From Bloch Sums to Mulliken Terms: Band Structures, Symmetry, and Orbital Character of Fifteen Semiconductors with Diamond and Zinc Blende Structures

Abstract

Band structure diagrams illustrate semiconductor electronic properties by describing the energy and symmetry of a material’s wavefunctions. Yet, the symmetry descriptors─space groups, high-symmetry points, and Bloch sums─can remain obscure to molecular inorganic chemists more familiar with Mulliken notation. This report relates Bloch sum and Mulliken term notation for diamond-type semiconductors, thus relaying the symmetry information embedded in band structure diagrams. We compare both unitary and binary (Group IV, III–V, II–VI) diamond-type lattice materials by appending Schoenflies point groups and Mulliken terms along the high-symmetry points. Migration from a unitary to binary diamond lattice induces a descent in symmetry at Γ from Oh (space group 227, Fd3̅m) to Td (space group 216, F4̅3m), respectively. The lattice symmetry changes from nonsymmorphic (Oh) to symmorphic (Td). The descent to Td symmetry alters the Schoenflies notation along several generalized high-symmetry k-points (L→Λ→Γ→Δ→X→W→K→Σ→Γ). Additionally, individual wavefunction symmetries were computationally generated and pictorially compared using VASP and the related analysis programs “Vaspkit” and “irvsp”. By recasting the band structures in the language of chemical group theory, we aim to better equip molecular chemists to interpret, hypothesize and predict properties such as quantum confinement, interfacial wavefunction propagation, and material|molecule hybridization.