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We report the temperature dependent THz to mid-infrared response ofhexagonal-ErMnO3 using absorption, reflectivity, and emissivity techniques from2 K to 1700 K. At low temperatures, lowest frequency vibrational modes coexistwith paramagnon excitations associated with well-defined crystal field RareEarth pure magnetic replicas in an intriguing phonon-magnetic convergence.Increasing the temperature, a number of vibrational bands close to the spacegroup predicted undergo profile broadening and softening. In particular, adistinctive set of bands in the 288-329 cm-1 (300 K) range has a componentwhose profile is carrier screened becoming nearly fully blurred in theintermediate phase between ~830 K and ~1500 K. Below TC ~830 K this asymmetricband further splits as spin phonon interaction and the tripling of the unitcell takes place revealing a delicate balance of long- and short-rangeinteractions. Ambient Raman scattering brings up evidence of a Fr\"ohlichresonance due to Coulomb interactions between carriers and the macroscopicfield of the longitudinal optical phonon mode. We found it is dynamicallycorrelated to the hexagonal c-axis negative thermal expansion. Mid-infrared(MIR) optical conductivity show that also plays a role in small polarons andmediates in high temperature bipolarones.At higher temperatures a low frequencyDrude contribution is triggered by electron hopping signaling aninsulator-metal phase transition at ~1600 K while the MIR response suggestscoexistence between single small polarons and bipolarons. We draw a parallelwith improper ferroelectrics sustaining a lattice incommensurate intermediatephase and unit cell tripling. We argue that in the h-RMnO3 (R=Rare Earth, Y)family of compounds the intermediate phase be considered incommensurate withonset at TINC ~1500 K and ferroelectric lock-in at TC ~830 K delimiting thisregime in h-ErMnO3.

physical review. b./physical review. b

h-ErMnO3 absorbance, reflectivity and, emissivity in the THz to  mid-infrared from 2 K to 1700 K: carrier screening, Fr\"ohlich resonance,  small polarons, and bipolarons