Non-primitive rectangular cells for tight-binding electronic structure calculations

Rectangular non-primitive unit cells are computationally convenient for use in nanodevice electronic structure and transport calculations. When these cells are used for calculations of structures with periodicity, the resulting bands are zone-folded and must be unfolded in order to identify important gaps and masses. Before the zone-unfolding method can be applied, one must first determine the allowed wavevectors for the specific non-primitive cell. Because most computationally convenient non-primitive cells do not have axes parallel to the primitive cell direct lattice vectors, finding the allowed wavevectors is generally a non-trivial task. (Solid state texts generally treat only the simplest case in which the non-primitive and primitive cell axes are all aligned.) Rectangular non-primitive cells with one axis aligned along a specific direction are especially useful for obtaining the approximate random-alloy bands for a bulk crystal, a critical verification step in any random-alloy nanostructure calculation. Here, we present an easily implemented method for determining a non-primitive rectangular cell for the FCC lattice with an axis aligned in a desired direction and the associated allowed primitive cell wavevectors. We illustrate its use by unfolding the bands of Ge.