Relationships between the band gaps of the zinc-blende and wurtzite modifications of semiconductors

While the direct band gaps of wurtzite (W) and zinc-blende (ZB) structures are rather similar, the W and ZB gaps can differ enormously (e.g., \ensuremath{\sim}1 eV in SiC) in indirect gap materials. This large difference is surprising given that the structural difference between wurtzite and zinc blende starts only in the third neighbor and that total energy differences are only \ensuremath{\sim}0.01 eV/atom. We show that zinc-blende compounds can be divided into five types (I--V) in terms of the order of their ${\mathrm{\ensuremath{\Gamma}}}_{1\mathit{c}}$, ${\mathit{X}}_{1\mathit{c}}$, and ${\mathit{L}}_{1\mathit{c}}$ levels and that this decides the character (direct, indirect, pseudodirect) of the wurtzite band gap. The observation of small ${\mathit{E}}_{\mathit{g}}^{\mathrm{W}}$-${\mathit{E}}_{\mathit{g}}^{\mathrm{ZB}}$ differences in direct band-gap systems (``type II,'' e.g., ZnS), and large differences in indirect gap systems (``type IV,'' e.g., SiC) are explained. We further show that while both type-III systems (e.g., AIN) and type-V systems (e.g., GaP) have an indirect gap in the zinc-blende form, their wurtzite form will have direct and pseudodirect band gaps, respectively. Furthermore, a direct-to-pseudodirect transition is predicted to occur in type-I (e.g., GaSb) systems.