Thickness-scaling of domain size in coexisting ferromagnetic-antiferromagnetic stripe domains

The domain size l in ferroic films is commonly believed to scale with the film thickness d following Kittel's law l ∝√d , as initially proposed by C. Kittel for ferromagnetic materials. Here, we establish an analytical model for the scaling behavior of alternating ferromagnetic (FM) and antiferromagnetic (AFM) stripe domains having unidentical widths in the presence of an applied magnetic field. The analytical model is derived based on domain thermodynamics where the change in the free energy of the configuration is evaluated when a parent AFM single domain structure transforms into the field-induced FM+AFM stripe domain structure due to the magnetostatic and domain wall energies arising in the mixed phase structure in addition to the bulk free energy change. Using MnBi4Te7 (MBT) as an example, an A-type antiferromagnetic material below the temperature of 13 K that undergoes a transition to the FM + AFM coexistence phase upon applying a magnetic field, our analytical model predicted, corroborated by phase-field simulations, a significant departure of the thickness-scaling of the ferromagnetic and antiferromagnetic domains from Kittel's scaling law. A careful analysis of the domain thermodynamics reveals that for a specific equilibrium FM/AFM phase fraction, the abnormal inverse thickness-scaling of domain size at small film thicknesses originates from the thickness dependence of the magnetostatic energy and its competition with the domain wall energy determines the equilibrium domain size at a given film thickness. In addition to the film thickness, the equilibrium domain width of each stripe domain is influenced by the relative equilibrium phase fractions of the FM and AFM phases in the two-phase mixture depending on the strength of the applied magnetic field and temperature. The control of the domain size in an FM/AFM coexistence with reduced dimensionality and magnetic field tuning offers unique possibilities for the realization of exotic topological phases and functional devices based on AFM/FM spintronics.