Supplementary Information : Ferroelectricity in the multiferroic hexagonal manganites

SHG has never been used for probing ferroelectric order at temperatures as high as 1400 K. At first, we therefore verified the relation between the SHG amplitude and the spontaneous polarization Ps on LiNbO3, an archetypal ferroelectric with a Curie temperature TC = 1403 K1. For this intensely investigated ferroelectric it is well established that the spontaneous polarization Ps follows the classical relation Ps ∝ (TC −T )0.5 obtained from Landau theory2. Figure S1 shows the temperature dependence of the SHG amplitude on a bulk single crystal of LiNbO3 in comparison to this scaling law. The agreement is excellent, including the value TC = (1404±1) K. In this reference ferroelectric E(2ω) is thus scaling exclusively with Ps across the entire temperature range. There is no “unspecific” decrease of the nonlinear conversion efficiency with temperature.