Effect of biquadratic exchange and anisotropy on the critical temperature of antiferromagnet

Abstract

The existence of biquadratic exchange had been well established by Anderson, Huang and Allen. Early investigations indicated that the biquadratic exchange term in spin Hamiltonian was found to have significant effects on the magnetic properties of compounds containing iron-group ions or rare-earth metal ions. Recently, studying the anisotropic magnetic excitation of iron-based superconducting materials indicated that the experimental phenomenon cannot be explained well only by a simple collinear Heisenberg antiferromagnetic model. However, if the nearest neighbor biquadratic spin coupling is introduced, the experiment can be well explained. Similar results had also been obtained in the investigation of the superconducting material FeSe. As a result, exploring the effect of biquadratic exchange interactions on the magnetic properties of magnetic systems is significance. In this paper, we use the double-time Green's function method to study the properties of the spin-1 Heisenberg antiferromagnets with biquadratic exchange interactions and anisotropy on a three-dimensional lattice. We derive the equation of motion of the Green's function by a standard procedure. In the course of this, the higher order Green functions have to be decoupled. For the higher order Green functions of different lattice points, a Tyablikov or random phase approximation decoupling are used to decuple. For the higher order Green functions of the same lattice points, we adopt the Anderson-Callen decoupling to decouple. Based on the above procedures, the analytic expressions of the magnetization and critical temperature are obtained. The effects of biquadratic exchange interactions and anisotropy on the critical temperature are discussed in detailed. Our results show that the critical temperature increases with the increase of the anisotropy for a given biquadratic interaction. Regardless of the value of the anisotropy, the critical temperature always decreases with the increase of the biquadratic interaction. As the value of anisotropic parameter <inline-formula><tex-math id="M1">\begin{document}$\eta = 0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200077_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="10-20200077_M1.png"/></alternatives></inline-formula>, there is an obvious linear relationship between the critical temperature and the biquadratic interaction. However, when the anisotropy of the system becomes weaker, there is no such linear relationship between them. By comparing similar ferromagnetic models, it is found that the results of this paper are significantly different from those of the ferromagnetic model. When the biquadratic interaction is equal to 0, our results agree with other theoretical results.