Why does silicon have an indirect band gap?

Emily Oliphant, Veda Mantena, Madison Brod, G. Jeffrey Snyder & Wenhao Sun

Materials Horizons (2025)

Chemical bonds give rise to electronic structure, but the complex bonding environments of crystals make it difficult to isolate which specific orbitals help to shape the overall electronic band dispersion. Here we present a systematic three-step approach to interpret how band dispersion arises from multiple orbital interactions in a 3D crystal. This process proceeds by determining 1) which orbitals contribute to a band, 2) how these orbitals bond across k-space, and 3) the strength of the orbital bonds. Applying this approach to silicon, we provide a revised interpretation for the origin of the silicon conduction band minimum at the low-symmetry Δ point along the Γ-X line. Specifically, we find that the dip in the silicon conduction band near X originates from a cosine shape along Γ-X arising from second nearest neighbor px–px bonding, combined with a positive linear slope due to first nearest neighbor s-s, s-px, and px–px interactions. Based on these insights, we illustrate how the bonding interactions can be tuned to morph the silicon band structure into the germanium band structure, and how short-range ordering can produce a direct band gap in a specific sequence of Si-Ge layers. Our 3-step process offers a general framework to extract the crystal chemistry origins of electronic structure features from DFT calculations, enabling a new paradigm of bonding-by-design.