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.
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