Magnon energy gap in a four-layer ferromagnetic superlattice

The magnon energy band in a four-layer ferromagnetic superlattice isstudied by using the linear spin-wave approach and Green's functiontechnique. It is found that three modulated energy gaps exist in themagnon energy band along $K_{x }$ direction perpendicular to thesuperlattice plane. The spin quantum numbers and the interlayerexchange couplings all affect the three energy gaps. The magnonenergy gaps of the four-layer ferromagnetic superlattice aredifferent from those of the three-layer one. For the four-layerferromagnetic superlattice, the disappearance of the magnon energygaps $Delta omega _{12} $, $Delta omega _{23} $ and $Deltaomega _{34} $ all correlates with the symmetry of this system. Thezero energy gap $Delta omega _{23}$ correlates with the symmetryof interlayer exchange couplings, while the vanishing of the magnonenergy gaps $Delta omega _{12} $ and $Delta omega _{34} $corresponds to a translational symmetry of $x$-direction in thelattice. When the parameters of the system deviate from thesesymmetries, the three energy gaps will increase.