Nonorthogonal tight-binding model for germanium

We present a pair of nonorthogonal tight-binding (TB) models for germanium within the NRL-TB approach. One uses an sp³ basis, and is optimized for total-energy calculations by fitting to the total energy and band structures of several high-symmetry lattice structures. The other uses an sp³d⁵ basis to accurately reproduce the diamond lattice band structure, including three conduction bands. We present tests of the sp³ TB model on bulk properties, including high-symmetry lattice structure energies and volumes and the diamond lattice elastic constants, phonons, and band structure. We also present results for point defect formation and relaxation energies and low index surface energies and stresses, many of which have not been calculated using the density-functional theory (DFT), as well as some medium size clusters. Taking advantage of the computational efficiency of the TB approach, we go beyond the capabilities of standard density-functional theory, combining it with molecular dynamics to simulate finite temperature properties of Ge. We get good agreement with experiment for the atomic mean-squared displacement and the melting point approximated using the Lindemann criterion, as well as the linear thermal-expansion coefficient. In another demonstration of the efficiency of the TB approach, we present results for the structure and electronic properties of a high angle twist grain boundary (GB). In agreement with DFT simulation we see a range of structures with comparable energies, all with electronic states deep in the band gap. In contrast to previous work we find some different geometries with perfect fourfold coordination of all atoms in the GB. Despite the perfect coordination, these structures also have deep electronic states in the gap, indicating that the GB will be electrically active.