Minimal tight-binding Hamiltonian for semiconductors

We present a minimal tight-binding Hamiltonian which describes properly both valence and conduction bands of tetrahedrally coordinated semiconductors. We find that, for a correct description of the electronic structure, it is necessary to include a realistic wave-function basis and at least the interaction matrix element between ${\mathrm{sp}}^3$ orbitals in second-neighbor atoms pointing towards the common neighbor. Using Slater orbitals as basis functions and a six-parameter Hamiltonian, the band structure of silicon is reproduced with a root-mean-square deviation with respect to the experimental one of 0.3 eV.