Magnon energy gap in a four-layer ferromagnetic superlattice

The magnon energy band in a four-layer ferromagnetic superlattice is studied by using the linear spin-wave approach and Green's function technique. It is found that three modulated energy gaps exist in the magnon energy band along $K_{x }$ direction perpendicular to the superlattice plane. The spin quantum numbers and the interlayer exchange couplings all affect the three energy gaps. The magnon energy gaps of the four-layer ferromagnetic superlattice are different from those of the three-layer one. For the four-layer ferromagnetic superlattice, the disappearance of the magnon energy gaps $\Delta \omega _{12} $, $\Delta \omega _{23} $ and $\Delta \omega _{34} $ all correlates with the symmetry of this system. The zero energy gap $\Delta \omega _{23}$ correlates with the symmetry of interlayer exchange couplings, while the vanishing of the magnon energy gaps $\Delta \omega _{12} $ and $\Delta \omega _{34} $ corresponds to a translational symmetry of $x$-direction in the lattice. When the parameters of the system deviate from these symmetries, the three energy gaps will increase.