Dynamics of Composite Domain Walls in Multiferroics in Magnetic Field and Their Instability

We study theoretically the dynamics of composite domain walls (DW) in multiferroic material GdFeO${}_3$ driven by magnetic field $H$. Two antiferromagnetic orders of Fe and Gd spins interact Gd-ion displacement in this system with coupling $c$ at low temperatures, and we have numerically simulated the corresponding time-dependent Ginzburg-Landau equations in which magnetic field $H$ couples to Fe-spin order parameter. We vary $H$ and $c$ systematically and calculate velocity $v$ and inner structure in a stationary state for magnetic DW and magneto-electric DW. DW mobility $v/H$ increases with $H$ and decreases with $c$ for both DWs, but their characteristics differ between the two. We have also studied analytically the smooth characteristics of magneto-electric DW by perturbation theory. Another finding is a splitting instability at large H. A magneto-electric DW splits into a pair of magnetic DW and electric DW when $c$ is large, while a magnetic DW splits when $c$ is small. Internal structure of composite DW deforms with increasing $c$ and $H$, and modulations in different order parameters separate in space. Their relative distances show a noticeable enhancement with approaching the splitting instability.