Refinement of the grand-canonical force theorem approach to magnetic anisotropy energy from first-principles

We developed a first-principles framework to estimate magnetic anisotropy energy (MAE) consisting of magnetocrystalline anisotropy energy (MCAE) and shape magnetic anisotropy energy (SMAE). For MCAE, raised due to the spin–orbit coupling (SOC), we refined the method of grand-canonical force theorem (GCFT) to solve the problem that the conventional GCFT sometimes shows discrepancies in MCAE with those of total energy difference methods in materials containing heavy elements with strong SOC. This refined version of GCFT (reGCFT), which incorporates the SOC-induced modulation of spin density, extends the applicability of the GCFT-based MCAE analysis, including atom- and k -resolved decompositions, to such materials. To demonstrate the reGCFT method, we applied it to the slab systems, Fe-based alloys with transition metals (TM): TM(5ML)/Fe/TM (TM = Ni, Pd, Ir, Pt, Au) and the ferromagnetic/ferroelectric interface: PtCoO/ZnO. We discussed reGCFT as a promising tool that shows good consistency with the total energy difference method for MCAE across the systems studied, enabling reliable evaluation and detailed analysis of MAE in combination with SMAE derived from the spin density.