Many-body Green’s function theory of ferromagnetic Heisenberg systems with single-ion anisotropies in more than one direction

Abstract

The behavior of ferromagnetic systems with single-ion anisotropies and magnetic fields in more than one direction is investigated with many-body Green's function theory generalizing earlier work with uniaxial anisotropies only. It turns out to be of advantage to construct Green's functions in terms of the spin operators ${S}^{x}$, ${S}^{y}$, and ${S}^{z}$, instead of the commonly used ${S}^{+}$, ${S}^{\ensuremath{-}}$, and ${S}^{z}$ operators. The exchange energy terms are decoupled by ramdom phase approximation and the single-ion anisotropy terms by a generalization of the Anderson-Callen decoupling. We stress that in the derivation of the formalism none of the three spatial axes is special, so that one is always able to select a reference direction along which a magnetization component is not zero. Analytical expressions are obtained for all three components of the magnetization and the expectation values $⟨{({S}^{x})}^{2}⟩$, $⟨{({S}^{y})}^{2}⟩$, and $⟨{({S}^{z})}^{2}⟩$ for any spin quantum number $S$. The formalism considers both in-plane and out-of-plane anisotropies. Numerical calculations illustrate the behavior of the magnetization for three-dimensional and two-dimensional systems for various parameters. In the two-dimensional (monolayer) case, the magnetic dipole-dipole coupling is included, and a comparison is made between in-plane and out-of-plane anisotropies.