Icosahedral order and disorder in tetrahedral semiconductors: Three-dimensional and six-dimensional views

Different approaches to tetrahedrally coordinated quasicrystals are discussed. It is shown that the BC8 phase, which is observed in silicon and germanium, has a hidden icosahedral symmetry of tetrahedral interatomic bonds. In this cubic crystal with the Ia3̄ space group, 75% of all bonds are directed almost exactly along fivefold icosahedral axes. This structure (16 atoms per unit cell) is the 1/0 approximant of a hypothetical icosahedral quasicrystal with a body-centered six-dimensional lattice. The 1/1 approximant is also constructed (64 atoms per unit cell with Ia3̄ space symmetry). A specific type of topological disorder, related with partial phason flips, is shown to be possible in both the approximants. The structure of the approximants and the atomic flips are analyzed both in physical and in perpendicular space and their energies are studied ab initio within the density functional theory. The considered phason disorder provides a new insight into crystalline-to-amorphous transformations.