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 transitionmetal 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 Delta F delta gamma = F delta - F gamma in a nonperturbative way, as the difference between the electronic free energies for different magnetization directions delta and gamma . The results for an Fe monolayer show, in qualitative agreement with experiments, that a transition from off-plane magnetization (delta = z) to in-plane magnetization (gamma = 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 Delta Fzx(T ) and the anisotropy Delta mLxz(T ) = (Lx) - (Lz) of the average orbital moment (L delta) as a function of temperature T, the easy axis corresponding to the magnetization direction delta yielding the largest (L delta). The microscopic origin of the SRT is revealed by identifying the various energy and entropy contributions to Delta Fzx(T ). At very low temperatures the magnetic anisotropy energy Delta Ezx = Ez - Ex favors the perpendicular magnetization direction. However, as T increases, the magnetic anisotropy entropy Delta Szx = Sz - Sx rapidly decreases, mainly because of the contribution of electron-hole excitations. In this way a more modest decrease of Delta Ezx is overridden, driving the temperature-induced reorientation transition to in-plane magnetization.