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 $\mathrm{\ensuremath{\Delta}}{F}_{\ensuremath{\delta}\ensuremath{\gamma}}={F}_{\ensuremath{\delta}}\ensuremath{-}F\ensuremath{\gamma}$ in a nonperturbative way, as the difference between the electronic free energies for different magnetization directions $\ensuremath{\delta}$ and $\ensuremath{\gamma}$. The results for an Fe monolayer show, in qualitative agreement with experiments, that a transition from off-plane magnetization ($\ensuremath{\delta}=z$) to in-plane magnetization ($\ensuremath{\gamma}=x$) takes place at a relatively low temperature ${T}_{\mathrm{SR}}\ensuremath{\simeq}\phantom{\rule{4pt}{0ex}}0.3{T}_{\mathrm{C}}$, where ${T}_{\mathrm{C}}$ refers to the film Curie temperature. A remarkable correlation is observed between $\mathrm{\ensuremath{\Delta}}{F}_{zx}(T)$ and the anisotropy $\mathrm{\ensuremath{\Delta}}{m}_{xz}^{L}(T)=\ensuremath{\langle}{L}_{x}\ensuremath{\rangle}\ensuremath{-}\ensuremath{\langle}{L}_{z}\ensuremath{\rangle}$ of the average orbital moment $\ensuremath{\langle}{L}_{\ensuremath{\delta}}\ensuremath{\rangle}$ as a function of temperature $T$, the easy axis corresponding to the magnetization direction $\ensuremath{\delta}$ yielding the largest $\ensuremath{\langle}{L}_{\ensuremath{\delta}}\ensuremath{\rangle}$. The microscopic origin of the SRT is revealed by identifying the various energy and entropy contributions to $\mathrm{\ensuremath{\Delta}}{F}_{zx}(T)$. At very low temperatures the magnetic anisotropy energy $\mathrm{\ensuremath{\Delta}}{E}_{zx}={E}_{z}\ensuremath{-}{E}_{x}$ favors the perpendicular magnetization direction. However, as $T$ increases, the magnetic anisotropy entropy $\mathrm{\ensuremath{\Delta}}{S}_{zx}={S}_{z}\ensuremath{-}{S}_{x}$ rapidly decreases, mainly because of the contribution of electron-hole excitations. In this way a more modest decrease of $\mathrm{\ensuremath{\Delta}}{E}_{zx}$ is overridden, driving the temperature-induced reorientation transition to in-plane magnetization.