What should be the thickness of a thin film where surface anisotropy on the shape anisotropy overcome?
it will vary from case to case depending on the material, synthesis procedure and physical conditions.
The surface anisotropy is closely related to the In- and out--plane strain acting on the films. According to the experiments performed on CoFe2O4 thin film on MgO substrate, below the threshold level of 240 nm, the thickness plays dominant role on the strain anisotropy compared to the shape anisotropy drastically, which amounts to about factors of 15 and 30 enhancement for 240 nm and 6o nm thick samples, respectively, as reported by Lisfi et. al. in :PHYSICAL REVIEW B 76, 054405 (2007).
İn their sample in-plane strain was tension, and out-plane strain as one aspects was compressive, and they were almost equal in magnitudes. For those thicknesses above the threshold value, in and out-plane strains dropped by a factor of about 20.
There is no precise answer for your question. Resulting orientation of magnetization in thin films depends on the effective anisotropy. In turn it contains several components: magnetocrystalline, magnetoelastic, surface (interface) and shape (dipolar). The last one always forces magnetization orientation in the sample plane. The remaining components may direct magnetization both in the film plane or in perpendicular direction. Magnetocrystalline anisotropy is associated with the crystalline structure for of the magnetic film. For example it differs for Co layer grown in hcp or fcc lattice. Magnetoelastic anisotropy is induced by strains generated e.g. by the lattice mismatch at the interfaces. However, depending on a material it may force magnetization in perpendicular or in-plane orientation for the same type of strain (tensile or compression). In some cases it might be considered as a component of interface anisotropy (if the strains are relaxed by dislocation in the vicinity of interfaces). At the interfaces or surfaces a broken symmetry or hybridization (depending of an adjacent material type) may modify the electron orbitals and change orientation (and strength) of the magnetic moment. Anisotropy can be also affected by the growth conditions (columnar structure or growth in the applied magnetic field). So you have many degrees of freedom. In Au/Co/Au systems perpendicular magnetization may exist up to 1.8 nm and in Pt/Co/Pt up to 2.2 nm of the Co layer thickness.
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