroelectricity (HIF), and a significant magnetoelectric (ME) coupling effect in Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$ has been reported theoretically and experimentally. However, unlike the case of Ca$_{\mathrm{3}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$, the switchable polarization has not been experimentally confirmed in Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$. From the systematic investigation of single-crystalline Ca$_{\mathrm{3}}$(Mn,Ti)$_{\mathrm{2}}$O$_{\mathrm{7}}$, we found a unique irregular domain structure in Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$, stemming from the random stacking of 90-degree $a$- and $b$-domains along the $c$-axis. We determined the structural transition temperatures, and also the phase diagram. We found that Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$ and Ca$_{\mathrm{3}}$Ti$_{\mathrm{2}}$O$_{\mathrm{7}}$ go through different routes upon structural phase transition, which results in different domain stackings along the c-axis. This random stacking of twins in Ca$_{\mathrm{3}}$Mn$_{\mathrm{2}}$O$_{\mathrm{7}}$ appears to make polarization switching difficult.