Rare-earth permanent magnets are important materials both technologically and scientifically. They are used in traction motors of hybrid vehicles and power generators to provide a sustainable solution to the energy problem. In those industrial applications, a recent challenge has been to soften a high-temperature degradation of the magnetic properties. Engineering of microstructure to better control the extrinsic properties has been pursued while we propose an intrinsic solution spotting a key energy scale of an exchange coupling between the anisotropic 4f-electron cloud in rare-earth atom and the magnetically polarized 3d-electron bands coming from the Fe-group elements. A simplified spin model derived basically from first principles shows that a slight enhancement of 4f-3d exchange coupling helps in partially avoiding the temperature degradation of the magnetic properties around the room temperature or higher [1]. Calculated temperature dependence of magnetic properties in our ab initio spin model is not entirely consistent with experimental results [1,2] of which reason can be at least partially tracked down to the fundamental issue in describing the nature of delocalized electrons on the basis of localized degrees of freedom. Other independent ab initio finite-temperature calculations for rare-earth permanent magnet compounds [3] provide a cross-check with ab initio description of itinerant electrons within Korringa-Kohn-Rostoker (KKR) method for the electronic structure calculation combined with coherent potential approximation (CPA) [4]. Finite-temperature scaling analysis between magnetization and uni-axial magnetic anisotropy energy elucidates the common nature and a difference between a spin model description, ab initio KKR-CPA, and the experimental results. Thus rare-earth permanent magnet makes a good playground for solid-state physics to address the nature of localized and itinerant magnetism. Extensions of the present models toward a finite-temperature description of magnetization reversal processes pose another challenge in statistical physics, and are under progress with several different approaches in implementing a scale-bridging scheme [5].

References
[1] MM, H. Akai, Y. Harashima, S. Doi, T. Miyake, J. Appl. Phys. 119, 213901 (2016).
[2] Y. Toga, MM, S. Miyashita, H. Akai, S. Doi, T. Miyake, A. Sakuma, Phys. Rev. B 94, 174433 (2016).
[3] MM, R. Banerjee, J. B. Staunton, Phys. Rev. B 90, 054421 (2014).
[4] H. Shiba, Prog. Theor. Phys. 46, 77 (1971); H. Akai, Physica 86-88B, 539 (1977).
[5] H. Sepehri-Amin, J. Thielsch, J. Fischbacher, T. Ohkubo, T. Schre, O. Guteisch, K. Hono, Act. Mater. 126, 1 (2017).