Long-range interactions in magnetic bilayer above the critical temperature

In this paper we have studied the stabilization of the long-range order in (z;x)-plane of two isotropic Heisenberg ferromagnetic monolayers coupled by a short-range exchange interaction (J⊥), by a long range dipole-dipole interactions and a magnetic field. We have applied a magnetic field along of the z-direction to study the thermodynamic properties above the critical temperature. The dispersion relation ω and the magnetization are given as function of dipolar anisotropy parameter defined as Ed=(gμ)2S/a3J∥ and for other Hamiltonian parameters, and they are calculated by the double-time Zubarev-Tyablikov Green's functions in the random-phase approximation (RPA). The results show that the system is unstable for values of Ed≥0.012 with external magnetic field ranging between H/J∥=0 and 10−3. The instability appears for Ed larger then Edc=0.0158 with H/J∥=10−5, Edc=0.02885 with H/J∥=10−4, and Edc=0.115 with H/J∥=10−3, i.e., a small magnetic field is sufficient to maintain the magnetic order in a greater range of the dipolar interaction.