Doping influence of spin dynamics and magnetoelectric effect in hexagonal Y$_{0.7}$Lu$_{0.3}$MnO$_3$

We use inelastic neutron scattering to study spin waves and their correlation with the magnetoelectric effect in Y$_{0.7}$Lu$_{0.3}$MnO$_3$. In the undoped YMnO$_3$ and LuMnO$_3$, the Mn trimerization distortion has been suggested to play a key role in determining the magnetic structure and the magnetoelectric effect. In Y$_{0.7}$Lu$_{0.3}$MnO$_3$, we find a much smaller in-plane (hexagonal $ab$-plane) single ion anisotropy gap that coincides with a weaker in-plane dielectric anomaly at $T_N$. Since both the smaller in-plane anisotropy gap and the weaker in-plane dielectric anomaly are coupled to a weaker Mn trimerization distortion in Y$_{0.7}$Lu$_{0.3}$MnO$_3$ comparing to YMnO$_3$ and LuMnO$_3$, we conclude that the Mn trimerization is responsible for the magnetoelectric effect and multiferroic phenomenon in Y$_{1-y}$Lu$_{y}$MnO$_{3}$.