The shift of the Curie temperature in bcc-Fe by phonon softening: first-principles phonon and Monte Carlo calculations.

Various estimation techniques based on spin-lattice models are widely used to estimate Curie temperatures in magnetic materials. In spin-lattice models, only magnetic couplings are explicitly treated, while the phonon effect could be incorporated implicitly into the exchange-coupling parameter. However, the validity of this approach is questionable for the cases where the coupling between the phonon and the magnetism is strong. For example, the phonon frequency typically decreases with a change in the magnetic order from ferromagnetic to paramagnetic state. This phonon softening makes paramagnetic states energetically stable, therefore, equilibrium magnetic states in finite temperatures deviate from the case where only magnetic terms are considered. We develop a theoretical scheme to describe the shift of the Curie temperature and apply it to bcc-Fe. Our scheme employs free energy minimization, therefore, the problem can be formulated as a simple optimization problem.