Absence of ferroelectricity in hexagonal InMnO$_3$

So far, it was believed that hexagonal (h-) InMnO$_3$ exhibit the same type of multiferroic order as the other compounds from the h-RMnO$_3$ family (R = Sc, Y, Dy - Lu), including, in particular, a unit-cell-tripling improper ferroelectric order. Here we present experimental evidence for the \textit{absence} of ferroelectricity in hexagonal InMnO$_3$ based on three different techniques: x-ray diffraction (XRD), piezoresponse force microscopy (PFM) and optical second harmonic generation (SHG). XRD data are ambiguous because they can be described likewise by the non-ferroelectric $P\overline{3}c$ structure and by the ferroelectric $P6_3cm$ structure present in the other h-RMnO$_3$ compounds. However, PFM at room temperature and SHG measurements at low temperature uniquely reveal the absence of ferroelectric order in InMnO$_3$. We therefore propose that InMnO$_3$ exhibits antiferrodistortive, but non-ferroelectric order according to the $P\overline{3}c$ symmetry. Density functional calculations show that the relative energy between the $P\overline{3}c$ and $P6_3cm$ structures is determined by a competition between electrostatic and covalency effects, with an \textit{absence} of covalency favoring the ferroelectric structure.