Simplified LCAO Method for Zincblende, Wurtzite, and Mixed Crystal Structures

Abstract

The tight-binding (LCAO) method is a convenient, if qualitative, way of comparing energy bands in the zincblende and wurtzite structures. Using this method it is shown that wurtzite states along $\ensuremath{\Gamma}\ensuremath{-}{\ensuremath{\Gamma}}^{\ensuremath{'}}$ ("$c$" direction in the zone) can be obtained by perturbing corresponding zincblende states along $\ensuremath{\Gamma}\ensuremath{-}\ensuremath{\Lambda}$ ([111] zone direction). The perturbation is the difference of crystal potential, ${V}^{\ensuremath{'}}=V(\mathrm{ZB})\ensuremath{-}V(\mathrm{Wur})$, which takes the zincblende structure into the wurtzite and results in a splitting and shifting of these corresponding states. For perpendicular to these directions the correspondence of vectors and of states is not so clear, although some comparison can still be made.The discussion of corresponding zincblende and wurtzite states is helpful in understanding the nature of the energy states in mixed crystals: (wurtzite structure [111] twinned on zincblende), and in faulted crystals: [111] stacking faults. We can show that barriers (discontinuities in energy surfaces) exist for electron propagation (current) parallel to the "$c$" axis of a twinned or faulted crystal due to two effects: a symmetry effect and a polar effect. The second effect is simply illustrated for rotation-twinned zincblende. For randomly faulted crystals, band gaps depend on $\ensuremath{\alpha}$, the probability of faulting. Quantitative theories of these effects remain to be developed.