We report on reversible electric-field-driven magnetic domain wall motion in a Cu/Ni multilayer on a ferroelectric BaTiO$_3$ substrate. In our heterostructure, strain-coupling to ferroelastic domains with in-plane and perpendicular polarization in the BaTiO$_3$ substrate causes the formation of domains with perpendicular and in-plane magnetic anisotropy, respectively, in the Cu/Ni multilayer. Walls that separate magnetic domains are elastically pinned onto ferroelectric domain walls. Using magneto-optical Kerr effect microscopy, we demonstrate that out-of-plane electric field pulses across the BaTiO$_3$ substrate move the magnetic and ferroelectric domain walls in unison. Our experiments indicate an exponential increase of domain wall velocity with electric field strength and opposite domain wall motion for positive and negative field pulses. Magnetic fields do not affect the velocity of magnetic domain walls, but independently tailor their internal spin structure, causing a change in domain wall dynamics at high velocities.
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