We present a finite-temperature second-order perturbation method incorporating spin-orbit coupling to investigate the temperature-dependent site-resolved contributions to the magnetocrystalline anisotropy energy (MAE), specifically K1(T), in FePt, MnAl, and FeNi alloys. Our developed method successfully reproduces the results obtained using the force theorem from our previous work. By employing this method, we identify the key sites responsible for the distinctive behaviors of MAE in these alloys, shedding light on the inadequacy of the spin model in capturing the temperature dependence of MAE in itinerant magnets. Moreover, we explore the lattice expansion effect on the temperature dependence of on-site contributions to K1(T) in FeNi. Our results not only provide insights into the limitations of the spin model in explaining the temperature dependence of MAE in itinerant ferromagnets but also highlight the need for further investigations. These findings contribute to a deeper understanding of the complex nature of MAE in itinerant magnetic systems. © 2023 American Physical Society.
