Prediction of the Curie temperature considering the dependence of the phonon free energy on magnetic states

Prediction of the Curie temperature is of significant importance for the design of ferromagnetic materials. One of the most widely used methods to estimate the Curie temperature from first principles relies on a spin Hamiltonian, for example, the Heisenberg Hamiltonian, and exchange coupling parameters obtained by first-principles calculations at zero temperature. Even though there have been attempts to include the effects of magnetism on phonons, the influence of magnetism-dependent phonons on magnetism has been disregarded in the theoretical estimation of the Curie temperature. Here, we propose a first-principles thermodynamic approach to minimise the total free energy considering both the influences of magnetism on phonons and the feedback effect from phonons to magnetism. By applying our scheme to body-centered cubic Fe, we find a significant reduction of the Curie temperature due to the feedback effect. This result indicates the importance of the feedback effect for a quantitative description of finite-temperature magnetism. In addition, we point out that the reduction in the theoretical Curie temperature arises in a wide range of ferromagnetic materials that exhibit phonon softening due to magnetic disordering.