Spin-fluctuation theory of temperature-driven spin reorientations in ferromagnetic transition metal thin films

An electronic spin-fluctuation theory of temperature-driven spin-reorientation transitions (SRTs) in transition-metal nanostructures is formulated in the framework of a realistic 𝑑-band model Hamiltonian including hybridizations, intra-atomic Coulomb interactions and spin-orbit coupling. Using the Hubbard-Stratonovich functional-integral transformation in the static approximation and the virtual crystal alloy analogy, we determine the temperature dependence of the magnetic anisotropy free energy Δ⁢𝐹𝛿⁢𝛾=𝐹𝛿−𝐹⁢𝛾 in a nonperturbative way, as the difference between the electronic free energies for different magnetization directions 𝛿 and 𝛾. The results for an Fe monolayer show, in qualitative agreement with experiments, that a transition from off-plane magnetization (𝛿=𝑧) to in-plane magnetization (𝛾=𝑥) takes place at a relatively low temperature 𝑇SR≃0.3⁢𝑇C, where 𝑇C refers to the film Curie temperature. A remarkable correlation is observed between Δ⁢𝐹𝑧⁢𝑥⁡(𝑇) and the anisotropy Δ⁢𝑚𝐿𝑥⁢𝑧⁡(𝑇)=⟨𝐿𝑥⟩−⟨𝐿𝑧⟩ of the average orbital moment ⟨𝐿𝛿⟩ as a function of temperature 𝑇, the easy axis corresponding to the magnetization direction 𝛿 yielding the largest ⟨𝐿𝛿⟩. The microscopic origin of the SRT is revealed by identifying the various energy and entropy contributions to Δ⁢𝐹𝑧⁢𝑥⁡(𝑇). At very low temperatures the magnetic anisotropy energy Δ⁢𝐸𝑧⁢𝑥=𝐸𝑧−𝐸𝑥 favors the perpendicular magnetization direction. However, as 𝑇 increases, the magnetic anisotropy entropy Δ⁢𝑆𝑧⁢𝑥=𝑆𝑧−𝑆𝑥 rapidly decreases, mainly because of the contribution of electron-hole excitations. In this way a more modest decrease of Δ⁢𝐸𝑧⁢𝑥 is overridden, driving the temperature-induced reorientation transition to in-plane magnetization.