Spectral origin of the $c$-axis dielectric constant anomalies in hexagonal HoMnO$_3$

The coupling between ferroelectric (P) and magnetic (M) order parameters in multiferroics is one of the most important problems both for basic understanding and applications. If this coupling in hexa-manganites occurs within a unit cell there should be a phonon, infrared-active in $E||c$ polarization, which modulates simultaneously $P_c$ and Mn--Mn superexchange in the $ab$-plane. The dielectric constant anomalies, observed at magnetic ordering/re-ordering temperatures, indicate existence of a spin-coupled phonon(s). To find the linking phonon we have measured temperature dependence of the $c$-axis infrared phonon spectrum of a hexagonal HoMnO$_3$ single crystal. These optical results and the dielectric anomalies will be discussed in terms of the coupling between the two order parameters in the multiferroic hexa-manganites.