  Domains in BaTiO$_3$ induces a regular modulation of uniaxial magnetic
anisotropy in CoFeB via an inverse magnetostriction effect. As a result, the
domain structures of the CoFeB wedge film and BaTiO$_3$ substrate correlate
fully and straight ferroelectric domain boundaries in BaTiO$_3$ pin magnetic
domain walls in CoFeB. We use x-ray photoemission electron microscopy and
magneto-optical Kerr effect microscopy to characterize the spin structure of
the pinned domain walls. In a rotating magnetic field, abrupt and reversible
transitions between two domain wall types occur, namely, narrow walls where the
magnetization vectors align head-to-tail and much broader walls with
alternating head-to-head and tail-to-tail magnetization configurations. We
characterize variations of the domain wall spin structure as a function of
magnetic field strength and CoFeB film thickness and compare the experimental
results with micromagnetic simulations.
