Mathematical expressions of germanium conduction band: Solved using anti-bond orbital model

The formation of α( = s, x, y, z)-like anti-bond orbitals (ABOs) is explained in detail: from atomic orbitals, sp3 tetragonal hybrid orbitals, sp3 tetragonal antibonding orbitals, and finally α-like ABOs. The mathematical formulas for ABO Hamiltonian matrix are derived in detail and illustrated with diagrams. The ABO model is used to explore the conduction band of germanium (Ge). Therefore, the mathematical expressions for its s-, x-, y-, and z-like bands and the ABO composition ratios of each band are obtained. Furthermore, the mathematical expressions for each α-like band derived in this paper are well-suited for computer simulations of advanced Ge-based nanoelectronic devices under conditions of higher electron energies.