Hysteresis Loops, Critical Fields and Energy Products for Exchange-spring Hard/soft/hard Trilayers

Macroscopic hysteresis loops and microscopic magnetic moment distributions have been determined by a three-dimensional (3D) model for exchange-coupled Sm-Co/${\alpha}-Fe$/Sm-Co trilayers with in-plane collinear easy axes. These results are carefully compared with the popular one-dimensional (1D) micromagnetic models and recent experimental data. It is found that the results obtained from the two methods match very well, especially for the remanence and coercivity, justifying the calculations. Both nucleation and coercive fields decrease monotonically as the soft layer thickness $L^s$ increases while the largest maximum energy product (roughly 50 MGOe) occurs when the thicknesses of hard and soft layers are 5 nm and 15 nm, respectively. Moreover, the calculated angular distributions in the thickness direction for the magnetic moments are similar. Nevertheless, the calculated nucleation and pinning fields as well as the energy products by 3D OOMMF are systematically smaller than those given by the 1D model, due mainly to the stray fields at the corners of the films. These demagnetization fields help the magnetic moments at the corners to deviate from the previous saturation state and facilitate the nucleation. Such an effect enhances as $L^s$ increases. When the thicknesses of hard and soft layers are 10 nm and 20 nm, respectively, the pinning field difference is as large as 30%, while the nucleation fields have opposite signs.