Evolution of the Chern Gap in Chern Magnet HoMn$_6$Sn$_{6-x}$Ge$_x$

The Chern gap is a unique topological feature that can host non-abelian particles. The Kagome lattice forms a chern insulator with no gap. Upon the inclusion of magnetism the Kagome system hosts a Chern gap at the K points in the lattice. In this work, the effect of Ge doping on HoMn$_6$Sn$_{6}$ is investigated. It is seen that with increased doping, a multi-stack Chern gap in formed in HoMn$_6$Sn$_{6-x}$Ge$_x$. In addition, the Chern gaps are much more pronounced and larger in energy in HoMn$_6$Ge$_{6}$ then HoMn$_6$Sn$_{6}$.