Size Dependence of Domain Pattern Transfer in Multiferroic Heterostructures

Magnetoelectric coupling in multiferroic heterostructures can produce large lateral modulations of magnetic anisotropy enabling the imprinting of ferroelectric domains into ferromagnetic films. Exchange and magnetostatic interactions within ferromagnetic films oppose the formation of such domains. Using micromagnetic simulations and a one-dimensional model, we demonstrate that competing energies lead to the breakdown of domain pattern transfer below a critical domain size. Moreover, rotation of the magnetic field results in abrupt transitions between two scaling regimes with different magnetic anisotropy. The theoretical predictions are confirmed by experiments on $\mathrm{CoFeB}/{\mathrm{BaTiO}}_3$heterostructures.

Figure 1

Schematic illustration of a thin-film multiferroic heterostructure exhibiting domain pattern transfer. The projections of the ferroelectric polarization onto the film plane are indicated by arrows in (a). Magnetoelectric coupling between the two ferroic subsystems induces uniaxial magnetic anisotropies in the ferromagnetic film, which are indicated by double-headed arrows in (b). Because of the lateral modulation of uniaxial anisotropy, magnetically uncharged (c) and charged (d) domain walls form when a magnetic field is applied perpendicular and parallel to the walls, respectively. Panels (c) and (d) illustrate the remanent magnetization state for these two configurations.

Figure 2

Simulated remanent magnetization state as a function of domain width for uncharged (left) and charged (right) domain walls, and corresponding hysteresis curves from the whole area for magnetic fields applied perpendicular (left) and parallel (right) to the stripe domains.

Figure 3

(a) Film remanence ($M_r$) and spin rotation $|{\varphi }_{\mathrm{\Delta }/2}-{\varphi }_{-(\mathrm{\Delta }/2)}|$ between domains in the remanent state (dashed line) as a function of domain width ($\mathrm{\Delta }$) for uncharged magnetic domain walls. (b) Variation of the domain wall width (${\delta }_{uc}$), magnetic anisotropy energy (blue dash-dotted line), exchange energy (black dashed line), and magnetostatic energy (green dotted line) with $\mathrm{\Delta }$ for uncharged domain walls. (c) Dependence of the spin rotation between domains and the width of uncharged domain walls on uniaxial magnetic anisotropy. The domain widths for which $|{\varphi }_{\mathrm{\Delta }/2}-{\varphi }_{-(\mathrm{\Delta }/2)}|$ decrease to $<30°$ (${\mathrm{\Delta }}_{30}$), $<20°$ (${\mathrm{\Delta }}_{20}$), and $<10°$ (${\mathrm{\Delta }}_{10}$) are also shown. Panels (d)–(f) contain the same information as (a)–(c) for charged magnetic domain walls. Note that different horizontal scales are used for $\mathrm{\Delta }$ to present data for uncharged and charged domain walls.

Figure 4

Simulated uncharged (open symbols) and charged (filled symbols) domain wall width as a function of uniaxial magnetic anisotropy. The circles and triangles indicate data for a magnetic film thickness of 10 and 50 nm, respectively. The lines are calculated using ${\delta }_{uc}\approx \pi \sqrt{A/2K_u}$ and ${\delta }_c\approx \pi {\mu }_0{M}_s^{2}t/8K_u$. The green cross corresponds to the parameters of the experimental $\mathrm{CoFeB}/{\mathrm{BaTiO}}_3$ sample (Fig. 5). Parameters of the simulation movies (Supplemental Material [23]) are indicated by numbers (1)–(3).

Figure 5

(a) Magneto-optical Kerr effect microscopy images of magnetic domains in a 50 nm thick CoFeB film on ${\mathrm{BaTiO}}_3$. The images on the left (right) are recorded with a magnetic field of 8 mT parallel (perpendicular) to the stripe domains. The width of the central stripe domain is $2.5\mu \mathrm{m}$ and the uniaxial magnetic anisotropy is $K_u=7.5\times {10}^3\mathrm{J}/{\mathrm{m}}^3$. The sensitive axis of magneto-optical Kerr effect contrast is indicated by $\omega$. (b) Variation of the spin rotation $|{\varphi }_{\mathrm{\Delta }/2}-{\varphi }_{-(\mathrm{\Delta }/2)}|$ between domains with the direction of a 6 mT magnetic field. The dashed black line and solid red line represent simulated and experimental data, respectively.