The effect of the interface atomic interdiffusion on the nucleation field and energy product in hard/soft exchange-coupled bilayers: A one-dimensional micromagnetic approach

A one-dimensional (1D) analytical model is extended to study the magnetic properties of hard/soft exchange-coupled bilayers with the interface layer, focusing on SmCo/Fe and Nd₂Fe₁₄B/Fe systems. The results show that increasing the interface layer thickness tⁱ enhances the nucleation field H_N, thereby increasing the maximum energy product (BH)_max, while simultaneously reducing the pinning field H_P. For parallel SmCo(20 nm)/Fe(10 nm) bilayers, magnetic property changes with increasing tⁱ predicted by the 1D model align well with experiments at higher annealing temperatures. Complementary three-dimensional (3D) micromagnetic simulations confirm that H_N increases with tⁱ over a wide range of interface exchange energy constants, which shows in excellent agreement with the 1D calculated results over the investigated tⁱ range. Meanwhile, H_N also increases with the interface exchange coupling coefficient for each specific tⁱ. These findings indicate that appropriate interface atomic interdiffusion enhances the exchange coupling effect between the hard and soft phases and improves magnetic performance. The 1D model also identifies optimal interface layer thicknesses for maximizing (BH)_max in both systems. Additionally, demagnetization curves of perpendicular Nd₂Fe₁₄B/Fe bilayers from the 1D model match 3D simulation results, verifying the model's accuracy. Importantly, the 1D model developed in this work is also applicable to hard/soft exchange-coupled trilayers with varying thicknesses and magnetic parameters in each layer. In contrast to our previous design, where atoms were assumed to diffuse from the hard phase into the soft phase, resulting in the interface layer being located on the soft phase side, this study considers mutual diffusion between the two phases, with the interface layer symmetrically distributed between them.