Spin-fluctuation theory of temperature-driven spin reorientations in ferromagnetic transition metal thin films
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abstract
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 ΔFδγ=Fδ-Fγ 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≃0.3TC, where TC refers to the film Curie temperature. A remarkable correlation is observed between ΔFzx(T) and the anisotropy ΔmxzL(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 ΔFzx(T). At very low temperatures the magnetic anisotropy energy ΔEzx=Ez-Ex favors the perpendicular magnetization direction. However, as T increases, the magnetic anisotropy entropy ΔSzx=Sz-Sx rapidly decreases, mainly because of the contribution of electron-hole excitations. In this way a more modest decrease of ΔEzx is overridden, driving the temperature-induced reorientation transition to in-plane magnetization.
