Correlation between magnetocapacitance effect and polarization flop direction in a slanted magnetic field in Tb$_{1-x}$Dy$_{x}$MnO$_{3}$

Recent extensive studies show that the cycloidal spin system can possess electric polarization through the spin-orbit coupling. The magnetoelectric coupling in multiferroics is enhanced by the clamping of helimagnetic and ferroelectric domain walls. For example, DyMnO$_{3}$ shows a gigantic magnetocapacitance effect caused by the microscopic motion of multiferroic domain walls at a magnetic field induced \textbf{\textit{P}}-flop transition. In contrast, the enhancement of capacitance at the \textbf{\textit{P}}-flop transition is much smaller in TbMnO$_{3}$. Here, we show the systematic control of magnetocapacitance effect in helimagetic ferroelectric Tb$_{1-x}$Dy$_{x}$MnO$_{3}$ as a function of the composition ratio $x$ and the intensity of the applied magnetic field. It has been also found that the rotation direction of \textbf{\textit{P}} in a slanted magnetic field changes with $x$ and $H$. The crossover between small and large enhancement in magnetocapacitance corresponds to the switch in the \textbf{\textit{P}}-flop direction. The correlation can be explained by assuming the mobility of domain wall would be dominated by the thickness of domain walls in a helical magnet.