Second-order Green's-function approach for the study of the sensitivity of the Curie temperature to single-ion anisotropy

Abstract

This paper utilizes the second-order Green's-function formalism to treat a Heisenberg ferromagnet in the presence of single-ion crystal-field anisotropy represented by the Hamiltonian: $H=\ensuremath{-}J\ensuremath{\Sigma}{i,\ensuremath{\delta}}^{}({\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{S}}}_{i+\ensuremath{\delta}})\ensuremath{-}D\ensuremath{\Sigma}{i}^{}{({S}_{i}^{z})}^{2}$. Second-order equations of motion for the Green's functions $〈〈{S}_{i}^{z}; {S}_{j}^{z}〉〉$ and $〈〈{{S}_{i}}^{+}; {{S}_{j}}^{\ensuremath{-}}〉〉$ are developed and the higher-order Green's functions are decoupled in the zeroth-order approximation in which the interspin correlations are not taken into account rigorously. The spin-correlation functions are derived and are solved self-consistently in the limit of zero spontaneous magnetization. The Curie temperature is thus obtained. Calculations are restricted to a simple-cubic lattice and positive values of $D$ only, and the sensitivity of the Curie temperature to the single-ion anisotropy is critically examined for a spin-1 lattice. It is seen that the results agree very closely with those of the Green's-function diagram technique. The results are also compared with those of the first-order Green's-function theory using the random-phase approximations of Lines using the correlated-effective-field approximation, and of the molecular-field theory.