Understanding the structure of metal encapsulated Si cages and nanotubes: Role of symmetry and $d$-band filling

Using ab initio calculations we study the stability of Si-based cages and nanotubes stabilized by encapsulated transition metal atoms (TMAs). It is demonstrated that the stabilization of these cages and nanotubes as well as their magnetic properties are strongly guided by a delicate interplay between the attainable symmetry of the system and the $d$-band filling of the encapsulated TMA. As a result, encapsulated TMAs of the early $3-d$ series lead to tubular stuctures of $C6$ symmetry and anti-ferromagnetic alignment between the magnetic moment of the TMA and that of the Si atoms. On the other hand, the encapsulated late $3-d$ elements lead to tubules of the $C5$ symmetry and to a ferromagnetic alignment of the metal and Si magnetic moments. Encapsulated Fe atoms (being near the middle of the $3-d$ series) lead to tubular structures of either $C6$ or $C5$ symmetry.