  Switching in magnetic materials gives rise to rich physical phenomena and
lies at the heart of their technological applications. Although domain wall
motion in ferro- and antiferromagnets has been studied, in spiral magnets it is
still poorly understood despite 20 years of active research since the discovery
of spiral multiferroics. The problem of the domain wall motion in a spiral
magnet is a compelling one, the more so the magnetic domain walls in cycloidal
spiral phase are also ferroelectric, thus enabling electric control of
magnetism, i.e. domain wall motion under the action of an external electric
field. Phase transition to a spiral phase leads to a formation of chiral
domains with opposite spin rotation senses, that are separated by chiral domain
walls. Spiral order breaks inversion symmetry and induces a ferroelectric
polarization, whose sign is determined by the chirality of the domain. Thus the
spiral order allows for the manipulation of spins via an external electric
field. Here we study domain wall motion in magnets with spiral ground state,
that are the most basic non-collinear magnets. We formulate a simplified
variational model and derive the equation of motion for the domain wall driven
by an external electric field. The results are corroborated with atomistic spin
dynamics simulations. The results suggest a linear dependence of the wall speed
on the external electric field, and a peculiar dependence on the system
geometry and domain structure.
