Why does silicon have an indirect band gap?

First-principles methods are routinely used to predict electronic band structures, but they rarely deliver intuition on the crystal chemistry origin of major qualitative band structure features like band gap magnitude, location of band extrema, effective masses, etc. However, all real-space bonding interactions can be derived from a tight-binding decomposition of the electronic structure, which explains the chemical and orbital origin of band structure features. Here, we examine the duality between real-space bonding interactions and reciprocal-space electronic structure by relating the chemical bond-type at distinct k-points; as well as by determining the reciprocal space band dispersion for distinct orbital interactions. Applying this to silicon, we present new mechanistic insights on how multiple orbital interactions combine to form the low-symmetry conduction band minimum along the Γ-X line. Specifically, we find that the linear nature of the 1st nearest-neighbor interactions combines with the cosine nature of the 2nd nearest-neighbor p_x–p_x bonding to form a minimum near X. Applying this method to understand band features can lead to the conceptual design of materials with superlative optoelectronic and magnetic properties.